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I came across a question which involved non linear tetra atomic gasses. In it, the value of specific heat capacity at constant volume was given as $9R,$ added up as

$$\frac 3 2 R + \frac 3 2 R + 6R = 9R$$

Now I know $6R$ is $C_V$ for triatomic gasses, but why do we add the two $\frac 3 2 R$ terms?

Also, how does being linear or non-linear affect the specific heat capacity of a gas? Explanations with diagrams showing it in 3-D coordinate system would be helpful.

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The heat capacity comprises contributions from translational motion (3 terms for each of $x, y, z$) plus (whole body) rotational motion ( 3 axis directions). Each motion accounts for $(1/2)R$ by equipartition theorem making the $(3/2)R + (3/2)R$ above.

Then there is a contribution from the fact that the molecule is vibrating. Non-linear molecules have $3N-6=6$ vibrational modes. Each vibration has $(1/2)R$ contribution from its potential energy and the same from its kinetic energy which in total is the $6R$ in your question.

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