Question

Two samples $A$ and $B$ of a gas initially of same pressure and temperature are compressed from a volume $V$ to a volume $\frac{V}{2}$ such that $A$ is compressed isothermally while $B$ is compressed adiabatically. The final

• A. pressure of $A$ is greater than that of $B$
• B. pressure of $A$ is equal to that of $B$
• C. pressure of $A$ is less than that of $B$
• D. pressure of $A$ is twice the pressure of $B$

Answer 1

Answer: (C)

pressure of $A$ is less than that of $B$

Ideal gas equation: $PV=RT$

Isothermal process:
Since temp T is constant $P_1{V}_1=P_2{V}_2$
$V_1=V$ as given
$V_2=\frac{V}{2}$ as given

$\therefore P_1\times V=P_2\times \frac{V}{2}$

$\therefore P_2=2P_1$

Adiabatic process:

$V_1=V$ as given
$V_2=\frac{V}{2}$ as given

$P{V}^{\gamma }=$constant

$P_1{V_1}^{\gamma }=P_2{V_2}^{\gamma }$

$\therefore P_1{V}^{\gamma }=P_2(\frac{V}{2}{)}^{\gamma }$

$\therefore P_2=P_1\times 2^{\gamma }$

Since $\gamma$ > 1, $P_2$ > $P_1$

Hence, $(P_2{)}_{isothermal}$ < $(P_2{)}_{adiabatic}$