Speed of Sound in Gas Mixture PYQ | JEE Main Physics 2025

 

❓ Question

The speed of sound in hydrogen is

$$v_{H_2}=1270\ \text{m/s}.$$

A mixture of hydrogen and oxygen is prepared such that

$$\frac{V_{H_2}}{V_{O_2}}=\frac{4}{1}.$$

Find the speed of sound in the mixture.

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✍️ Solution

Let the volume of oxygen be

$$V.$$

Then,

$$\text{Volume of }H_2=4V.$$

Hence,

$$\text{Total volume}=5V.$$

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Step 1: Density of the mixture

At the same temperature and pressure,

$$\text{Density}\propto\text{Molar mass}.$$

Since

$$M_{H_2}=2,\qquad M_{O_2}=32,$$
$$\rho_{O_2}=16\rho_{H_2}.$$

Let

$$\rho_{H_2}=d.$$

Then

$$\rho_{O_2}=16d.$$

Now,

$$\text{Mass of mixture} = 4V(d)+V(16d) =20Vd.$$

Therefore,

$$\rho_{\text{mix}} = \frac{20Vd}{5V} =4d.$$

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Step 2: Use the speed of sound relation

For an ideal gas,

$$v=\sqrt{\frac{\gamma P}{\rho}}.$$

Since hydrogen and oxygen are both diatomic gases,

$$\gamma$$

is the same, and for the mixture at the same pressure,

$$v\propto\frac{1}{\sqrt{\rho}}.$$

Hence,

$$\frac{v_{\text{mix}}}{v_{H_2}} = \sqrt{\frac{\rho_{H_2}}{\rho_{\text{mix}}}} = \sqrt{\frac{d}{4d}} = \frac12.$$

Therefore,

$$v_{\text{mix}} = \frac{1270}{2} = 635\ \text{m/s}.$$

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✅ Final Answer

$$\boxed{635\ \text{m/s}}$$