Taking the composition of air to be $75 %$ of nitrogen and $25 %$ of oxygen by weight, calculate the velocity of sound through air at STP.

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Solution

The molecular weight of a mixture is given by
$\frac{m_{1} + m_{2} + m_{3} + . . .}{M} = \frac{m_{1}}{M_{1}} + \frac{m_{2}}{M_{2}} +$ ...
$∴ \frac{75 + 25}{M} = \frac{75}{28} + \frac{25}{32}$
or, $M = 28.9$
$γ$ of a mixture is given by
$= \frac{n_{1} + n_{2}}{γ - 1} = \frac{n_{1}}{γ_{1} - 1} + \frac{n_{2}}{γ_{2} - 1} +$ ...
$\Rightarrow \frac{\frac{75}{28} + \frac{25}{32}}{γ - 1} = \frac{\frac{75}{28}}{1.4 - 1} + \frac{\frac{25}{32}}{1.4 - 1}$
$\Rightarrow \frac{3.46}{γ - 1} = 6.70 + 1.95$
$\Rightarrow γ = 1.4$
$∵ v = \sqrt{\frac{γ P}{ρ}}$
By the perfect gas equation
$P = \frac{ρ R T}{M}$
$∴ \frac{P}{ρ} = (\frac{R T}{M})$
$v = \sqrt{\frac{γ R T}{M}}$
$= \sqrt{\frac{1.4 \times 8.3 \times 1000 \times 273}{28.9}} = 331.3 m s^{- 1}$ .

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