In physics, the second law of thermodynamics is a general principle that describes constraints on the direction of heat transfer and limits to the potential of heat engines to convert energy into useful work. While the first law of thermodynamics tells us that energy is conserved in all conversion processes, the second law describes the limits to the efficiency of that process, and the qualitative change in energy that occurs.

The Second Law and Heat Engines

The second law is frequently illustrated by its application to heat engines. A heat engine is a device that converts heat energy into mechanical energy and exhausts energy as waste heat. The second law in this case can be stated as: no process is possible whose sole result is the absorption of heat from a reservoir and the conversion of this heat into work. This is also known as the Kelvin-Planck statement of the second law.

Heat Engine Diagram

In terms of the engine diagram, 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. Q_C is also known as “waste heat.”

The efficiency of this energy conversion process is given by $\eta=W/Q_H$. The second law tells us that $\eta$ is always less than 100%, i.e., there can be no perfect heat engine.

The Second Law, Refrigerators, and Heat Pumps

Another way to look at the second law is from the cooling side. A refrigerator is basically a cyclic heat engine run backwards. It takes heat in from a cold reservoir, does work on it, and rejects heat to a hot reservoir. Its net effect is thus to make the cold reservoir colder (refrigeration) by removing heat from inside it to the warmer warm reservoir.

Refrigerator diagram

Technically a refrigerator cycle is also a heat pump cycle, defined as a cycle in which heat is removed from a low-temperature space or source and rejected to a high-temperature sink with the help of external mechanical work.

In this application the second law can be stated as: is impossible to construct a cyclic refrigerator whose sole effect is the transfer of energy from a cold reservoir to a warm reservoir without the input of energy by work. This is also known as the Clausius statement of the second law of thermodynamics.

In terms of the engine diagram, it is impossible to extract an amount of heat Q_C from a cold reservoir and move it to a hot reservoir without an input of work W. In other words, energy will not flow spontaneously from a low temperature object to a higher temperature object. The statements about refrigerators also apply to air conditioners and heat pumps.

The effectiveness of a refrigerator is measured by the coefficient of performance (COP), which is the ratio of the amount of heat removed at the lower temperature to the work put into the system:

$COP=Q_c/W$

Note that the effectiveness will be greater than 1 only if the absolute temperature of the cold reservoir is warmer than half that of the hot reservoir. Thus, refrigeration to extremely cold temperatures is very difficult.

Cold weather heat pump: A cold weather heat pump. Source: Bonneville Power Administration.

In a heat pump used to provide thermal comfort, the effectiveness is the ratio of the energy delivered to the high-temperature reservoir to the work required to force the machine around its cycle (the energy consumed and paid for):

$COP_{HEATING}=Q_H/W=Q_H/(Q_H-Q_C)$

This formulation indicates that effectiveness is always greater than 1. Electrically powered heat pumps make economic sense only if the effectiveness of the heat pump times the efficiency of the electrical generation and transmission process exceeds 1. Otherwise, only part of the fuel burned to produce the electricity would have to be burned to provide the heat needed. (Modern natural gas furnaces can easily transfer more than 95% of the combustion heat to the heated space.) As the temperature of the cold reservoir (the outside temperature) declines, the effectiveness of the heat pump decreases toward 1. Because large electrical generators and high-voltage transmission lines deliver about one-third as much electrical energy as the heat value of the fuel they consume, as soon as the performance is less than about 3, it would be cheaper to burn the original fuel directly for the heat, rather than generate electricity to operate a heat pump. This limits the geographical regions where heat pumps make economic sense.

Reversibility and Ideal Engines

All natural processes are irreversible-they cannot be run backwards. Irreversibility can be explained in terms of entropy and the second law of thermodynamics. Consider an isolated system. The second law says that any process that would reduce the entropy of the isolated system is impossible. Suppose a process takes place within the isolated system in what we shall call the forward direction. If the change in state of the system is such that the entropy increases for the forward process, then for the backward process (that is, for the reverse change in state) the entropy would decrease. The backward process is therefore impossible, and we therefore say that the forward process is irreversible.

This is related to the well-known “arrow of time” in thermodynamics: as one goes "forward" in time, the second law of thermodynamics says that the entropy of an isolated system can only increase or remain the same; it cannot decrease. Hence, from one perspective, entropy measurement is thought of as a kind of clock.

Carnot Cycle

The Carnot cycle is a work cycle (of expansion and compression) of an idealized reversible heat engine that does work without loss of heat. The Carnot cycle is the theoretically most efficient engine cycle that all other cycles can be compared to it. While the second law of thermodynamics states that not all the heat supplied to a heat engine can be used to do work, the Carnot efficiency defines the upper limit on the fraction of the heat that can be so used. It is named for Nicolas Léonard Sadi Carnot, the French military engineer who provided the first authoritative theoretical account of heat engines.

Ideal Carnot cycle: The ideal Carnot cycle.

Carnot diagram

The Carnot cycle consists of two reversible adiabatic processes and two reversible isothermal processes. These steps are defined below, and in the accompanying temperature-entropy (T-S) and pressure-volume (P-V) diagrams.

1. Reversible isothermal expansion of the gas at the "hot" temperature, TH (isothermal heat addition). During this step (A to B on diagram) the expanding gas causes the piston to do work on the surroundings. The gas expansion is propelled by absorption of heat from the high temperature reservoir.
2. Isentropic (Reversible adiabatic) expansion of the gas. For this step (B to C on diagram) we assume the piston and cylinder are thermally insulated, so that no heat is gained or lost. The gas continues to expand, doing work on the surroundings. The gas expansion causes it to cool to the "cold" temperature, T_C.
3. Reversible isothermal compression of the gas at the "cold" temperature, T_C (isothermal heat rejection) (C to D on diagram). Now the surroundings do work on the gas, causing heat to flow out of the gas to the low temperature reservoir.
4. Isentropic compression of the gas. (D to A on diagram). Once again we assume the piston and cylinder are thermally insulated. During this step, the surroundings do work on the gas, compressing it and causing the temperature to rise to T_H. At this point the gas is in the same state as at the start of step 1.

By definition, the isothermal segments (AB and CD) occur when there is perfect thermal contact between the working fluid and one of the reservoirs, so that whatever heat is needed to maintain constant temperature will flow into or out of the working fluid, from or to the reservoir.

Entropy

In classical thermodynamics, the science of heat, entropy is a fundamental concept that measures the amount of energy that is unavailable to do work in a system. For a very small change in the independent parameters of the system such as volume, in which $\Delta Q$ amount of heat flows into it via a reversible process, the change in entropy $\Delta S$ of the system is given by:

$\Delta S= \Delta Q/T$

This leads to the following statement of the second law in terms of entropy:

In any cyclic process the entropy will either increase or remain the same.

In statistical mechanics, entropy is the amount of uncertainty or “mixedupness” that remains about a system after its observable macroscopic properties have been taken into account. The general form of the definition of entropy in statistical mechanics:

$s=k\cdot log(N)$

where N which is the total number of microstates available to the system. N is not the total number of particles, but rather is the total number of microstates that the particles could occupy, with the constraint that all such microstate collections would show the same macrostate. Form this perspective, the second law can be stated as :

In an isolated system the entropy of the system cannot decrease, that is $\Delta S\geq 0$

Sources

• Wolfs, Frank, Lecture Notes for Physics 121, University of Rochester, Accessed 21 December 2007.
• Spakovszky, Z. S. and I. A. Waitz, Lecture Notes for Thermodynamics and Propulsion, Massachusetts Institute of Technology, Accessed 21 December, 2007.
• Brown, Robert G. Lecture Notes for Physics 51, Duke University, Accessed 21 December 2007.
• Breinig,M. , Lecture Notes for Physics 136, the University of Tennessee, Accessed 21 December 2007.
• Piccard, R., Lecure Notes for Physics 202, Ohio University, Accessed 21 December 2007.

Further reading

• Beads of doubt, BBC News Online science, Thursday, 18 July, 2002.
• Plambeck, James A. The Second Law of Thermodynamics, Intute: Science, Engineering and Technology.
• Lambert, Frank L. Shakespeare and Thermodynamics:Dam the Second Law!