How much work must be done by a Carnot refrigerator to transfer 1.0 J as heat (a) from a reserv

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How much work must be done by a Carnot refrigerator to...

How much work must be done by a Carnot refrigerator to transfer $1.0 \mathrm{~J}$ as heat (a) from a reservoir at $7.0^{\circ} \mathrm{C}$ to one at $27^{\circ} \mathrm{C},$ (b) from a reservoir at $-73^{\circ} \mathrm{C}$ to one at $27^{\circ} \mathrm{C},$ (c) from a reservoir at $-173^{\circ} \mathrm{C}$ to one at $27^{\circ} \mathrm{C},$ and $(\mathrm{d})$ from a reservoir at $-223^{\circ} \mathrm{C}$ to one at $27^{\circ} \mathrm{C} ?$

Key Concepts

Carnot Cycle

The Carnot cycle is a theoretical thermodynamic cycle that represents the most efficient heat engine or refrigerator possible based on reversible processes. It sets the maximum possible performance limit and is used as a benchmark to compare real devices. In the context of refrigerators, the Carnot cycle provides the ideal coefficient of performance (COP), which relates the amount of heat removed from the cold reservoir to the work input, and helps determine the minimum work required for a given heat transfer.

Coefficient of Performance (COP) for Refrigerators

The coefficient of performance (COP) for a refrigerator is a measure of its effectiveness and is defined as the ratio of the heat removed from the cold reservoir to the work input. For a Carnot refrigerator, the COP is given by COP = T_cold / (T_hot - T_cold), where temperatures are in absolute units (Kelvin). This concept is crucial because it determines the relationship between the work required and the amount of heat transferred, enabling the calculation of the minimum work needed for a given refrigeration task.

Absolute Temperature Conversion

In thermodynamics, temperatures must be expressed in absolute units (Kelvin) when used in formulas such as those for the coefficient of performance. Converting from Celsius to Kelvin by adding 273.15 is essential in problems involving thermal cycles like the Carnot refrigerator. This conversion ensures accuracy in calculations involving temperature differences, which are critical in determining the performance and efficiency limits of thermodynamic processes.

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