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Consider the following question:

A closed container contains a mixture of Chlorine gas and Argon gas. What is the ratio of the average kinetic energy of a molecule of the two gases?

According to the kinetic theory of gases, the kinetic energy of any molecule, regardless of whether it is monoatomic, diatomic or polyatomic, is given by $\frac{3}{2}k_bT$, where $k_b$ is the Boltzmann Constant. Hence the ratio should be $1:1$. This is also the answer and the explanation provided by my textbook.

However, a diatomic gas has $5$ degrees of freedom ($3$ translational $+ 2$ rotational), and according to the Law of Equipartition of Energy, each degree of freedom contributes an energy of $\frac{1}{2}k_bT$. So, shouldn't the kinetic energy of a diatomic gas like Chlorine actually be $\frac{5}{2}k_bT$, in which case the ratio will be $5:3$?

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  • $\begingroup$ Wrong answer in that textbook. $\endgroup$
    – user137289
    Commented Jul 14, 2018 at 9:26

2 Answers 2

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The kinetic energy of a molecule in a diatomic gas is, as you correctly stated,

5/2(NkT) = 5/2(nRT)

However, this is only an approximation and applies in intermediate temperatures. At lower temperatures, the only contribution to kinetic energy is due to the translational motion. At higher energies, two additional contributions (kinetic and potential) come from vibration.

(Excuse the shabby format, I answered this on the mobile app)

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  • $\begingroup$ Here's a source that aligns with my answer, to help explain: nuclear-power.net/nuclear-engineering/thermodynamics/… $\endgroup$
    – Mike H
    Commented Mar 19, 2018 at 16:04
  • $\begingroup$ Thanks! I still have a question though. If, at low temperatures, diatomic gases have an internal energy of 3/2kT, then wouldn't the heat capacity ratio of air be closer to 5/3, like in monoatomic gases, as opposed to 7/5 which is the accepted value? $\endgroup$
    – Tarun
    Commented Mar 19, 2018 at 16:17
  • $\begingroup$ The heat capacity ratio of air is 7/5, and I believe this is due to the fact air is only MOSTLY diatomic, but I am not entirely sure. $\endgroup$
    – Mike H
    Commented Mar 19, 2018 at 18:02
  • $\begingroup$ Exactly. 99% of air is diatomic which is why its heat capacity ratio is 7/5. However, if your explanation holds true, even the diatomic gases present in air will have the same energy (3/2kT) as a monatomic gas (which has heat capacity ratio of 5/3) at low temperature and hence air will also have a heat capacity ratio close to 5/3, which is not the case. $\endgroup$
    – Tarun
    Commented Mar 19, 2018 at 18:15
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    $\begingroup$ @Tarun -- Keep in mind that low-temp means very low: $kT<{{\hbar }^{2}}/I$. That’s colder than a witch's tit, colder than a bucket of penguin shit, colder than Oymyakon in January. The Cp/Cv ratio would indeed be 5/3 in gas phase, but not in liquid nitrogen. $\endgroup$
    – Bert Barrois
    Commented Jul 14, 2018 at 9:56
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as it is per molecule it is is constant but in case they ask average energy per molecule PER DEGREE OF FREEDOM that time u must use the formula n 1/2ktwhere n is degree of freedom of gas

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