Question

A monoatomic gas at pressure $P$ and volume $V$ expands isothermally to a volume $2V$ and then adiabatically to a volume $16V$. Then the final pressure of the gas is (take $\gamma =\frac{5}{3})$

• A. $64P$
• B. $32P$
• C. $\frac{P}{64}$
• D. $16P$

Answer 1

The correct option is C $\frac{P}{64}$

Let $P_1$ be the pressure after isothermal expansion.
So,
$\Rightarrow PV=Constant$
$\Rightarrow PV=P_{1}2V$
$\Rightarrow P_1=\frac{P}{2}$

Let $P_f$ be the final pressure after adiabatical expansion.
So,
$\Rightarrow P{V}^{\gamma }=Constant.\Rightarrow P_1(2V{)}^{\gamma }=P_f(16V{)}^{\gamma }\Rightarrow P_f=\frac{P}{2}(\frac{1}{2^3}{)}^{\frac{5}{3}}\Rightarrow P_f=\frac{P}{2}(\frac{1}{2^3}{)}^{\frac{5}{3}}=\frac{P}{\left(2\right)(2{)}^5}\Rightarrow P_f=\frac{P}{64}$