# Contradiction in the Carnot Cycle?

The carnot cycle is reversible, because the temperature difference between heat reservoir and fluid is extremely low, so that heat can flow in both directions.

Also, the carnot cycle is described as the ideal engine with the maximum efficiency

At the same time, it is always stated that the efficiency is better, if the temperature difference is as high as possible.

Can someone solve this apparent contradiction for me? I know it's not a real contradiction and I am just dumb for not getting it, but I really need help here.

Best regards

You are talking about two different temperature differences.

The first is the temperature difference between the heat engine and the thermal reservoirs during the isothermal expansion and compression processes. Those differences must be kept infinitesimal, meaning the engine and reservoir temperatures are essentially the same, in order to make the processes reversible.

The second is the temperature difference between the isothermal processes connecting the two reversible adiabatic processes. That difference determines the net heat and thus the net work done over the cycle. The greater the difference the greater the amount of work done per cycle, maximizing the efficiency.

The efficiency of the cycle is

$$\eta=1-\frac{T_L}{T_H}$$

Where $$T_H$$ is the temperature of the high reservoir and heat engine during the reversible isothermal expansion where heat is added, and $$T_L$$ is the temperature of the low temperature reservoir and heat engine during the reversible isothermal compression where heat is rejected.

There is no contradiction. The Carnot cycle has 4 stages consisting of two isothermal stages, one at a temperature $$T_1$$ and the other at a temperature $$T_0 < T_1$$, say, and two adiabatic stages. All four stages are to be reversible processes. During the isothermal stages the "engine" absorbs some thermal energy at the higher temperature from a thermal reservoir at temperature $$T_1$$ and rejects some thermal energy at the lower temperature into a thermal reservoir whose temperature is $$T_0$$. The temperature difference between these isothermal stages can be arbitrarily large since the process connecting them is adiabatic. On the contrary, during an isothermal thermal energy exchange the difference in temperature between the engine and the reservoir is infinitesimal to allow the isothermal process be reversible but during the adiabatic stage the engines own temperature can change an arbitrary finite amount since there is no thermal energy exchange that would generate entropy thus making the process irreversible.