Question

During an adiabatic process, the pressure P of a fixed mass of an ideal gas changes by \( \Delta P \) and its volume V changes by \( \Delta V \). If \( \gamma = C_P/C_V \), then \( \Delta V/V \) is given by:

• A. \( -\Delta p/p \)
• B. \( -\gamma \Delta P/P \)
• C. \( -\frac{\Delta P}{\gamma P} \)
• D. \( \frac{\Delta P}{\gamma^2 P} \)

Hint:

Always remember: \( PV^\gamma = \text{constant} \) and differentiate logarithmically.

Answer 1

The Correct Option is C

Concept: Adiabatic relation: \[ PV^\gamma = \text{constant} \]

Step 1: Differentiate.
\[ d(PV^\gamma) = 0 \] \[ V^\gamma dP + \gamma P V^{\gamma-1} dV = 0 \]

Step 2: Simplify.
Divide by \(PV^\gamma\): \[ \frac{dP}{P} + \gamma \frac{dV}{V} = 0 \]

Step 3: Relation.
\[ \frac{dV}{V} = -\frac{1}{\gamma} \frac{dP}{P} \] \[ {\frac{\Delta V}{V} = -\frac{\Delta P}{\gamma P}} \]