Question

The speed of sound in hydrogen at 0°C is 1280 m s⁻¹. The density of hydrogen at STP is 0.089 kg m⁻³. Calculate the molar heat capacities C_p and C_v of hydrogen.

Answer 1

Given:
Velocity of sound in hydrogen, V = 1280 m/s
Temperature, T = 0°C = 273 K

Density of H₂ = 0.089 kg/m³
R = 8.3 J/mol-K
At STP,
P = 10⁵ Pa
We know:
$$\begin{gathered} V_{sound}\mathit{=}\sqrt{\frac{\gamma p}{\rho }} \\ 1280=\sqrt{\frac{\gamma \times {10}^5}{0.089}} \\ \text{Or}\gamma =\frac{1280\times 1280\times 0.089}{{10}^5} \\ =1.46 \\ \frac{C_p}{C_v}\mathit{=}\gamma \mathrm{or}\mathit{}{C}_p\mathit{-}{C}_v\mathit{=}R \\ {C}_v\mathit{=}\frac{R}{\gamma \mathit{-}\mathit{1}}=\frac{8.3}{1.46-1} \\ =18.0\mathrm{J}/\mathrm{mol}-\mathrm{K} \\ {C}_p\mathit{=}\gamma C_v=1.46\times 18.0 \\ =26.28\approx 26.3/\mathrm{mol}-\mathrm{K} \end{gathered}$$