Question

For an adiabatic expansion of an ideal gas, the fractional change in its pressure is equal to (where $\mathrm{\lambda }$ is the ratio of specific heats)

• A. $-\mathrm{\gamma }(\mathrm{dV}/\mathrm{V})$
• B. $\mathrm{dV}/\mathrm{V}$
• C. $1/\mathrm{\gamma }(\mathrm{dV}/\mathrm{V})$
• D. $-\mathrm{\gamma }(\mathrm{V}/\mathrm{dV})$

Answer 1

The correct option is A

$-\mathrm{\gamma }(\mathrm{dV}/\mathrm{V})$

Explanation for the correct options:

A) $-\mathrm{\gamma }(\mathrm{dV}/\mathrm{V})$

Fractional change in pressure $=(\mathrm{dP}/\mathrm{P})$

For adiabatic expansion : ${\mathrm{PV}}^{\mathrm{\gamma }}=$ constant

$\ln P+\gamma \ln V=$constant

Differentiating $\ln P+\gamma \ln V=$constant both sides, we get

$(\mathrm{dP}/\mathrm{P})+\mathrm{\gamma }(\mathrm{dV}/\mathrm{V})=0$

$=\mathrm{dP}/\mathrm{P}=-\mathrm{\gamma }(\mathrm{dV}/\mathrm{V})$

For an adiabatic expansion of an ideal gas, the fractional change in its pressure is equal to $-\mathrm{\gamma }(\mathrm{dV}/\mathrm{V})$

Explanation for the incorrect options:

B) The above explanation clearly shows that option B is incorrect. $\mathrm{dV}/\mathrm{V}$

C) The above explanation clearly shows that option C is incorrect. $1/\mathrm{\gamma }(\mathrm{dV}/\mathrm{V})$

D) The above explanation clearly shows that option D is incorrect.$-\mathrm{\gamma }(\mathrm{V}/\mathrm{dV})$

Hence, the correct option is A $-\mathrm{\gamma }(\mathrm{dV}/\mathrm{V})$. On differentiating the adiabatic expansion of ideal gas we did not get the B, C, and D options. So, these are incorrect options.