Carnot's Theorem and Absolute Temperature

The Universal Limit

Imagine two heat engines operating between the same hot reservoir at temperature \(T_H\) and the same cold reservoir at temperature \(T_C\). One engine uses steam, the other uses an ideal gas. Intuitively, one might expect the steam engine to be more efficient due to the phase change, or the ideal gas engine to be superior due to its simplicity. The argument, which goes back to Carnot (1824) and was put in its modern form by Clausius and Kelvin, is that if one engine were more efficient than the other, we could couple them: let the more efficient engine run forward, producing work, and use that work to drive the less efficient engine in reverse, acting as a refrigerator. The net result would be a transfer of heat from the cold reservoir to the hot reservoir with no other change, violating the Clausius statement of the Second Law. Therefore, no engine can be more efficient than a reversible engine operating between the same two temperatures. This logical necessity implies that the maximum efficiency is a universal function of the reservoir temperatures alone.

Defining the Absolute Temperature Scale

Kelvin took this further: the ratio of heat rejected to heat absorbed by a reversible engine is a function of the reservoir temperatures only. Let \(Q_H\) be the heat absorbed from the hot reservoir and \(Q_C\) be the heat rejected to the cold reservoir. For a reversible engine, we can write:

By considering a third reservoir at temperature \(T_0\) and using the transitivity of the efficiency limit, Kelvin showed that this function must be separable. Specifically, the ratio depends on the ratio of two temperature functions, \(F(T_C)/F(T_H)\). By defining the absolute temperature \(T\) such that \(F(T) \propto T\), we arrive at the relation:

This equation is the cornerstone of the absolute temperature scale. It defines temperature not by the expansion of mercury or the resistance of platinum, but by the fundamental thermodynamic property of heat exchange in a reversible cycle. This scale is independent of any specific substance and is now known as the thermodynamic temperature scale, which aligns with the Kelvin scale.

Deriving the Carnot Efficiency

The efficiency \(\eta\) of any heat engine is defined as the work output divided by the heat input: \(\eta = W/Q_H\). From the First Law of Thermodynamics, for a cyclic process, the work done is the difference between the heat absorbed and the heat rejected: \(W = Q_H - Q_C\). Substituting this into the efficiency definition gives:

For a reversible engine, we substitute the Carnot relation \(Q_C/Q_H = T_C/T_H\). This yields the maximum possible efficiency:

Any irreversible engine operating between the same reservoirs will have a lower efficiency, meaning \(Q_C/Q_H > T_C/T_H\). Thus, the general inequality for any heat engine is:

This result is strikingly simple. It shows that efficiency approaches unity only if \(T_C\) approaches absolute zero or \(T_H\) approaches infinity. Since neither is physically achievable, all real engines have efficiencies strictly less than 100%.

Implications for Refrigerators and Heat Pumps

The logic of Carnot's theorem extends directly to refrigerators and heat pumps, which are heat engines run in reverse. For a refrigerator, the coefficient of performance (COP) is defined as the heat removed from the cold reservoir divided by the work input: \(\text{COP}_{\text{ref}} = Q_C/W\). Using the First Law, \(W = Q_H - Q_C\), so:

For a reversible refrigerator, \(Q_H/Q_C = T_H/T_C\). Therefore, the maximum COP is:

Similarly, for a heat pump, which delivers heat to the hot reservoir, the COP is \(\text{COP}_{\text{hp}} = Q_H/W\). The maximum value is:

These expressions demonstrate that the performance of cooling and heating devices is also bounded by the absolute temperatures of the reservoirs. A smaller temperature difference between \(T_H\) and \(T_C\) results in a higher COP, which is why heat pumps are more efficient in mild climates than in freezing conditions.