Consider a mixture of n moles of helium gas and 2 n
Question:

Consider a mixture of $n$ moles of helium gas and $2 n$ moles of oxygen gas (molecules taken to be rigid) as an ideal gas.

Its $C_{P} / C_{V}$ value will be:

1. (1) $19 / 13$

2. (2) $67 / 45$

3. (3) $40 / 27$

4. (4) $23 / 15$

Correct Option: 1

Solution:

(1) Helium is a monoatomic gas and Oxygen is a diatomic gas.

For helium, $C_{V_{1}}=\frac{3}{2} R$ and $C_{R_{1}}=\frac{5}{2} R$

For oxygen, $C_{V_{2}}=\frac{5}{2} R$ and $C_{P_{2}}=\frac{7}{2} R$

$\gamma=\frac{N_{1} C_{P}+N_{2} C_{P_{2}}}{N_{1} C_{V_{1}}+N_{2} C_{V_{2}}}$

$\Rightarrow \gamma=\frac{n \cdot \frac{5}{2} R+2 n \cdot \frac{7}{2} R}{n \cdot \frac{3}{2} R+2 n \cdot \frac{5}{2} R}=\frac{19 n R \times 2}{2(13 n R)}$

$\therefore\left(\frac{C_{P}}{C_{V}}\right)_{\text {mixture }}=\frac{19}{13}$