Question:
A vessel contains \(0.15\,\text{m}^3\) of a gas at pressure \(8\) bar and temperature \(140^\circ\text{C}\) with \(c_p=3R\) and \(c_v=2R\). It expands adiabatically till pressure falls to \(1\) bar. The work done during this process is ____ kJ. (R is gas constant)}
A vessel contains \(0.15\,\text{m}^3\) of a gas at pressure \(8\) bar and temperature \(140^\circ\text{C}\) with \(c_p=3R\) and \(c_v=2R\). It expands adiabatically till pressure falls to \(1\) bar. The work done during this process is ____ kJ. (R is gas constant)}
Updated On: Apr 10, 2026
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Correct Answer: 120
Solution and Explanation
Concept:
For an adiabatic process:
\[
PV^\gamma = \text{constant}
\]
where
\[
\gamma = \frac{C_p}{C_v}
\]
Work done in an adiabatic expansion:
\[
W = \frac{P_1V_1 - P_2V_2}{\gamma - 1}
\]
Step 1:Find \(\gamma\)} \[ \gamma = \frac{3R}{2R} = \frac{3}{2} = 1.5 \]
Step 2:Use adiabatic relation} \[ P_1V_1^\gamma = P_2V_2^\gamma \] \[ \frac{V_2}{V_1} = \left(\frac{P_1}{P_2}\right)^{1/\gamma} \] \[ = 8^{2/3} \] \[ = 4 \] \[ V_2 = 4 \times 0.15 = 0.6\,\text{m}^3 \]
Step 3:Convert pressure to SI} \[ P_1 = 8 \times 10^5 \text{ Pa}, \quad P_2 = 1 \times 10^5 \text{ Pa} \]
Step 4:Calculate work} \[ W = \frac{P_1V_1 - P_2V_2}{\gamma - 1} \] \[ = \frac{(8\times10^5)(0.15) - (1\times10^5)(0.6)}{0.5} \] \[ = \frac{120000 - 60000}{0.5} \] \[ = \frac{60000}{0.5} \] \[ = 120000 \text{ J} \] \[ = 120 \text{ kJ} \] \[ \boxed{120} \]
Step 1:Find \(\gamma\)} \[ \gamma = \frac{3R}{2R} = \frac{3}{2} = 1.5 \]
Step 2:Use adiabatic relation} \[ P_1V_1^\gamma = P_2V_2^\gamma \] \[ \frac{V_2}{V_1} = \left(\frac{P_1}{P_2}\right)^{1/\gamma} \] \[ = 8^{2/3} \] \[ = 4 \] \[ V_2 = 4 \times 0.15 = 0.6\,\text{m}^3 \]
Step 3:Convert pressure to SI} \[ P_1 = 8 \times 10^5 \text{ Pa}, \quad P_2 = 1 \times 10^5 \text{ Pa} \]
Step 4:Calculate work} \[ W = \frac{P_1V_1 - P_2V_2}{\gamma - 1} \] \[ = \frac{(8\times10^5)(0.15) - (1\times10^5)(0.6)}{0.5} \] \[ = \frac{120000 - 60000}{0.5} \] \[ = \frac{60000}{0.5} \] \[ = 120000 \text{ J} \] \[ = 120 \text{ kJ} \] \[ \boxed{120} \]
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