Question

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

• A. $800m{s}^{-1}$
• B. $400.8m{s}^{-1}$
• C. $600m{s}^{-1}$
• D. $1200m{s}^{-1}$

Answer 1

Answer: (B)

Molecular weight of mixture
$M_{\text{mix }}=\frac{n_1{M}_1+n_2{M}_2}{n_1+n_2}$
$=\frac{1\times 4+2\times 32}{1\times 2}=\frac{68}{3}$
$=\frac{68}{3}\times 10^{-3}kgmo{l}^{-1}$
for helıum, $C_{V_1}=\frac{3}{2}R$
For oxygen, $C_{V_2}=\frac{5}{2}R$
${\left(C_V\right)}_{\text{mix }}=\frac{n_1{C}_{V_1}+n_2{C}_{V_2}}{n_1+n_2}$
$=\frac{1\times \frac{3R}{2}+2\times \frac{5R}{2}}{1+2}=\frac{13R}{6}$
Now,${\left(C_p\right)}_{\text{mix }}={\left(C_V\right)}_{\text{mix }}+R$
$=\frac{13R}{6}+R=\frac{19R}{6}$
$\therefore {\gamma }_{\text{mix }}=\frac{{\left(C_p\right)}_{\text{mix }}}{{\left(C_V\right)}_{\text{mix }}}=\frac{19}{13}$
Sound, $V=\sqrt{\frac{{\gamma }_{\text{mix }}RT}{M_{\text{mix }}}}$
$=\sqrt{\frac{19}{13}\times \frac{8.31\times 300}{\frac{68}{3}\times 10^{-3}}}$
$=400.8m{s}^{-1}$