A cycle followed by an engine is shown in the figure. Find heat exchanged by the engine, with the surroundings for each section of the cycle considering C_{v} = \left (\frac{3}{2} \right )R.

AB: constant volume

BC: constant pressure

CD: adiabatic

DA: constant pressure

Answers (1)

Ans.

(a) For A\rightarrow B, dV=0

dW=\int PdV=\int P\times 0=0

dQ=dU+dW=dU+0

dQ=dU

dQ=nC_vdT 

n=1;C_v=\frac{3}{2}R 

dQ=1\left (\frac{3}{2} R \right )(T_B-T_A)

dU=dQ_1=\frac{3}{2}\left (RT_B-RT_A \right )=\frac{3}{2}(P_BV_B-P_AV_A)

(b) B\rightarrow C

dQ_2=dU+dW=C_vdT+P_BdV

dQ_2=\frac{3}{2}R\left (T_C-T_B \right )+P_B\left (V_C-V_B \right )

=\frac{3}{2}[P_CV_C]-\frac{3}{2}{P_BV_B}-P_BV_B+P_BV_C 

V_A=V_B and P_B=P_{C}

dQ_2=\frac{5}{2}P_BV_C-\frac{5}{2}P_BV_A=\frac{5}{2}P_B[V_C-V_A ]

(c ) C\rightarrow D, adiabatic change

dQ_{3}=0

  1. In diagram D\rightarrow A

\Delta P=0

Compression of gas at constant pressure takes place. Therefore, heat exchange is similar to part (b)

dQ_{4}=\frac{5}{2}P_A(V_A-V_D)

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