Heat pump vs. refrigerator relationship For the same device viewed as a heat pump and as a refrigerator, the coefficients of performance satisfy:

Introduction / Context:
The same vapour-compression machine can be interpreted either as a refrigerator (cooling device) or as a heat pump (heating device). Their C.O.P.s are closely linked.

Given Data / Assumptions:

• Q_L = cooling effect, Q_H = heating effect, W = compressor work.
• (C.O.P.)_R = Q_L / W; (C.O.P.)_HP = Q_H / W.
• First law for the cycle: Q_H = Q_L + W.

Concept / Approach:
Substitute the energy balance into the definitions to relate the performance ratios directly.

Step-by-Step Solution:
Start: Q_H = Q_L + W.Divide by W: Q_H/W = Q_L/W + 1.Therefore, (C.O.P.)_HP = (C.O.P.)_R + 1.

Verification / Alternative check:
Check a numerical example: if (C.O.P.)_R = 3, then (C.O.P.)_HP = 4. This matches the formula.

Why Other Options Are Wrong:
Options A and C add or subtract the wrong constant; D ignores the energy balance; E is not generally true.

Common Pitfalls:
Forgetting that a heat pump’s useful output includes both the cooling effect and the work input converted to heat at the hot side.

Final Answer:
(C.O.P.)_HP = (C.O.P.)_R + 1