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Question

# A polyatomic ideal gas has $24$ vibrational modes. What is the value of $\mathrm{\gamma }$?

A

$1.03$

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B

$1.30$

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C

$10.3$

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D

$1.37$

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Solution

## The correct option is A $1.03$Explanation for correct optionSince each vibrational mode has $2$ degrees of freedom hence total vibrational degrees of freedom$=48$$\mathrm{f}=3\mathrm{T}+3\mathrm{R}+24×2\mathrm{V}\phantom{\rule{0ex}{0ex}}=54\phantom{\rule{0ex}{0ex}}\mathrm{\gamma }=1+\frac{2}{\mathrm{f}}\phantom{\rule{0ex}{0ex}}\mathrm{f}=\mathrm{number}\mathrm{of}\mathrm{degrees}\mathrm{of}\mathrm{freedom}\mathrm{due}\mathrm{to}\mathrm{translation},\mathrm{rotation},\mathrm{vibration}\phantom{\rule{0ex}{0ex}}\mathrm{\gamma }=1+\left(2/54\right)\phantom{\rule{0ex}{0ex}}=1+1/27=28/27\phantom{\rule{0ex}{0ex}}=1.03$Hence option(A) is correct option.

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