A freezer with a coefficient of performance 30% that of a Carnot refrigerator keeps the inside temperature at -22℃ in a 25℃ room. 3.0 L of water at 20℃ are placed in the freezer. How long does it take for the water to freeze if the freezer's compressor does work at the rate of 200 W while the water is freezing?

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Convert all temperatures to Kelvin. The inside temperature of the freezer is \(-22 + 273.15 = 251.15\,\text{K}\), and the room temperature is \(25 + 273.15 = 298.15\,\text{K}\).

Calculate the coefficient of performance (COP) of a Carnot refrigerator using the formula \(\text{COP}_{\text{Carnot}} = \frac{T_{\text{cold}}}{T_{\text{hot}} - T_{\text{cold}}}\), where \(T_{\text{cold}}\) is the inside temperature of the freezer and \(T_{\text{hot}}\) is the room temperature.

Determine the actual coefficient of performance of the freezer. Since the freezer's COP is 30% of the Carnot COP, use \(\text{COP}_{\text{freezer}} = 0.3 \times \text{COP}_{\text{Carnot}}\).

Calculate the heat that must be removed to freeze the water. First, find the mass of the water using its density (\(\rho = 1000\,\text{kg/m}^3\)) and volume (\(3.0\,\text{L} = 0.003\,\text{m}^3\)). Then, calculate the heat removed in two steps: (1) cooling the water from \(20\,\degree\text{C}\) to \(0\,\degree\text{C}\) using \(Q_1 = mc\Delta T\), where \(c = 4186\,\text{J/kg·K}\), and (2) freezing the water using \(Q_2 = mL_f\), where \(L_f = 334,000\,\text{J/kg}\). The total heat removed is \(Q = Q_1 + Q_2\).

Determine the time required to freeze the water. The work done by the compressor is related to the heat removed by \(W = \frac{Q}{\text{COP}_{\text{freezer}}}\). Since the compressor works at a rate of \(200\,\text{W}\), the time is given by \(t = \frac{W}{200}\).

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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, defined as the ratio of heat removed from the cold reservoir to the work input. In this context, the COP of the freezer is given as 30% of that of an ideal Carnot refrigerator, which sets a theoretical maximum efficiency based on the temperatures of the hot and cold reservoirs.

Heat Transfer and Latent Heat

When water freezes, it undergoes a phase change from liquid to solid, which requires the removal of heat known as latent heat of fusion. For water, this value is approximately 334 kJ/kg. Understanding how much heat must be removed to freeze the water is crucial for calculating the time required for the freezing process.

Power and Energy Relationship

Power is the rate at which energy is transferred or converted, measured in watts (W). In this scenario, the freezer's compressor operates at 200 W, meaning it can remove 200 joules of energy per second. To determine how long it takes to freeze the water, one must relate the total energy required to freeze the water to the power output of the freezer.