Carnot Cycle

Heat engines like steam engine, gasoline engine, diesel engine or steam turbine convert heat to mechanical energy. They are based on cyclic processes in which the same states are passed through again and again. The Carnot process, proposed and studied by Nicolas Léonard Sadi Carnot in 1824, is a special cyclic process significant for the theory of heat engines. It is an idealized process for which the efficiency can be calculated rather easily.

An ideal gas (shown here in green) passes successively through the following processes:

• Isothermal expansion: The gas is in contact with a hot thermal reservoir (not shown) and expands at a constant temperature. Heat is transferred from the hot reservoir to the gas; the gas releases the same amount of energy to the environment in the form of work. The internal energy of the gas remains constant during this process.
• Isentropic (reversible adiabatic) expansion: The gas is now separated from the hot reservoir. It continues to expand and therefore still releases work to the environment. However, since no more heat is supplied, temperature and internal energy decrease during this transformation.
• Isothermal compression: The gas is now in contact with a cold thermal reservoir (not shown). It is compressed at a constant temperature. Work is transferred to the gas from outside; the gas emits the same amount of energy to the cold reservoir in the form of heat. The internal energy of the gas remains constant.
• Isentropic (reversible adiabatic) compression: The gas is now separated from the cold reservoir, but is still compressed. Work is transferred to the gas from outside. However, since no more heat is released, temperature and internal energy increase during this partial process.

This app alternatively shows the experimental setup with two small diagrams, a large P-V diagram (relationship between pressure and volume) or a large T-S diagram (relationship between temperature and entropy). The following sizes can be entered in the input fields:

• Amount of substance: The amount of substance is a measure for the number of particles (mostly molecules, in the case of noble gases atoms) and is expressed in moles. One mole (1 mol) contains approximately 6 × 10²³ particles. Values from 0,1 mol to 1 mol are accepted.
• Adiabatic index: This index depends on the number of degrees of freedom. Strongly simplified, it is 5/3 ≈ 1,67 for noble gases (monatomic gases), 7/5 = 1,4 for diatomic gases and 8/6 ≈ 1,33 for gases whose molecules have three or more atoms. Values from 1,3 to 1,7 can be entered.
• Minimal and maximal temperature: You have to enter the absolute temperature in Kelvin. Values between 100 K (corresponding to −173°C) and 1000 K (corresponding to 727°C) are allowed.
• Minimal und maximal volume: Values from 1 dm³ to 10 dm³ are permitted.

Sometimes the entered values must be changed by the program. The minimal and maximal pressure are calculated with the gas equation and displayed in the unit kPa (kilopascal).

Efficiency of a Carnot engine:

η  .... efficiency
T_C  ... absolute temperature of the cold reservoir
T_H  ... absolute temperature of the hot reservoir

The efficiency of a real heat engine operating between two thermal reservoirs is always lower than the efficiency of a corresponding Carnot engine.