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 Cp and Cv of hydrogen.

Answer 1

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, The molar heat capacities Cp and Cv of hydrogen is $26.3 / m o l-K$

Explanation:

Step 1:

Given data,

Sound velocity, in hydrogen, $V=1280 \mathrm{m} / \mathrm{s}$

$T=0^{\circ} \mathrm{C}=273 \mathrm{K}$

$\mathrm{H}_{2}$  Density = $0.089 \mathrm{kg} / \mathrm{m}^{3}$

$R=8.3 \mathrm{J} / \mathrm{mol}-\mathrm{K}$

At STP,

$P=10^{5} \mathrm{Pa}$

Step 2:

We know that

$V_{s o u n d}=\sqrt{\frac{\gamma p}{\rho}}$

$1280=\sqrt{\frac{\gamma \times 10^{5}}{0.089}}$

$1280 \times 1280=\frac{\gamma \times 10^{5}}{0.089}$

$1280 \times 1280 \times 0.089=\gamma \times 10^{5}$

$\frac{1280 \times 1280 \times 0.089}{10^{5}}=\gamma$

$\gamma=1.46$

Step 3:

$\frac{C_{p}}{C_{v}}=\gamma \text { or } C_{p}-C_{v}=R$

$C_{v}=\frac{R}{\gamma-1}=\frac{8.3}{1.46-1}=\frac{18.0 \mathrm{J}}{\mathrm{mol}}-\mathrm{K}$

$C_{p}=\gamma C_{v}=1.46 \times 18.0=26.28 \approx 26.3 / \mathrm{mol}-K$