Question

Let \( T_H \) and \( T_L \) denote the absolute temperatures of high and low temperature reservoirs, respectively. The coefficient of performance of a reversible refrigerator operating between these two reservoirs is

• A. \( \frac{T_H}{T_L - T_H} \)
• B. \( \frac{1}{1 - \frac{T_L}{T_H}} \)
• C. \( \frac{T_L}{T_H - 1} \)
• D. \( \frac{T_L}{T_H + 1} \)

Hint:

For a Carnot refrigerator, the coefficient of performance depends only on the temperatures of the hot and cold reservoirs. A smaller temperature difference between the two reservoirs results in a higher COP.

Answer 1

The Correct Option is A

The coefficient of performance (COP) of a refrigerator is defined as the ratio of the heat extracted from the cold reservoir \( Q_L \) to the work input \( W \). For a reversible refrigerator operating between two thermal reservoirs at temperatures \( T_H \) and \( T_L \), the COP is maximized when the system operates according to the Carnot cycle.
The Carnot refrigerator has the highest possible COP for a given temperature difference. The COP of a Carnot refrigerator is given by: \[ COP = \frac{T_L}{T_H - T_L} \] where:
- \( T_L \) is the temperature of the cold reservoir,
- \( T_H \) is the temperature of the hot reservoir.
To understand this formula, note that as the temperature difference \( (T_H - T_L) \) decreases, the COP increases, which means that the refrigerator becomes more efficient as the difference between the temperatures of the reservoirs decreases.
Now, examining the options:
- Option (A): This is the correct answer. The COP for a Carnot refrigerator operating between two reservoirs is \( \frac{T_H}{T_L - T_H} \), derived from the fundamental thermodynamic relationship for reversible refrigerators.
- Option (B): This is incorrect because it doesn’t represent the correct relationship for the COP of a Carnot refrigerator. It would be applicable in a different context but not for a reversible refrigerator.
- Option (C): This is incorrect. It represents an invalid expression for the COP in this context.
- Option (D): This is also incorrect as it does not describe the COP of a reversible refrigerator.
Thus, the correct formula for the COP of a reversible refrigerator is: \[ \boxed{\frac{T_H}{T_L - T_H}} \]