Question 34

Ratio of specific heats $$\gamma$$ of a mixture of gases is given by $$\gamma = 1 + \frac{R(n_1 + n_2)}{n_1C_{v1} + n_2C_{v2}}$$ where $$C_{v1}$$ and $$C_{v2}$$ are the specific heats at constant volume of the two gases in the mixture and $$\gamma$$ is the ratio of specific heats, $$n_1$$ and $$n_2$$ are the number of moles of the gas in the given mixture.<br>Find the velocity of sound in a mixture by equal volumes Hydrogen and Helium at $$27^\circ\text{C}$$. Specific heats at constant volume for Hydrogen and Helium are $$\frac{5R}{2}$$ and $$\frac{3R}{2}$$ respectively.


Correct Answer: 1117

Solution

Equal volumes at the same temperature and pressure contain equal numbers of moles, so $$n_1 = n_2 = n$$ and $$\gamma = 1 + \frac{2nR}{n\frac{5R}{2} + n\frac{3R}{2}} = 1 + \frac{2R}{4R} = 1.5$$. The mean molar mass is $$\frac{2 + 4}{2} = 3\ \text{g/mol} = 3 \times 10^{-3}\ \text{kg/mol}$$. Hence $$v = \sqrt{\frac{\gamma RT}{M}} = \sqrt{\frac{1.5 \times 8.314 \times 300}{3 \times 10^{-3}}} \approx 1117\ \text{m/s}$$.

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