Question

The speed of sound in a mixture of $1$ mole of helium and $2$ moles of oxygen at $27{}^{\circ }\text{C}$
will be

• A. $501\text{m/s}$
• B. $201\text{m/s}$
• C. $401\text{m/s}$
• D. $301\text{m/s}$

Answer 1

The correct option is C $401\text{m/s}$

Given,
Number of mole of Helium, $n_1=1$
Number of mole of Oxygen, $n_2=2$
So, Molar heat capacity at constant volume of Helium per mole,
$C_{v1}=\frac{f_{1}R}{2}=\frac{3R}{2}$
[$f$ of monoatomic gas is $3$]
Similarly, for Oxygen gas,
Molar heat capacity at constant volume per mole,
$C_{v2}=\frac{f_{2}R}{2}=\frac{5R}{2}$
[$f$ of diatomic gas is $5$]
For the mixture of two non-reactive gases,
Molar heat capacity at constant volume,
$C_v=\frac{n_1{C}_{v1}+n_2{C}_{v2}}{n_1+n_2}$
$\Rightarrow C_v=\frac{1\times \frac{3R}{2}+2\times \frac{5R}{2}}{1+2}=\frac{13R}{6}$
Molar heat capacity of mixture at constant pressure,
$C_p=C_v+R$
$\Rightarrow C_p=\frac{19R}{6}$
Now,
$\gamma$ of mixture,
$\gamma =\frac{C_p}{C_v}=\frac{19}{13}$
Molar mass of mixture of two gases,
$M=\frac{n_1{M}_1+n_2{M}_2}{n_1+n_2}$
$\Rightarrow M=\frac{1\times 4+2\times 32}{1+2}=\frac{68}{3}\text{g}$
Speed of sound in the mixture of gas will be,
$v=\sqrt{\frac{\gamma RT}{M}}$
$\Rightarrow v=\sqrt{\frac{\frac{19}{13}\times 8.3\times (27+273)}{\frac{68}{3}\times {10}^{-3}}}$
$\Rightarrow v=401\text{m/s}$