Question

Two monoatomic ideal gases 1 and 2 has molecular weights \(\mathrm{m}_{1}\) and \(\mathrm{m}_{2}\). Both are kept in two different containers at the same temperature. The ratio of velocity of sound wave in gas 1 and 2 is $\ldots \ldots \ldots$ (A) \(\sqrt{\left(m_{2} / m_{1}\right)}\) (B) \(\sqrt{\left(m_{1} / m_{2}\right)}\) (C) \(\left(\mathrm{m}_{1} / \mathrm{m}_{2}\right)\) (D) \(\left(\mathrm{m}_{2} / \mathrm{m}_{1}\right)\)

Short answer

The ratio of the velocity of sound wave in gas 1 and 2 is \(\sqrt{\frac{m_{2}}{m_{1}}}\).

Step-by-step solution

Write the formula for the velocity of sound in ideal gases

The formula for the velocity of sound in a monoatomic ideal gas is given by: \(v = \sqrt{\frac{\gamma P}{\rho}}\) Where \(v\) is the velocity of sound, \(\gamma\) is the adiabatic constant (for a monoatomic ideal gas, \(\gamma = \frac{5}{3}\), \(P\) is the pressure of the gas, and \(\rho\) is the density of the gas.

Write the density in terms of molecular weight

The density of a gas is related to its molecular weight as follows: \(\rho = \frac{mN}{V}\) Where \(m\) is the molecular weight of the gas, \(N\) is the amount of substance, and \(V\) is the volume of the gas.

Substitute the density expression in the formula for the velocity of sound

By substituting the expression for density (\(\rho = \frac{mN}{V}\)) into the formula for the velocity of sound, we have: \(v = \sqrt{\frac{\gamma P}{\frac{mN}{V}}}\)

Simplifying the formula for the velocity of sound

Now if we simplify the above formula, we get: \(v = \sqrt{\frac{\gamma PV}{mN}}\)

Relate the pressure, volume, amount of substance, and temperature for ideal gases

According to the ideal gas law: \(PV = NRT\) Where \(R\) is the gas constant, and \(T\) is the temperature.

Find the expressions for the velocities of sound in both gases

For the gases 1 and 2 at the same temperature, the ideal gas law becomes: \(P_1V_1 = N_1RT\) \(P_2V_2 = N_2RT\) Now, we can find the expressions for the velocities of sound in both gases using these equations and the simplified formula for the velocity of sound: \(v_1 = \sqrt{\frac{\frac{5}{3}P_1V_1}{m_1N_1}} = \sqrt{\frac{\frac{5}{3}N_1RT}{m_1N_1}} = \sqrt{\frac{5R}{3m_1}T}\) Similarly for gas 2, \(v_2 = \sqrt{\frac{5R}{3m_2}T}\)

Find the ratio of velocities

To find the ratio of velocities of sound in gas 1 and 2, divide \(v_1\) by \(v_2\): \(\frac{v_1}{v_2} = \frac{\sqrt{\frac{5R}{3m_1}T}}{\sqrt{\frac{5R}{3m_2}T}} = \sqrt{\frac{\frac{5R}{3m_1}T}{\frac{5R}{3m_2}T}} = \sqrt{\frac{m_2}{m_1}}\) The ratio is equal to \(\sqrt{\frac{m_2}{m_1}}\). Therefore, the correct answer is (A).