Question

Three mole of an ideal gas $(C_p=7/2R)$ at pressure $P_A$ and temperature $T_A$ is isothermally expanded to twice its initial volume. It is then compressed at constant pressure to its original volume. Finally the gas is compressed at constant volume to its original pressure $P_A$.
(a) Sketch $P-T$ curves for complete process.
(b) Calculate the net work done by the gas and net heat supplied to gas during the complete process.

Answer 1

Assume the process starts from $P_A$ and $T_A$ and $V_A$

$A\to B$

Isothermal expansion, $T_A$ remains constant.

Since $PV=constantat{T}_A$

$P_A\to \frac{P_A}{2}{V}_A\to 2V_{A}T\to constant$

$B\to C$

Compression at constant pressure. Isobaric process

Since $\left(V\alpha T\right)T_A\to \frac{T_A}{2}2V_A\to V_{A}P\to constant$

$C\to A$

Isochoric process

$P\alpha T\frac{P_A}{2}\to P_A\frac{T_A}{2}\to T_{A}V\to constant$

For cyclic process, work done

= area of closed curve

$=\frac{1}{2}(T_A-\frac{T_A}{2})(P_A-\frac{P_A}{2})=\frac{1}{2}\left(\frac{T_A}{2}\right)\left(\frac{P_A}{2}\right)=\frac{T_A{P}_A}{8}$

and $△U=0$ (for cyclic process)

Hence heat supplied $=\frac{T_A{P}_A}{8}$