Question

Hydrogen is at temperature $T$ and helium is at temperature $2T$. The internal energy of both gases is the same. The ratio of number of moles of hydrogen and helium gases is

• A. $\frac{6}{5}$
• B. $\frac{5}{6}$
• C. $\frac{3}{2}$
• D. $\frac{2}{3}$

Answer 1

The Correct Option is A

Internal energy of a gas, $U = nC_VT$ where $n$ is the number of the moles of a gas and $C_v$ is the molar specific heat at constant volume and $T$ is the temperature of the gas. As hydrogen is a diatomic gas, $\therefore\left(C_{V}\right)_{H_2} = \frac{5}{2}R$ As helium is a monatomic gas, $\therefore\left(C_{V}\right)_{He} = \frac{3}{2}R$ According to the question $U_{H_2} = U_{He} $ $\Rightarrow n_{H_2} \times\left(C_{V}\right)_{H_2} \times T_{H_2} $ $ = n_{He}\times\left(C_{V}\right)_{He} \times T_{He}$ $ \therefore n_{H_2} \times\frac{5}{2}R \times T $ $ = n_{He} \times\frac{3}{2}R \times 2T $ $ \frac{n_{H_2}}{n_{He}} = \frac{6}{5}$