Question

Three moles of an ideal gas ($C_p =\frac{7}{2}R)$ 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 gas is compressed at constant
volume to its original pressure p$_A$.
(a) Sketch p-V and p-Tdiagrams for the complete process.
(b) Calculate the net work done by the gas, and net heat
supplied to the gas during the complete process.

Answer 1

(a) Thep-V andp-T diagrams are shown below $(pV)_C =\frac{(pV)_A}{2} \, \, \Rightarrow \, \, \, \therefore \, \, \, T_C=\frac{T_A}{2}$ (b) Process A -B T = constant $p \propto \frac{1}{V},V$ is doubled. Therefore, p will become half. Further , $V_A =\frac{nRT_A}{p_A} =\frac{3RT_A}{p_A}$ $ \, \, \, \, \, \, \Delta U_{AB}=0$ $\therefore \, \, \, \, \, Q_{AB}=W_{AB}-nRT_A ln \bigg(\frac{2V_A}{V_A}\bigg)$ $ \, \, \, \, \, \, \, =3RT_A ln(2) =2.08 RT_A$ Process B-C $Q_{BC} =nV_p (T_C-T_B)$ $=(3) \bigg(\frac{7}{2}R\bigg) \bigg(\frac{T_A}{2}T_A\bigg)=-\frac{21}{4}RT_A$ =-5.25 $RT_A$ Process C-A V = constant $\therefore \, \, \, \, W_{CA} =0$ ' or $ \, \, \, \, Q_{CA} =\Delta U_{CA} =nC_V (T_A -T_C)$ $ \, \, \, \, \, \, \, \, \, \, (3)\bigg(\frac{5}{2}R\bigg)\bigg(T_A -\frac{T_A}{2}\bigg)=3.75 RT_A$ In a cyclic process, $\Delta$U = 0 $\therefore \, \, \, Q_{net} =W_{net}=0.58 RT_A$