Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 35A / a / Solution

The Carnot theorem states that no engine between two fixed heat reservoirs can be more efficient than a reversible engine, and that all reversible engines between those reservoirs have the same efficiency. Thus the heat ratio of a reversible engine depends only on the two reservoirs. Its multiplicative consistency for three reservoirs permits a state variable $T$, unique up to an overall scale, such thatFixing one reference value defines the thermodynamic temperature scale, and the reversible efficiency is

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