(II) What is the temperature inside an ideal refrigerator–freezer that operates with a COP = 7.0 in a 22°C room?

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Understand the problem: The Coefficient of Performance (COP) for a refrigerator is defined as the ratio of the heat removed from the cold reservoir (Q_c) to the work input (W). Mathematically, COP = Q_c / W. For an ideal refrigerator, the COP can also be expressed in terms of the temperatures of the cold reservoir (T_c) and the hot reservoir (T_h) as COP = T_c / (T_h - T_c), where temperatures are in Kelvin.

Convert the given room temperature (T_h) from Celsius to Kelvin. Use the formula T(K) = T(°C) + 273.15. For the room temperature of 22°C, calculate T_h in Kelvin.

Rearrange the formula for COP to solve for the cold reservoir temperature (T_c). The formula becomes T_c = COP × (T_h - T_c).

Substitute the given COP value (7.0) and the calculated T_h into the equation. Solve algebraically for T_c. This will involve isolating T_c on one side of the equation.

Once T_c is calculated in Kelvin, convert it back to Celsius using the formula T(°C) = T(K) - 273.15 to find the temperature inside the refrigerator-freezer.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Coefficient of Performance (COP)

The Coefficient of Performance (COP) is a measure of the efficiency of a refrigeration system. It is 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.

Thermodynamics of Refrigeration

Refrigeration operates based on the principles of thermodynamics, particularly the second law, which states that heat naturally flows from hot to cold. Refrigerators use a refrigerant that absorbs heat from the interior and releases it outside, effectively lowering the temperature inside. Understanding this process is crucial for calculating the internal temperature based on external conditions and COP.

Absolute Temperature Scale

The absolute temperature scale, measured in Kelvin (K), is essential for thermodynamic calculations. To convert Celsius to Kelvin, you add 273.15. This scale is used in thermodynamic equations to ensure that temperature values are always positive, which is necessary for accurate calculations involving energy and heat transfer.