Question

The value of $\gamma =\frac{C_P}{C_V}$ is $\frac{4}{3}$ for an adiabatic process of an ideal gas for which the internal energy is given by $dU=n(VdP+PdV)$. The value of $n$ is

• A. $\frac{2}{3}$
• B. $3$
• C. $4$
• D. $\frac{3}{4}$

Answer 1

The correct option is B $3$

Given $\gamma =\frac{C_P}{C_V}=\frac{4}{3}$
and $dU=n(VdP+PdV)$
In a simple adiabatic process,
$\Delta Q=\Delta U+\Delta W$
where $\Delta Q=0=\Delta U+\Delta Wor=\Delta U+PdV...\left(1\right)$

In a adiabatic process:$P{V}^{\gamma }=CP{V}^{4/3}=C$

Taking $log$ on both sides,
$lnP+\frac{4}{3}lnV=C3lnP+4lnV=C$

Differentiating the above equation
$3\frac{dP}{P}+4\frac{dV}{V}=0$
$3VdP+4PdV=03VdP+3PdV+PdV=0...\left(2\right)$

On comparing $1$ and $2$,
$dU=3(VdP+PdV)$
Hence, $n=3$

So, option B is correct.