# Carnot Engines and Efficiency

A heat engine can perform work by bridging a thermal difference. To do so, it allows energy to flow from a hot reservoir. The First Law of Thermodynamics requires that no energy be created or destroyed. Part of that energy is extracted as work  W while the remainder is exhausted to a cold reservoir as waste heat ΔQcold.

When heat is removed from the hot reservoir, entropy S is likewise removed:

$$\Delta S_{hot} = \frac{\Delta Q}{T_{hot}}$$.

The maximum amount of work that can be performed cannot exceed the amount energy withdrawn from the hot reservoir. Yet the First Law is silent as to how much of that heat energy can be converted into work.

According to the Second Law of Thermodynamics, the entropy of a system cannot decrease, even in an idealized system such as a Carnot Engine. Therefore, the entropy added to the cold reservoir will be:

$$\Delta S_{cold} = \frac{\Delta Q}{T_{cold}}$$.

The heat added to the cold reservoir is simply:

$$\Delta Q_{cold} = \Delta Q_{hot} – W$$.