Question 749: Thermodynamics & KTG

Question

The ratio of the velocity of sound in hydrogen gas $\left(\gamma=\frac{7}{5}\right)$ to that in helium gas $\left(\gamma=\frac{5}{3}\right)$ at the same temperature is

Accepted Answer

The velocity of sound in a gas is given by the formula:
$$ v = \sqrt{\frac{\gamma R T}{M}} $$
where:
$ v $ is the velocity of sound,
$ \gamma $ is the adiabatic index (ratio of specific heats $ C_p/C_v $),
$ R $ is the universal gas constant,
$ T $ is the absolute temperature,
$ M $ is the molar mass of the gas.
To find the ratio of the velocities of sound in hydrogen and helium at the same temperature, we'll first write the expression for each gas:
For hydrogen ($\gamma = \frac{7}{5}$, $M_H = 2$, as hydrogen is $H_2$):
$$ v_H = \sqrt{\frac{\frac{7}{5} \cdot R \cdot T}{2}} = \sqrt{\frac{7RT}{10}} $$
For helium ($\gamma = \frac{5}{3}$, $M_{He} = 4$):
$$ v_{He} = \sqrt{\frac{\frac{5}{3} \cdot R \cdot T}{4}} = \sqrt{\frac{5RT}{12}} $$
Now, let's find the ratio $\frac{v_H}{v_{He}}$:
$$ \frac{v_H}{v_{He}} = \frac{\sqrt{\frac{7RT}{10}}}{\sqrt{\frac{5RT}{12}}} $$
This simplifies to:
$$ \frac{v_H}{v_{He}} = \sqrt{\frac{7RT}{10} \cdot \frac{12}{5RT}} = \sqrt{\frac{7 \cdot 12}{10 \cdot 5}} = \sqrt{\frac{84}{50}} = \sqrt{\frac{42}{25}} $$
Therefore, the ratio is:
$$ \sqrt{42} : 5 $$
Option D is the correct answer.

Answer

$1: 1$

Answer

$7: 3$

Answer

$21: 25$

Answer

$\sqrt{42}: 5$

Thermodynamics & KTG

The ratio of the velocity of sound in hydrogen gas \left(\gamma=\frac{7}{5}\right) to that in helium gas \left(\gamma=\frac{5}{3}\right) at the same temperature is

Choose the correct answer: