Question

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

• A. $1.03$
• B. $1.30$
• C. $10.3$
• D. $1.37$

Answer 1

The correct option is A

$1.03$

Explanation for correct option
Since each vibrational mode has $2$ degrees of freedom hence total vibrational degrees of freedom$=48$

$$\begin{gathered} \mathrm{f}=3\mathrm{T}+3\mathrm{R}+24\times 2\mathrm{V} \\ =54 \\ \mathrm{\gamma }=1+\frac{2}{\mathrm{f}} \\ \mathrm{f}=\mathrm{number}\mathrm{of}\mathrm{degrees}\mathrm{of}\mathrm{freedom}\mathrm{due}\mathrm{to}\mathrm{translation},\mathrm{rotation},\mathrm{vibration} \\ \mathrm{\gamma }=1+(2/54) \\ =1+1/27=28/27 \\ =1.03 \end{gathered}$$

Hence option(A) is correct option.