Explanation

(b) Molecular weight of mixture
$M_{mix} = \frac{n_{1}M_{1} + n_{2}M_{2}}{n_{1} + n_{2}} = \frac{1 \times 4 + 2 \times 32}{1 + 2} = \frac{68}{3}$
$= \frac{68}{3} \times 10^{3}kg\ mol^{- 1}$
For helium $C_{v_{1}} = \frac{3}{2}R$
For oxygen $C_{v_{2}} = \frac{5}{2}R$
$\left( C_{v} \right)_{mix} = \frac{n_{1}C_{v_{1}} + n_{2}C_{v_{2}}}{n_{1} + n_{2}}$
$= \frac{1 \times \frac{3R}{2} + 2 \times \frac{5R}{2}}{1 + 2} = \frac{13R}{6}$
$\left( C_{p} \right)_{mix} = \left( C_{v} \right)_{mix} + R$
$= \frac{13R}{6} + R = \frac{19R}{6}$
$y_{mix} = \sqrt{\frac{\left( C_{p} \right)_{mix}}{\left( C_{v} \right)_{mix}}} = \frac{19}{13}$
$v = \sqrt{\frac{\gamma_{mix}RT}{M_{mix}}}$
$= \sqrt{\frac{19}{13} \times \frac{8.31 \times 300}{\frac{68}{3} \times 10^{- 3}}} = 400.9\ ms^{- 1}$