Why is the following situation impossible? An inventor comes...

Why is the following situation impossible? An inventor comes to a patent office with the claim that her heat engine, which employs water as a working substance, has a thermodynamic efficiency of $0.110 .$ Although this efficiency is low compared with typical automobile engines, she explains that her engine operates between an energy reservoir at room temperature and a water-ice mixture at atmospheric pressure and therefore requires no fuel other than that to make the ice. The patent is approved, and working prototypes of the engine prove the inventor's efficiency claim.

Key Concepts

Second Law of Thermodynamics

The second law of thermodynamics governs the direction of energy transfers and states that heat cannot spontaneously flow from a colder body to a hotter body. It also establishes that no engine can convert heat completely into work without losses, and thus places a fundamental upper limit on the efficiency of heat engines. This law is the underlying basis for the Carnot efficiency limit and ensures that all practical engines must operate below this theoretical maximum.

Absolute Temperature Scale Conversion

Using the absolute temperature scale (Kelvin) is essential when calculating the maximum efficiency of heat engines. Efficiency calculations based on the Carnot cycle rely on the absolute temperatures of the reservoirs, not the Celsius values. In the context of engine design, the efficiency is calculated by converting room temperature and water-ice mixture temperatures into Kelvin, which reveals if the claimed efficiency exceeds what is theoretically possible under the given conditions.

Carnot Efficiency

Carnot efficiency is the theoretical maximum efficiency that any heat engine operating between two temperature reservoirs can achieve. It is determined solely by the absolute temperatures of the hot and cold reservoirs and is given by the expression 1 - (T_cold/T_hot). This principle implies that no real engine can surpass the Carnot limit, making any claim of an engine achieving efficiency greater than this limit physically impossible.

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