Three moles of an ideal gas at a 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 the P-V and P-T diagrams for the complete process.
(b) Calculate the net work done by the gas, and net heat supplied to the gas during the complete process.

Explanation

Process `A rarr B : (P_(A), V_(A), T_(A)) rarr (P_(B), 2V_(A) = V_(B), T_(B) = T_(A))`
`:. (P_(A) V_(A))/(T_(A)) = (P_(B) 2V_(A))/(T_(A)) implies P_(B) = P_(A)//2`
`Delta U = 0 ( :' Delta T = 0)` and `Delta W = int_(V_(A))^(V_(B)) pdV = RT_(A) 1n 2`
`:. Delta Q = Delta U + Delta W = Delta W = RT_(A) 1n 2`
Process `B rarr C : (P_(A)//2, T_(A), 2V_(A)) rarr (P_(A)//2, T_(C), V_(A))`
`:. ((P_(A)//2) 2V_(A))/(T_(A)) = ((P_(A)//2) V_(A))/(T_(C )) = gt T_(C ) = T_(A) // 2`
`:. Delta W = int_(2V_(A))^(V_(A)) pdV = (1)/(2) P_(A) (V_(A) - 2V_(A))`
`= - (1)/(2) P_(A) V_(A) = - (1)/(2) RT_(A)`
`:. Delta Q = Delta U + Delta W = - (5)/(4) RT_(A) - (1)/(2) RT_(A) = - (7)/(4) RT_(A)`
Process `C rarr A: ((1)/(2) P_(A), (1)/(2) T_(A), V_(A)) rarr (P_(A), T, V_(A))`
`((1)/(2) P_(A) xx V_(A))/((1)/(2) T_(A)) = (P_(A) V_(A))/(T) implies T = T_(A)`
`Delta U = C_(v) (T_(A) - (1)/(2) T_(A)) = (5)/(2) R (T_(A) // 2) = (5)/(4) RT_(A)`
`Delta W = 0 :. Delta Q = Delta U + Delta W = ((5)/(4)) RT_(A)`
`:.` Net work done by the gas is
`3[RT_(A) 1n 2 - (1)/(2) RT_(A) + 0] = 3RT_(A) (1n 2 - (1)/(2))`
Net heat abosrbed is
`3[RT_(A) 1n 2 - (7)/(4) RT_(A) + (5)/(4) RT_(A)] = 3 RT_(A) (1n 2 - (1)/(2))`