A Carnot refrigerator is operating between thermal reservoirs with temperatures of 27.9°C and 0.0°C.
a) How much work will need to be input to extract 13.7 J of heat from the colder reservoir?
b) How much work will be needed if the colder reservoir is at -15.7°C?
Convert temperatures to kelvin and apply the Carnot efficiency formula.
Work(out)/Heat(in) = 1 - (Tcold/Thot)
Help will be provided when effort is demonstrated.
-0.0988
idontknow
To solve these questions, we can use the formulas for the efficiency and work of a Carnot refrigerator. The efficiency of a Carnot refrigerator is given by the equation:
Efficiency = 1 - (Tc/Th)
Where Tc is the absolute temperature of the colder reservoir and Th is the absolute temperature of the hotter reservoir.
a) Let's calculate the work needed to extract 13.7 J of heat from the colder reservoir when Tc = 0.0°C and Th = 27.9°C.
1. Convert the temperatures to Kelvin:
Tc = 273.15 K (0.0°C + 273.15)
Th = 273.15 + 27.9 = 301.05 K
2. Calculate the efficiency:
Efficiency = 1 - (Tc/Th) = 1 - (273.15/301.05) = 1 - 0.907 = 0.093
3. Use the efficiency formula to calculate the work:
Work = heat extracted / efficiency = 13.7 J / 0.093 = 147.31 J (rounded to two decimal places)
Therefore, the work needed to extract 13.7 J of heat from the colder reservoir is approximately 147.31 J.
b) Now, let's calculate the work needed if the colder reservoir is at -15.7°C.
1. Convert the temperatures to Kelvin:
Tc = 273.15 - 15.7 = 257.45 K
Th remains the same: Th = 273.15 + 27.9 = 301.05 K
2. Calculate the efficiency:
Efficiency = 1 - (Tc/Th) = 1 - (257.45/301.05) = 1 - 0.854 = 0.146
3. Use the efficiency formula to calculate the work:
Work = heat extracted / efficiency = 13.7 J / 0.146 = 93.84 J (rounded to two decimal places)
Therefore, the work needed to extract 13.7 J of heat from the colder reservoir at -15.7°C is approximately 93.84 J.