Physics Problem: Ideal Refrigerator and Hot Reservoir Temperature
Question:
Suppose you want to operate an ideal refrigerator at -18.5 °C, and you would like it to have a coefficient of performance of at least 8.25. What is the maximum hot reservoir temperature Th for such a refrigerator?
Answer:
To solve this problem, we use the formula for the coefficient of performance (COP) of an ideal Carnot refrigerator:
COP = Tc / (Th − Tc)
Where:
- Tc is the temperature of the cold reservoir in Kelvin
- Th is the temperature of the hot reservoir in Kelvin
Step 1: Convert cold reservoir temperature to Kelvin.
Tc = -18.5°C + 273.15 = 254.65 K
Step 2: Use the COP formula to solve for Th.
8.25 = 254.65 / (Th − 254.65)
Step 3: Solve for the denominator.
Th − 254.65 = 254.65 / 8.25 ≈ 30.87 K
Step 4: Add this to Tc to find Th.
Th = 254.65 + 30.87 = 285.52 K
Step 5: Convert Th back to Celsius.
Th(°C) = 285.52 − 273.15 = 12.37°C
Final Answer:
The maximum hot reservoir temperature is approximately 12.4 °C.