For an adiabatic expansion of an ideal gas,

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):

1. $-\gamma \frac{\mathrm{dV}}{\mathrm{V}}$

2. $-\gamma \frac{\mathrm{V}}{\mathrm{dV}}$

3. $-\frac{1}{\gamma} \frac{\mathrm{dV}}{\mathrm{V}}$

4. $\frac{\mathrm{dV}}{\mathrm{V}}$

Correct Option: 1

Solution:

(1)

$\mathrm{PV} \gamma=$ constant

Differentiating

$\frac{\mathrm{dP}}{\mathrm{dV}}=-\frac{\gamma \mathrm{P}}{\mathrm{V}}$

$\frac{\mathrm{dP}}{\mathrm{P}}=-\frac{\gamma \mathrm{dV}}{\mathrm{V}}$