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,

${\displaystyle \Delta Q=\Delta U+\Delta W=\Delta U+0=\Delta U}$

= ${\displaystyle nCv\Delta T=nCv(T_B-T_A)}$

= ${\displaystyle \frac{3}{2}R(T_B-T_A)}$  .....(∵ n = 1)

= ${\displaystyle \frac{3}{2}(R{T}_B-R{T}_A)}$

= ${\displaystyle \frac{3}{2}(P_A{V}_B-P-A{V}_A)}$

= ${\displaystyle \frac{3}{2}(P_B-P_A)V_A}$  .....[∵ V_B = V_A]

b. For process BC, P = constant

${\displaystyle \Delta Q=\Delta U+\Delta W}$

= ${\displaystyle \frac{3}{2}R(T_C-T_B)+P_B(V_C-V_B)}$

= ${\displaystyle \frac{3}{2}(P_C{V}_C-P+B{V}_B)+P_B(V_C-V_B)}$

= ${\displaystyle \frac{5}{2}{P}_B(V_C-V_B)}$

Heat exchanged = ${\displaystyle \frac{5}{2}{P}_B(V_C-V_A)}$  .....(∵ 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, ${\displaystyle \Delta Q=\frac{5}{2}{P}_A(V_A-V_D)}$.