Question

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

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

Answer 1

Answer: (B)

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