Question

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

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

Answer 1

The correct option is B $\frac{b+1}{b}$

Given, $U=a+bPV$
For an ideal gas, $PV=nRT$
$\therefore U=a+bnRT$
Differentiation of the above equation,
$dU=0+bnRdT$.....(i)
$\Rightarrow \frac{dU}{dT}=bnR$
In general we know, $dU=n{C}_{v}dT\Rightarrow \frac{dU}{dT}=n{C}_v$.....(ii)
We know for an adiabatic process, $C_v=\frac{R}{\gamma -1}$
Equating (i) and (ii) and substituting the value of $C_v$ we get, $\frac{1}{\gamma -1}=b$
$\Rightarrow \gamma =\frac{b+1}{b}$