Consider a polyatomic gas which has 3 degrees of translational motion, 3 degrees of rotational motion, and k vibrational modes. Obtain a relation for Gamma.

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Solution

$P o l y a t o m i c g a s h a v i n g 3 d e g r e e s o f t r a n s l a t i o n a l m o t i o n a n d 3 d e g r e e s o f r o t a t i o n a l m o t i o n a n d k v i b r a t i o n a l m o t i o n, h a s t o t a l d e g r e e s o f f r e e d o m = N = 3 \times 3 - k = (9 - k) d e g r e e s o f f r e e d o m T h e r e f o r e E = (9 - k) \times \frac{1}{2} R T T h e r e f o r e C_{v} = \frac{d E}{d T} = (9 - k) \times \frac{1}{2} R a n d C_{p} = C_{v} + R = (9 - k) \times \frac{1}{2} R + R = \frac{(11 - k)}{2} R T h e r e f o r e γ = \frac{C_{p}}{C_{v}} = \frac{\frac{(11 - k)}{2} R}{\frac{(9 - k)}{2} R} = \frac{(11 - k)}{(9 - k)}$

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\frac{C_{p}}{C_{v}}

C_{p} / C_{v}

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γ

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β

β = \frac{C_{p}}{C_{v}}

β

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