An ideal refrigerator or ideal heat pump is equivalent to a...

An ideal refrigerator or ideal heat pump is equivalent to a Carnot engine running in reverse. That is, energy $\left|Q_{c}\right|$ is taken in from a cold reservoir and energy $\left|Q_{h}\right|$ is rejected to a hot reservoir. (a) Show that the work that must be supplied to run the refrigerator or heat pump is $$ W=\frac{T_{h}-T_{c}}{T_{c}}\left|Q_{c}\right| $$ (b) Show that the coefficient of performance of the ideal refrigerator is $$ \mathrm{COP}=\frac{T_{e}}{T_{h}-T_{c}} $$

An ideal refrigerator or ideal heat pump is equivalent to a Carnot engine running in reverse. That is, energy $\left|Q_{c}\right|$ is taken in from a cold reservoir and energy $\left|Q_{h}\right|$ is rejected
to a hot reservoir. (a) Show that the work that must be supplied to run the refrigerator or heat pump is
$$
W=\frac{T_{h}-T_{c}}{T_{c}}\left|Q_{c}\right|
$$
(b) Show that the coefficient of performance of the ideal refrigerator is
$$
\mathrm{COP}=\frac{T_{e}}{T_{h}-T_{c}}
$$

Key Concepts

Carnot Cycle

The Carnot cycle is an idealized thermodynamic cycle that provides the maximum possible efficiency a heat engine can achieve when operating between two temperatures. It consists of two isothermal processes and two adiabatic processes, and it is the benchmark for determining the limits of possible efficiency in real engines. When reversed, the cycle becomes a refrigerator or heat pump, providing insight into the best-case performance for these devices.

Reversible Processes

Reversible processes are idealized processes that occur infinitely slowly so that the system remains in thermodynamic equilibrium throughout. In the context of the Carnot cycle, all steps are reversible, which means there are no irreversible losses. This idealization is crucial for deriving theoretical limits such as maximum efficiency and optimal coefficient of performance (COP) in heat engines and refrigerators.

Energy Conservation in Thermodynamic Cycles

The first law of thermodynamics, which is a statement of energy conservation, plays a central role in analyzing thermodynamic cycles such as the Carnot cycle. It explains how the heat energies absorbed from the hot and cold reservoirs and the work done are interrelated. Specifically, for a refrigerator or heat pump operating on a reversed Carnot cycle, the work input is equal to the difference between the energy rejected and the energy absorbed, allowing derivation of expressions for work based on the temperatures of the reservoirs.

Coefficient of Performance (COP)

The coefficient of performance (COP) is a measure commonly used for refrigerators and heat pumps to describe how effectively they use work to transfer heat. It is defined as the ratio of the desired effect (like the amount of heat removed from the cold reservoir in a refrigerator) to the work input required to achieve that effect. In ideal, reversible processes like the reversed Carnot cycle, the COP can be expressed solely in terms of the temperatures of the thermal reservoirs, highlighting the fundamental thermodynamic limits imposed by the second law.

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