# What is the kinetic energy of one molecule of a diatomic gas?

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 $5:3$?

• Wrong answer in that textbook. – Pieter Jul 14 '18 at 9:26

## 1 Answer

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)

• Here's a source that aligns with my answer, to help explain: nuclear-power.net/nuclear-engineering/thermodynamics/… – Mike H Mar 19 '18 at 16:04
• 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? – Tarun Mar 19 '18 at 16:17
• 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. – Mike H Mar 19 '18 at 18:02
• 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. – Tarun Mar 19 '18 at 18:15
• I'll get back to you in a couple years when I learn this stuff forreal :) – Mike H Mar 19 '18 at 18:16