Question

A container of fixed volume have a mixture of one mole of hydrogen and one mole of helium in equilibrium at temperature $T$. Assuming the gases are ideal, the correct statement(s) is/are

• A. The average energy per mole of the gas mixture is $2RT$
• B. The ratio of speed of sound in the gas mixture to that in helium gas is $\frac{\sqrt{6}}{5}$.
• C. The ratio of the $\text{RMS}$ speed of helium atoms that of hydrogen molecules is $\frac{1}{2}$
• D. The ratio of the $\text{RMS}$ speed of helium atoms to that of hydrogen molecules is $\frac{1}{\sqrt{2}}$

Answer 1

Answer: (A), (B), (D)

The ratio of the $\text{RMS}$ speed of helium atoms to that of hydrogen molecules is $\frac{1}{\sqrt{2}}$

We know that internal energy of the ideal gas is given by,

$U=\frac{nfRT}{2}$

where, $f\to$ degree of freedom
$n\to$ number of moles

So, Energy per mole

$E=\frac{\frac{n_1{f}_{1}RT}{2}+\frac{n_2{f}_{2}RT}{2}}{n_1+n_2}$

From the data given in the question,

$E=\frac{1\times 5\times RT+1\times 3\times RT}{1+1}\Rightarrow E=4RT$

Now, average energy per mole, $E_{av}=\frac{E}{2}=2RT$

Hence, option $\left(a\right)$ is correct.

${\gamma }_{mix}=\frac{n_1{C}_{p_1}+n_2{C}_{p_2}}{n_1{C}_{v_1}+n_2{C}_{v_2}}=\frac{1\times \frac{7R}{2}+1\times \frac{5R}{2}}{1\times \frac{5R}{2}+1\times \frac{3R}{2}}=\frac{3}{2}$

$[C_p-C_v=R,C_v=\frac{fR}{2}]$

$M_{mix}=\frac{n_1{M}_1+n_2{M}_2}{n_1+n_2}=\frac{1\times 2+1\times 4}{1+1}=3$

Speed of sound $v=\sqrt{\frac{\gamma RT}{M}}$

At same temperature,

$v\propto \sqrt{\frac{\gamma }{M}}$

So,
$\frac{v_{mix}}{v_{He}}=\sqrt{\frac{{\gamma }_{mix}}{{\gamma }_{He}}\times \frac{{\mu }_{He}}{{\mu }_{mix}}}=\sqrt{\frac{3\times 3}{2\times 5}\times \frac{4}{3}}=\sqrt{\frac{6}{5}}$

Hence, option $\left(b\right)$ is correct.

$\text{RMS}$ speed , $v_{rms}=\sqrt{\frac{3RT}{M}}$

At same temperature

$v_{rms}\propto \sqrt{\frac{1}{M}}$

So, $\frac{v_{He}}{v_{H_2}}=\sqrt{\frac{M_{H_2}}{M_{He}}}=\sqrt{\frac{2}{4}}=\sqrt{\frac{1}{2}}=\frac{1}{\sqrt{2}}$

Hence, option (c) is wrong and option (d) is correct.

Why this question? Concept involved -(1)Internal Energy of ideal gas (2) Speed of sound and RMS speed. Importance in JEE -Asked in advance $2015$