Question

Sound travels in a mixture of two moles of helium and $n$ moles of hydrogen If rms speed of gas molecules in the mixture is $\sqrt{2}$ times the speed of sound, then the value of $n$ will be

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The Correct Option is B

To determine the value of \(n\), we first need to understand the relationship between the root mean square (rms) speed of the gas molecules and the speed of sound in the mixture.

The root mean square speed \(v_{\text{rms}}\) of gas molecules is given by:

\(v_{\text{rms}} = \sqrt{\frac{3kT}{M}}\)

where \(k\) is the Boltzmann constant, \(T\) is temperature, and \(M\) is the molar mass of the gas.

The speed of sound \(v_s\) in a gas is given by:

\(v_s = \sqrt{\frac{\gamma RT}{M}}\)

where \(\gamma\) is the heat capacity ratio, \(R\) is the universal gas constant, and \(M\) is the molar mass of the gas.

Given that the rms speed is \(\sqrt{2}\) times the speed of sound in the mixture, we have:

\(\sqrt{\frac{3kT}{M}} = \sqrt{2} \times \sqrt{\frac{\gamma RT}{M}}\)

Squaring both sides yields:

\(\frac{3kT}{M} = 2 \times \frac{\gamma RT}{M}\)

Canceling common terms and simplifying, we get:

\(3k = 2\gamma R\)

We know \(k = \frac{R}{N_A}\), where \(N_A\) is Avogadro's number. Thus, substituting we get:

\(3 \frac{R}{N_A} = 2\gamma R\)

\(3 = 2\gamma N_A\)

\(\gamma = \frac{3}{2N_A}\)

For the mixture, comprising two moles of helium (monatomic) and \(n\) moles of hydrogen (diatomic), the effective molar mass \(M\) of the mixture can be calculated using:

\(M_{\text{mixture}} = \frac{2 \times 4 + n \times 2}{2+n}\)

For the gases in the mixture:

• Helium has \(\gamma = \frac{5}{3}\)
• Hydrogen has \(\gamma = \frac{7}{5}\)

The net value of \(\gamma\) for the mixture is determined by calculating the weighted average:

\(\gamma_{\text{mixture}} = \frac{2 \times \frac{5}{3} + n \times \frac{7}{5}}{2 + n}\)

Setting \(\gamma_{\text{mixture}} = \frac{3}{2}\) and solving:

\(\frac{2 \times \frac{5}{3} + n \times \frac{7}{5}}{2 + n} = \frac{3}{2}\)

Solving the above equation, we get \(n = 2\).

Thus, the correct answer is 2.