Suppose that you want to freeze 1.2 kg of water for a party, and only have 10 minutes to do it. Relying on nothing but your keen physical abilities, a dixie cup, duct tape and a piece of string, you manage to build a refrigerator unit which can do the job.
The temperature inside the refrigeration unit is 273 K, and the temperature outside is Thot = 301 K. Since you are a brilliant UW student, assume that your refrigerator has the maximum possible efficiency. Assume also that the initial temperature of the water is 273 K. The Latent Heat of Fusion of water is 3.33×105 J/kg.
(a) What is the heat flow ( Qcold / time ) out of the cold well per second in J/s?
Hcold = Qcold / time = J/s *
666 OK
(b) What is the change in entropy in the water per second in J/K s?
ΔS / time = J/K s
HELP: Change in entropy = Change in energy / Absolute Temperature.
HELP: To get the change in entropy of the water, use the total heat of fusion for the 1.2 kg of water. The water freezes at 273 K, and it all happens over a period of 10×60 seconds.
(c) What is the heat flow ( Qhot / time ) into the hot well per second in J/s?
Hhot = Qhot / time = J/s
(d) How much POWER must you provide to drive your refrigerator, in Watts? (That is, work done per second?)
Power = W
(a) To find the heat flow out of the cold well per second (Qcold/time), we can use the equation:
Qcold/time = Hcold
Given that the Latent Heat of Fusion of water is 3.33×10^5 J/kg and we need to freeze 1.2 kg of water in 10 minutes (600 seconds), we can calculate Hcold as follows:
Hcold = (3.33×10^5 J/kg) * (1.2 kg) / (600 s)
Hcold = 666 J/s
Therefore, the heat flow out of the cold well per second is 666 J/s.
(b) To find the change in entropy in the water per second (ΔS/time), we can use the equation:
ΔS/time = (Change in energy) / (Absolute Temperature)
The change in energy is the total heat of fusion for the 1.2 kg of water, which is given by:
Change in energy = (Latent Heat of Fusion) * (Mass of Water)
Change in energy = (3.33×10^5 J/kg) * (1.2 kg)
The absolute temperature is 273 K, and the process happens over a period of 10 minutes, which is equivalent to 10 × 60 seconds. Substituting these values into the equation, we get:
ΔS/time = [(3.33×10^5 J/kg) * (1.2 kg)] / [273 K * (10 × 60 s)]
ΔS/time = 0.242 J/K s
Therefore, the change in entropy in the water per second is 0.242 J/K s.
(c) To find the heat flow into the hot well per second (Qhot/time), we can use the equation:
Qhot/time = Hhot
Since the refrigerator has maximum efficiency, the heat flow into the hot well per second is equal to the heat flow out of the cold well per second:
Qhot/time = Qcold/time = Hcold = 666 J/s
Therefore, the heat flow into the hot well per second is 666 J/s.
(d) To find the power required to drive the refrigerator (W), we can use the equation:
Power = W = Qhot/time - Qcold/time
Substituting the values we found earlier, we get:
Power = 666 J/s - 666 J/s
Power = 0 W
Therefore, no power is required to drive the refrigerator in this scenario.
To answer the questions, we need to use the formulas and principles of thermodynamics. Let's break down each question:
(a) What is the heat flow (Qcold / time) out of the cold well per second in J/s?
The formula for heat flow is Q = mcΔT, where Q is the heat flow, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.
In this case, since we are freezing the water, the change in temperature is -273 K, since the water will reach 0 K (the freezing point).
We know the mass of the water is 1.2 kg, assuming the specific heat capacity is the same as water, which is 4.18 J/(g*K), we convert it to kilograms: c = 4.18 J/(g*K) * (1 kg / 1000 g) = 4.18 J/(kg*K).
Substituting the values into the formula, we get:
Qcold = mcΔT = (1.2 kg) * (4.18 J/(kg*K)) * (-273 K) = -1344.504 J.
To find the heat flow per second, we divide by time:
Hcold = Qcold / time = -1344.504 J / (10 minutes * 60 seconds/minute) = -2.241 J/s.
(b) What is the change in entropy in the water per second in J/K s?
The formula for the change in entropy is ΔS = ΔQ / T, where ΔS is the change in entropy, ΔQ is the change in energy, and T is the absolute temperature.
In this case, the change in energy is the total heat of fusion, which is given as 3.33×10^5 J/kg, multiplied by the mass of the water (1.2 kg).
The absolute temperature is 273 K, as the water freezes at that temperature and remains constant during the process.
Substituting the values into the formula, we get:
ΔS = (1.2 kg) * (3.33×10^5 J/kg) / (273 K) = 1461.538 J/K.
To find the change in entropy per second, we divide by time:
ΔS / time = 1461.538 J/K / (10 minutes * 60 seconds/minute) = 2.436 J/K s.
(c) What is the heat flow (Qhot / time) into the hot well per second in J/s?
Since the efficiency of the refrigerator is maximum, it means that the heat flow into the hot well (Qhot) is equal to the heat flow out of the cold well (Qcold). So, Qhot = Qcold.
Therefore, Qhot / time = Hcold = -2.241 J/s.
(d) How much POWER must you provide to drive your refrigerator, in Watts?
The power required is given by the formula Power = Work / time, where Power is the power required, Work is the work done, and time is the duration of the process.
In this case, the work done is equal to the heat flow out of the cold well, since the efficiency is maximum. So, Work = Qcold.
Therefore, Power = Work / time = Qcold / time = -2.241 J/s.
Since we are only interested in the magnitude of the power, we can take the absolute value:
Power = |-2.241 J/s| = 2.241 J/s.
So, the power required to drive the refrigerator is 2.241 Watts.
Note: The negative sign in some of the calculations indicates that energy is being lost or flowing out of the system.