Question

In an adiabatic process with the ratio of two specific heat, \( \gamma = \frac{3}{2} \), pressure is increased by \( \frac{2}{3} \% \), then decrease in the volume will be

• A. \( \frac{4}{9}\% \)
• B. \( \frac{2}{3}\% \)
• C. 4\%
• D. \( \frac{9}{4}\% \)

Hint:

In adiabatic processes, use the relation \( PV^\gamma = \text{constant} \) and apply logarithmic differentiation to find small changes.

Answer 1

The Correct Option is A

For an adiabatic process: \[ PV^\gamma = \text{constant} \Rightarrow \ln P + \gamma \ln V = \text{constant} \] Differentiating: \[ \frac{\Delta V}{V} = -\frac{1}{\gamma} \cdot \frac{\Delta P}{P} \Rightarrow \frac{\Delta V}{V} = -\frac{1}{\frac{3}{2}} \cdot \frac{2}{3} = -\frac{4}{9} \] So, volume decreases by \( \frac{4}{9}\% \).