Question

A container of fixed volume has 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 $\sqrt{6/5}$.
• C.
  the ratio of the rms speed of helium atoms to that of hydrogen molecules is 1/2.
• D.
  the ratio of the rms speed of helium atoms to that of hydrogen molecules is 1/$\sqrt{2}$.

Answer 1

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

the ratio of the rms speed of helium atoms to that of hydrogen molecules is 1/$\sqrt{2}$.

Given,
($1$ mole of $H)+(1$ mole of $He$)
molar mass of the mixture=$M_{mix}=\frac{n_1{m}_1+n_2{m}_2}{n_1+n_2}$ ..(1)
here, $n_1$=no. of moles of $H$
$n_2$=no. of moles of $He$
$m_1$= mass of $H$
$m_2$= mass of $He$

$M_{mix}=\frac{1\times 2+1\times 4}{1+1}=3mol/g$

Ratio of specific heat of mixture=
${\gamma }_{mix}=\frac{\frac{n_{1}C{p}_1+n_{2}C{p}_2}{n_1+n_2}}{\frac{n_{1}C{v}_1+n_{2}C{v}_2}{n_1+n_2}}$ ..(2)
where, $Cp=$ specific heat capacity

Here, $H$ is a diatomic gas,
for $H$,
$C_p=\frac{7}{2R},C_v=\frac{5}{2R}$

and $He$ is a monoatomic gas,
for $He$,
$C_p=\frac{5}{2R},C_v=\frac{3}{2R}$
on putting both values at eq(2).

${\gamma }_{mix}=\frac{\frac{7}{2R}+\frac{5}{2R}}{\frac{5}{2R}+\frac{3}{2R}}=\frac{3}{2}{\gamma }_{mix}=\frac{3}{2}$

Now, calculation of velocity of sound in mixture=
$v_H=\sqrt{{\gamma }_{mix}RT/M_H}$
$v_{He}=\sqrt{{\gamma }_{mix}RT/M_{He}}$
So, $\frac{v_H}{v_{He}}=\frac{\frac{\sqrt{3/2RT}}{3}}{\frac{\sqrt{5/3RT}}{4}}=\sqrt{\frac{6}{5}}$
Hence, option B is correct.

Calculation of r.m.s. speed of mixture,
in a ideal gas condition rms speed is given by $\sqrt{3RT/M}$
So,
$\frac{V_{He}}{V_H}=\sqrt{\frac{M_H}{M_{He}}}$
where, $M_H=2,M_{He}=4$
so, $\frac{V_{He}}{V_H}=\sqrt{\frac{2}{4}}=\frac{1}{\sqrt{2}}$
Hence, option (D) is correct.

Now total internal energy of gas,

$U=\frac{internalenergyofH+internalenergyofHe}{no.ofmoles}$
$U=\frac{\frac{5}{2}RT+\frac{3}{2}RT}{1+1}=2RT$
Hence, option A is correct.