The Speed of Sound — Question 4

Question 4

A gas-composition comparison can overwhelm a density intuition

Two ideal gases at the same temperature have $(\gamma_A,M_A)=(1.40,0.028\,\mathrm{kg/mol})$ and $(\gamma_B,M_B)=(5/3,0.0040\,\mathrm{kg/mol})$. Use $c=\sqrt{\gamma RT/M}$ with $R=8.314\,\mathrm{J/(mol\,K)}$ and $T=300\,\mathrm K$.

1. Compute both speeds and their ratio.

2. What temperature must gas B have to match the speed of gas A at $300\,\mathrm K$ within this constant-$\gamma$ idealization?

Question 4 - Solution

A gas-composition comparison can overwhelm a density intuition

(a) Method and calculation

The material factors differ in both heat-capacity ratio and molar mass:$$c_A=\sqrt{\frac{1.40(8.314)(300)}{0.028}}=353.2\,\mathrm{m/s},$$$$c_B=\sqrt{\frac{(5/3)(8.314)(300)}{0.0040}}=1019.4\,\mathrm{m/s}.$$$$c_B/c_A=\sqrt{\frac{(5/3)(0.028)}{1.40(0.0040)}}=2.887.$$

(b) Interpretation and further calculation

Equality of squared speeds requires$$\frac{\gamma_B T_B}{M_B}=\frac{\gamma_A(300)}{M_A},\qquad T_B=300\frac{1.40}{5/3}\frac{0.0040}{0.028}=36.0\,\mathrm K.$$This is a prediction of the supplied model. Real-gas phase and property changes would need checking before applying it to an actual cooled sample.

Physical check

The ratio is independent of the common initial temperature. A larger molecular mass generally reduces speed when other model parameters are fixed.