A refrigerator has a coefficient of performance of 2.25, runs on an input of 135 W of electrical power, and keeps its inside compartment at 5°C. If you put a dozen 1.0-L plastic bottles of water at 31°C into this refrigerator, how long will it take for them to be cooled down to 5°C? (Ignore any heat that leaves the plastic.)

Verified step by step guidance

1. Calculate the total mass of water in the bottles. Since the density of water is approximately 1 kg/L, multiply the volume of water by its density.
2. Calculate the amount of heat that needs to be removed from the water to cool it from 31°C to 5°C. Use the specific heat capacity of water, which is approximately 4186 J/kg°C, and the formula Q = mcΔT, where m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.
3. Determine the rate at which heat is removed by the refrigerator. The coefficient of performance (COP) of the refrigerator is given as 2.25, which relates the heat removed from the refrigerator compartment (Q_c) to the work input (W) by the formula COP = Q_c/W. Rearrange this formula to find Q_c = COP × W.
4. Calculate the time required to remove the calculated amount of heat. Use the formula t = Q/Q_c, where Q is the total heat to be removed (from step 2) and Q_c is the rate of heat removal per second (from step 3).
5. Convert the time from seconds to a more convenient unit if necessary, such as minutes or hours, to find out how long it will take for the bottles of water to cool down to 5°C.

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 heat removed from the refrigerated space to the work input. A higher COP indicates a more efficient refrigerator. In this case, a COP of 2.25 means that for every unit of work (energy) input, the refrigerator removes 2.25 units of heat from its interior.

Heat Transfer and Specific Heat Capacity

Heat transfer refers to the movement of thermal energy from one object to another, which occurs until thermal equilibrium is reached. The specific heat capacity is the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius. For water, this value is approximately 4.18 J/g°C, which is crucial for calculating how much energy is needed to cool the bottles from 31°C to 5°C.

Energy Balance and Time Calculation

Energy balance involves equating the energy removed from the water bottles to the energy input by the refrigerator. To find the time required to cool the bottles, one must calculate the total heat energy that needs to be removed and then divide this by the effective cooling power of the refrigerator, which can be derived from its COP and input power. This calculation will yield the time it takes to achieve the desired temperature.