Question

In an adiabatic process, $R=\frac{2}{3}{C}_v$. The pressure of the gas will be proportional to

• A. $T^{5/3}$
• B. $T^{5/2}$
• C. $T^{5/4}$
• D. $T^{5/6}$

Answer 1

The correct option is B $T^{5/2}$

We know that for an adiabatic process,
$P{V}^{\gamma }=constant$
where $\gamma =\frac{C_p}{C_v}$
Given, $R=\frac{3}{2}{C}_v\Rightarrow C_v=\frac{3}{2}R$
Also for an adiabatic process, $C_v=\frac{R}{\gamma -1}$
Equating the expressions of $C_v$ we get,
$\frac{1}{\gamma -1}=\frac{3}{2}\Rightarrow \gamma =\frac{5}{3}$
In general for an adiabatic process, $P\propto T^{\frac{\gamma }{\gamma -1}}$
Subtituting the value of $\gamma$ in the above equation we get, $P\propto T^{\frac{5}{2}}$