To find the average rate of heat extraction by the refrigerator, we need to calculate the total heat removed during the entire process of converting water at 20 °C to ice at -10 °C. This involves three main steps: cooling the water from 20 °C to 0 °C, freezing the water at 0 °C to ice, and then cooling the ice from 0 °C to -10 °C. After calculating the total heat removed, we can then determine the average rate of heat extraction in watts.
Step 1: Cooling Water from 20 °C to 0 °C
First, we need to calculate the heat removed when cooling the water. The formula for heat transfer is:
Q = mcΔT
- m = mass of water = 100 g
- c = specific heat capacity of water = 4.2 J/g·K
- ΔT = change in temperature = (0 - 20) °C = -20 K
Now, substituting the values:
Q₁ = 100 g × 4.2 J/g·K × (-20 K) = -8400 J
Step 2: Freezing Water at 0 °C to Ice
Next, we calculate the heat removed during the phase change from water to ice. The formula for latent heat is:
Q = mL
- L = specific latent heat of ice = 336 J/g
Substituting the values:
Q₂ = 100 g × 336 J/g = 33600 J
Step 3: Cooling Ice from 0 °C to -10 °C
Finally, we calculate the heat removed when cooling the ice. Again, we use the heat transfer formula:
- c = specific heat capacity of ice = 2.1 J/g·K
- ΔT = change in temperature = (-10 - 0) °C = -10 K
Substituting the values:
Q₃ = 100 g × 2.1 J/g·K × (-10 K) = -2100 J
Total Heat Removed
Now, we can find the total heat extracted by summing up all the heat values:
Total Q = Q₁ + Q₂ + Q₃
Total Q = -8400 J + (-33600 J) + (-2100 J) = -44100 J
Calculating Average Rate of Heat Extraction
To find the average rate of heat extraction, we divide the total heat removed by the time taken. The time is given as 73.5 minutes, which we need to convert to seconds:
Time = 73.5 min × 60 s/min = 4410 s
Now, we can calculate the average rate:
Average Power (P) = Total Q / Time
P = -44100 J / 4410 s ≈ -10 W
Final Result
The average rate of heat extraction by the refrigerator is approximately 10 watts. The negative sign indicates that heat is being removed from the system, which is expected in refrigeration processes.
