Consider an arbitrary heat engine which operates between two...

Consider an arbitrary heat engine which operates between two reservoirs, each of which has the same finite temperature-independent heat capacity $c$. The reservoirs have initial temperatures $T_1$ and $T_2$, where $T_2>T_1$, and the engine operates until both reservoirs have the same final temperature $T_3$. (a) Give the argument which shows that $T_3>\sqrt{T_1 T_2}$. (b) What is the maximum amount of work obtainable from the engine?

Consider an arbitrary heat engine which operates between two reservoirs, each of which has the same finite temperature-independent heat capacity $c$. The reservoirs have initial temperatures $T_1$ and $T_2$, where $T_2>T_1$, and the engine operates until both reservoirs have the same final temperature $T_3$.
(a) Give the argument which shows that $T_3>\sqrt{T_1 T_2}$.
(b) What is the maximum amount of work obtainable from the engine?

Key Concepts

Finite Thermal Reservoirs

In thermodynamics, when dealing with reservoirs whose heat capacities are finite and constant, any heat exchange between them results in their temperatures changing. This is markedly different from the idealized concept of infinite reservoirs, where the temperature is assumed constant. The finite nature of the reservoirs is central to the problem because as heat is transferred from the hot to the cold reservoir, their temperatures approach each other, and the final equilibrium temperature is determined by the conservation of energy along with the specifics of the heat exchange process.

Entropy and Reversibility

Entropy is a measure of the disorder or randomness of a system, and its balance is essential when assessing the reversibility of a process. For a reversible (ideal) process, the total entropy change of the system and surroundings is zero. In the context of a finite reservoir heat engine, if the process were reversible, one would set the sum of the entropy changes of the hot and cold reservoirs to zero, leading to a specific final temperature (namely, the geometric mean of the initial temperatures, ?(T1 T2)). However, any real (irreversible) process will generate extra entropy, implying that the final actual temperature must be higher than the reversible limit of ?(T1 T2).

Maximum Work Extraction

The concept of maximum work extraction in thermodynamics refers to the ideal limit where, starting from a non-equilibrium state, one extracts the greatest possible work while the process is carried out reversibly—that is, without any entropy production. In the case of the two finite reservoirs, the maximum obtainable work is the difference between the initial total internal energy and the final internal energy (achieved at the final common temperature determined by reversible conditions). The key insight is that any deviation from the reversible (ideal) process, marked by extra entropy generation, results in less work being extractable.

Heat Engine Efficiency and Finite Processes

Heat engine efficiency is defined as the ratio of work output to heat input, and it is limited by the Carnot efficiency when operating reversibly between two temperature reservoirs. In scenarios where the reservoirs have finite heat capacities, the temperatures continuously change during operation, and it becomes impossible for an arbitrary (irreversible) engine to achieve the ideal Carnot limit at every infinitesimal step of the process. As a result of these finite-size effects and the inevitability of irreversibilities (e.g., friction, finite rate processes), the final equilibrium temperature reached by the reservoirs exceeds the reversible case value of ?(T1 T2), and the maximum work obtainable approaches that of the reversible process only under idealized conditions.

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