Second Law of Thermodynamics

Limitations of the First Law

The first law tells us that heat can be converted into work and work into heat. It is merely a quantitative statement of the equivalence of heat and work, and it has these limitations.

• It does not tell us whether any particular process can actually occur.
• According to the first law, heat may on its own flow from an object at higher temperature to one at lower temperature, and it may equally on its own flow from an object at lower temperature to one at higher temperature. Practically, heat cannot flow from an object at lower temperature to another at higher temperature. The first law does not predict this practical observation.
• According to the first law, all (100 per cent) of the heat available could be converted into work, and similarly all the work could be converted into heat. Practically this is impossible.

Two further practical observations point the same way. It is impossible to build a heat engine of 100 per cent efficiency, that is, one that completely converts heat into work. And it is impossible for a refrigerator to remove heat without any work being done on the system, so the coefficient of performance can never be infinite. These observations form the basis of the second law of thermodynamics, which is a general principle that puts constraints upon the direction of heat transfer and upon the efficiency a heat engine can achieve.

An Analogy for the Limit on Efficiency

Figure 4.24 (a) shows an object A at height $h$ above the ground and an identical object B at height $h/2$. The potential energy of B is half that of A, so only half as much energy can be extracted from B. Similarly (Fig. 4.24 (b)), a heat engine that receives heat at 800 K and rejects it at 400 K can have a maximum efficiency of 50 per cent.

First Form: Kelvin-Planck (Engine) Statement

It is impossible to extract an amount of heat $Q_H$ from a hot reservoir and use it all to do work $W$. Some amount of heat $Q_C$ must be exhausted to a cold reservoir. This prohibits the possibility of a perfect heat engine.

It is sometimes called the Engine Law or the Engine Statement of the second law.

Diagram (Fig. 4.25): the energy flow diagram of a perfect heat engine would be a circle taking a pipeline $Q_H$ from the hot reservoir and sending it entirely out as $W$, with no branch to a cold reservoir. The second law says this diagram is impossible: every real engine must have a branch of finite width going down into the cold reservoir.

Second Form: Clausius (Refrigerator) Statement

It is not possible for heat to flow from a colder body to a warmer body without any work having been done to accomplish this flow.

It is sometimes called the Refrigerator Law or the Refrigerator Statement. It means that energy will not flow spontaneously from an object at low temperature to an object at a higher temperature, which rules out the possibility of a perfect refrigerator. The statements about refrigerators apply equally to air conditioners and heat pumps, which work on the same principles.

Diagram (Fig. 4.26): a practical refrigerator (Fig. 4.26 (a)) needs a work pipeline $W$ entering the circle in addition to the heat $Q_C$ drawn from the cold reservoir. A perfect refrigerator (Fig. 4.26 (b)) would move heat from the cold reservoir to the hot reservoir with no work input at all; the second law says this is impossible.

A Caution About the Word "Spontaneous"

When we say that energy will not flow spontaneously from a cold object to a hot object, we are referring to the net transfer of energy. Energy can transfer from a cold object to a hot object by the transfer of energetic particles or by electromagnetic radiation. However, in any spontaneous process the net transfer of energy will be from the hot object to the cold object. Work is required to transfer net energy from a cold object to a hot object.

Standard Results

• Kelvin-Planck statement: it is impossible to extract an amount of heat $Q_H$ from a hot reservoir and use it all to do work $W$; some heat $Q_C$ must be exhausted to a cold reservoir. This prohibits a perfect heat engine.
• Clausius statement: it is not possible for heat to flow from a colder body to a warmer body without any work having been done to accomplish this flow. This prohibits a perfect refrigerator.
• Consequences: $\eta < 1$ always, and $K$ can never be infinite.
• The second law is what sets the limit on the efficiency of a heat engine; the limiting value is the Carnot efficiency $\eta = 1 - T_C/T_H$.