Question

The pressure and volume of an ideal gas are related as \( PV^{3/2} = K \) (constant). The work done when the gas is taken from state A (\( P_1, V_1, T_1 \)) to state B (\( P_2, V_2, T_2 \)) is:

• A. \( 2(P_1V_1 - P_2V_2) \)
• B. \( 2(P_2V_2 - P_1V_1) \)
• C. \( 2(\sqrt{P_1V_1} - \sqrt{P_2V_2}) \)
• D. \( 2(P_2\sqrt{V_2} - P_1\sqrt{V_1}) \)

Answer 1

The Correct Option is A, B

Given the relation \(PV^{3/2} = \text{constant}\), the work done \(W\) is defined as:

\[ W = \int P \, dV \]

Substituting \(P = \frac{K}{V^{3/2}}\) from the given relation, where \(K\) is a constant:

\[ W = \int_{V_1}^{V_2} \frac{K}{V^{3/2}} \, dV \]

Upon integration, the work done is calculated as:

\[ W = \left[ -\frac{2K}{V^{1/2}} \right]_{V_1}^{V_2} = 2(P_1V_1 - P_2V_2) \]

- If the work done by the gas is required:

\[ W = 2(P_1V_1 - P_2V_2) \quad \text{(Option 1)} \]

- If the work done on the gas (by an external agent) is required:

\[ W = 2(P_2V_2 - P_1V_1) \quad \text{(Option 2)} \]