Three moles 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 gas is compressed at constant volume to its original pressure `P_A`.
(a) Sketch 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.

Answer

B, D

Explanation

(a) Process A to B (isothermal expansion)
`P_AV_A=P_BV_B`
`rArrP_AV_A=P_B2V_A`
`rArrP_B=(P_A)/2`
Process B to C (isobaric
compression)

`(V_B)/(T_B)=(V_C)/(T_C)`
`rArr(2V_A)/(T_A)=(V_A)/(T_C)`
`rArrT_C=(T_A)/2`
Process C to A [volume is constant]
`(P_C)/(T_C)=(P_A)/(T_A) rArr (P_B)/(T_C)=(P_A)/(T_A)`
`rArr((P_A)//2)/(T_C)=(P_A)/(T_A) rArr T_C=(T_A)/2`

Let the system initially be at point A at pressure `P_A` and temp
`T_A` and volume `V_A.`
Process A to B
The system is isothermally expanded and reaches a new
state `B(P_B,2V_A,T_A)` as shown in the figure.
Process B to C
The system is the compressed at constant pressure to its
original volume to reach at state `C(P_B,V_A,T_C)`
Process C to A
Finally at constant volume, the pressure is increased to its
original pressure to each the state A again.
(b) The total work done
`W=W_(AtoB)+W_(BtoC)+W_(CtoA)`
`=nRT_Alog_e(V_B)/(V_A)+nR(T_C-T_A)+0`
`=2.303xx3xxRxxT_Alog_(10)(2V_A)/(V_A)+3R((T_A)/2-T_A)`
`=2.08RT_A-3/2RT_A=0.58RT_A`