Heat pump, Clausius statement of the second law of Thermodynamics

A heat pump or refrigerator is a cyclically operating device which absorbs heat from a cold thermal reservoir and releases it to a warmer thermal reservoir. This process does not occur spontaneously so in order to operate, a refrigerator requires a work input (W<0).

The following figure schematically represents the operation of a heat pump:

The working fluid of the heat pump cyclically undergoes a series of processes so that the refrigerator can run continuously.

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During the cycle, the working fluid absorbs heat from the cold thermal reservoir at temperature T2 (in blue in the figure), and transfers it to a warmer thermal reservoir at temperature T1 (in yellow). A thermal reservoir is a thermodynamic system that can transfer heat indefinitely at constant temperature. The operation of a heat pump requires work to be done by an external energy source.

Depending on the design of heat pump, the processes undergone by the working fluid will be different, but they will always consist of a anticlockwise cycle, because work (negative) will have to be done by the surroundings (W<0):

The coefficient of performance (COP) of the heat pump is the ratio of heat absorbed from the cold thermal reservoir to the absolute value of work required:

In addition, as energy is conserved, theheat pump must satisfy the first law of Thermodynamics. The internal energy variation per cycle of the working fluid is zero (because the internal energy is a state function), therefore:

Taking absolute values, we have:

The purpose of a heat pump is to transfer the largest possible amount of heat from the cold  to the hot thermal reservoir for a given amount of work input; there is however a limit to the amount of heat that can be transferred. This fact is known as the Clausius statement of the second law of Thermodynamics:

Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.

The second law of Thermodynamics is an empirical law that is not deduced from any previous law.

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A consequence of the second law of Thermodynamics is that the absolute value of Q2 (the heat absorbed from the cold thermal reservoir) cannot equal Q1 (the amount of heat discharged to the hot thermal reservoir). If it were the case, the heat pump could operate without being supplied with energy. But we all know that a fridge consumes energy to operate. Heat does not spontaneously flow from cold to hot bodies; for that to happen, we need to provide energy.

The question therefore is: what is the maximum coefficient of performance for a heat pump operating between two given thermal reservoirs? And the answer to this question is the Carnot refrigerator or Carnot heat pump.

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