gamma_{A} is the specific heat ratio of a mon | vedclass

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(D) For a gas with $f$ degrees of freedom,the specific heat ratio is given by $\gamma = 1 + \frac{2}{f} = \frac{f+2}{f}$.For monoatomic gas $A$,the de

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$3$

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(D) For a gas with $f$ degrees of freedom,the specific heat ratio is given by $\gamma = 1 + \frac{2}{f} = \frac{f+2}{f}$.For monoatomic gas $A$,the degrees of freedom $f_{A} = 3$. Thus,$\gamma_{A} = \frac{3+2}{3} = \frac{5}{3}$.For polyatomic gas $B$,the degrees of freedom $f_{B} = 3 	ext{ (translational)} + 3 	ext{ (rotational)} + 2 	imes 1 	ext{ (vibrational)} = 8$. Thus,$\gamma_{B} = \frac{8+2}{8} = \frac{10}{8} = \frac{5}{4}$.Now,calculate the ratio: $\frac{\gamma_{A}}{\gamma_{B}} = \frac{5/3}{5/4} = \frac{5}{3} 	imes \frac{4}{5} = \frac{4}{3}$.Given $\frac{\gamma_{A}}{\gamma_{B}} = 1 + \frac{1}{n}$,we have $1 + \frac{1}{n} = \frac{4}{3}$.$\frac{1}{n} = \frac{4}{3} - 1 = \frac{1}{3}$.Therefore,$n = 3$.

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