The proof of the Carnot theorem is a
proof by contradiction, based on a situation like the right figure where two heat engines with different
efficiencies are operating between two
thermal reservoirs at different temperature. The relatively hotter reservoir is called the hot reservoir and the other reservoir is called the cold reservoir. A (not necessarily
reversible) heat engine M with a greater efficiency \eta_{_M} is driving a reversible heat engine L with a less efficiency \eta_{_L}, causing the latter to act as a
heat pump. The requirement for the engine L to be reversible is necessary to explain work W and heat Q associated with it by using its known efficiency. However, since \eta_{_M}>\eta_{_L}, the net heat flow would be backwards, i.e., into the hot reservoir: :Q^\text{out}_\text{h} = Q where Q represents heat, \text{in} denotes input to an object, \text{out} for output from an object, and h for the hot thermal reservoir. If heat Q^\text{out}_\text{h} flows from the hot reservoir then it has the sign of + while if Q^\text{in}_\text{h} flows from the hot reservoir then it has the sign of -. This expression can be easily derived by using the definition of the
efficiency of a heat engine, \eta=W/Q_\text{h}^\text{in}, where work and heat in this expression are net quantities per engine cycle, and the conservation of energy for each engine as shown below. The sign convention of work W, with which the sign of + for work done by an engine to its surroundings, is employed. The above expression means that heat into the hot reservoir from the engine pair (can be considered as a single engine) is greater than heat into the engine pair from the hot reservoir (i.e., the hot reservoir continuously gets energy). A reversible heat engine with a low efficiency delivers more heat (energy) to the hot reservoir for a given amount of work (energy) to this engine when it is being driven as a heat pump. All these mean that heat can transfer from cold to hot places without external work, and such a heat transfer is impossible by the
second law of thermodynamics. • It may seem odd that a hypothetical reversible heat pump with a low efficiency is used to violate the second law of thermodynamics, but the
figure of merit for refrigerator units is not the efficiency, W/Q_\text{h}^\text{out}, but the
coefficient of performance (COP), which is Q_\text{c}^\text{out}/W where this W has the sign opposite to the above (+ for work done to the engine). Let's find the values of work Wand heat Q depicted in the right figure in which a reversible heat engine L with a less efficiency \eta_{_L} is driven as a heat pump by a heat engine M with a more efficiency \eta_{_M}. The definition of the
efficiency is \eta = W/Q_\text{h}^\text{out} for each engine and the following expressions can be made: :\eta_M= \frac{W_M}{Q^{\text{out},M}_\text{h}} = \frac{\eta_M Q}{Q}=\eta_M, : \eta_L = \frac{W_L}{Q^{\text{out},L}_\text{h}} = \frac{-\eta_M Q}{-\frac{\eta_M}{\eta_L}Q} = \eta_L. The denominator of the second expression, Q^{\text{out},L}_\text{h} = -\frac{\eta_M}{\eta_L}Q, is made to make the expression to be consistent, and it helps to fill the values of work and heat for the engine L. For each engine, the absolute value of the energy entering the engine, E_\text{abs}^\text{in} , must be equal to the absolute value of the energy leaving from the engine, E_\text{abs}^\text{out} . Otherwise, energy is continuously accumulated in an engine or the conservation of energy is violated by taking more energy from an engine than input energy to the engine: :E_\text{M,abs}^\text{in} = Q = (1-\eta_M)Q + \eta_M Q = E_\text{M,abs}^\text{out}, :E_\text{L,abs}^\text{in} = \eta_M Q + \eta_M Q \left(\frac{1}{\eta_L}- 1 \right ) = \frac{\eta_M}{\eta_L}Q = E_\text{L,abs}^\text{out}. In the second expression, \left| Q^{\text{out},L}_\text{h} \right| = \left| - \frac{\eta_M}{\eta_L}Q \right| is used to find the term \eta_M Q \left(\frac{1}{\eta_L}- 1 \right ) describing the amount of heat taken from the cold reservoir, completing the absolute value expressions of work and heat in the right figure. Having established that the right figure values are correct, Carnot's theorem may be proven for irreversible and the reversible heat engines as shown below. :\Delta S = \int_a^b \frac {dQ_\text{rev}}T as the entropy change, that is made during a transition from a
thermodynamic equilibrium state a to a state b in a
V-T (Volume-Temperature) space, is the same over all reversible process paths between these two states. If this integral were not path independent, then entropy would not be a
state variable.
Irreversible engines Consider two engines, M and L, which are irreversible and reversible respectively. We construct the machine shown in the right figure, with M driving L as a heat pump. Then if M is more efficient than L, the machine will violate the second law of thermodynamics. Since a Carnot heat engine is a reversible heat engine, and all reversible heat engines operate with the same efficiency between the same reservoirs, we have the first part of Carnot's theorem: :
No irreversible heat engine is more efficient than a Carnot heat engine operating between the same two thermal reservoirs. ==Definition of thermodynamic temperature==