Consider two ideal diatomic gases A and B at some temperature T.

Question:

Consider two ideal diatomic gases $A$ and $B$ at some temperature T. Molecules of the gas A are rigid, and have a mass $\mathrm{m}$. Molecules of the gas B have an additional

vibrational mode, and have a mass $\frac{m}{4}$. The ratio of the

specific heats $\left(C_{V}^{A}\right.$ and $\left.C_{V}^{B}\right)$ of gas $A$ and $B$, respectively is:

1. (1) $7: 9$

2. (2) $5: 9$

3. (3) $3: 5$

4. (4) $5: 7$

Correct Option: , 4

Solution:

(4) Specific heat of gas at constant volume

$C_{v}=\frac{1}{2} f R ; f=$ degree of freedom

For gas A (diatomic)

$\mathrm{f}=5(3$ translational $+2$ rotational $)$

$\therefore C_{v}^{A}=\frac{5}{2} R$

For gas B (diatomic) in addition to ( 3 translational + 2 rotational) 2 vibrational degree of freedom.

$\therefore \quad C_{v}^{B}=\frac{7}{2} R$ Hence $\frac{C_{v}^{A}}{C_{v}^{B}}=\frac{\frac{5}{2} R}{\frac{7}{2} R}=\frac{5}{7}$