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

Step 1: Understanding the Concept:
For an ideal gas, the isothermal Bulk modulus is always equal to its instantaneous pressure. The work done in an isothermal process depends on the initial pressure, initial volume, and the ratio of final to initial volume.
Step 2: Key Formula or Approach:
Isothermal Bulk Modulus: \(B = P\)
Work done in an isothermal process: \[ W = nRT \ln\left(\frac{V_f}{V_i}\right) = P_i V_i \ln\left(\frac{V_f}{V_i}\right) \]
Step 3: Detailed Explanation:
The initial Bulk modulus is \(B\), so the initial pressure \(P_0 = B\). The initial volume is \(V_i = V_0\). The final volume is \(V_f = \frac{V_0}{3}\). Substitute these directly into the work done formula: \[ W = P_0 V_0 \ln\left(\frac{V_f}{V_i}\right) \] \[ W = B V_0 \ln\left(\frac{V_0 / 3}{V_0}\right) \] \[ W = B V_0 \ln\left(\frac{1}{3}\right) \]
Step 4: Final Answer:
The work done by the gas is \(BV_0\ell n(1/3)\).