Question

An inventor claims to have developed a refrigerator that maintains the refrigerated space at \(40^{\circ} \mathrm{F}\) while operating in a room where the temperature is \(85^{\circ} \mathrm{F}\) and that has a COP of \(13.5 .\) Is this claim reasonable?

Short answer

Answer: ___________ [Use the provided step-by-step solution to calculate the Carnot COP, compare it with the claimed COP, and conclude if the claim is reasonable or not.]

Step-by-step solution

Convert temperatures to Kelvin

Before we proceed with calculations, we need to convert the given temperatures from Fahrenheit to Kelvin. We use the following formula to convert Fahrenheit to Kelvin: $$ K = \frac{5}{9}(°F - 32) + 273.15 $$ Calculate the Kelvin temperatures for the refrigerated space and the room: $$ T_{cold} = \frac{5}{9}(40 - 32) + 273.15 $$ $$ T_{hot} = \frac{5}{9}(85 - 32) + 273.15 $$

Calculate Carnot COP

The Carnot COP is the theoretical maximum efficiency of any heat engine or refrigerator, and it is determined by the ratio of the temperatures of the cold and hot reservoirs. The formula for Carnot COP of a refrigerator is: $$ COP_{Carnot} = \frac{T_{cold}}{T_{hot} - T_{cold}} $$ Substitute the values of \(T_{cold}\) and \(T_{hot}\) obtained in Step 1 into the formula, and calculate the Carnot COP.

Compare Carnot COP with the claimed COP

Now that we have calculated the Carnot COP, we can compare it with the claimed COP of \(13.5\). If the claimed COP is indeed less than or equal to the Carnot COP, then the claim is reasonable. Otherwise, the claim is not reasonable. Compare the calculated Carnot COP with the claimed COP of \(13.5\).

Conclude if the inventor's claim is reasonable

Based on the comparison of the Carnot COP and the claimed COP in Step 3, we can conclude if the inventor's claim of developing a refrigerator that maintains the refrigerated space at \(40^{\circ} \mathrm{F}\) while operating in a room at \(85^{\circ} \mathrm{F}\) with a COP of \(13.5\) is reasonable or not.