Question

Calculate the speed of sound in oxygen from the following data. The mass of $22.4litre$ of oxygen at STP ($T=273K$ and $P=1.0\times {10}^{5}N{m}^{-2}$ ) is $32g$, the molar heat capacity of oxygen at constant volume is $C_v=2.5R$ and that at constant pressure is$C_p=3.5R$

Answer 1

Step 1. Given data

Temperature, $T=273K$

Pressure, $P=1.0\times {10}^{5}N{m}^{-2}$

Molecular weight, $$\begin{gathered} M=32g \\ M=32\times {10}^{-3}kgmo{l}^{-1} \end{gathered}$$

Heat capacity at constant pressure, $C_p=3.5R$

Heat capacity at constant volume, $C_v=2.5R$

Volume , $$\begin{gathered} V=22.4l \\ V=22.4\times {10}^{-3}{m}^3 \end{gathered}$$

Step 2. Finding the adiabatic constant, $\gamma$

By using the formula of the adiabatic constant,

$$\begin{gathered} \gamma =\frac{C_p}{C_v} \\ \gamma =\frac{3.5R}{2.5R} \\ \gamma =1.4 \end{gathered}$$ [ Where, $R=8.314$ is gas constant]

Step 3. Finding the speed of the sound in oxygen, $v$

By using the formula of speed,

$$\begin{gathered} v=\sqrt{\frac{\gamma RT}{M}} \\ v=\sqrt{\frac{1.4\times 8.314\times 273}{32\times {10}^{-3}}} \\ v=\sqrt{99300} \\ v=315.12m{s}^{-1} \end{gathered}$$

Therefore, the speed of the sound is $315.12m{s}^{-1}$