Question

An ideal system operating with n moles of an ideal monatomic gas performs the cycle shown in the p-V diagram to the right. All parts of the cycle are carried out slowly and reversibly. Starting at state a, where its volume and pressure are $(V_0,p_0)$, the gas undergoes an isothermal expansion until it reaches state b where its volume and pressure are $(8V_0,p_0/8)$. In process $b\to c$ the gas is cooled at constant volume to $(8V_0,p_0/32)$. Process $c\to a$ is an adiabatic compression back to state a.

What is the heat added to the gas in process $b\to c$ ?

• A. $-\frac{9}{8}{p}_0{V}_0$
• B. $\frac{3}{2}{p}_0{V}_0$
• C. $\frac{9}{8}{p}_0{V}_0$
• D. 0

Answer 1

The correct option is A $-\frac{9}{8}{p}_0{V}_0$

$dq=dE=n{C}_{v}dT=n{C}_v(T_2-T_1)$ .......... (1)
$C_{v}foramonoatomicgas=\frac{3}{2}R$ .......... (2)
$T_2(temp.atC)\Rightarrow \frac{p_0}{32}\times 8V_0$ = $nR{T}_2$
$\Rightarrow T_2$ = $\frac{p_0{V}_0}{4nR}$ .......... (3)
$T_1(temp.atB)\Rightarrow \frac{p_0}{8}\times 8V_0$ = $nR{T}_2$
$\Rightarrow T_1$ = $\frac{p_0{V}_0}{nR}$ .......... (4)
Put the value of equation (2), (3), (4) in equation (1).
$dq=n\frac{3}{2}R[\frac{p_0{V}_0}{4nR}-\frac{p_0{V}_0}{nR}]$
$dq=-\frac{9}{8}{p}_0{V}_0$
$b\to c$ is an isometric process (dW = 0).