Question

At constant pressure, if the work done by a gas is 40% of the increase in the internal energy of the gas, then the specific heat capacity of the gas at constant volume is
(Universal gas constant \(R = 8.3 \, \text{J mol}^{-1} \text{K}^{-1}\))

• A. \(29.05 \, \text{J mol}^{-1} \text{K}^{-1}\)
• B. \(25.35 \, \text{J mol}^{-1} \text{K}^{-1}\)
• C. \(20.75 \, \text{J mol}^{-1} \text{K}^{-1}\)
• D. \(32.55 \, \text{J mol}^{-1} \text{K}^{-1}\)

Hint:

Use the relation \(Q = \Delta U + W\) at constant pressure and express all quantities in terms of \(\Delta U\) to find \(C_V\).

Answer 1

The Correct Option is C

Let the increase in internal energy be \(\Delta U\). Given, \(W = 0.4 \Delta U\) At constant pressure: \[ Q = \Delta U + W = \Delta U + 0.4 \Delta U = 1.4 \Delta U \] So, \[ \gamma = \frac{C_P}{C_V} = \frac{Q}{\Delta U} = 1.4 \Rightarrow \frac{C_V + R}{C_V} = 1.4 \Rightarrow \frac{8.3}{C_V} = 0.4 \Rightarrow C_V = \frac{8.3}{0.4} = 20.75 \, \text{J mol}^{-1} \text{K}^{-1} \]