Question

Consider two ideal diatomic gases $A$ and $B$ at some temperature $T$. Molecules of the gas $A$ are rigid, and have a mass $m$. Molecules of the gas $B$ have an additional vibrational mode, and have a mass $\frac{m}{4}$. The ratio of the specific heats $(C_V{)}_A$ and $(C_V{)}_B$ of the gases $A$ and $B$ respectively, is:

• A. $7:9$
• B. $5:9$
• C. $3:5$
• D. $5:7$

Answer 1

The correct option is D $5:7$

Specific heat of gas at constant volume
$C_V=\frac{1}{2}fRf\to$ degree of freedom

For gas $A$ (diatomic)
$f=5(3\text{translational}+2\text{rotational})$
$\therefore (C_V{)}_A=\frac{5}{2}R$

For gas $B$ (diatomic) $f=7(3\text{translational}+2\text{rotational}),2\text{vibrational}$ degree of freedom.

$\therefore (C_V{)}_B=\frac{7}{2}R$

Now, $\frac{(C_V{)}_A}{(C_V{)}_B}=\frac{\frac{5}{2}R}{\frac{7}{2}R}=\frac{5}{7}$

Hence, $\left(D\right)$ is the correct answer.