# Teoria Cinetica dei Gas e Meccanica Statistica

> Una lezione universitaria che costruisce il ponte tra il moto delle molecole e le grandezze della termodinamica. Dalle ipotesi del caos molecolare si ricavano la pressione, la velocità quadratica media e il significato microscopico della temperatura, l'energia cinetica media pari a tre mezzi di k\_B T. Il teorema di equipartizione spiega i calori specifici dei gas monoatomici e biatomici, il congelamento quantistico dei gradi di libertà e la legge di Dulong e Petit. Seguono la distribuzione di Maxwell con la sua verifica a dischi rotanti, il fattore di Boltzmann, il libero cammino medio, la formula S = k\_B ln Omega con il terzo principio, e infine la misura di gamma con il metodo di Rüchardt, la formazione delle nubi e il vento di Föhn.

- Canonical watch page: [Teoria Cinetica dei Gas e Meccanica Statistica](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica)
- Publisher: [Academa, Inc.](https://academa.ai)
- Subject: Physics
- Published: 2026-10-11T15:52:01.841Z
- Updated: 2026-10-11T15:52:01.841Z
- Duration: PT684S (11 minutes 24 seconds)
- Chapters: 5
- Views: 0
- Language: en-US
- Access: Free
- Video stream: [HLS content](https://academa.ai/media/l/01M4NSAEBMXACPWE3N2HSQGAPK/0/dark/master.m3u8)
- Embed: [Player](https://academa.ai/embed/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica)
- Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M4NSAEBMXACPWE3N2HSQGAPK/0/semantic.json)
- Thumbnail: [Image](https://academa.ai/media/l/01M4NSAEBMXACPWE3N2HSQGAPK/0/dark/poster.jpg)

## Description

Teoria cinetica dei gas: temperatura come energia media, equipartizione, distribuzione di Maxwell, libero cammino medio ed entropia S = k ln Ω.

## Chapters

- [00:00–02:18.843 · Dal microscopico al macroscopico](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=0)
- [02:18.843–04:37.529 · Equipartizione e calori specifici](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=138.8426875)
- [04:37.529–07:14.104 · Distribuzioni e libero cammino medio](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=277.52864583333337)
- [07:14.104–09:30.517 · Entropia statistica](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=434.10439583333334)
- [09:30.517–11:24 · Misurare gamma, e l'atmosfera](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=570.516875)

## Transcript

### [00:00 · Dal microscopico al macroscopico](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=0)

Un gas è fatto di un numero enorme di molecole, eppure lo descriviamo con poche grandezze: pressione, volume, temperatura ed energia interna. La teoria cinetica è il primo ponte tra questi due mondi, e il primo esempio di meccanica statistica. Alla fine di questa lezione due formule avranno un significato preciso. La prima dice che la temperatura misura l'energia cinetica media di una molecola. La seconda dice che l'entropia conta i modi microscopici di realizzare uno stato. Le ricaveremo entrambe. Cominciamo dalla prima. Prendiamo un recipiente pieno di gas. Le molecole si muovono in tutte le direzioni, e nessuna direzione è privilegiata: è questa l'ipotesi del caos molecolare. In media, quindi, ogni componente della velocità si annulla. Ma i quadrati non si annullano: lungo ciascun asse il valore medio del quadrato vale un terzo di quello del modulo. Supponiamo poi che le molecole non si attraggano a distanza, e che si urtino soltanto in modo elastico, come palline rigide. Guardiamo una molecola che urta la parete di destra. Rimbalza invertendo la componente orizzontale, e cede alla parete una quantità di moto pari a due m v x. Sommando su tutti gli urti e usando le ipotesi di prima, si trova che P per V vale un terzo di N m per la media del quadrato della velocità. Confrontando con i gas perfetti, n R T, moto molecolare e temperatura si legano. Da qui segue la velocità quadratica media: la radice di tre R T diviso la massa molare A. Per l'azoto a trecento kelvin vale circa cinquecentoventi metri al secondo, mentre l'idrogeno sfiora i duemila. Dividendo per il numero di Avogadro, l'energia cinetica media di una singola molecola risulta tre mezzi di kappa B per T. Qui kappa B è la costante di Boltzmann, R diviso il numero di Avogadro. Questa è la prima delle nostre due formule: la temperatura assoluta misura l'energia cinetica media di traslazione. Moltiplicando per il numero di molecole si ottiene l'energia interna di n moli di gas monoatomico: tre mezzi n R T.

### [02:18.843 · Equipartizione e calori specifici](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=138.8426875)

Dall'energia interna discendono i calori specifici molari. C v è la derivata di U rispetto a T, divisa per il numero di moli: per un gas monoatomico vale tre mezzi R. A pressione costante il gas compie anche lavoro, e si aggiunge un R: C p vale cinque mezzi R. Il loro rapporto, gamma, vale cinque terzi, circa uno virgola sei sette. I gas nobili lo confermano con grande precisione: elio, neon e argon danno tutti circa uno virgola sei sette. Ecco il calore specifico di un gas biatomico, come l'idrogeno, al variare della temperatura, misurato in unità di R. La curva sale a gradini, e per capirla serve il teorema di equipartizione. Ogni termine quadratico indipendente nell'energia di una molecola contribuisce in media con un mezzo kappa B T. Un gas monoatomico ha solo i tre termini della traslazione, e infatti in basso la curva vale tre mezzi. Una molecola biatomica può anche ruotare attorno a due assi. La rotazione attorno all'asse dei nuclei non conta, perché lì il momento di inerzia è trascurabile. Cinque termini in tutto: C v vale cinque mezzi R, C p sette mezzi R, e gamma sette quinti, cioè uno virgola quattro. Ma perché i gradini? I livelli di energia sono quantizzati, e un modo partecipa solo quando kappa B T supera la sua spaziatura. Le rotazioni si sbloccano per prime; le vibrazioni, con due termini in più, solo a temperature molto alte. Passiamo ai solidi. Ogni atomo del reticolo oscilla attorno alla sua posizione di equilibrio in tre dimensioni: tre termini di energia cinetica e tre di energia potenziale elastica, sei in tutto. Quindi U vale tre n R T, e C v vale tre R, circa venticinque joule per mole e per kelvin: è la legge di Dulong e Petit. In ascissa c'è la temperatura divisa per una temperatura caratteristica del solido, theta. Ad alta temperatura tutti i solidi tendono proprio a questo valore. A bassa temperatura, invece, il calore specifico crolla verso zero, perché anche le vibrazioni del reticolo si congelano.

### [04:37.529 · Distribuzioni e libero cammino medio](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=277.52864583333337)

Finora abbiamo usato solo valori medi. Ma le molecole non hanno tutte la stessa velocità: si distribuiscono secondo la curva di Maxwell, qui per l'azoto a trecento kelvin. Il fattore v quadro conta le direzioni possibili, mentre il termine esponenziale punisce le velocità alte. Il massimo cade alla velocità più probabile, la radice di due R T su A. Un po' più a destra sta la velocità quadratica media, radice di tre R T su A. Come si verifica questa curva? Si scalda un vapore in un forno a temperatura T e lo si lascia uscire da un piccolo foro, formando un fascio collimato. Il fascio incontra due dischi rotanti, distanti L, con le fenditure sfalsate di un angolo theta. Passa solo la molecola che vola da un disco al successivo nel tempo in cui i dischi ruotano di theta. Quindi la velocità selezionata vale omega L su theta. Variando omega si scorrono tutte le velocità, e il rivelatore conta le molecole: l'istogramma ottenuto coincide con la curva di Maxwell. La stessa logica vale per livelli di energia discreti. Uno stato eccitato sta sopra il fondamentale di una quantità delta E. Il rapporto tra le popolazioni dei due livelli è il fattore di Boltzmann: e alla meno delta E su kappa B T. A temperatura ambiente kappa B T vale circa zero virgola zero due cinque elettronvolt, un quarantesimo. Se delta E è molto più grande, lo stato eccitato resta quasi vuoto. Per questo, per allineare gli spin nucleari in un campo magnetico, Madame Wu dovette raffreddare i suoi campioni sotto un milionesimo di kelvin. Ultima grandezza: il libero cammino medio, cioè la distanza media percorsa da una molecola tra due urti. Pensiamo le molecole come sfere rigide di diametro d. Avanzando di un tratto h, la molecola urta tutte quelle il cui centro cade entro una distanza d dalla sua traiettoria: un cilindro di volume pi greco d quadro per h. Con densità numerica n, il numero di urti è n pi greco d quadro h. Il cammino medio è h diviso questo numero, e il moto delle altre molecole introduce un fattore radice di due. Per un gas perfetto n vale P su kappa B T. A trecento kelvin e un'atmosfera, con d di circa due angstrom, si ottengono circa duecentotrenta nanometri, mille volte il diametro. A cinquecento metri al secondo, ogni molecola subisce miliardi di urti al secondo.

### [07:14.104 · Entropia statistica](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=434.10439583333334)

Veniamo alla seconda formula. Uno stato macroscopico, fissato da poche variabili come P, V e T, è realizzato da moltissimi microstati: le posizioni e le velocità di tutte le molecole. Prendiamo sei molecole distinguibili in una scatola divisa a metà. Se stanno tutte a sinistra, esiste un solo modo: un microstato. Spostiamone una a destra: può essere una qualsiasi delle sei, quindi sei microstati. Con due a destra i modi diventano quindici, e con tre diventano venti. In generale il numero di modi è il coefficiente binomiale, sei su k. Già con sei molecole, la divisione a metà è venti volte più probabile del caso estremo. Con dieci alla ventitré molecole, il macrostato uniforme ha immensamente più microstati di qualsiasi altro: è l'unico che osserviamo davvero. Due sistemi indipendenti: le loro entropie si sommano, perché l'entropia è estensiva. I loro microstati invece si moltiplicano, perché ogni microstato di uno si combina con ogni microstato dell'altro. La sola funzione che trasforma prodotti in somme è il logaritmo. Dunque S deve essere una costante C per il logaritmo di omega. Fissiamo C con l'espansione libera di Joule: togliamo il setto, e ogni molecola ha a disposizione il doppio del volume. I microstati crescono di un fattore due alla N, e l'entropia di N C logaritmo di due. La termodinamica dà per la stessa espansione n R logaritmo di due. Confrontando, C è R diviso il numero di Avogadro: proprio la costante di Boltzmann. Ecco la seconda formula: S uguale kappa B per il logaritmo di omega. Disordine vuol dire esattamente questo: più volume o più temperatura offrono più posizioni e più velocità, cioè più microstati. E allo zero assoluto? Quando la temperatura tende a zero, il sistema scivola nel suo unico stato fondamentale. Omega tende a uno, e quindi l'entropia tende a zero. È il terzo principio della termodinamica, il teorema di Nernst. Una sua conseguenza è che lo zero assoluto non si raggiunge con un numero finito di trasformazioni.

### [09:30.517 · Misurare gamma, e l'atmosfera](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=570.516875)

Gamma si può misurare con un esperimento semplice, ideato da Rüchardt. Un grande recipiente rigido di volume V contiene il gas, e nel suo collo verticale di sezione S scorre senza attrito una sferetta di massa m. Spostiamo la sferetta di un tratto y. Il gas si comprime e si espande così in fretta da non scambiare calore: la trasformazione è adiabatica, e la pressione cambia di meno gamma P per S y su V. La forza di richiamo è proporzionale allo spostamento: la sferetta oscilla di moto armonico. Chiamo tau il periodo, per non confonderlo con la temperatura. Misurando tau, insieme a m, V, P e S, si ricava gamma. Per l'aria si trova circa uno virgola quattro, proprio il valore di un gas biatomico. Le trasformazioni adiabatiche governano anche il tempo atmosferico. Una massa di aria calda e umida sale lungo il versante di una montagna: in quota la pressione è minore, e la massa si espande. Si espande a spese della propria energia interna, e la temperatura scende. Il vapore raggiunge la saturazione e condensa in goccioline: nasce la nube. Sul versante opposto succede il contrario. L'aria ormai secca scende verso la pianura e viene compressa adiabaticamente: si scalda di circa dieci gradi ogni chilometro, e arriva calda e asciutta. È il vento di Föhn. Ricapitoliamo i due ponti promessi all'inizio. La temperatura è energia cinetica media, tre mezzi kappa B T per molecola: da qui discendono equipartizione, calori specifici e velocità molecolari. L'entropia invece conta i microstati, kappa B per il logaritmo di omega, e porta con sé il terzo principio. Così il moto disordinato di miliardi di miliardi di molecole diventa la termodinamica che misuriamo.

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## Complete audiovisual record

Immutable source: [semantic.json](https://academa.ai/media/l/01M4NSAEBMXACPWE3N2HSQGAPK/0/semantic.json)

Record version: 1. Render attempt: 0.

### How to read this timeline

Each scene owns its object identifiers. A beat's board is the complete board when listed, empty when marked empty, and unchanged from the nearest earlier listed board in the same scene when marked unchanged. Action times are absolute positions in the published video.

### Scene 1: [Dal microscopico al macroscopico](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=0)

Span: 00:00–02:18.843 (0s–138.8426875s).

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##### [00:00](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=0)

Narration: Un gas è fatto di un numero enorme di molecole, eppure lo descriviamo con poche grandezze: pressione, volume, temperatura ed energia interna. La teoria cinetica è il primo ponte tra questi due mondi, e il primo esempio di meccanica statistica.

Board: Empty.

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##### [00:17.122](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=17.1225)

Narration: Alla fine di questa lezione due formule avranno un significato preciso. La prima dice che la temperatura misura l'energia cinetica media di una molecola. La seconda dice che l'entropia conta i modi microscopici di realizzare uno stato.

Board: Unchanged from the preceding beat in this scene.

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##### [00:32.462](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=32.4625)

Narration: Le ricaveremo entrambe. Cominciamo dalla prima.

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##### [00:36.597](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=36.5975)

Narration: Prendiamo un recipiente pieno di gas. Le molecole si muovono in tutte le direzioni, e nessuna direzione è privilegiata: è questa l'ipotesi del caos molecolare.

Board: Empty.

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- [00:41.849](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=41.8494627029571): arrows\_4 is shown on the screen, written out.
- [00:42.066](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=42.065950270609456): arrows\_5 is shown on the screen, written out.

##### [00:47.455](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=47.455)

Narration: In media, quindi, ogni componente della velocità si annulla. Ma i quadrati non si annullano: lungo ciascun asse il valore medio del quadrato vale un terzo di quello del modulo.

Board: box — a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 3.0), aspect=(4.0, 3.0)); head\_kin — a Heading that says "Le ipotesi del caos molecolare"; walls — a Polygon \[gray\] drawn in box (vertices=((0.2, 0.2), (3.8, 0.2), (3.8, 2.8), (0.2, 2.8)), filled=False); dots — a Point \[blue\] drawn in box (location=(0.9, 0.9)); dots\_2 — a Point \[blue\] drawn in box (location=(1.6, 2.1)); dots\_3 — a Point \[blue\] drawn in box (location=(2.4, 1.4)); dots\_4 — a Point \[blue\] drawn in box (location=(1.3, 1.6)); dots\_5 — a Point \[blue\] drawn in box (location=(3.0, 2.2)); arrows — a Vector \[green\] drawn in box (start=(0.9, 0.9), end=(1.4, 1.25)); arrows\_2 — a Vector \[green\] drawn in box (start=(1.6, 2.1), end=(1.1500000000000001, 2.4)); arrows\_3 — a Vector \[green\] drawn in box (start=(2.4, 1.4), end=(2.6999999999999997, 0.8999999999999999)); arrows\_4 — a Vector \[green\] drawn in box (start=(1.3, 1.6), end=(0.9, 1.25)); arrows\_5 — a Vector \[green\] drawn in box (start=(3.0, 2.2), end=(3.2, 2.6))

Actions:
- [00:48.04](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=48.04): box moves to a new place on the board.
- [00:48.04](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=48.04): iso1 is shown on the screen, written out.
- [00:52.16](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=52.16): iso2 is shown on the screen, written out.
- [00:57.12](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=57.12): iso2 (the "frac(1, 3) overline(v^2)" part) is emphasized.
- [00:58.785](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=58.785): iso2 (the "frac(1, 3) overline(v^2)" part) is no longer emphasized.

##### [00:59.385](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=59.385)

Narration: Supponiamo poi che le molecole non si attraggano a distanza, e che si urtino soltanto in modo elastico, come palline rigide.

Board: iso1 — a Math \[text\] that says "$overline(v\_x) = overline(v\_y) = overline(v\_z) = 0$"; iso2 — a Math \[text\] that says "$overline(v\_x^2) = overline(v\_y^2) = overline(v\_z^2) = frac(1, 3) overline(v^2)$"; box — a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 3.0), aspect=(4.0, 3.0)); head\_kin — a Heading that says "Le ipotesi del caos molecolare"; walls — a Polygon \[gray\] drawn in box (vertices=((0.2, 0.2), (3.8, 0.2), (3.8, 2.8), (0.2, 2.8)), filled=False); dots — a Point \[blue\] drawn in box (location=(0.9, 0.9)); dots\_2 — a Point \[blue\] drawn in box (location=(1.6, 2.1)); dots\_3 — a Point \[blue\] drawn in box (location=(2.4, 1.4)); dots\_4 — a Point \[blue\] drawn in box (location=(1.3, 1.6)); dots\_5 — a Point \[blue\] drawn in box (location=(3.0, 2.2)); arrows — a Vector \[green\] drawn in box (start=(0.9, 0.9), end=(1.4, 1.25)); arrows\_2 — a Vector \[green\] drawn in box (start=(1.6, 2.1), end=(1.1500000000000001, 2.4)); arrows\_3 — a Vector \[green\] drawn in box (start=(2.4, 1.4), end=(2.6999999999999997, 0.8999999999999999)); arrows\_4 — a Vector \[green\] drawn in box (start=(1.3, 1.6), end=(0.9, 1.25)); arrows\_5 — a Vector \[green\] drawn in box (start=(3.0, 2.2), end=(3.2, 2.6))

Actions:
- [01:1.8](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=61.79999999999999): urti is shown on the screen, written out.

##### [01:7.447](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=67.44749999999999)

Narration: Guardiamo una molecola che urta la parete di destra. Rimbalza invertendo la componente orizzontale, e cede alla parete una quantità di moto pari a due m v x.

Board: iso1 — a Math \[text\] that says "$overline(v\_x) = overline(v\_y) = overline(v\_z) = 0$"; iso2 — a Math \[text\] that says "$overline(v\_x^2) = overline(v\_y^2) = overline(v\_z^2) = frac(1, 3) overline(v^2)$"; urti — a Text \[text\] that says "Nessuna forza a distanza, solo urti elastici."; box — a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 3.0), aspect=(4.0, 3.0)); head\_kin — a Heading that says "Le ipotesi del caos molecolare"; walls — a Polygon \[gray\] drawn in box (vertices=((0.2, 0.2), (3.8, 0.2), (3.8, 2.8), (0.2, 2.8)), filled=False); dots — a Point \[blue\] drawn in box (location=(0.9, 0.9)); dots\_2 — a Point \[blue\] drawn in box (location=(1.6, 2.1)); dots\_3 — a Point \[blue\] drawn in box (location=(2.4, 1.4)); dots\_4 — a Point \[blue\] drawn in box (location=(1.3, 1.6)); dots\_5 — a Point \[blue\] drawn in box (location=(3.0, 2.2)); arrows — a Vector \[green\] drawn in box (start=(0.9, 0.9), end=(1.4, 1.25)); arrows\_2 — a Vector \[green\] drawn in box (start=(1.6, 2.1), end=(1.1500000000000001, 2.4)); arrows\_3 — a Vector \[green\] drawn in box (start=(2.4, 1.4), end=(2.6999999999999997, 0.8999999999999999)); arrows\_4 — a Vector \[green\] drawn in box (start=(1.3, 1.6), end=(0.9, 1.25)); arrows\_5 — a Vector \[green\] drawn in box (start=(3.0, 2.2), end=(3.2, 2.6))

Actions:
- [01:7.447](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=67.44749999999999): arrows is hidden from the screen.
- [01:7.447](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=67.44749999999999): arrows\_2 is hidden from the screen.
- [01:7.447](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=67.44749999999999): arrows\_3 is hidden from the screen.
- [01:7.447](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=67.44749999999999): arrows\_4 is hidden from the screen.
- [01:7.447](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=67.44749999999999): arrows\_5 is hidden from the screen.
- [01:9.2](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=69.19999999999999): incoming is shown on the screen, written out.
- [01:10.88](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=70.88): outgoing is shown on the screen, written out.
- [01:14.24](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=74.23999999999998): impulse is shown on the screen, written out.
- [01:18.315](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=78.315): box moves to a new place on the board.
- [01:18.315](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=78.315): head\_kin is hidden from the screen — left the board.
- [01:18.315](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=78.315): iso1 is hidden from the screen — left the board.
- [01:18.315](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=78.315): iso2 is hidden from the screen — left the board.
- [01:18.315](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=78.315): urti is hidden from the screen — left the board.

##### [01:18.915](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=78.91499999999999)

Narration: Sommando su tutti gli urti e usando le ipotesi di prima, si trova che P per V vale un terzo di N m per la media del quadrato della velocità. Confrontando con i gas perfetti, n R T, moto molecolare e temperatura si legano.

Board: box — a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 3.0), aspect=(4.0, 3.0)); walls — a Polygon \[gray\] drawn in box (vertices=((0.2, 0.2), (3.8, 0.2), (3.8, 2.8), (0.2, 2.8)), filled=False); dots — a Point \[blue\] drawn in box (location=(0.9, 0.9)); dots\_2 — a Point \[blue\] drawn in box (location=(1.6, 2.1)); dots\_3 — a Point \[blue\] drawn in box (location=(2.4, 1.4)); dots\_4 — a Point \[blue\] drawn in box (location=(1.3, 1.6)); dots\_5 — a Point \[blue\] drawn in box (location=(3.0, 2.2)); incoming — a Vector \[yellow\] labelled "arrow(v)" drawn in box (start=(2.7, 0.8), end=(3.75, 1.3)); outgoing — a Vector \[yellow\] labelled "arrow(v)'" drawn in box (start=(3.75, 1.3), end=(2.7, 1.8)); impulse — a Math \[text\] that says "$Delta p\_x = 2 m v\_x$" drawn in box

Actions:
- [01:18.915](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=78.91499999999999): head\_der is shown on the screen, written out.
- [01:18.915](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=78.91499999999999): deriv is shown on the screen, written out.
- [01:29.88](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=89.88000000000001): deriv (the "n R T" part) is emphasized.
- [01:34.385](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=94.38499999999999): deriv (the "n R T" part) is no longer emphasized.

##### [01:34.985](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=94.98499999999999)

Narration: Da qui segue la velocità quadratica media: la radice di tre R T diviso la massa molare A. Per l'azoto a trecento kelvin vale circa cinquecentoventi metri al secondo, mentre l'idrogeno sfiora i duemila.

Board: box — a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 3.0), aspect=(4.0, 3.0)); walls — a Polygon \[gray\] drawn in box (vertices=((0.2, 0.2), (3.8, 0.2), (3.8, 2.8), (0.2, 2.8)), filled=False); dots — a Point \[blue\] drawn in box (location=(0.9, 0.9)); dots\_2 — a Point \[blue\] drawn in box (location=(1.6, 2.1)); dots\_3 — a Point \[blue\] drawn in box (location=(2.4, 1.4)); dots\_4 — a Point \[blue\] drawn in box (location=(1.3, 1.6)); dots\_5 — a Point \[blue\] drawn in box (location=(3.0, 2.2)); incoming — a Vector \[yellow\] labelled "arrow(v)" drawn in box (start=(2.7, 0.8), end=(3.75, 1.3)); outgoing — a Vector \[yellow\] labelled "arrow(v)'" drawn in box (start=(3.75, 1.3), end=(2.7, 1.8)); impulse — a Math \[text\] that says "$Delta p\_x = 2 m v\_x$" drawn in box; head\_der — a Heading that says "Pressione, velocità e temperatura"

Actions:
- [01:34.985](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=94.98499999999999): incoming is hidden from the screen.
- [01:34.985](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=94.98499999999999): outgoing is hidden from the screen.
- [01:34.985](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=94.98499999999999): impulse is hidden from the screen.
- [01:36.08](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=96.08000000000001): deriv is shown on the screen, written out.
- [01:42.64](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=102.64000000000001): deriv is shown on the screen, written out.

##### [01:48.997](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=108.99749999999999)

Narration: Dividendo per il numero di Avogadro, l'energia cinetica media di una singola molecola risulta tre mezzi di kappa B per T. Qui kappa B è la costante di Boltzmann, R diviso il numero di Avogadro.

Board: box — a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 3.0), aspect=(4.0, 3.0)); walls — a Polygon \[gray\] drawn in box (vertices=((0.2, 0.2), (3.8, 0.2), (3.8, 2.8), (0.2, 2.8)), filled=False); dots — a Point \[blue\] drawn in box (location=(0.9, 0.9)); dots\_2 — a Point \[blue\] drawn in box (location=(1.6, 2.1)); dots\_3 — a Point \[blue\] drawn in box (location=(2.4, 1.4)); dots\_4 — a Point \[blue\] drawn in box (location=(1.3, 1.6)); dots\_5 — a Point \[blue\] drawn in box (location=(3.0, 2.2)); head\_der — a Heading that says "Pressione, velocità e temperatura"

Actions:
- [01:52.92](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=112.92000000000003): deriv is shown on the screen, written out.
- [01:58.28](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=118.28000000000003): deriv (the "frac(3, 2) k\_B T" part) is emphasized.
- [02:1.575](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=121.57499999999999): deriv (the "frac(3, 2) k\_B T" part) is no longer emphasized.

##### [02:2.175](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=122.17499999999998)

Narration: Questa è la prima delle nostre due formule: la temperatura assoluta misura l'energia cinetica media di traslazione. Moltiplicando per il numero di molecole si ottiene l'energia interna di n moli di gas monoatomico: tre mezzi n R T.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [02:3.36](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=123.36000000000003): deriv is indicated — a transient flash.
- [02:9.84](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=129.84000000000003): deriv is shown on the screen, written out.
- [02:14.72](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=134.72000000000003): A box is drawn around deriv.
- [02:17.801](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=137.80102083333333): box is hidden from the screen — left the board.
- [02:17.801](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=137.80102083333333): walls is hidden from the screen — box left the board.
- [02:17.801](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=137.80102083333333): dots is hidden from the screen — box left the board.
- [02:17.801](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=137.80102083333333): dots\_2 is hidden from the screen — box left the board.
- [02:17.801](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=137.80102083333333): dots\_3 is hidden from the screen — box left the board.
- [02:17.801](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=137.80102083333333): dots\_4 is hidden from the screen — box left the board.
- [02:17.801](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=137.80102083333333): dots\_5 is hidden from the screen — box left the board.
- [02:17.801](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=137.80102083333333): deriv is hidden from the screen — left the board.
- [02:17.801](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=137.80102083333333): head\_der is hidden from the screen — left the board.

### Scene 2: [Equipartizione e calori specifici](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=138.8426875)

Span: 02:18.843–04:37.529 (138.8426875s–277.52864583333337s).

#### Objects

- cv\_curve: a FunctionPlot \[blue\] drawn in heat (function=\<function\>)
- diat1: a Math \[text\] that says "$f = 3 + 2 = 5 arrow.r.double C\_v = frac(5, 2) R$"
- diat2: a Math \[text\] that says "$C\_p = frac(7, 2) R, quad gamma = frac(7, 5) = 1.4$"
- equip\_def: a Panel that says "Ogni termine quadratico indipendente dell'energia di una molecola contribuisce in media con $frac(1, 2) k\_B T$, cioè $frac(1, 2) R T$ per mole."
- gas\_table: a Table \[text\] that says "Gas $gamma$ misurato He 1.66 Ne 1.64 Ar 1.67" (rows=(('Gas', '$gamma$ misurato'), ('He', '1.66'), ('Ne', '1.64'), (…, header=True)
- head\_eq: a Heading that says "Equipartizione e gradi di libertà"
- head\_mono: a Heading that says "Il gas monoatomico"
- head\_sol: a Heading that says "I solidi: Dulong e Petit"
- heat: an Axes (x\_range=(0.0, 10.0), y\_range=(0.0, 4.0), x\_ticks\_every=2.0)
- limit\_line: a Line \[gray\] labelled "3 R" drawn in solid\_axes (start=(0.0, 25.0), end=(2.0, 25.0), dashed=True)
- mono: a Derivation \[text\] that says "$C\_v &= frac(1, n) frac(dif U, dif T) = frac(3, 2) R \\ C\_p &= C\_v + R = frac(5, 2) R \\ gamma &= frac(C\_p, C\_v) = frac(5, 3) approx 1.67$"
- plateau\_r: a PlotPoint \[yellow\] labelled "frac(5, 2)" drawn in heat (target='cv\_curve', x=4.8)
- plateau\_t: a PlotPoint \[yellow\] labelled "frac(3, 2)" drawn in heat (target='cv\_curve', x=0.8)
- plateau\_v: a PlotPoint \[yellow\] labelled "frac(7, 2)" drawn in heat (target='cv\_curve', x=9.4)
- point: a Point \[yellow\] drawn in solid\_axes (location=(0.1, 0.41))
- sol0: a Math \[text\] that says "$f = 3 + 3 = 6 arrow.r.double U = 3 n R T$"
- sol1: a Math \[text\] that says "$C\_v = 3 R approx 25 thin upright("J/(mol K)")$"
- solid\_axes: an Axes (x\_range=(0.0, 2.0), y\_range=(0.0, 30.0), x\_ticks\_every=0.5)
- solid\_curve: a FunctionPlot \[blue\] drawn in solid\_axes (function=\<function\>)

#### Beats

##### [02:18.843](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=138.8426875)

Narration: Dall'energia interna discendono i calori specifici molari. C v è la derivata di U rispetto a T, divisa per il numero di moli: per un gas monoatomico vale tre mezzi R.

Board: Empty.

Actions:
- [02:18.843](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=138.8426875): head\_mono is shown on the screen, written out.
- [02:23.883](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=143.8826875): mono is shown on the screen, written out.

##### [02:31.363](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=151.36268750000002)

Narration: A pressione costante il gas compie anche lavoro, e si aggiunge un R: C p vale cinque mezzi R. Il loro rapporto, gamma, vale cinque terzi, circa uno virgola sei sette.

Board: head\_mono — a Heading that says "Il gas monoatomico"

Actions:
- [02:31.883](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=151.8826875): mono is shown on the screen, written out.
- [02:39.323](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=159.32268750000003): mono is shown on the screen, written out.

##### [02:44.565](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=164.5651875)

Narration: I gas nobili lo confermano con grande precisione: elio, neon e argon danno tutti circa uno virgola sei sette.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [02:45.163](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=165.1626875): gas\_table is shown on the screen, written out.
- [02:48.043](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=168.0426875): gas\_table is shown on the screen, written out.
- [02:48.492](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=168.49152616917766): gas\_table is shown on the screen, written out.
- [02:48.842](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=168.84158768832393): gas\_table is shown on the screen, written out.
- [02:52.343](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=172.3426875): gas\_table is hidden from the screen — left the board.
- [02:52.343](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=172.3426875): head\_mono is hidden from the screen — left the board.
- [02:52.343](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=172.3426875): mono is hidden from the screen — left the board.

##### [02:53.543](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=173.5426875)

Narration: Ecco il calore specifico di un gas biatomico, come l'idrogeno, al variare della temperatura, misurato in unità di R. La curva sale a gradini, e per capirla serve il teorema di equipartizione.

Board: Empty.

Actions:
- [02:53.543](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=173.5426875): head\_eq is shown on the screen, written out.
- [02:53.543](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=173.5426875): heat is shown on the screen, written out.
- [02:57.963](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=177.96268750000002): cv\_curve is shown on the screen, drawn.
- [03:3.643](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=183.64268750000002): heat moves to a new place on the board.
- [03:3.643](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=183.64268750000002): equip\_def is shown on the screen, written out.

##### [03:5.955](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=185.95518750000002)

Narration: Ogni termine quadratico indipendente nell'energia di una molecola contribuisce in media con un mezzo kappa B T. Un gas monoatomico ha solo i tre termini della traslazione, e infatti in basso la curva vale tre mezzi.

Board: equip\_def — a Panel that says "Ogni termine quadratico indipendente dell'energia di una molecola contribuisce in media con $frac(1, 2) k\_B T$, cioè $frac(1, 2) R T$ per mole."; heat — an Axes (x\_range=(0.0, 10.0), y\_range=(0.0, 4.0), x\_ticks\_every=2.0); head\_eq — a Heading that says "Equipartizione e gradi di libertà"; cv\_curve — a FunctionPlot \[blue\] drawn in heat (function=\<function\>)

Actions:
- [03:17.363](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=197.36268750000002): plateau\_t is shown on the screen, written out.

##### [03:20.1](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=200.1001875)

Narration: Una molecola biatomica può anche ruotare attorno a due assi. La rotazione attorno all'asse dei nuclei non conta, perché lì il momento di inerzia è trascurabile.

Board: equip\_def — a Panel that says "Ogni termine quadratico indipendente dell'energia di una molecola contribuisce in media con $frac(1, 2) k\_B T$, cioè $frac(1, 2) R T$ per mole."; heat — an Axes (x\_range=(0.0, 10.0), y\_range=(0.0, 4.0), x\_ticks\_every=2.0); head\_eq — a Heading that says "Equipartizione e gradi di libertà"; cv\_curve — a FunctionPlot \[blue\] drawn in heat (function=\<function\>); plateau\_t — a PlotPoint \[yellow\] labelled "frac(3, 2)" drawn in heat (target='cv\_curve', x=0.8)

Actions:
- [03:22.003](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=202.0026875): plateau\_r is shown on the screen, written out.

##### [03:30.053](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=210.0526875)

Narration: Cinque termini in tutto: C v vale cinque mezzi R, C p sette mezzi R, e gamma sette quinti, cioè uno virgola quattro.

Board: equip\_def — a Panel that says "Ogni termine quadratico indipendente dell'energia di una molecola contribuisce in media con $frac(1, 2) k\_B T$, cioè $frac(1, 2) R T$ per mole."; heat — an Axes (x\_range=(0.0, 10.0), y\_range=(0.0, 4.0), x\_ticks\_every=2.0); head\_eq — a Heading that says "Equipartizione e gradi di libertà"; cv\_curve — a FunctionPlot \[blue\] drawn in heat (function=\<function\>); plateau\_t — a PlotPoint \[yellow\] labelled "frac(3, 2)" drawn in heat (target='cv\_curve', x=0.8); plateau\_r — a PlotPoint \[yellow\] labelled "frac(5, 2)" drawn in heat (target='cv\_curve', x=4.8)

Actions:
- [03:30.323](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=210.3226875): diat1 is shown on the screen, written out.
- [03:35.043](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=215.0426875): diat2 is shown on the screen, written out.

##### [03:40.205](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=220.20518750000002)

Narration: Ma perché i gradini? I livelli di energia sono quantizzati, e un modo partecipa solo quando kappa B T supera la sua spaziatura. Le rotazioni si sbloccano per prime; le vibrazioni, con due termini in più, solo a temperature molto alte.

Board: equip\_def — a Panel that says "Ogni termine quadratico indipendente dell'energia di una molecola contribuisce in media con $frac(1, 2) k\_B T$, cioè $frac(1, 2) R T$ per mole."; diat1 — a Math \[text\] that says "$f = 3 + 2 = 5 arrow.r.double C\_v = frac(5, 2) R$"; diat2 — a Math \[text\] that says "$C\_p = frac(7, 2) R, quad gamma = frac(7, 5) = 1.4$"; heat — an Axes (x\_range=(0.0, 10.0), y\_range=(0.0, 4.0), x\_ticks\_every=2.0); head\_eq — a Heading that says "Equipartizione e gradi di libertà"; cv\_curve — a FunctionPlot \[blue\] drawn in heat (function=\<function\>); plateau\_t — a PlotPoint \[yellow\] labelled "frac(3, 2)" drawn in heat (target='cv\_curve', x=0.8); plateau\_r — a PlotPoint \[yellow\] labelled "frac(5, 2)" drawn in heat (target='cv\_curve', x=4.8)

Actions:
- [03:41.083](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=221.08268750000002): cv\_curve is indicated — a transient flash.
- [03:51.323](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=231.32268750000003): plateau\_v is shown on the screen, written out.
- [03:55.453](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=235.45268750000002): diat1 is hidden from the screen — left the board.
- [03:55.453](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=235.45268750000002): diat2 is hidden from the screen — left the board.
- [03:55.453](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=235.45268750000002): equip\_def is hidden from the screen — left the board.
- [03:55.453](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=235.45268750000002): head\_eq is hidden from the screen — left the board.
- [03:55.453](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=235.45268750000002): heat is hidden from the screen — left the board.
- [03:55.453](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=235.45268750000002): cv\_curve is hidden from the screen — heat left the board.
- [03:55.453](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=235.45268750000002): plateau\_t is hidden from the screen — heat left the board.
- [03:55.453](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=235.45268750000002): plateau\_r is hidden from the screen — heat left the board.
- [03:55.453](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=235.45268750000002): plateau\_v is hidden from the screen — heat left the board.

##### [03:56.653](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=236.6526875)

Narration: Passiamo ai solidi. Ogni atomo del reticolo oscilla attorno alla sua posizione di equilibrio in tre dimensioni: tre termini di energia cinetica e tre di energia potenziale elastica, sei in tutto.

Board: Empty.

Actions:
- [03:56.653](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=236.6526875): head\_sol is shown on the screen, written out.
- [03:56.653](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=236.6526875): solid\_axes is shown on the screen, written out.
- [04:4.843](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=244.84268750000004): solid\_axes moves to a new place on the board.
- [04:4.843](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=244.84268750000004): sol0 is shown on the screen, written out.

##### [04:9.523](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=249.52268750000002)

Narration: Quindi U vale tre n R T, e C v vale tre R, circa venticinque joule per mole e per kelvin: è la legge di Dulong e Petit.

Board: sol0 — a Math \[text\] that says "$f = 3 + 3 = 6 arrow.r.double U = 3 n R T$"; solid\_axes — an Axes (x\_range=(0.0, 2.0), y\_range=(0.0, 30.0), x\_ticks\_every=0.5); head\_sol — a Heading that says "I solidi: Dulong e Petit"

Actions:
- [04:14.883](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=254.88268750000006): sol1 is shown on the screen, written out.
- [04:17.683](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=257.68268750000004): limit\_line is shown on the screen, written out.

##### [04:19.873](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=259.8726875)

Narration: In ascissa c'è la temperatura divisa per una temperatura caratteristica del solido, theta. Ad alta temperatura tutti i solidi tendono proprio a questo valore. A bassa temperatura, invece, il calore specifico crolla verso zero, perché anche le vibrazioni del reticolo si congelano.

Board: sol0 — a Math \[text\] that says "$f = 3 + 3 = 6 arrow.r.double U = 3 n R T$"; sol1 — a Math \[text\] that says "$C\_v = 3 R approx 25 thin upright("J/(mol K)")$"; solid\_axes — an Axes (x\_range=(0.0, 2.0), y\_range=(0.0, 30.0), x\_ticks\_every=0.5); head\_sol — a Heading that says "I solidi: Dulong e Petit"; limit\_line — a Line \[gray\] labelled "3 R" drawn in solid\_axes (start=(0.0, 25.0), end=(2.0, 25.0), dashed=True)

Actions:
- [04:25.683](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=265.68268750000004): solid\_curve is shown on the screen, drawn.
- [04:33.003](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=273.0026875000001): point is shown on the screen, grown.
- [04:34.823](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=274.82267522252926): point is hidden from the screen.
- [04:36.487](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=276.4869791666667): head\_sol is hidden from the screen — left the board.
- [04:36.487](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=276.4869791666667): sol0 is hidden from the screen — left the board.
- [04:36.487](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=276.4869791666667): sol1 is hidden from the screen — left the board.
- [04:36.487](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=276.4869791666667): solid\_axes is hidden from the screen — left the board.
- [04:36.487](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=276.4869791666667): limit\_line is hidden from the screen — solid\_axes left the board.
- [04:36.487](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=276.4869791666667): solid\_curve is hidden from the screen — solid\_axes left the board.

### Scene 3: [Distribuzioni e libero cammino medio](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=277.52864583333337)

Span: 04:37.529–07:14.104 (277.52864583333337s–434.10439583333334s).

#### Objects

- atom: a Point \[yellow\] drawn in beam\_fig (location=(\<VariableNumber pos = 5.3\>, 1.5), marker\_radius=0.08)
- beam: a Line \[gray\] drawn in beam\_fig (start=(1.0, 1.5), end=(5.4, 1.5), dashed=True)
- beam\_eq: a Math \[text\] that says "$t = frac(L, v) = frac(theta, omega) arrow.r.double v = frac(omega L, theta)$"
- beam\_fig: a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0))
- beam\_text: a Text \[text\] that says "Il rivelatore ricostruisce la curva di Maxwell."
- bz\_formula: a Math \[text\] that says "$frac(N\_1, N\_0) = e^(- frac(Delta E, k\_B T))$"
- bz\_note: a Math \[text\] that says "$k\_B T approx 0.025 thin upright("eV") approx frac(1, 40) thin upright("eV")$"
- detector: a Polygon \[green\] drawn in beam\_fig (vertices=((5.4, 1.1), (5.8, 1.1), (5.8, 1.9), (5.4, 1.9)))
- disks: a Line \[blue\] drawn in beam\_fig (start=(2.0, 0.4), end=(2.0, 1.35))
- disks\_2: a Line \[blue\] drawn in beam\_fig (start=(2.0, 1.65), end=(2.0, 2.6))
- disks\_3: a Line \[blue\] drawn in beam\_fig (start=(4.2, 0.4), end=(4.2, 1.35))
- disks\_4: a Line \[blue\] drawn in beam\_fig (start=(4.2, 1.65), end=(4.2, 2.6))
- gap\_L: a Line \[gray\] labelled "L" drawn in beam\_fig (start=(2.0, 0.2), end=(4.2, 0.2))
- gap\_arrow: an Arrow \[yellow\] labelled "Delta E" drawn in levels\_fig (start=(0.25, 0.6), end=(0.25, 2.2))
- head\_beam: a Heading that says "Fasci molecolari e dischi rotanti"
- head\_bz: a Heading that says "Il fattore di Boltzmann"
- head\_mfp: a Heading that says "Il libero cammino medio"
- head\_mx: a Heading that says "La distribuzione di Maxwell"
- hits: a Point \[red\] drawn in mfp\_fig (location=(1.8, 1.8))
- hits\_2: a Point \[red\] drawn in mfp\_fig (location=(3.2, 1.25))
- hits\_3: a Point \[red\] drawn in mfp\_fig (location=(4.6, 1.7))
- hlen: a Line \[gray\] labelled "h" drawn in mfp\_fig (start=(0.4, 0.6), end=(5.6, 0.6))
- lab0: a Math \[text\] that says "$E\_0$" drawn in levels\_fig
- lab1: a Math \[text\] that says "$E\_1$" drawn in levels\_fig
- lev0: a Line \[text\] drawn in levels\_fig (start=(0.5, 0.6), end=(3.5, 0.6))
- lev1: a Line \[text\] drawn in levels\_fig (start=(0.5, 2.2), end=(3.5, 2.2))
- levels\_fig: a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 3.0), aspect=(4.0, 3.0))
- lower: a Point \[blue\] drawn in levels\_fig (location=(0.9, 0.8))
- lower\_2: a Point \[blue\] drawn in levels\_fig (location=(1.4, 0.8))
- lower\_3: a Point \[blue\] drawn in levels\_fig (location=(1.9, 0.8))
- lower\_4: a Point \[blue\] drawn in levels\_fig (location=(2.4, 0.8))
- lower\_5: a Point \[blue\] drawn in levels\_fig (location=(2.9, 0.8))
- lower\_6: a Point \[blue\] drawn in levels\_fig (location=(3.3, 0.8))
- mfp: a Derivation \[text\] that says "$N &= n thin pi d^2 h \\ lambda &= frac(h, sqrt(2) thin N) = frac(1, sqrt(2) thin n pi d^2) \\ n &= frac(P, k\_B T) \\ lambda &= frac(k\_B T, sqrt(2) thin P pi d^2) \\ &approx 230 thin upright("nm") approx 10^3 thin d$"
- mfp\_fig: a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0))
- mx0: a Math \[text\] that says "$f(v) prop v^2 e^(- frac(m v^2, 2 k\_B T))$"
- mx1: a Math \[text\] that says "$v\_(upright("mp")) = sqrt(frac(2 R T, A))$"
- mx2: a Math \[text\] that says "$v\_(upright("rms")) = sqrt(frac(3 R T, A))$"
- mx\_axes: an Axes (x\_range=(0.0, 1500.0), y\_range=(0.0, 1.2), x\_ticks\_every=500.0)
- mx\_curve: a FunctionPlot \[blue\] drawn in mx\_axes (function=\<function\>)
- omega\_label: a Math \[text\] that says "$omega$" drawn in beam\_fig
- others: a Point \[gray\] drawn in mfp\_fig (location=(1.2, 2.5))
- others\_2: a Point \[gray\] drawn in mfp\_fig (location=(2.6, 0.3))
- others\_3: a Point \[gray\] drawn in mfp\_fig (location=(3.9, 2.6))
- others\_4: a Point \[gray\] drawn in mfp\_fig (location=(5.0, 0.3))
- oven: a Polygon \[red\] drawn in beam\_fig (vertices=((0.2, 0.9), (1.0, 0.9), (1.0, 2.1), (0.2, 2.1)))
- oven\_label: a Math \[text\] that says "$T$" drawn in beam\_fig
- p\_mp: a PlotPoint \[yellow\] labelled "v\_(upright("mp"))" drawn in mx\_axes (target='mx\_curve', x=422.0)
- p\_rms: a PlotPoint \[red\] labelled "v\_(upright("rms"))" drawn in mx\_axes (target='mx\_curve', x=517.0)
- pos: a VariableNumber (initial\_value=1.1)
- probe: a Point \[yellow\] drawn in mfp\_fig (location=(\<VariableNumber px = 5.3\>, 1.5), marker\_radius=0.09)
- px: a VariableNumber (initial\_value=0.5)
- radius\_d: a Line \[red\] labelled "d" drawn in mfp\_fig (start=(0.7, 1.5), end=(0.7, 2.0))
- track: a Line \[gray\] drawn in mfp\_fig (start=(0.4, 1.5), end=(5.6, 1.5), dashed=True)
- tube: a Polygon \[blue\] drawn in mfp\_fig (vertices=((0.4, 1.0), (5.6, 1.0), (5.6, 2.0), (0.4, 2.0)), fill\_opacity=0.15)
- upper: a Point \[blue\] drawn in levels\_fig (location=(1.4, 2.4))
- upper\_2: a Point \[blue\] drawn in levels\_fig (location=(2.4, 2.4))

#### Beats

##### [04:37.529](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=277.52864583333337)

Narration: Finora abbiamo usato solo valori medi. Ma le molecole non hanno tutte la stessa velocità: si distribuiscono secondo la curva di Maxwell, qui per l'azoto a trecento kelvin.

Board: Empty.

Actions:
- [04:37.529](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=277.52864583333337): head\_mx is shown on the screen, written out.
- [04:37.529](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=277.52864583333337): mx\_axes is shown on the screen, written out.
- [04:43.636](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=283.6356458333334): mx\_curve is shown on the screen, drawn.
- [04:45.369](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=285.36864583333335): mx\_axes moves to a new place on the board.
- [04:45.369](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=285.36864583333335): mx0 is shown on the screen, written out.

##### [04:48.956](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=288.9561458333334)

Narration: Il fattore v quadro conta le direzioni possibili, mentre il termine esponenziale punisce le velocità alte.

Board: mx0 — a Math \[text\] that says "$f(v) prop v^2 e^(- frac(m v^2, 2 k\_B T))$"; mx\_axes — an Axes (x\_range=(0.0, 1500.0), y\_range=(0.0, 1.2), x\_ticks\_every=500.0); head\_mx — a Heading that says "La distribuzione di Maxwell"; mx\_curve — a FunctionPlot \[blue\] drawn in mx\_axes (function=\<function\>)

Actions:
- [04:49.329](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=289.3286458333334): mx0 (the "v^2" part) is emphasized.
- [04:53.009](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=293.0086458333334): mx0 (the "e^(- frac(m v^2, 2 k\_B T))" part) is emphasized.
- [04:53.009](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=293.0086458333334): mx0 (the "v^2" part) is no longer emphasized.
- [04:55.529](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=295.52864583333337): mx0 (the "e^(- frac(m v^2, 2 k\_B T))" part) is no longer emphasized.

##### [04:56.129](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=296.1286458333334)

Narration: Il massimo cade alla velocità più probabile, la radice di due R T su A. Un po' più a destra sta la velocità quadratica media, radice di tre R T su A.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [04:56.569](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=296.5686458333334): p\_mp is shown on the screen, written out.
- [04:59.209](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=299.2086458333334): mx1 is shown on the screen, written out.
- [05:2.009](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=302.0086458333334): p\_rms is shown on the screen, written out.
- [05:3.129](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=303.1286458333334): mx2 is shown on the screen, written out.
- [05:6.451](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=306.4511458333334): head\_mx is hidden from the screen — left the board.
- [05:6.451](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=306.4511458333334): mx0 is hidden from the screen — left the board.
- [05:6.451](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=306.4511458333334): mx1 is hidden from the screen — left the board.
- [05:6.451](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=306.4511458333334): mx2 is hidden from the screen — left the board.
- [05:6.451](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=306.4511458333334): mx\_axes is hidden from the screen — left the board.
- [05:6.451](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=306.4511458333334): mx\_curve is hidden from the screen — mx\_axes left the board.
- [05:6.451](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=306.4511458333334): p\_mp is hidden from the screen — mx\_axes left the board.
- [05:6.451](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=306.4511458333334): p\_rms is hidden from the screen — mx\_axes left the board.

##### [05:7.651](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=307.6511458333334)

Narration: Come si verifica questa curva? Si scalda un vapore in un forno a temperatura T e lo si lascia uscire da un piccolo foro, formando un fascio collimato.

Board: Empty.

Actions:
- [05:7.651](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=307.6511458333334): head\_beam is shown on the screen, written out.
- [05:7.651](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=307.6511458333334): beam\_fig is shown on the screen, written out.
- [05:11.369](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=311.3686458333334): oven is shown on the screen, written out.
- [05:11.369](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=311.3686458333334): oven\_label is shown on the screen, written out.
- [05:15.609](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=315.60864583333336): beam is shown on the screen, written out.

##### [05:17.364](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=317.36364583333335)

Narration: Il fascio incontra due dischi rotanti, distanti L, con le fenditure sfalsate di un angolo theta. Passa solo la molecola che vola da un disco al successivo nel tempo in cui i dischi ruotano di theta.

Board: beam\_fig — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0)); head\_beam — a Heading that says "Fasci molecolari e dischi rotanti"; oven — a Polygon \[red\] drawn in beam\_fig (vertices=((0.2, 0.9), (1.0, 0.9), (1.0, 2.1), (0.2, 2.1))); oven\_label — a Math \[text\] that says "$T$" drawn in beam\_fig; beam — a Line \[gray\] drawn in beam\_fig (start=(1.0, 1.5), end=(5.4, 1.5), dashed=True)

Actions:
- [05:18.929](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=318.92864583333335): disks is shown on the screen, written out.
- [05:19.017](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=319.0173772856854): disks\_2 is shown on the screen, written out.
- [05:19.106](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=319.10610873803745): disks\_3 is shown on the screen, written out.
- [05:19.195](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=319.1948401903895): disks\_4 is shown on the screen, written out.
- [05:19.289](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=319.28864583333336): omega\_label is shown on the screen, written out.
- [05:19.889](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=319.8886458333334): gap\_L is shown on the screen, written out.
- [05:23.849](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=323.84864583333336): detector is shown on the screen, written out.
- [05:23.849](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=323.84864583333336): atom is shown on the screen, written out.
- [05:25.249](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=325.2486458333334): atom is redrawn as the numbers it depends on change.
- [05:25.249](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=325.2486458333334): pos ticks to 5.3.

##### [05:29.934](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=329.93364583333334)

Narration: Quindi la velocità selezionata vale omega L su theta. Variando omega si scorrono tutte le velocità, e il rivelatore conta le molecole: l'istogramma ottenuto coincide con la curva di Maxwell.

Board: beam\_fig — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0)); head\_beam — a Heading that says "Fasci molecolari e dischi rotanti"; oven — a Polygon \[red\] drawn in beam\_fig (vertices=((0.2, 0.9), (1.0, 0.9), (1.0, 2.1), (0.2, 2.1))); oven\_label — a Math \[text\] that says "$T$" drawn in beam\_fig; beam — a Line \[gray\] drawn in beam\_fig (start=(1.0, 1.5), end=(5.4, 1.5), dashed=True); disks — a Line \[blue\] drawn in beam\_fig (start=(2.0, 0.4), end=(2.0, 1.35)); disks\_2 — a Line \[blue\] drawn in beam\_fig (start=(2.0, 1.65), end=(2.0, 2.6)); disks\_3 — a Line \[blue\] drawn in beam\_fig (start=(4.2, 0.4), end=(4.2, 1.35)); disks\_4 — a Line \[blue\] drawn in beam\_fig (start=(4.2, 1.65), end=(4.2, 2.6)); omega\_label — a Math \[text\] that says "$omega$" drawn in beam\_fig; gap\_L — a Line \[gray\] labelled "L" drawn in beam\_fig (start=(2.0, 0.2), end=(4.2, 0.2)); detector — a Polygon \[green\] drawn in beam\_fig (vertices=((5.4, 1.1), (5.8, 1.1), (5.8, 1.9), (5.4, 1.9))); atom — a Point \[yellow\] drawn in beam\_fig (location=(\<VariableNumber pos = 5.3\>, 1.5), marker\_radius=0.08)

Actions:
- [05:30.169](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=330.16864583333336): beam\_fig moves to a new place on the board.
- [05:30.169](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=330.16864583333336): beam\_eq is shown on the screen, written out.
- [05:37.209](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=337.2086458333334): detector is indicated — a transient flash.
- [05:40.249](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=340.24864583333334): beam\_text is shown on the screen, written out.
- [05:42.239](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=342.2386458333334): beam\_eq is hidden from the screen — left the board.
- [05:42.239](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=342.2386458333334): beam\_fig is hidden from the screen — left the board.
- [05:42.239](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=342.2386458333334): oven is hidden from the screen — beam\_fig left the board.
- [05:42.239](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=342.2386458333334): oven\_label is hidden from the screen — beam\_fig left the board.
- [05:42.239](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=342.2386458333334): beam is hidden from the screen — beam\_fig left the board.
- [05:42.239](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=342.2386458333334): disks is hidden from the screen — beam\_fig left the board.
- [05:42.239](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=342.2386458333334): disks\_2 is hidden from the screen — beam\_fig left the board.
- [05:42.239](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=342.2386458333334): disks\_3 is hidden from the screen — beam\_fig left the board.
- [05:42.239](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=342.2386458333334): disks\_4 is hidden from the screen — beam\_fig left the board.
- [05:42.239](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=342.2386458333334): omega\_label is hidden from the screen — beam\_fig left the board.
- [05:42.239](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=342.2386458333334): gap\_L is hidden from the screen — beam\_fig left the board.
- [05:42.239](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=342.2386458333334): detector is hidden from the screen — beam\_fig left the board.
- [05:42.239](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=342.2386458333334): atom is hidden from the screen — beam\_fig left the board.
- [05:42.239](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=342.2386458333334): beam\_text is hidden from the screen — left the board.
- [05:42.239](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=342.2386458333334): head\_beam is hidden from the screen — left the board.

##### [05:43.439](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=343.43864583333334)

Narration: La stessa logica vale per livelli di energia discreti. Uno stato eccitato sta sopra il fondamentale di una quantità delta E.

Board: Empty.

Actions:
- [05:43.439](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=343.43864583333334): head\_bz is shown on the screen, written out.
- [05:43.439](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=343.43864583333334): levels\_fig is shown on the screen, written out.
- [05:43.439](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=343.43864583333334): lev0 is shown on the screen, written out.
- [05:43.439](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=343.43864583333334): lab0 is shown on the screen, written out.
- [05:47.449](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=347.4486458333334): lev1 is shown on the screen, written out.
- [05:47.449](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=347.4486458333334): lab1 is shown on the screen, written out.
- [05:50.169](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=350.16864583333336): gap\_arrow is shown on the screen, written out.

##### [05:51.616](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=351.61614583333335)

Narration: Il rapporto tra le popolazioni dei due livelli è il fattore di Boltzmann: e alla meno delta E su kappa B T.

Board: levels\_fig — a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 3.0), aspect=(4.0, 3.0)); head\_bz — a Heading that says "Il fattore di Boltzmann"; lev0 — a Line \[text\] drawn in levels\_fig (start=(0.5, 0.6), end=(3.5, 0.6)); lab0 — a Math \[text\] that says "$E\_0$" drawn in levels\_fig; lev1 — a Line \[text\] drawn in levels\_fig (start=(0.5, 2.2), end=(3.5, 2.2)); lab1 — a Math \[text\] that says "$E\_1$" drawn in levels\_fig; gap\_arrow — an Arrow \[yellow\] labelled "Delta E" drawn in levels\_fig (start=(0.25, 0.6), end=(0.25, 2.2))

Actions:
- [05:52.689](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=352.68864583333334): lower is shown on the screen, written out.
- [05:52.775](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=352.77496296888324): lower\_2 is shown on the screen, written out.
- [05:52.861](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=352.8612801044331): lower\_3 is shown on the screen, written out.
- [05:52.948](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=352.947597239983): lower\_4 is shown on the screen, written out.
- [05:53.034](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=353.03391437553285): lower\_5 is shown on the screen, written out.
- [05:53.12](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=353.1202315110827): lower\_6 is shown on the screen, written out.
- [05:53.649](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=353.6486458333334): upper is shown on the screen, written out.
- [05:53.752](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=353.7522263959932): upper\_2 is shown on the screen, written out.
- [05:54.369](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=354.36864583333335): bz\_formula is shown on the screen, written out.

##### [05:59.364](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=359.36364583333335)

Narration: A temperatura ambiente kappa B T vale circa zero virgola zero due cinque elettronvolt, un quarantesimo. Se delta E è molto più grande, lo stato eccitato resta quasi vuoto.

Board: bz\_formula — a Math \[text\] that says "$frac(N\_1, N\_0) = e^(- frac(Delta E, k\_B T))$"; levels\_fig — a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 3.0), aspect=(4.0, 3.0)); head\_bz — a Heading that says "Il fattore di Boltzmann"; lev0 — a Line \[text\] drawn in levels\_fig (start=(0.5, 0.6), end=(3.5, 0.6)); lab0 — a Math \[text\] that says "$E\_0$" drawn in levels\_fig; lev1 — a Line \[text\] drawn in levels\_fig (start=(0.5, 2.2), end=(3.5, 2.2)); lab1 — a Math \[text\] that says "$E\_1$" drawn in levels\_fig; gap\_arrow — an Arrow \[yellow\] labelled "Delta E" drawn in levels\_fig (start=(0.25, 0.6), end=(0.25, 2.2)); lower — a Point \[blue\] drawn in levels\_fig (location=(0.9, 0.8)); lower\_2 — a Point \[blue\] drawn in levels\_fig (location=(1.4, 0.8)); lower\_3 — a Point \[blue\] drawn in levels\_fig (location=(1.9, 0.8)); lower\_4 — a Point \[blue\] drawn in levels\_fig (location=(2.4, 0.8)); lower\_5 — a Point \[blue\] drawn in levels\_fig (location=(2.9, 0.8)); lower\_6 — a Point \[blue\] drawn in levels\_fig (location=(3.3, 0.8)); upper — a Point \[blue\] drawn in levels\_fig (location=(1.4, 2.4)); upper\_2 — a Point \[blue\] drawn in levels\_fig (location=(2.4, 2.4))

Actions:
- [06:0.249](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=360.2486458333334): bz\_note is shown on the screen, written out.
- [06:10.089](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=370.0886458333334): upper is hidden from the screen.
- [06:10.089](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=370.0886458333334): upper\_2 is hidden from the screen.

##### [06:11.371](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=371.37114583333334)

Narration: Per questo, per allineare gli spin nucleari in un campo magnetico, Madame Wu dovette raffreddare i suoi campioni sotto un milionesimo di kelvin.

Board: bz\_formula — a Math \[text\] that says "$frac(N\_1, N\_0) = e^(- frac(Delta E, k\_B T))$"; bz\_note — a Math \[text\] that says "$k\_B T approx 0.025 thin upright("eV") approx frac(1, 40) thin upright("eV")$"; levels\_fig — a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 3.0), aspect=(4.0, 3.0)); head\_bz — a Heading that says "Il fattore di Boltzmann"; lev0 — a Line \[text\] drawn in levels\_fig (start=(0.5, 0.6), end=(3.5, 0.6)); lab0 — a Math \[text\] that says "$E\_0$" drawn in levels\_fig; lev1 — a Line \[text\] drawn in levels\_fig (start=(0.5, 2.2), end=(3.5, 2.2)); lab1 — a Math \[text\] that says "$E\_1$" drawn in levels\_fig; gap\_arrow — an Arrow \[yellow\] labelled "Delta E" drawn in levels\_fig (start=(0.25, 0.6), end=(0.25, 2.2)); lower — a Point \[blue\] drawn in levels\_fig (location=(0.9, 0.8)); lower\_2 — a Point \[blue\] drawn in levels\_fig (location=(1.4, 0.8)); lower\_3 — a Point \[blue\] drawn in levels\_fig (location=(1.9, 0.8)); lower\_4 — a Point \[blue\] drawn in levels\_fig (location=(2.4, 0.8)); lower\_5 — a Point \[blue\] drawn in levels\_fig (location=(2.9, 0.8)); lower\_6 — a Point \[blue\] drawn in levels\_fig (location=(3.3, 0.8))

Actions:
- [06:12.449](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=372.4486458333334): bz\_formula is indicated — a transient flash.
- [06:19.566](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=379.5661458333334): bz\_formula is hidden from the screen — left the board.
- [06:19.566](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=379.5661458333334): bz\_note is hidden from the screen — left the board.
- [06:19.566](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=379.5661458333334): head\_bz is hidden from the screen — left the board.
- [06:19.566](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=379.5661458333334): levels\_fig is hidden from the screen — left the board.
- [06:19.566](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=379.5661458333334): lev0 is hidden from the screen — levels\_fig left the board.
- [06:19.566](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=379.5661458333334): lab0 is hidden from the screen — levels\_fig left the board.
- [06:19.566](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=379.5661458333334): lev1 is hidden from the screen — levels\_fig left the board.
- [06:19.566](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=379.5661458333334): lab1 is hidden from the screen — levels\_fig left the board.
- [06:19.566](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=379.5661458333334): gap\_arrow is hidden from the screen — levels\_fig left the board.
- [06:19.566](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=379.5661458333334): lower is hidden from the screen — levels\_fig left the board.
- [06:19.566](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=379.5661458333334): lower\_2 is hidden from the screen — levels\_fig left the board.
- [06:19.566](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=379.5661458333334): lower\_3 is hidden from the screen — levels\_fig left the board.
- [06:19.566](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=379.5661458333334): lower\_4 is hidden from the screen — levels\_fig left the board.
- [06:19.566](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=379.5661458333334): lower\_5 is hidden from the screen — levels\_fig left the board.
- [06:19.566](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=379.5661458333334): lower\_6 is hidden from the screen — levels\_fig left the board.

##### [06:20.766](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=380.7661458333334)

Narration: Ultima grandezza: il libero cammino medio, cioè la distanza media percorsa da una molecola tra due urti. Pensiamo le molecole come sfere rigide di diametro d.

Board: Empty.

Actions:
- [06:20.766](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=380.7661458333334): head\_mfp is shown on the screen, written out.
- [06:20.766](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=380.7661458333334): mfp\_fig is shown on the screen, written out.
- [06:20.766](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=380.7661458333334): track is shown on the screen, written out.
- [06:26.129](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=386.1286458333334): probe is shown on the screen, written out.
- [06:28.449](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=388.4486458333334): others is shown on the screen, written out.
- [06:28.541](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=388.54067969484845): others\_2 is shown on the screen, written out.
- [06:28.633](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=388.63271355636357): others\_3 is shown on the screen, written out.
- [06:28.725](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=388.7247474178787): others\_4 is shown on the screen, written out.
- [06:30.129](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=390.1286458333334): radius\_d is shown on the screen, written out.

##### [06:31.661](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=391.66114583333336)

Narration: Avanzando di un tratto h, la molecola urta tutte quelle il cui centro cade entro una distanza d dalla sua traiettoria: un cilindro di volume pi greco d quadro per h.

Board: mfp\_fig — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0)); head\_mfp — a Heading that says "Il libero cammino medio"; track — a Line \[gray\] drawn in mfp\_fig (start=(0.4, 1.5), end=(5.6, 1.5), dashed=True); probe — a Point \[yellow\] drawn in mfp\_fig (location=(\<VariableNumber px = 5.3\>, 1.5), marker\_radius=0.09); others — a Point \[gray\] drawn in mfp\_fig (location=(1.2, 2.5)); others\_2 — a Point \[gray\] drawn in mfp\_fig (location=(2.6, 0.3)); others\_3 — a Point \[gray\] drawn in mfp\_fig (location=(3.9, 2.6)); others\_4 — a Point \[gray\] drawn in mfp\_fig (location=(5.0, 0.3)); radius\_d — a Line \[red\] labelled "d" drawn in mfp\_fig (start=(0.7, 1.5), end=(0.7, 2.0))

Actions:
- [06:31.929](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=391.9286458333334): probe is redrawn as the numbers it depends on change.
- [06:31.929](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=391.9286458333334): px ticks to 5.3.
- [06:32.649](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=392.6486458333334): hlen is shown on the screen, written out.
- [06:34.169](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=394.1686458333334): hits is shown on the screen, written out.
- [06:34.355](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=394.35500757969515): hits\_2 is shown on the screen, written out.
- [06:34.539](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=394.5392487439363): hits\_3 is shown on the screen, written out.
- [06:39.129](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=399.1286458333334): tube is shown on the screen, faded in.

##### [06:42.639](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=402.6386458333334)

Narration: Con densità numerica n, il numero di urti è n pi greco d quadro h. Il cammino medio è h diviso questo numero, e il moto delle altre molecole introduce un fattore radice di due.

Board: mfp\_fig — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0)); head\_mfp — a Heading that says "Il libero cammino medio"; track — a Line \[gray\] drawn in mfp\_fig (start=(0.4, 1.5), end=(5.6, 1.5), dashed=True); probe — a Point \[yellow\] drawn in mfp\_fig (location=(\<VariableNumber px = 5.3\>, 1.5), marker\_radius=0.09); others — a Point \[gray\] drawn in mfp\_fig (location=(1.2, 2.5)); others\_2 — a Point \[gray\] drawn in mfp\_fig (location=(2.6, 0.3)); others\_3 — a Point \[gray\] drawn in mfp\_fig (location=(3.9, 2.6)); others\_4 — a Point \[gray\] drawn in mfp\_fig (location=(5.0, 0.3)); radius\_d — a Line \[red\] labelled "d" drawn in mfp\_fig (start=(0.7, 1.5), end=(0.7, 2.0)); hlen — a Line \[gray\] labelled "h" drawn in mfp\_fig (start=(0.4, 0.6), end=(5.6, 0.6)); hits — a Point \[red\] drawn in mfp\_fig (location=(1.8, 1.8)); hits\_2 — a Point \[red\] drawn in mfp\_fig (location=(3.2, 1.25)); hits\_3 — a Point \[red\] drawn in mfp\_fig (location=(4.6, 1.7)); tube — a Polygon \[blue\] drawn in mfp\_fig (vertices=((0.4, 1.0), (5.6, 1.0), (5.6, 2.0), (0.4, 2.0)), fill\_opacity=0.15)

Actions:
- [06:43.489](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=403.4886458333334): mfp\_fig moves to a new place on the board.
- [06:43.489](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=403.4886458333334): mfp is shown on the screen, written out.
- [06:48.209](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=408.2086458333334): mfp is shown on the screen, written out.

##### [06:54.944](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=414.94364583333333)

Narration: Per un gas perfetto n vale P su kappa B T. A trecento kelvin e un'atmosfera, con d di circa due angstrom, si ottengono circa duecentotrenta nanometri, mille volte il diametro.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [06:55.729](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=415.7286458333334): mfp is shown on the screen, written out.
- [06:59.609](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=419.6086458333334): mfp is shown on the screen, written out.
- [07:2.969](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=422.9686458333334): mfp is shown on the screen, written out.

##### [07:7.696](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=427.6961458333334)

Narration: A cinquecento metri al secondo, ogni molecola subisce miliardi di urti al secondo.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [07:11.009](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=431.0086458333334): A box is drawn around mfp.
- [07:13.063](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=433.0627291666667): head\_mfp is hidden from the screen — left the board.
- [07:13.063](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=433.0627291666667): mfp is hidden from the screen — left the board.
- [07:13.063](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=433.0627291666667): mfp\_fig is hidden from the screen — left the board.
- [07:13.063](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=433.0627291666667): track is hidden from the screen — mfp\_fig left the board.
- [07:13.063](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=433.0627291666667): probe is hidden from the screen — mfp\_fig left the board.
- [07:13.063](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=433.0627291666667): others is hidden from the screen — mfp\_fig left the board.
- [07:13.063](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=433.0627291666667): others\_2 is hidden from the screen — mfp\_fig left the board.
- [07:13.063](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=433.0627291666667): others\_3 is hidden from the screen — mfp\_fig left the board.
- [07:13.063](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=433.0627291666667): others\_4 is hidden from the screen — mfp\_fig left the board.
- [07:13.063](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=433.0627291666667): radius\_d is hidden from the screen — mfp\_fig left the board.
- [07:13.063](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=433.0627291666667): hlen is hidden from the screen — mfp\_fig left the board.
- [07:13.063](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=433.0627291666667): hits is hidden from the screen — mfp\_fig left the board.
- [07:13.063](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=433.0627291666667): hits\_2 is hidden from the screen — mfp\_fig left the board.
- [07:13.063](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=433.0627291666667): hits\_3 is hidden from the screen — mfp\_fig left the board.
- [07:13.063](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=433.0627291666667): tube is hidden from the screen — mfp\_fig left the board.

### Scene 4: [Entropia statistica](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=434.10439583333334)

Span: 07:14.104–09:30.517 (434.10439583333334s–570.516875s).

#### Objects

- binom\_eq: a Math \[text\] that says "$Omega = binom(6, k)$"
- head\_3: a Heading that says "Il terzo principio"
- head\_micro: a Heading that says "Macrostati e microstati"
- head\_sb: a Heading that says "La formula di Boltzmann"
- joule\_box: a Polygon \[gray\] drawn in joule\_fig (vertices=((0.2, 0.2), (5.8, 0.2), (5.8, 2.8), (0.2, 2.8)), filled=False)
- joule\_fig: a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0))
- macro\_table: a Table \[text\] that says "Sinistra Destra $Omega$ 6 0 1 5 1 6 4 2 15 3 3 20" (rows=(('Sinistra', 'Destra', '$Omega$'), ('6', '0', '1'), ('5', '1',…, header=True)
- moving: a Point \[yellow\] labelled "4" drawn in joule\_fig (location=(\<VariableNumber xa = 3.9\>, 1.8))
- moving\_2: a Point \[yellow\] labelled "5" drawn in joule\_fig (location=(\<VariableNumber xb = 4.6\>, 0.6))
- moving\_3: a Point \[yellow\] labelled "6" drawn in joule\_fig (location=(\<VariableNumber xc = 5.2\>, 2.4))
- nernst: a Math \[text\] that says "$T arrow.r 0 thin arrow.r.double thin Omega arrow.r 1 thin arrow.r.double thin S = k\_B ln 1 = 0$"
- nernst\_def: a Panel that says "Per $T arrow.r 0$ l'entropia di ogni sistema tende a un valore costante, che si pone uguale a zero."
- partition: a Line \[gray\] drawn in joule\_fig (start=(3.0, 0.2), end=(3.0, 2.8), dashed=True)
- s\_final: a Math \[text\] that says "$S = k\_B ln Omega$"
- sder: a Derivation \[text\] that says "$S\_(A+B) &= S\_A + S\_B \\ Omega\_(A+B) &= Omega\_A dot Omega\_B \\ S &= C ln Omega \\ Delta S &= C ln 2^N = N C ln 2 \\ Delta S &= n R ln 2 arrow.r.double C = frac(R, N\_A) = k\_B$"
- static: a Point \[blue\] labelled "1" drawn in joule\_fig (location=(0.8, 0.9))
- static\_2: a Point \[blue\] labelled "2" drawn in joule\_fig (location=(1.5, 2.2))
- static\_3: a Point \[blue\] labelled "3" drawn in joule\_fig (location=(2.3, 1.4))
- xa: a VariableNumber (initial\_value=0.9)
- xb: a VariableNumber (initial\_value=1.7)
- xc: a VariableNumber (initial\_value=2.5)

#### Beats

##### [07:14.104](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=434.10439583333334)

Narration: Veniamo alla seconda formula. Uno stato macroscopico, fissato da poche variabili come P, V e T, è realizzato da moltissimi microstati: le posizioni e le velocità di tutte le molecole.

Board: Empty.

Actions:
- [07:14.104](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=434.10439583333334): head\_micro is shown on the screen, written out.
- [07:14.104](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=434.10439583333334): joule\_fig is shown on the screen, written out.
- [07:14.104](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=434.10439583333334): joule\_box is shown on the screen, written out.
- [07:22.904](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=442.90439583333335): static is shown on the screen, written out.
- [07:23.071](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=443.0711875312578): static\_2 is shown on the screen, written out.
- [07:23.238](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=443.2379792291823): static\_3 is shown on the screen, written out.
- [07:23.405](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=443.40477092710677): moving is shown on the screen, written out.
- [07:23.572](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=443.5715626250313): moving\_2 is shown on the screen, written out.
- [07:23.738](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=443.73835432295573): moving\_3 is shown on the screen, written out.

##### [07:27.539](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=447.53939583333334)

Narration: Prendiamo sei molecole distinguibili in una scatola divisa a metà. Se stanno tutte a sinistra, esiste un solo modo: un microstato.

Board: joule\_fig — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0)); head\_micro — a Heading that says "Macrostati e microstati"; joule\_box — a Polygon \[gray\] drawn in joule\_fig (vertices=((0.2, 0.2), (5.8, 0.2), (5.8, 2.8), (0.2, 2.8)), filled=False); static — a Point \[blue\] labelled "1" drawn in joule\_fig (location=(0.8, 0.9)); static\_2 — a Point \[blue\] labelled "2" drawn in joule\_fig (location=(1.5, 2.2)); static\_3 — a Point \[blue\] labelled "3" drawn in joule\_fig (location=(2.3, 1.4)); moving — a Point \[yellow\] labelled "4" drawn in joule\_fig (location=(\<VariableNumber xa = 3.9\>, 1.8)); moving\_2 — a Point \[yellow\] labelled "5" drawn in joule\_fig (location=(\<VariableNumber xb = 4.6\>, 0.6)); moving\_3 — a Point \[yellow\] labelled "6" drawn in joule\_fig (location=(\<VariableNumber xc = 5.2\>, 2.4))

Actions:
- [07:28.304](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=448.30439583333333): joule\_fig moves to a new place on the board.
- [07:28.304](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=448.30439583333333): macro\_table is shown on the screen, written out.
- [07:30.544](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=450.54439583333334): partition is shown on the screen, written out.
- [07:33.624](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=453.6243958333333): macro\_table is shown on the screen, written out.

##### [07:36.204](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=456.20439583333336)

Narration: Spostiamone una a destra: può essere una qualsiasi delle sei, quindi sei microstati. Con due a destra i modi diventano quindici, e con tre diventano venti.

Board: joule\_fig — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0)); head\_micro — a Heading that says "Macrostati e microstati"; joule\_box — a Polygon \[gray\] drawn in joule\_fig (vertices=((0.2, 0.2), (5.8, 0.2), (5.8, 2.8), (0.2, 2.8)), filled=False); static — a Point \[blue\] labelled "1" drawn in joule\_fig (location=(0.8, 0.9)); static\_2 — a Point \[blue\] labelled "2" drawn in joule\_fig (location=(1.5, 2.2)); static\_3 — a Point \[blue\] labelled "3" drawn in joule\_fig (location=(2.3, 1.4)); moving — a Point \[yellow\] labelled "4" drawn in joule\_fig (location=(\<VariableNumber xa = 3.9\>, 1.8)); moving\_2 — a Point \[yellow\] labelled "5" drawn in joule\_fig (location=(\<VariableNumber xb = 4.6\>, 0.6)); moving\_3 — a Point \[yellow\] labelled "6" drawn in joule\_fig (location=(\<VariableNumber xc = 5.2\>, 2.4)); partition — a Line \[gray\] drawn in joule\_fig (start=(3.0, 0.2), end=(3.0, 2.8), dashed=True)

Actions:
- [07:36.424](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=456.42439583333334): moving is redrawn as the numbers it depends on change.
- [07:36.424](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=456.42439583333334): xa ticks to 3.9.
- [07:40.824](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=460.82439583333337): macro\_table is shown on the screen, written out.
- [07:42.104](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=462.10439583333334): moving\_2 is redrawn as the numbers it depends on change.
- [07:42.104](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=462.10439583333334): xb ticks to 4.6.
- [07:43.784](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=463.78439583333335): macro\_table is shown on the screen, written out.
- [07:44.824](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=464.82439583333337): moving\_3 is redrawn as the numbers it depends on change.
- [07:44.824](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=464.82439583333337): xc ticks to 5.2.
- [07:45.704](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=465.70439583333336): macro\_table is shown on the screen, written out.

##### [07:46.802](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=466.80189583333333)

Narration: In generale il numero di modi è il coefficiente binomiale, sei su k. Già con sei molecole, la divisione a metà è venti volte più probabile del caso estremo.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [07:47.184](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=467.1843958333333): binom\_eq is shown on the screen, written out.
- [07:54.504](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=474.5043958333333): macro\_table (the "row=5" part) is emphasized.
- [07:56.879](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=476.8793958333333): macro\_table (the "row=5" part) is no longer emphasized.

##### [07:57.479](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=477.47939583333334)

Narration: Con dieci alla ventitré molecole, il macrostato uniforme ha immensamente più microstati di qualsiasi altro: è l'unico che osserviamo davvero.

Board: binom\_eq — a Math \[text\] that says "$Omega = binom(6, k)$"; joule\_fig — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0)); head\_micro — a Heading that says "Macrostati e microstati"; joule\_box — a Polygon \[gray\] drawn in joule\_fig (vertices=((0.2, 0.2), (5.8, 0.2), (5.8, 2.8), (0.2, 2.8)), filled=False); static — a Point \[blue\] labelled "1" drawn in joule\_fig (location=(0.8, 0.9)); static\_2 — a Point \[blue\] labelled "2" drawn in joule\_fig (location=(1.5, 2.2)); static\_3 — a Point \[blue\] labelled "3" drawn in joule\_fig (location=(2.3, 1.4)); moving — a Point \[yellow\] labelled "4" drawn in joule\_fig (location=(\<VariableNumber xa = 3.9\>, 1.8)); moving\_2 — a Point \[yellow\] labelled "5" drawn in joule\_fig (location=(\<VariableNumber xb = 4.6\>, 0.6)); moving\_3 — a Point \[yellow\] labelled "6" drawn in joule\_fig (location=(\<VariableNumber xc = 5.2\>, 2.4)); partition — a Line \[gray\] drawn in joule\_fig (start=(3.0, 0.2), end=(3.0, 2.8), dashed=True)

Actions:
- [08:0.344](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=480.34439583333335): macro\_table (the "row=5" part) is indicated — a transient flash.
- [08:6.024](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=486.02439583333336): joule\_fig moves to a new place on the board.
- [08:6.024](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=486.02439583333336): binom\_eq is hidden from the screen — left the board.
- [08:6.024](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=486.02439583333336): head\_micro is hidden from the screen — left the board.
- [08:6.024](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=486.02439583333336): macro\_table is hidden from the screen — left the board.

##### [08:7.224](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=487.22439583333335)

Narration: Due sistemi indipendenti: le loro entropie si sommano, perché l'entropia è estensiva. I loro microstati invece si moltiplicano, perché ogni microstato di uno si combina con ogni microstato dell'altro.

Board: joule\_fig — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0)); joule\_box — a Polygon \[gray\] drawn in joule\_fig (vertices=((0.2, 0.2), (5.8, 0.2), (5.8, 2.8), (0.2, 2.8)), filled=False); static — a Point \[blue\] labelled "1" drawn in joule\_fig (location=(0.8, 0.9)); static\_2 — a Point \[blue\] labelled "2" drawn in joule\_fig (location=(1.5, 2.2)); static\_3 — a Point \[blue\] labelled "3" drawn in joule\_fig (location=(2.3, 1.4)); moving — a Point \[yellow\] labelled "4" drawn in joule\_fig (location=(\<VariableNumber xa = 3.9\>, 1.8)); moving\_2 — a Point \[yellow\] labelled "5" drawn in joule\_fig (location=(\<VariableNumber xb = 4.6\>, 0.6)); moving\_3 — a Point \[yellow\] labelled "6" drawn in joule\_fig (location=(\<VariableNumber xc = 5.2\>, 2.4)); partition — a Line \[gray\] drawn in joule\_fig (start=(3.0, 0.2), end=(3.0, 2.8), dashed=True)

Actions:
- [08:7.224](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=487.22439583333335): head\_sb is shown on the screen, written out.
- [08:7.224](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=487.22439583333335): sder is shown on the screen, written out.
- [08:13.464](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=493.46439583333336): sder is shown on the screen, written out.

##### [08:20.467](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=500.46689583333335)

Narration: La sola funzione che trasforma prodotti in somme è il logaritmo. Dunque S deve essere una costante C per il logaritmo di omega.

Board: joule\_fig — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0)); joule\_box — a Polygon \[gray\] drawn in joule\_fig (vertices=((0.2, 0.2), (5.8, 0.2), (5.8, 2.8), (0.2, 2.8)), filled=False); static — a Point \[blue\] labelled "1" drawn in joule\_fig (location=(0.8, 0.9)); static\_2 — a Point \[blue\] labelled "2" drawn in joule\_fig (location=(1.5, 2.2)); static\_3 — a Point \[blue\] labelled "3" drawn in joule\_fig (location=(2.3, 1.4)); moving — a Point \[yellow\] labelled "4" drawn in joule\_fig (location=(\<VariableNumber xa = 3.9\>, 1.8)); moving\_2 — a Point \[yellow\] labelled "5" drawn in joule\_fig (location=(\<VariableNumber xb = 4.6\>, 0.6)); moving\_3 — a Point \[yellow\] labelled "6" drawn in joule\_fig (location=(\<VariableNumber xc = 5.2\>, 2.4)); partition — a Line \[gray\] drawn in joule\_fig (start=(3.0, 0.2), end=(3.0, 2.8), dashed=True); head\_sb — a Heading that says "La formula di Boltzmann"

Actions:
- [08:24.704](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=504.70439583333336): sder is shown on the screen, written out.

##### [08:29.362](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=509.36189583333334)

Narration: Fissiamo C con l'espansione libera di Joule: togliamo il setto, e ogni molecola ha a disposizione il doppio del volume. I microstati crescono di un fattore due alla N, e l'entropia di N C logaritmo di due.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [08:32.224](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=512.2243958333333): partition is hidden from the screen.
- [08:37.544](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=517.5443958333334): sder is shown on the screen, written out.

##### [08:42.964](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=522.9643958333334)

Narration: La termodinamica dà per la stessa espansione n R logaritmo di due. Confrontando, C è R diviso il numero di Avogadro: proprio la costante di Boltzmann.

Board: joule\_fig — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0)); joule\_box — a Polygon \[gray\] drawn in joule\_fig (vertices=((0.2, 0.2), (5.8, 0.2), (5.8, 2.8), (0.2, 2.8)), filled=False); static — a Point \[blue\] labelled "1" drawn in joule\_fig (location=(0.8, 0.9)); static\_2 — a Point \[blue\] labelled "2" drawn in joule\_fig (location=(1.5, 2.2)); static\_3 — a Point \[blue\] labelled "3" drawn in joule\_fig (location=(2.3, 1.4)); moving — a Point \[yellow\] labelled "4" drawn in joule\_fig (location=(\<VariableNumber xa = 3.9\>, 1.8)); moving\_2 — a Point \[yellow\] labelled "5" drawn in joule\_fig (location=(\<VariableNumber xb = 4.6\>, 0.6)); moving\_3 — a Point \[yellow\] labelled "6" drawn in joule\_fig (location=(\<VariableNumber xc = 5.2\>, 2.4)); head\_sb — a Heading that says "La formula di Boltzmann"

Actions:
- [08:43.344](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=523.3443958333334): sder is shown on the screen, written out.

##### [08:53.679](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=533.6793958333334)

Narration: Ecco la seconda formula: S uguale kappa B per il logaritmo di omega. Disordine vuol dire esattamente questo: più volume o più temperatura offrono più posizioni e più velocità, cioè più microstati.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [08:53.944](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=533.9443958333334): s\_final is shown on the screen, written out.
- [08:58.904](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=538.9043958333334): A box is drawn around s\_final.
- [09:7.419](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=547.4193958333334): head\_sb is hidden from the screen — left the board.
- [09:7.419](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=547.4193958333334): joule\_fig is hidden from the screen — left the board.
- [09:7.419](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=547.4193958333334): joule\_box is hidden from the screen — joule\_fig left the board.
- [09:7.419](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=547.4193958333334): static is hidden from the screen — joule\_fig left the board.
- [09:7.419](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=547.4193958333334): static\_2 is hidden from the screen — joule\_fig left the board.
- [09:7.419](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=547.4193958333334): static\_3 is hidden from the screen — joule\_fig left the board.
- [09:7.419](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=547.4193958333334): moving is hidden from the screen — joule\_fig left the board.
- [09:7.419](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=547.4193958333334): moving\_2 is hidden from the screen — joule\_fig left the board.
- [09:7.419](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=547.4193958333334): moving\_3 is hidden from the screen — joule\_fig left the board.
- [09:7.419](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=547.4193958333334): s\_final is hidden from the screen — left the board.
- [09:7.419](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=547.4193958333334): sder is hidden from the screen — left the board.

##### [09:8.619](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=548.6193958333333)

Narration: E allo zero assoluto? Quando la temperatura tende a zero, il sistema scivola nel suo unico stato fondamentale. Omega tende a uno, e quindi l'entropia tende a zero.

Board: Empty.

Actions:
- [09:8.619](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=548.6193958333333): head\_3 is shown on the screen, written out.
- [09:11.424](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=551.4243958333334): nernst is shown on the screen, written out.

##### [09:19.644](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=559.6443958333334)

Narration: È il terzo principio della termodinamica, il teorema di Nernst. Una sua conseguenza è che lo zero assoluto non si raggiunge con un numero finito di trasformazioni.

Board: nernst — a Math \[text\] that says "$T arrow.r 0 thin arrow.r.double thin Omega arrow.r 1 thin arrow.r.double thin S = k\_B ln 1 = 0$"; head\_3 — a Heading that says "Il terzo principio"

Actions:
- [09:20.144](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=560.1443958333334): nernst\_def is shown on the screen, written out.
- [09:29.475](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=569.4752083333333): head\_3 is hidden from the screen — left the board.
- [09:29.475](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=569.4752083333333): nernst is hidden from the screen — left the board.
- [09:29.475](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=569.4752083333333): nernst\_def is hidden from the screen — left the board.

### Scene 5: [Misurare gamma, e l'atmosfera](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=570.516875)

Span: 09:30.517–11:23.604 (570.516875s–683.6044583333334s).

#### Objects

- adiab\_eq: a Math \[text\] that says "$T P^(frac(1 - gamma, gamma)) = upright("cost")$"
- atm\_list: a Block \[text\] that says "Nubi: l'aria che sale si espande e si raffredda. Föhn: l'aria che scende si comprime e si scalda."
- ball: a Point \[red\] labelled "m" drawn in flask\_fig (location=(2.0, (3.6 + y)), marker\_radius=0.15)
- cloud: a Circle \[gray\] drawn in mount\_fig (center=(2.3, 2.3), radius=0.3, filled=True)
- fin\_ls: a Tex \[text\] that says "Entropia: conteggio dei microstati"
- fin\_lt: a Tex \[text\] that says "Temperatura: energia media"
- fin\_s: a Math \[text\] that says "$S = k\_B ln Omega$"
- fin\_t: a Math \[text\] that says "$overline(E)\_c = frac(3, 2) k\_B T$"
- flask\_fig: a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 5.0), aspect=(4.0, 5.0))
- head\_atm: a Heading that says "Adiabatiche in atmosfera"
- head\_fin: a Heading that says "I due ponti"
- head\_r: a Heading that says "Il metodo di Rüchardt"
- mount\_fig: a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0))
- mountain: a Polygon \[green\] drawn in mount\_fig (vertices=((0.2, 0.3), (3.0, 2.5), (5.8, 0.3)))
- neck\_l: a Line \[gray\] drawn in flask\_fig (start=(1.75, 2.6), end=(1.75, 4.7))
- neck\_label: a Math \[text\] that says "$S$" drawn in flask\_fig
- neck\_r: a Line \[gray\] drawn in flask\_fig (start=(2.25, 2.6), end=(2.25, 4.7))
- parcel: a Point \[yellow\] labelled "15 degree upright("C")" drawn in mount\_fig (location=(\<VariableNumber px = 5.3\>, \<VariableNumber py = 1.05\>))
- px: a VariableNumber (initial\_value=0.7)
- py: a VariableNumber (initial\_value=1.05)
- rder: a Derivation \[text\] that says "$Delta P &= - gamma P frac(S y, V) \\ m frac(dif^2 y, dif t^2) &= - frac(gamma P S^2, V) y \\ tau &= 2 pi sqrt(frac(m V, gamma P S^2)) \\ gamma &= frac(4 pi^2 m V, tau^2 P S^2)$"
- temp: a VariableNumber (initial\_value=15.0, format\_spec='.0f')
- vessel: a Polygon \[blue\] drawn in flask\_fig (vertices=((0.5, 0.3), (3.5, 0.3), (3.5, 2.6), (0.5, 2.6)), fill\_opacity=0.15)
- vessel\_label: a Math \[text\] that says "$V, thin P$" drawn in flask\_fig
- y: a VariableNumber

#### Beats

##### [09:30.517](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=570.516875)

Narration: Gamma si può misurare con un esperimento semplice, ideato da Rüchardt. Un grande recipiente rigido di volume V contiene il gas, e nel suo collo verticale di sezione S scorre senza attrito una sferetta di massa m.

Board: Empty.

Actions:
- [09:30.517](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=570.516875): head\_r is shown on the screen, written out.
- [09:30.517](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=570.516875): flask\_fig is shown on the screen, written out.
- [09:35.557](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=575.556875): vessel is shown on the screen, written out.
- [09:36.757](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=576.756875): vessel\_label is shown on the screen, written out.
- [09:39.157](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=579.156875): neck\_l is shown on the screen, written out.
- [09:39.157](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=579.156875): neck\_r is shown on the screen, written out.
- [09:40.117](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=580.116875): neck\_label is shown on the screen, written out.
- [09:42.877](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=582.876875): ball is shown on the screen, written out.

##### [09:44.969](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=584.969375)

Narration: Spostiamo la sferetta di un tratto y. Il gas si comprime e si espande così in fretta da non scambiare calore: la trasformazione è adiabatica, e la pressione cambia di meno gamma P per S y su V.

Board: flask\_fig — a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 5.0), aspect=(4.0, 5.0)); head\_r — a Heading that says "Il metodo di Rüchardt"; vessel — a Polygon \[blue\] drawn in flask\_fig (vertices=((0.5, 0.3), (3.5, 0.3), (3.5, 2.6), (0.5, 2.6)), fill\_opacity=0.15); vessel\_label — a Math \[text\] that says "$V, thin P$" drawn in flask\_fig; neck\_l — a Line \[gray\] drawn in flask\_fig (start=(1.75, 2.6), end=(1.75, 4.7)); neck\_r — a Line \[gray\] drawn in flask\_fig (start=(2.25, 2.6), end=(2.25, 4.7)); neck\_label — a Math \[text\] that says "$S$" drawn in flask\_fig; ball — a Point \[red\] labelled "m" drawn in flask\_fig (location=(2.0, (3.6 + y)), marker\_radius=0.15)

Actions:
- [09:45.197](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=585.196875): ball is redrawn as the numbers it depends on change.
- [09:45.197](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=585.196875): y ticks to -0.5.
- [09:54.397](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=594.396875): flask\_fig moves to a new place on the board.
- [09:54.397](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=594.396875): rder is shown on the screen, written out.

##### [09:58.734](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=598.734375)

Narration: La forza di richiamo è proporzionale allo spostamento: la sferetta oscilla di moto armonico. Chiamo tau il periodo, per non confonderlo con la temperatura.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [09:59.157](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=599.156875): rder is shown on the screen, written out.
- [10:2.677](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=602.676875): ball is redrawn as the numbers it depends on change.
- [10:2.677](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=602.676875): y ticks to 0.5.
- [10:3.557](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=603.556875): ball is redrawn as the numbers it depends on change.
- [10:3.557](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=603.556875): y ticks to -0.5.
- [10:4.517](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=604.516875): rder is shown on the screen, written out.
- [10:5.477](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=605.4768750000001): ball is redrawn as the numbers it depends on change.
- [10:5.477](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=605.4768750000001): y ticks to 0.0.

##### [10:8.714](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=608.714375)

Narration: Misurando tau, insieme a m, V, P e S, si ricava gamma. Per l'aria si trova circa uno virgola quattro, proprio il valore di un gas biatomico.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [10:8.957](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=608.956875): rder is shown on the screen, written out.
- [10:14.557](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=614.556875): rder is indicated — a transient flash.
- [10:18.429](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=618.429375): flask\_fig is hidden from the screen — left the board.
- [10:18.429](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=618.429375): vessel is hidden from the screen — flask\_fig left the board.
- [10:18.429](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=618.429375): vessel\_label is hidden from the screen — flask\_fig left the board.
- [10:18.429](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=618.429375): neck\_l is hidden from the screen — flask\_fig left the board.
- [10:18.429](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=618.429375): neck\_r is hidden from the screen — flask\_fig left the board.
- [10:18.429](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=618.429375): neck\_label is hidden from the screen — flask\_fig left the board.
- [10:18.429](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=618.429375): ball is hidden from the screen — flask\_fig left the board.
- [10:18.429](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=618.429375): head\_r is hidden from the screen — left the board.
- [10:18.429](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=618.429375): rder is hidden from the screen — left the board.

##### [10:19.629](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=619.629375)

Narration: Le trasformazioni adiabatiche governano anche il tempo atmosferico. Una massa di aria calda e umida sale lungo il versante di una montagna: in quota la pressione è minore, e la massa si espande.

Board: Empty.

Actions:
- [10:19.629](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=619.629375): head\_atm is shown on the screen, written out.
- [10:19.629](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=619.629375): mount\_fig is shown on the screen, written out.
- [10:19.629](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=619.629375): mountain is shown on the screen, written out.
- [10:23.917](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=623.916875): parcel is shown on the screen, written out.
- [10:25.517](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=625.516875): parcel is redrawn as the numbers it depends on change.
- [10:25.517](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=625.516875): px ticks to 3.0.
- [10:25.517](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=625.516875): py ticks to 2.85.
- [10:25.517](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=625.516875): temp ticks to 3.0.
- [10:28.477](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=628.4768750000001): mount\_fig moves to a new place on the board.
- [10:28.477](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=628.4768750000001): adiab\_eq is shown on the screen, written out.

##### [10:31.517](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=631.516875)

Narration: Si espande a spese della propria energia interna, e la temperatura scende. Il vapore raggiunge la saturazione e condensa in goccioline: nasce la nube.

Board: adiab\_eq — a Math \[text\] that says "$T P^(frac(1 - gamma, gamma)) = upright("cost")$"; mount\_fig — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0)); head\_atm — a Heading that says "Adiabatiche in atmosfera"; mountain — a Polygon \[green\] drawn in mount\_fig (vertices=((0.2, 0.3), (3.0, 2.5), (5.8, 0.3))); parcel — a Point \[yellow\] labelled "15 degree upright("C")" drawn in mount\_fig (location=(\<VariableNumber px = 5.3\>, \<VariableNumber py = 1.05\>))

Actions:
- [10:31.877](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=631.876875): atm\_list is shown on the screen, written out.
- [10:34.677](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=634.676875): atm\_list (the "si espande e si raffredda" part) is emphasized.
- [10:38.757](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=638.756875): cloud is shown on the screen, written out.
- [10:40.237](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=640.236875): atm\_list (the "si espande e si raffredda" part) is no longer emphasized.

##### [10:41.419](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=641.4193750000001)

Narration: Sul versante opposto succede il contrario. L'aria ormai secca scende verso la pianura e viene compressa adiabaticamente: si scalda di circa dieci gradi ogni chilometro, e arriva calda e asciutta. È il vento di Föhn.

Board: adiab\_eq — a Math \[text\] that says "$T P^(frac(1 - gamma, gamma)) = upright("cost")$"; atm\_list — a Block \[text\] that says "Nubi: l'aria che sale si espande e si raffredda. Föhn: l'aria che scende si comprime e si scalda."; mount\_fig — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.0), aspect=(6.0, 3.0)); head\_atm — a Heading that says "Adiabatiche in atmosfera"; mountain — a Polygon \[green\] drawn in mount\_fig (vertices=((0.2, 0.3), (3.0, 2.5), (5.8, 0.3))); parcel — a Point \[yellow\] labelled "15 degree upright("C")" drawn in mount\_fig (location=(\<VariableNumber px = 5.3\>, \<VariableNumber py = 1.05\>)); cloud — a Circle \[gray\] drawn in mount\_fig (center=(2.3, 2.3), radius=0.3, filled=True)

Actions:
- [10:45.437](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=645.436875): parcel is redrawn as the numbers it depends on change.
- [10:45.437](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=645.436875): px ticks to 5.3.
- [10:45.437](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=645.436875): py ticks to 1.05.
- [10:45.437](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=645.436875): temp ticks to 23.0.
- [10:47.237](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=647.236875): atm\_list (the "si comprime e si scalda" part) is emphasized.
- [10:53.837](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=653.836875): atm\_list (the "si comprime e si scalda" part) is no longer emphasized.
- [10:54.902](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=654.901875): adiab\_eq is hidden from the screen — left the board.
- [10:54.902](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=654.901875): atm\_list is hidden from the screen — left the board.
- [10:54.902](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=654.901875): head\_atm is hidden from the screen — left the board.
- [10:54.902](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=654.901875): mount\_fig is hidden from the screen — left the board.
- [10:54.902](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=654.901875): mountain is hidden from the screen — mount\_fig left the board.
- [10:54.902](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=654.901875): parcel is hidden from the screen — mount\_fig left the board.
- [10:54.902](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=654.901875): cloud is hidden from the screen — mount\_fig left the board.

##### [10:56.102](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=656.1018750000001)

Narration: Ricapitoliamo i due ponti promessi all'inizio. La temperatura è energia cinetica media, tre mezzi kappa B T per molecola: da qui discendono equipartizione, calori specifici e velocità molecolari.

Board: Empty.

Actions:
- [10:56.102](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=656.1018750000001): head\_fin is shown on the screen, written out.
- [10:59.197](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=659.196875): fin\_lt is shown on the screen, written out.
- [10:59.381](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=659.3809125586855): fin\_t is shown on the screen, written out.
- [11:5.277](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=665.276875): A box is drawn around fin\_t.

##### [11:9.264](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=669.264375)

Narration: L'entropia invece conta i microstati, kappa B per il logaritmo di omega, e porta con sé il terzo principio. Così il moto disordinato di miliardi di miliardi di molecole diventa la termodinamica che misuriamo.

Board: fin\_lt — a Tex \[text\] that says "Temperatura: energia media"; fin\_t — a Math \[text\] that says "$overline(E)\_c = frac(3, 2) k\_B T$"; head\_fin — a Heading that says "I due ponti"

Actions:
- [11:10.517](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=670.516875): fin\_ls is shown on the screen, written out.
- [11:10.701](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=670.700904497551): fin\_s is shown on the screen, written out.
- [11:15.077](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=675.076875): A box is drawn around fin\_s.
- [11:22.563](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=682.5627916666667): fin\_ls is hidden from the screen — left the board.
- [11:22.563](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=682.5627916666667): fin\_lt is hidden from the screen — left the board.
- [11:22.563](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=682.5627916666667): fin\_s is hidden from the screen — left the board.
- [11:22.563](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=682.5627916666667): fin\_t is hidden from the screen — left the board.
- [11:22.563](https://academa.ai/@pallone/lectures/teoria-cinetica-dei-gas-e-meccanica-statistica?t=682.5627916666667): head\_fin is hidden from the screen — left the board.