Question

The relation between internal energy $U$ , pressure $P$ and volume $V$ of gas in an adiabatic process is $\text{U}=a+\text{bPV}$ , where $a$ and $b$ are constants. What is the effective value of adiabatic constant $\gamma$ ?

• A. $\frac{\text{a}}{\text{b}}$
• B. $\frac{\text{b}+1}{\text{b}}$
• C. $\frac{a+1}{a}$
• D. $\frac{\text{b}}{\text{a}}$

Answer 1

Answer: (B)

For an adiabatic process,
$0=dU+PdV$
or $d(a+bPV)+PdV=0$
or $bPdV+bVdP+pdV=0$
or $(b+1)PdV+bVdP=0$
or $\left(\text{b}+1\right)\frac{\text{dV}}{\text{V}}+\text{b}\frac{\text{dP}}{\text{P}}=0$
$(b+1)logV+blogP=$ constant
${\text{V}}^{\text{b}+1}{\text{p}}^{\text{b}}=\text{constant}$
or ${\text{PV}}^{\frac{\text{b}+1}{\text{b}}}=\text{constant}$
$\therefore \gamma =\frac{\text{b}+1}{\text{b}}$