Question

Given that for a gas $B_i$ = isothermal bulk modulus, $B_a$ = adiabatic bulk modulus, pressure $=p$ and $\gamma =$ the ratio of the specific heat at constant pressure to that constant volume. Which of the following relations is correct?

• A. $B_a=p$
• B. $B_i=p$
• C. $B_i=\gamma B_a$
• D. None of the above.

Answer 1

The correct option is B $B_i=p$

For isothermal process:

$B_i=-V.\frac{dP}{dV}atconstantTPV=nRTP=\frac{nRT}{V}\frac{dP}{dV}=-\frac{nRT}{V^2}{B}_i=-V.-\frac{nRT}{V^2}{B}_i=\frac{nRT}{V}=P$

For adiabatic process:

$d\left(P{V}^x\right)=xP{V}^{(x-1)}dV+V^{x}dP=0$, since $P{V}^x$ is constant.

Thus, $-\frac{dV}{V}=\frac{dP}{xP}$ and plugging this in as above gives $B=xP$.