The working gas of a weird refrigerator undergoes the p - V cycle described in t

Question

The working gas of a weird refrigerator undergoes the p - V cycle described in the diagram below. It has an ideal gas with specific heat ratio as its working substance. What is the coefficient of performance eta of this refrigerator in terms of the radios and (Note, the coefficient of performance is the total heat absorbed by the refrigerator in its heat- divided by the net work done on the refrigerator in a cycle of operation, and here heat its absorbed in both stages b & c.) The working gas of a weird refrigerator undergoes the p - V cycle described in the diagram below. It has an ideal gas with specific heat ratio as its working substance. What is the coefficient of performance eta of this refrigerator in terms of the radios and (Note, the coefficient of performance is the total heat absorbed by the refrigerator in its heat- divided by the net work done on the refrigerator in a cycle of operation, and here heat its absorbed in both stages b & c.)

Explanation / Answer

First calculating work done on the system.
Work will be done on the system in case of compression.
for cycles b and d work done would be zero, as change in volume is zero.
and work done = PdV =0
So compression in case of cycle a,
So work done = P2*(dV) = P2 * (V2-V1)
For heat calculation in cycle b and c
for cycle b
w =0
We know,
q=U +w

so q=U = nCvdt = (Cv/R)*(V1)*(P2 - P1)

for cycle c,

Since pressure is constant, q = nCpdt = (Cp/R)*(P1)*(V2-V1)

so, coefficient of performance = ((Cp/R)*(P1)*(V2-V1) + (Cv/R)*(V1)*(P2 - P1)) / P2(V2-V1)

and Cp/Cv =

but P2(V2-V1)= H - U (since it is for cycle b)

so,  P2(V2-V1)= (Cp/R)*(P1)*(V2-V1) -   (Cv/R)*(P1)*(V2-V1)

SO,  coefficient of performance = (P1(V2-V1)+V1(P2-P1))/(P1(V2-V1) - P1(V2-V1))

OR,   coefficient of performance = (((V2/V1) -1) + ((P2/P1) -1)) / ((V2/V1) -1)(-1)

OR,   coefficient of performance = (/(-1)) + ((P2/P1) -1)/((-1)*((V2/V1) -1))

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