Question

Let speed of sound waves in hydrogen gas at room temperature is $v_0$. What will be the speed of sound waves in a room which contains an equimolar mixture of hydrogen and 'He' at same temperature :-

• A. $\sqrt{\frac{7}{5}}v_0$
• B. $\sqrt{\frac{5}{7}}v_0$
• C. $\sqrt{\frac{5}{2}}v_0$
• D. None

Answer 1

Answer: (B)

$\sqrt{\frac{5}{7}}v_0$

Answer 2

The speed of sound in an ideal gas is given by $v = \sqrt{\frac{\gamma RT}{M}}$.

For hydrogen ($H_2$), $\gamma_{H_2} = 7/5$ and $M_{H_2} \approx 2 \, g/mol$. $v_0 = \sqrt{\frac{(7/5) RT}{2}}$.

For helium ($He$), $\gamma_{He} = 5/3$ and $M_{He} \approx 4 \, g/mol$.

For an equimolar mixture of hydrogen and helium: Effective molar mass $M_{mix} = 0.5 \times M_{H_2} + 0.5 \times M_{He} = 0.5 \times 2 + 0.5 \times 4 = 1 + 2 = 3 \, g/mol$.

The specific heats are $C_{v, H_2} = \frac{R}{\gamma_{H_2}-1} = \frac{R}{7/5-1} = \frac{5R}{2}$ and $C_{v, He} = \frac{R}{\gamma_{He}-1} = \frac{R}{5/3-1} = \frac{3R}{2}$. The specific heat at constant volume for the mixture is $C_{v, mix} = 0.5 \times C_{v, H_2} + 0.5 \times C_{v, He} = 0.5 \times \frac{5R}{2} + 0.5 \times \frac{3R}{2} = \frac{R}{2} (\frac{5}{2} + \frac{3}{2}) = \frac{R}{2} (4) = 2R$. The specific heat at constant pressure for the mixture is $C_{p, mix} = C_{v, mix} + R = 2R + R = 3R$. The adiabatic index for the mixture is $\gamma_{mix} = \frac{C_{p, mix}}{C_{v, mix}} = \frac{3R}{2R} = \frac{3}{2}$.

The speed of sound in the mixture is $v_{mix} = \sqrt{\frac{\gamma_{mix} RT}{M_{mix}}} = \sqrt{\frac{(3/2) RT}{3}}$.

The ratio of the speeds is: $\frac{v_{mix}}{v_0} = \frac{\sqrt{\frac{(3/2) RT}{3}}}{\sqrt{\frac{(7/5) RT}{2}}} = \sqrt{\frac{(3/2)/3}{(7/5)/2}} = \sqrt{\frac{1/2}{7/5}} = \sqrt{\frac{1}{2} \times \frac{5}{7}} = \sqrt{\frac{5}{7}}$. Therefore, $v_{mix} = \sqrt{\frac{5}{7}} v_0$.