Name: gas particle speed distributions and temperature
Uploaded: 2020-09-03T13:38:27.000Z
Duration: 3 min 7 s
Channel: JoVE

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NC K-12 Science Standard Course of Study

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Kinetic Molecular Theory: Molecular Velocities, Temperature, and Kinetic Energy

Wszystkie cząstki gazu mają energię kinetyczną, która jest funkcją masy cząstki w kilogramach i prędkości, czyli wielkości jej prędkości, w metrach na sekundę.

Przy każdym zderzeniu zmieniają się prędkości poszczególnych cząstek gazu. Dlatego zbiór cząstek gazu ma w rzeczywistości rozkład lub zakres prędkości i energii kinetycznych. 

Oznacza to, że w każdej chwili niektóre cząsteczki poruszają się wolniej niż inne, jednak średnia energia kinetyczna pozostaje taka sama. 

Średnia energia kinetyczna jest związana ze średnią kwadratów prędkości lub prędkością średniokwadratową - obie te wartości pozostają stałe w danej temperaturze dla danego gazu.

Teraz średnia energia kinetyczna jednego mola gazu jest wyrażona przez wprowadzenie  stałej Avogadro, NA. Iloczyn masy na cząstkę i stałej Avogadro cząstek na mol jest równy masie molowej gazu w kilogramach na mol.

Przypomnijmy sobie z kinetycznej teorii molekularnej, że średnia energia kinetyczna mola gazu jest wprost proporcjonalna do temperatury. Za pomocą złożonych pochodnych ustala się, że stała proporcjonalności wynosi 3/2 R.

Połączenie tych dwóch równań, przegrupowanie wyrazów i wzięcie pierwiastka kwadratowego po obu stronach wiąże pierwiastek kwadratowy prędkości średniokwadratowej - zwanej również prędkością średnią kwadratową lub prędkością RMS - masę molową i bezwzględną temperaturę gazu.

Prędkość skuteczna jest odwrotnie proporcjonalna do masy molowej i wprost proporcjonalna do temperatury.

Załóżmy, że dwa gazy – hel i argon – mają tę samą temperaturę. Ponieważ hel ma niższą masę molową, równanie wskazuje, że hel musi mieć wyższą prędkość skuteczną niż argon.

Podobną obserwację poczyniono na wykresie rozkładu prędkości molekularnych dla trzech gazów – helu, argonu i chloru – w tej samej temperaturze. 

Zauważ, że chociaż wszystkie gazy mają tę samą średnią energię kinetyczną, najlżejszy gaz, hel, ma zarówno najwyższą prędkość RMS, jak i najszerszy rozkład prędkości, odpowiadający najszerszemu zakresowi prędkości molekularnych.

Wykres rozkładu prędkości dla dowolnego gazu – powiedzmy argonu – w różnych temperaturach pokazuje wzrost prędkości RMS i poszerzenie rozkładu prędkości w wyższych temperaturach.

Krótko mówiąc, gazy poruszają się szybciej w wyższych temperaturach. Na przykład cząsteczki gazu powodujące aromat z gorącego jedzenia poruszają się szybciej niż cząsteczki z zimnego jedzenia. W ten sposób gorące jedzenie jest wykrywane szybciej niż zimne.

kinetyczna teoria molekularna: prędkości molekularne, temperatura i energia kinetyczna

Kinetyczna teoria molekularna jakościowo wyjaśnia zachowania opisane przez różne prawa gazowe. Postulaty tej teorii można zastosować w sposób bardziej ilościowy w celu wyprowadzenia każdego z tych praw.

Łącznie cząsteczki w próbce gazu mają średnią energię kinetyczną i średnią prędkość; ale indywidualnie poruszają się z różnymi prędkościami. Cząsteczki często ulegają zderzeniom sprężystym, w których pęd jest zachowany. Ponieważ zderzające się cząsteczki są odchylane z różnymi prędkościami, poszczególne cząsteczki mają bardzo zróżnicowane prędkości. Ze względu na ogromną liczbę cząsteczek i występujących zderzeń, rozkład prędkości molekularnej i średnia prędkość są stałe. Ten rozkład prędkości molekularnej jest znany jako rozkład Maxwella-Boltzmanna i przedstawia względną liczbę cząsteczek w masowej próbce gazu, która posiada daną prędkość.

Energię kinetyczną (Ek) cząstki o masie (m) i prędkości (u) wyraża się wzorem:

<img alt="Eq1" src="https://www.jove.com/files/core/CDNSource/lessons/1043/1043_Equations-01.png" style="height: 40px;">


Wyrażenie masy w kilogramach i prędkości w metrach na sekundę da wartości energii w jednostkach dżuli (J = kg·m2/s2). Aby poradzić sobie z dużą liczbą cząsteczek gazu, używamy średnich zarówno prędkości, jak i energii kinetycznej. W KMT pierwiastek średniej kwadratowej prędkości cząstki, urms, definiuje się jako pierwiastek kwadratowy średniej kwadratów prędkości, gdzie n = liczba cząstek:

<img alt="Eq2" src="https://www.jove.com/files/core/CDNSource/lessons/1043/1043_Equations-02.png" style="height: 40px;">


Średnia energia kinetyczna mola cząstek KEavg jest wówczas równa:

<img alt="Eq3" src="https://www.jove.com/files/core/CDNSource/lessons/1043/1043_Equations-03.png" style="height: 40px;">


gdzie M jest masą molową wyrażoną w kg/mol. Wartość KEavg mola cząsteczek gazu jest również wprost proporcjonalna do temperatury gazu i można ją opisać równaniem:

<img alt="Eq4" src="https://www.jove.com/files/core/CDNSource/lessons/1043/1043_Equations-04.png" style="height: 40px;">


gdzie R jest stałą gazową, a T jest temperaturą w stopniach Kelvina. W tym równaniu odpowiednią formą stałej gazowej jest 8,314 J/mol⋅K (8,314 kg·m2/s2·mol·K). Te dwa oddzielne równania dla KEavg można połączyć i uporządkować, aby uzyskać zależność między prędkością molekularną a temperaturą:

<img alt="Eq5" src="https://www.jove.com/files/core/CDNSource/lessons/1043/1043_Equations-05.png" style="height: 40px;">
<img alt="Eq6" src="https://www.jove.com/files/core/CDNSource/lessons/1043/1043_Equations-06.png" style="height: 40px;">

Jeśli temperatura gazu wzrasta, wzrasta jego KEavg, więcej cząsteczek ma wyższe prędkości, a mniej cząsteczek ma niższe prędkości. Wówczas rozkład przesuwa się w stronę wyższych prędkości, to znaczy w prawo. Jeśli temperatura spada, KEavg maleje, więcej cząsteczek ma niższe prędkości, mniej cząsteczek ma wyższe prędkości, a rozkład przesuwa się w stronę niższych prędkości, to znaczy w lewo.

W danej temperaturze wszystkie gazy mają takie same wartości KEavg cząsteczek. Prędkość molekularna gazu jest bezpośrednio związana z masą cząsteczkową. Gazy złożone z lżejszych cząsteczek mają więcej cząstek o dużej prędkości i wyższe wartości urms, a rozkład prędkości osiąga szczyt przy stosunkowo wyższych prędkościach. Gazy składające się z cięższych cząsteczek mają więcej cząstek o niskiej prędkości, niższe wartości urms i rozkład prędkości, który osiąga szczyt przy stosunkowo niższych prędkościach.

Kinetyczna teoria molekularna: prędkości molekularne, temperatura i energia kinetyczna

Kinetyczna teoria molekularna jakościowo wyjaśnia zachowania opisane przez różne prawa gazów. Postulaty tej teorii można zastosować w ...

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gas particle speed distributions and temperature

Gas particle speed distributions and temperature are key ideas in the kinetic molecular theory. This theory explains the gas laws by describing how gas molecules move, collide, and share energy.

Gas molecules do not all move at the same speed. A gas sample has an average speed and an average kinetic energy, but individual molecules move at different speeds. Their collisions are elastic, which means momentum is conserved. Because so many molecules collide, the overall speed distribution stays constant.

That distribution is called the Maxwell-Boltzmann distribution. It shows the relative number of molecules in a gas sample at each speed. Kinetic energy depends on both mass and speed, and it is measured in joules when mass is in kilograms and speed is in meters per second. For large numbers of molecules, chemists use averages for speed and kinetic energy.

The root mean square velocity, or u rms, is one way to describe average molecular speed. It is the square root of the average of the squared velocities. The average kinetic energy for one mole of gas also depends on molar mass, and it can be written in terms of the gas constant and the kelvin temperature. In this equation, the correct gas constant is 8.314 J/mol·K.

These relationships show how speed changes with temperature. When temperature increases, average kinetic energy increases. More molecules move at higher speeds, fewer move at lower speeds, and the distribution shifts to the right. When temperature decreases, average kinetic energy decreases, more molecules move slowly, and the distribution shifts to the left.

At the same temperature, gases have the same average kinetic energy, even if their molecules have different masses. Molecular mass still affects speed. Lighter gases have more high-speed particles and a higher u rms, while heavier gases have more low-speed particles and a lower u rms. Their speed distributions peak at different velocities, with lighter gases peaking at higher speeds and heavier gases peaking at lower speeds.

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Gas Particle Speed Distributions and Temperature

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