Prove that if you had a refrigerator whose COP was better than the ideal value (4.9), you could hook it up to an ordinary Carnot engine to make an engine that produces no waste heat.

Let us write the expression of the ideal value of COP for Carnot engine

$\mathrm{COP}=\frac{T_{\text{cold}}}{T_{\text{hol}}{T}_{\text{coll}}}$

Here, $T_{\text{cold}}$is temperature of the cold reservoir and $T_{\text{hot}}$is temperature of hot reservoir.

A refrigerator that has a coefficient of performance greater than the maximum Carnot value of COP requires an amount of work that is less than the work done in a Carnot engine.

A Carnot engine extracts heat from the hot reservoir and transfers it to the cold reservoir. The refrigerator, when hooked up to the Carnot engine, causes the composite system to yield a net surplus of work.

The compound system does not produce any waste heat because the refrigerator extracts the same amount of heat that the Carnot engine transfers.

Thus, it is verified that a refrigerator with COP better than the ideal value can be hooked up to a Carnot engine to make an engine not producing waste heat.