Question

In a certain gas, the ratio of the speed of sound and root mean square speed is $\sqrt{\frac{9}{5}}$. The molar heat capacity of the gas in a process given by PT = constant is (Take R = 2 cal/mole K). Treat the gas as ideal.

• A. R 2
• B. 3R 2
• C. 5R 2
• D. 7R 2

Answer 1

Answer: (D)

7R/2

Answer 2

The ratio of the speed of sound ($v_s$) to the root mean square speed ($v_{rms}$) is given by: $v_s = \sqrt{\frac{\gamma RT}{M}}$ $v_{rms} = \sqrt{\frac{3RT}{M}}$ $\frac{v_s}{v_{rms}} = \sqrt{\frac{\gamma}{3}}$

Given $\frac{v_s}{v_{rms}} = \sqrt{\frac{9}{5}}$. So, $\sqrt{\frac{\gamma}{3}} = \sqrt{\frac{9}{5}}$ $\frac{\gamma}{3} = \frac{9}{5} \implies \gamma = \frac{27}{5} = 5.4$.

Determine the polytropic index 'n' for the process PT = constant: For an ideal gas, $PV = RT$. The given process is $PT = \text{constant}$. Substitute $T = PV/R$ into $PT = \text{constant}$: $P \left(\frac{PV}{R}\right) = \text{constant}$ $P^2V = \text{constant} \times R = \text{new constant}$ To match the standard polytropic form $PV^n = \text{constant}$, we can take the square root of both sides: $(P^2V)^{1/2} = \text{constant}' \implies P V^{1/2} = \text{constant}'$. Thus, the polytropic index is $n = 1/2$.

Calculate the molar heat capacity (C): The molar heat capacity for a polytropic process $PV^n = \text{constant}$ is given by: $C = C_V + \frac{R}{1-n}$ Since $C_V = \frac{R}{\gamma-1}$, substitute this: $C = \frac{R}{\gamma-1} + \frac{R}{1-n} = R \left( \frac{1}{\gamma-1} + \frac{1}{1-n} \right) = R \frac{1-n+\gamma-1}{(\gamma-1)(1-n)} = R \frac{\gamma-n}{(\gamma-1)(1-n)}$

Substitute $\gamma = 5/3$ and $n = 1/2$: $C = R \frac{5/3 - 1/2}{(5/3 - 1)(1 - 1/2)}$ $C = R \frac{(10-3)/6}{(2/3)(1/2)}$ $C = R \frac{7/6}{1/3}$ $C = R \times \frac{7}{6} \times 3 = \frac{7R}{2}$