A restaurant refrigerator has a coefficient of performance of 5.0. If the temperature in the kitchen outside the refrigerator is 32°C, what is the lowest temperature that could be obtained inside the refrigerator if it were ideal?

Verified step by step guidance

1. Understand the concept of the coefficient of performance (COP) for a refrigerator. The COP is defined as the ratio of the heat removed from the cold reservoir (Qc) to the work input (W). For an ideal refrigerator, the COP is also related to the temperatures of the cold reservoir (Tc) and the hot reservoir (Th) by the formula: $\mathrm{COP}=\frac{\mathrm{Tc}}{\mathrm{Th}-\mathrm{Tc}}$, where temperatures are in Kelvin.
2. Convert the given temperature of the kitchen (Th) from Celsius to Kelvin. Use the formula: $T=T{C}_0+273.15$. For Th = 32°C, calculate Th in Kelvin.
3. Rearrange the COP formula to solve for the cold reservoir temperature (Tc). The rearranged formula is: $\mathrm{Tc}=\frac{\mathrm{COP}\cdot \mathrm{Th}}{1+\mathrm{COP}}$.
4. Substitute the given COP value (5.0) and the calculated Th (in Kelvin) into the rearranged formula to find Tc. Perform the algebraic operations to isolate Tc.
5. Convert the calculated Tc from Kelvin back to Celsius using the formula: $T{C}_0=T-273.15$. This will give the lowest possible temperature inside the refrigerator in Celsius.

Key Concepts

Coefficient of Performance (COP)

The Coefficient of Performance (COP) is a measure of the efficiency of a refrigeration system, defined as the ratio of the heat removed from the refrigerated space to the work input required to remove that heat. A higher COP indicates a more efficient refrigerator, meaning it can remove more heat for each unit of work done. In this case, a COP of 5.0 suggests that for every unit of energy consumed, the refrigerator removes five units of heat.

Carnot Efficiency

Carnot efficiency is a theoretical maximum efficiency for a heat engine or refrigerator operating between two temperatures. It is based on the second law of thermodynamics and is calculated using the temperatures of the hot and cold reservoirs. For refrigeration, the lowest achievable temperature inside the refrigerator can be determined by applying the Carnot principle, which states that no real refrigerator can be more efficient than a Carnot refrigerator operating between the same two temperatures.

Absolute Temperature Scale

The absolute temperature scale, measured in Kelvin (K), is crucial for thermodynamic calculations. It starts at absolute zero, where all molecular motion ceases, and is related to the Celsius scale by the equation K = °C + 273.15. When calculating the lowest temperature achievable in a refrigeration cycle, it is essential to convert Celsius temperatures to Kelvin to ensure accurate application of thermodynamic principles, particularly when using the Carnot efficiency formula.