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 , $V_A=\frac{nR{T}_A}{p_A}=\frac{3R{T}_A}{p_A}$ $\Delta U_{AB}=0$ $\therefore Q_{AB}=W_{AB}-nR{T}_{A}ln\left(\frac{2V_A}{V_A}\right)$ $=3R{T}_{A}ln(2)=2.08R{T}_A$ Process B-C $Q_{BC}=n{V}_p(T_C-T_B)$ $=(3)\left(\frac{7}{2}R\right)\left(\frac{T_A}{2}{T}_A\right)=-\frac{21}{4}R{T}_A$ =-5.25 $R{T}_A$ Process C-A V = constant $\therefore W_{CA}=0$ ' or $Q_{CA}=\Delta U_{CA}=n{C}_V(T_A-T_C)$ $(3)\left(\frac{5}{2}R\right)(T_A-\frac{T_A}{2})=3.75R{T}_A$ In a cyclic process, $\Delta$U = 0 $\therefore Q_{net}=W_{net}=0.58R{T}_A$