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  4. The speed of sound in a mixture of 1 mole of helium and 2 moles of oxygen at 27° C is

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

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Solution:

Molecular weight of mixture
$M_{\text {mix }} =\frac{n_{1} M_{1}+n_{2} M_{2}}{n_{1}+n_{2}}$
$=\frac{1 \times 4+2 \times 32}{1 \times 2}=\frac{68}{3}$
$=\frac{68}{3} \times 10^{-3} kg\, mol ^{-1}$
for helıum, $C_{V_{1}} =\frac{3}{2} R$
For oxygen, $C_{V_{2}} =\frac{5}{2} R$
$\left(C_{V}\right)_{\text {mix }}=\frac{n_{1} C_{V_{1}}+n_{2} C_{V_{2}}}{n_{1}+n_{2}}$
$=\frac{1 \times \frac{3 R}{2}+2 \times \frac{5 R}{2}}{1+2}=\frac{13 R}{6}$
Now,$\left(C_{p}\right)_{\text {mix }} =\left(C_{V}\right)_{\text {mix }}+R$
$=\frac{13 R}{6}+R=\frac{19 R}{6}$
$\therefore \gamma_{\text {mix }} =\frac{\left(C_{p}\right)_{\text {mix }}}{\left(C_{V}\right)_{\text {mix }}}=\frac{19}{13}$
Sound, $V =\sqrt{\frac{\gamma_{\text {mix }} R T}{M_{\text {mix }}}}$
$=\sqrt{\frac{19}{13} \times \frac{8.31 \times 300}{\frac{68}{3} \times 10^{-3}}}$
$=400.8\, ms ^{-1}$