# Specific heat for tetraatomic gas molecules

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.

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.