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

Answer: (A)

Internal energy of a gas, $U=n{C}_{V}T$
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
$U_{H_2}=U_{He}$
$\Rightarrow n_{H_2}\times {\left(C_V\right)}_{H_2}\times T_{H_2}$
$=n_{He}\times {\left(C_V\right)}_{He}\times T_{He}$
$\therefore n_{H_2}\times \frac{5}{2}R\times T$
$=n_{He}\times \frac{3}{2}R\times 2T$
$\frac{n_{H_2}}{n_{He}}=\frac{6}{5}$