# Why Do Different Methods Yield Different Efficiencies in a Reversible Cycle?

• L0r3n20
In summary, the conversation discusses the Carnot theorem and a problem involving a reversible cycle and an ideal gas. It is explained that the efficiency of a reversible engine is based on heat transfer at only the two given temperatures, but in the problem, heat is added at intermediate temperatures, resulting in a lower efficiency. This is why the two efficiency calculations are not the same.
L0r3n20
I just read about Carnot theorem (the highest efficiency is the one of reversible machines and all reversible machines working between two given temperatures have the same efficiency).

Then I found a problem where I have a reversible cycle made of an isochoric, adiabatic and isotherm. I report here the data. Computing the efficiency as work/absorbed heat I get 0.12 while if I use 1-Tc/Th I have 0.25. Why are the twos not the same?

The problem states:
"A monoatomic ideal gas is in a state A where the temperature is 300K, p= 2atm and V = 20L. Through an isochoric process the temperature rise up to 400K (state B); through an adiabatic process the gas arrive in a state C where Tc = Ta = 300K and finally through an isotherm process it gets back to A. Compute the efficiency of the cycle.

Good question. When people say that "all reversible engines working between two given temperatures have the same efficiency", they are talking about cases where heat is transferred to or from the engine only at the two given temperatures. In the example with the isochoric process, the temperature of the engine will continuously increase as heat is added during the isochoric process. So, the engine will take in heat at temperatures intermediate between the two given temperatures. This cycle has less efficiency compared to a cycle where all of the heat is added at the highest given temperature.

Delta2 and L0r3n20
Thanks! That's exactly what I needed To understand!

## 1. What is the Carnot Cycle?

The Carnot Cycle is a theoretical thermodynamic cycle that describes the most efficient way to convert heat into work. It consists of four reversible processes: isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression.

## 2. What is the Carnot Theorem?

The Carnot Theorem states that the maximum efficiency of a heat engine operating between two thermal reservoirs is independent of the working substance and is solely determined by the temperatures of the reservoirs.

## 3. How is the efficiency of the Carnot Cycle calculated?

The efficiency of the Carnot Cycle is calculated by taking the difference between the hot and cold reservoir temperatures and dividing it by the hot reservoir temperature. This value is then subtracted from 1 to give the maximum efficiency of the cycle.

## 4. What are the assumptions of the Carnot Cycle?

The Carnot Cycle assumes that all processes are reversible, the working substance is an ideal gas, and there are no internal losses such as friction or heat transfer.

## 5. What is the significance of the Carnot Cycle and Theorem?

The Carnot Cycle and Theorem are important concepts in thermodynamics as they establish a theoretical limit for the efficiency of heat engines. They also demonstrate the relationship between temperature and energy, and how this affects the efficiency of energy conversion processes.

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