23. The Second Law of Thermodynamics

A refrigerator has a coefficient of performance of 2.4. Each cycle, it takes in 3×10⁴ J of heat from the cold reservoir. How much is expelled to the hot reservoir?

(II) An ideal (Carnot) engine has an efficiency of 33%. If it were possible to run it backward as a heat pump, what would be its coefficient of performance?

(I) If an ideal refrigerator keeps its contents at 2.5°C when the house temperature is 22°C, what is its COP?

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

Refrigeration units can be rated in “tons.” A 1-ton air conditioning system can remove sufficient energy to freeze 1 ton (2000 pounds = 909 kg) of 0°C water into 0°C ice in one 24-h day. Assume the hot part of a day averages 35°C and the interior of a house is maintained at 22°C by the continuous operation of a 6-ton air conditioning system for 6 hours a day. How much does this cooling cost the homeowner per day, and per month?Assume the work done by the refrigeration unit is powered by electricity that costs $0.13 per kWh and that the unit’s coefficient of performance is only 18% of an ideal refrigerator. 1 kWh = 3.60 x 10⁶ J .

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?

A dehumidifier removes water vapor from air and has been referred to as a “refrigerator with an open door.” The humid air is pulled in by a fan and passes over a cold coil, whose temperature is less than the dew point, and some of the air’s water condenses. After this water is extracted, the air is warmed back to its original temperature and sent into the room. In a well-designed dehumidifier, the heat that is removed by the cooling coil mostly comes from the condensation of water vapor to liquid, and this heat is used to re-warm the air. Estimate how much water is removed in 1.0 h by an ideal dehumidifier, if the temperature of the room is 25°C, the water condenses at 8°C, and the dehumidifier does work at the rate of 550 W of electrical power.

Trees offsetting CO₂ . Trees can help a bit to offset the buildup of CO₂ due to burning coal and other fossil fuels. CO₂ can be absorbed by tree foliage. Trees use the carbon to grow, and release O₂ into the atmosphere. Suppose a refrigerator uses 600 kWh of electricity per year (about 2 x 10⁹ J) from a 33% efficient coal-fired power plant. Burning 1 kg of coal releases about 2 x 10⁷ J of energy. Assume coal is all carbon, which when burned in air becomes CO₂.

(a) How much coal is burned per year to run this refrigerator?

(b) Assuming a forest can capture 1700 kg of carbon per hectare ( = 10, 000 m²) per year, estimate how many square meters of forest are needed to capture the carbon (in the form now of CO₂) emitted by the refrigerator in (a).