Question

A cycle followed by an engine (made of one mole of an ideal gas in a cylinder with a piston) is shown in figure. Find heat exchanged by the engine, with the surroundings for each section of the cycle. (C_v = (3/2)R)

1. AB : constant volume
2. BC : constant pressure
3. CD : adiabatic
4. DA : constant pressure

Answer 1

a. By using the first law of thermodynamics, we can find the amount of heat associated with each process

For process AB

Volume is constant, hence work done dW = 0

According to first law of thermodynamics,

`ΔQ = ΔU + ΔW = ΔU + 0 = ΔU`

= `nCvΔT = nCv(T_B - T_A)`

= `3/2 R(T_B - T_A)`  .....(∵ n = 1)

= `3/2 (RT_B - RT_A)`

= `3/2 (P_AV_B - P-AV_A)`

= `3/2 (P_B - P_A)V_A`  .....[∵ V_B = V_A]

b. For process BC, P = constant

`ΔQ = ΔU + ΔW`

= `3/2 R(T_C - T_B) + P_B(V_C - V_B)`

= `3/2 (P_CV_C - P+BV_B) + P_B (V_C - V_B)`

= `5/2 P_B (V_C - V_B)`

Heat exchanged = `5/2 P_B (V_C - V_A)`  .....(∵ P_B = P_C and P_B = V_A)

c. For process CD, Q_CD = 0  .....(As the change is adiabatic.)

d. In process DA involves compression of gas from V_D to V_A at constant pressure P_A.

∴ Heat exchanged can be calculated in a similar way as process BC.

Hence, `ΔQ = 5/2 P_A(V_A - V_D)`.