An apparatus that liquefies helium is in a room maintained at 300 K. If the helium in the appar

step 1 For this problem on the topic of entropy, we have an apparatus that can liquefy helium in a room

An apparatus that liquefies helium is in a room maintained...

An apparatus that liquefies helium is in a room maintained at $300 \mathrm{~K}$. If the helium in the apparatus is at $4.0 \mathrm{~K},$ what is the minimum ratio $Q_{\mathrm{to}} / Q_{\text {trom }},$ where $Q_{\mathrm{to}}$ is the energy delivered as heat to the room and $Q_{\text {from }}$ is the energy removed as heat from the helium?

Key Concepts

Carnot Refrigerator

This concept pertains to an idealized refrigeration cycle operating between two thermal reservoirs. In such a cycle, all processes are reversible, and the cycle sets the theoretical upper limit (or best performance) for real refrigerators. It is used to derive the minimum work input required for a given amount of heat removal from the cold reservoir at a specified low temperature, which is crucial in problems involving low-temperature liquefaction.

Coefficient of Performance (COP) for Refrigeration

The COP of a refrigerator is defined as the ratio of the heat removed from the cold reservoir to the work input required to remove that heat. For a Carnot refrigerator operating between a cold temperature T_c and a hot temperature T_h, the COP is given by T_c/(T_h ? T_c). This relation helps determine the relative expenditure of energy in moving heat from the cold side to the hot side, and it directly relates the energy delivered as heat to the room to the heat removed from the low-temperature helium.

Reversible Processes and the Second Law of Thermodynamics

Reversible processes are idealized scenarios where a system can be returned to its initial state without net changes in entropy, and they provide the maximum efficiency possible under thermodynamic constraints. The Second Law of Thermodynamics sets the limits on the direction of heat flow and the efficiency of energy conversion processes. In the context of refrigeration, these principles ensure that the minimum work required to transfer a given amount of heat is defined by the temperature difference between the reservoirs, leading to the ratio of heat transfers being determined by their absolute temperatures.

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