Question

For an adiabatic expansion of an ideal gas, the fractional change in its pressure is equal to (where \(\gamma\) is the ratio of specific heats):

• A. \(-\gamma \frac{dV}{V}\)
• B. \(-\frac{dV}{V}\)
• C. \(\gamma \frac{dV}{V}\)
• D. \(\frac{dV}{V}\)

Hint:

In adiabatic processes, pressure drops significantly faster than in isothermal processes (\(\frac{dP}{P} = -\frac{dV}{V}\)) for the same volume change.

Answer 1

The Correct Option is A

Step 1: For an adiabatic process: \(PV^\gamma = \text{constant}\).
Step 2: Differentiate both sides: \[d(PV^\gamma) = 0 \implies V^\gamma dP + P(\gamma V^{\gamma-1} dV) = 0\]
Step 3: Rearrange to find the fractional change in pressure \(\frac{dP}{P}\): \[V^\gamma dP = -P\gamma V^{\gamma-1} dV \implies \frac{dP}{P} = -\gamma \frac{V^{\gamma-1}}{V^\gamma} dV = -\gamma \frac{dV}{V}\]