Refrigerator A works between `-10^@C` and `27^@C`, while refrigerator B works between `-27^@C` and `17^@C`, both removing heat equal to 2000J from the freezer. Which of the two is the better refrigerator?

Explanation

To determine which refrigerator is better, we need to calculate the coefficient of performance (COP) for both refrigerators. The COP for a refrigerator is given by the formula:

\[ \text{COP} = \frac{Q_L}{W} \]

where \( Q_L \) is the heat removed from the cold reservoir (the freezer), and \( W \) is the work done by the refrigerator. However, we can also express the COP in terms of the temperatures of the hot and cold reservoirs:

\[ \text{COP} = \frac{T_C}{T_H - T_C} \]

where:
- \( T_C \) is the absolute temperature of the cold reservoir (in Kelvin),
- \( T_H \) is the absolute temperature of the hot reservoir (in Kelvin).

### Step 1: Convert temperatures from Celsius to Kelvin

For Refrigerator A:
- Hot reservoir temperature \( T_H = 27^\circ C = 27 + 273 = 300 \, K \)
- Cold reservoir temperature \( T_C = -10^\circ C = -10 + 273 = 263 \, K \)

For Refrigerator B:
- Hot reservoir temperature \( T_H = 17^\circ C = 17 + 273 = 290 \, K \)
- Cold reservoir temperature \( T_C = -27^\circ C = -27 + 273 = 246 \, K \)

### Step 2: Calculate the COP for Refrigerator A

Using the formula for COP:

\[ \text{COP}_A = \frac{T_C}{T_H - T_C} = \frac{263}{300 - 263} = \frac{263}{37} \]

Calculating this gives:

\[ \text{COP}_A \approx 7.1 \]

### Step 3: Calculate the COP for Refrigerator B

Using the same formula:

\[ \text{COP}_B = \frac{T_C}{T_H - T_C} = \frac{246}{290 - 246} = \frac{246}{44} \]

Calculating this gives:

\[ \text{COP}_B \approx 5.6 \]

### Step 4: Compare the COPs

Now we compare the COPs of both refrigerators:

- COP of Refrigerator A: \( 7.1 \)
- COP of Refrigerator B: \( 5.6 \)

Since \( \text{COP}_A > \text{COP}_B \), this indicates that Refrigerator A is more efficient than Refrigerator B.

### Conclusion

Refrigerator A is the better refrigerator.

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