Question

Bulk modulus of an ideal gas for isothermal process initially is \(B\). Gas is compressed from volume \(V_0\) to \( \frac{V_0}{3} \) isothermally. Find the work done by gas.

• A. \(BV_0\ln3\)
• B. \(\frac{BV_0}{3}\ln3\)
• C. \(BV_0\ln\left(\frac{1}{3}\right)\)
• D. \(3BV_0\ln\left(\frac{1}{2}\right)\)

Answer 1

The Correct Option is C

Concept: For an isothermal process in an ideal gas, \[ W = nRT \ln\left(\frac{V_2}{V_1}\right) \] For an ideal gas undergoing isothermal change, bulk modulus \[ B = P \] Thus initial pressure \(P_0 = B\). Step 1: Use work done expression for isothermal process.} \[ W = nRT \ln\left(\frac{V_2}{V_1}\right) \] But \[ nRT = P_0V_0 \] \[ W = P_0V_0 \ln\left(\frac{V_2}{V_1}\right) \]
Step 2: Substitute volume change.} \[ V_1 = V_0, \qquad V_2 = \frac{V_0}{3} \] \[ W = P_0V_0 \ln\left(\frac{V_0/3}{V_0}\right) \] \[ W = P_0V_0 \ln\left(\frac{1}{3}\right) \]
Step 3: Replace \(P_0\) by \(B\).} \[ W = BV_0 \ln\left(\frac{1}{3}\right) \] Final Result \[ W = BV_0 \ln\left(\frac{1}{3}\right) \]