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

Step 1: Understanding the Concept:
In an adiabatic process, the state equation is \( PV^\gamma = \text{constant} \). We can differentiate this equation to find the relationship between small changes in pressure and volume.
: Key Formula or Approach:
Differentiating \( PV^\gamma = K \).
Step 2: Detailed Explanation:
Differentiating both sides with respect to the variables:
\[ d(PV^\gamma) = 0 \]
Using the product rule:
\[ V^\gamma dP + P (\gamma V^{\gamma-1} dV) = 0 \]
Divide the entire equation by \( V^\gamma \):
\[ dP + \frac{\gamma P dV}{V} = 0 \]
Rearranging to find \( \frac{dV}{V} \):
\[ \frac{\gamma P dV}{V} = -dP \]
\[ \frac{dV}{V} = -\frac{dP}{\gamma P} \]
Substituting \( \Delta \) for infinitesimal \( d \):
\[ \frac{\Delta V}{V} = -\frac{\Delta P}{\gamma P} \]
Step 3: Final Answer:
The ratio \( \Delta V/V \) is \( -\Delta P/\gamma P \).