QJEEM23Pha9e1: JEE Main 2023 Physics - Class 11 JEEM23Pha9e1 Solution: The ratio of speed of sound in hydrogen gas to the speed of…

The speed of sound in a gas is given by the formula:

$$v=\sqrt{\frac{\gamma RT}{M}}$$

where $$v$$ is the speed of sound, $$\gamma$$ is the adiabatic index, $$R$$ is the universal gas constant, $$T$$ is the temperature, and $$M$$ is the molar mass of the gas.

For diatomic gases, such as hydrogen (H₂) and oxygen (O₂), the adiabatic index $$\gamma$$ is the same, approximately equal to $$\frac{7}{5}$$, and the temperature is given as the same for both gases.

Let's denote the speed of sound in hydrogen as $$v_{\text{H}}$$ and in oxygen as $$v_{\text{O}}$$. The ratio of the speeds can be calculated as:

$$\frac{v_{\text{H}}}{v_{\text{O}}}=\sqrt{\frac{M_{\text{O}}}{M_{\text{H}}}}$$

The molar mass of hydrogen (H₂) is $$2\text{g/mol}$$, and the molar mass of oxygen (O₂) is $$32\text{g/mol}$$.

Now, we can calculate the ratio of the speeds:

$$\frac{v_{\text{H}}}{v_{\text{O}}}=\sqrt{\frac{32}{2}}=\sqrt{16}=4$$

This means the ratio of the speed of sound in hydrogen gas to the speed of sound in oxygen gas at the same temperature is $$4:1$$.

Therefore, the correct answer is $$4:1$$.