Q. 4.17 - An Introduction to Thermal Physics

Question

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.

Step-by-Step Solution

Answer

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.

Step 1 : Concept Introduction

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.

Step 2 : Explanation

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.

Step 3 : Conclusion

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.