Second law of thermodynamics

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Laws of thermodynamics
Zeroth law of thermodynamics
First law of thermodynamics
Second law of thermodynamics
Third law of thermodynamics
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The second law of thermodynamics is an expression of the universal law of increasing entropy. In simple terms, it is an expression of the fact that over time, differences in temperature, pressure, and density tend to even out in a physical system which is isolated from the outside world. Entropy is a measure of how far along this evening-out process has progressed.

The most common enunciation of second law of thermodynamics is essentially due to Rudolf Clausius:

The entropy of an isolated system not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium.

There are many statements of the second law which use different terms, but are all equivalent. (Fermi, 1936) Another statement by Clausius is:

"Heat cannot of itself pass from a colder to a hotter body."

An equivalent statement by Lord Kelvin is:

"A transformation whose only final result is to convert heat, extracted from a source at constant temperature, into work, is impossible."

The second law is only applicable to macroscopic systems. The second law is actually a statement about the probable behavior of an isolated system. As larger and larger systems are considered, the probability of the second law being practically true becomes more and more certain. For any system with a mass of more than a few picograms, the second law is true to within a few parts in a million.

[edit] Overview

In a general sense, the second law says that temperature differences between systems in contact with each other tend to even out and that work can be obtained from these non-equilibrium differences, but that loss of heat occurs, in the form of entropy, when work is done.^[1] Pressure differences, density differences, and particularly temperature differences, all tend to equalize if given the opportunity. This means that an isolated system will eventually come to have a uniform temperature. A heat engine is a mechanical device that provides useful work from the difference in temperature of two bodies:

Heat engine diagram

During the 19th century, the second law was synthesized, essentially, by studying the dynamics of the Carnot heat engine in coordination with James Joule's Mechanical equivalent of heat experiments. Since any thermodynamic engine requires such a temperature difference, it follows that no useful work can be derived from an isolated system in equilibrium; there must always be an external energy source and a cold sink. The second law is often invoked as the reason why perpetual motion machines cannot exist.

[edit] Informal descriptions

The second law can be stated in various succinct ways, including:

It is impossible to produce work in the surroundings using a cyclic process connected to a single heat reservoir (Kelvin, 1851).
It is impossible to carry out a cyclic process using an engine connected to two heat reservoirs that will have as its only effect the transfer of a quantity of heat from the low-temperature reservoir to the high-temperature reservoir (Clausius, 1854).
If thermodynamic work is to be done at a finite rate, free energy must be expended.^[2]

[edit] Mathematical descriptions

In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:^[3]

$\int \frac{\delta Q}{T} = -N$

where N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most infamous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24th, in which, in the end of his presentation, Clausius concludes:

The entropy of the universe tends to a maximum.

This statement is the best-known phrasing of the second law. Moreover, owing to the general broadness of the terminology used here, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, to which this statement applies, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This, of course, is not true; this statement is only a simplified version of a more complex description.

In terms of time variation, the mathematical statement of the second law for a closed system undergoing an adiabatic transformation is:

$\frac{dS}{dt} \ge 0$

where

S is the entropy and

t is time.

It should be noted that statistical mechanics gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a microstate of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/√N where N is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically nil. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.

[edit] Available useful work

See also: Available useful work (thermodynamics)

An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an unlimited heat reservoir at temperature T_R and pressure P_R — so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain T_R; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain P_R.

Whatever changes dS and dS_R occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy S_tot of the isolated total system must increase:

$dS_{\mathrm{tot}}= dS + dS_R \ge 0$

According to the First Law of Thermodynamics, the change dU in the internal energy of the sub-system is the sum of the heat δq added to the sub-system, less any work δw done by the sub-system, plus any net chemical energy entering the sub-system d ∑μ_iRN_i, so that:

$dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,$

where μ_iR are the chemical potentials of chemical species in the external surroundings.

Now the heat leaving the reservoir and entering the sub-system is

$\delta q = T_R (-dS_R) \le T_R dS$

where we have first used the definition of entropy in classical thermodynamics (alternatively, the definition of temperature in statistical thermodynamics); and then the Second Law inequality from above.

It therefore follows that any net work δw done by the sub-system must obey

$\delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,$

It is useful to separate the work done δw done by the subsystem into the useful work δw_u that can be done by the sub-system, over and beyond the work P_R dV done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work that can be done:

$\delta w_u \le -d (U - T_R S + P_R V - \sum \mu_{iR} N_i )\,$

It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the availability or exergy X of the subsystem,

$X = U - T_R S + P_R V - \sum \mu_{iR} N_i$

The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,

$d X + \delta w_u \le 0 \,$

i.e. the change in the subsystem's exergy plus the useful work done by the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done on the system) must be less than or equal to zero.

[edit] Special cases: Gibbs and Helmholtz free energies

When no useful work is being extracted from the sub-system, it follows that

$d X \le 0 \,$

with the exergy X reaching a minimum at equilibrium, when dX=0.

If no chemical species can enter or leave the sub-system, then the term ∑ μ_iR N_i can be ignored. If furthermore the temperature of the sub-system is such that T is always equal to T_R, then this gives:

$X = U - TS + P_R V + \mathrm{const.} \,$

If the volume V is constrained to be constant, then

$X = U - TS + \mathrm{const.'} = A + \mathrm{const.'}\,$

where A is the thermodynamic potential called Helmholtz free energy, A=U-TS. Under constant volume conditions therefore, dA ≤ 0 if a process is to go forward; and dA=0 is the condition for equilibrium.

Alternatively, if the sub-system pressure P is constrained to be equal to the external reservoir pressure P_R, then

$X = U - TS + PV + \mathrm{const.} = G + \mathrm{const.}\,$

where G is the Gibbs free energy, G=U-TS+PV. Therefore under constant pressure conditions dG ≤ 0 if a process is to go forwards; and dG=0 is the condition for equilibrium.

[edit] Application

In sum, if a proper infinite-reservoir-like reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in X for an irreversible process and no change for a reversible process.

$dS_{tot} \ge 0$ is equivalent to $dX + \delta w_u \le 0$

This expression together with the associated reference state permits a design engineer working at the macroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (Also, see process engineer). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (See second law efficiency.)

This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systems ecology, and other disciplines.

[edit] History

See also: History of entropy

The first theory on the conversion of heat into mechanical work is due to Nicolas Léonard Sadi Carnot in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.

Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was the first to formulate the second law in 1850, in this form: heat does not spontaneously flow from cold to hot bodies. While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which considered heat as a liquid. From there he was able to infer the law of Sadi Carnot and the definition of entropy (1865).

Established in the 19th century, the Kelvin-Planck statement of the Second Law says, "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.

The Ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.

In quantum mechanics, the ergodicity approach can also be used. However, there is an alternative explanation, which involves Quantum collapse - it is a straightforward result that quantum measurement increases entropy of the ensemble. Thus, the Second Law is intimately related to quantum measurement theory and quantum collapse - and none of them is completely understood.

[edit] Criticisms

Owing to the somewhat ambiguous nature of the formulation of the second law, i.e. the postulate that the quantity heat divided by temperature increases in spontaneous natural processes, it has occasionally been subject to criticism as well as attempts to dispute or disprove it. Clausius himself even noted the abstract nature of the second law. In his 1862 memoir, for example, after mathematically stating the second law by saying that integral of the differential of a quantity of heat divided by temperature must be greater than or equal to zero for every cyclical process which is in any way possible:^[3]

$\int \frac{dQ}{T} \ge 0$

Clausius then states:

Although the necessity of this theorem admits of strict mathematical proof if we start form the fundamental proposition above quoted it thereby nevertheless retains an abstract form, in which it is with difficulty embraced by the mind, and we feel compelled to seek for the precise physical cause, of which this theorem is a consequence.

This statement, curiously, from one perspective, may be said to be equally valid to this very day. That is, owing to ambiguous nature of heat and temperature themselves, as they relate to sciences such as chemistry and particle physics, entropy is still an ambiguous quantity. As such, scientists are forever trying to find loop-holes in the second law:

[edit] Perpetual motion of the second kind

Before 1850, heat was regarded as an indestructible particle of matter. This was called the “material hypothesis”, as based principally on the views of Isaac Newton. It was on these views, partially, that in 1824 Sadi Carnot formulated the initial version of the second law. It soon was realized, however, that if the heat particle was conserved, and as such not changed in the cycle of an engine, that it would be possible to send the heat particle cyclically through the working fluid of the engine and use it to push the piston and then return the particle, unchanged, to its original state. In this manner perpetual motion could be created and used as an unlimited energy source. Thus, historically, people have always been attempting to create a perpetual motion machine so to disprove the second law.

[edit] Maxwell's Demon

In 1871, James Clerk Maxwell proposed a thought experiment, now called Maxwell's demon, which challenged the second law. This experiment reveals the importance of observability in discussing the second law. In other words, it requires a certain amount of energy to collect the information necessary for the demon to "know" the whereabouts of all the particles in the system. This energy requirement thus negates the challenge to the second law. Moreover, to reconcile this apparent paradox from another perspective, one may resort to a questionable use of information entropy.

[edit] Time's Arrow

Main article: Entropy (arrow of time)

The second law is a law about macroscopic irreversibility, and so may appear to violate the principle of T-symmetry. Boltzmann first investigated the link with microscopic reversibility. In his H-theorem he gave an explanation, by means of statistical mechanics, for dilute gases in the zero density limit where the ideal gas equation of state holds. He derived the second law of thermodynamics not from mechanics alone, but also from the probability arguments. His idea was to write an equation of motion for the probability that a single particle has a particular position and momentum at a particular time. One of the terms in this equation accounts for how the single particle distribution changes through collisions of pairs of particles. This rate depends of the probability of pairs of particles. Boltzmann introduced the assumption of molecular chaos to reduce this pair probability to a product of single particle probabilities. From the resulting Boltzmann equation he derived his famous H-theorem, which implies that on average the entropy of an ideal gas can only increase.

The assumption of molecular chaos in fact violates time reversal symmetry. It assumes that particle momenta are uncorrelated before collisions. If you replace this assumption with "anti-molecular chaos," namely that particle momenta are uncorrelated after collision, then you can derive an anti-Boltzmann equation and an anti-H-Theorem which implies entropy decreases on average. Thus we see that in reality Boltzmann did not succeed in solving Loschmidt's paradox. The molecular chaos assumption is the key element that introduces the arrow of time.

[edit] Applications to living systems

Main article: Entropy and life

The second law of thermodynamics has been proven mathematically for thermodynamic systems, where entropy is defined in terms of heat divided by the absolute temperature. The second law is often applied to other situations, such as the complexity of life, or orderliness.^[4] Some, however, object to this application, on possibly philosophical or theological grounds, reasoning that thermodynamics does not apply to the process of life. In sciences such as biology and biochemistry, however, the application of thermodynamics is well-established, e.g. biological thermodynamics. The general viewpoint on this subject is summarized well by biological thermodynamicist Donald Haynie, where as he states: "Any theory claiming to describe how organisms originate and continue to exist by natural causes must be compatible with the first and second laws of thermodynamics.^[5]

[edit] Small systems

In statistical thermodynamics, which uses probability theory to calculated thermodynamic variables, such as entropy, the second law only holds for ensemble averages and the probability for single systems to violate it increases with discreasing size. The fluctuation theorem describes this behaviour.

[edit] Complex systems

It is occasionally claimed that the second law is incompatible with autonomous self-organisation, or even the coming into existence of complex systems. The entry self-organisation explains how this claim is a misconception.

In fact, as hot systems cool down in accordance with the second law, it is not unusual for them to undergo spontaneous symmetry breaking, i.e. for structure to spontaneously appear as the temperature drops below a critical threshold. Complex structures, such as Bénard cells, also spontaneously appear where there is a steady flow of energy from a high temperature input source to a low temperature external sink. It is conjectured that such systems tend to evolve into complex, structured, critically unstable "edge of chaos" arrangements, which very nearly maximise the rate of energy degradation (the rate of entropy production).^[6]

[edit] Quotes

Wikiquote has a collection of quotations related to:

Second law of thermodynamics

The law that entropy always increases, holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations — then so much the worse for Maxwell's equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.

--Sir Arthur Stanley Eddington, The Nature of the Physical World (1927)

There are almost as many formulations of the second law as there have been discussions of it.

--Philosopher / Physicist P.W. Bridgman, (1941)

[edit] Miscellany

Flanders and Swann produced a setting of a statement of the Second Law of Thermodynamics to music, called "First and Second Law".
The economist Nicholas Georgescu-Roegen showed the significance of the Entropy Law in the field of economics (see his work The Entropy Law and the Economic Process (1971), Harvard University Press).

[edit] See also

[edit] References

^ Mendoza, E. (1988). Reflections on the Motive Power of Fire – and other Papers on the Second Law of Thermodynamics by E. Clapeyron and R. Clausius. New York: Dover Publications. ISBN 0-486-44641-7.
^ Stoner, C.D. (2000). Inquiries into the Nature of Free Energy and Entropy - in Biochemical Thermodynamics. Entropy, Vol 2.
^ ^a ^b Clausius, R. (1865). "Mechanical Theory of Heat - with its Application to the Steam Engine and the Physical Properties of Bodies." London: John van Voorst.
^ Hammes, Gordon, G. (2000). Thermodynamics and Kinetics for the Biological Sciences. New York: John Wiley & Sons. ISBN 0-471-37491-1.
^ Haynie, Donald, T. (2001). Biological Thermodynamics. Cambridge: Cambridge University Press. ISBN 0-521-79549-4.
^ Kauffman, Stuart (1995). At Home in the Universe - the Search for the Laws of Self-Organization and Complexity. Oxford University Press. ISBN 0-19-509599-5.

Fermi, Enrico [1936] (1956). Thermodynamics. New York: Dover Publications, Inc. ISBN 0-486-60361-X.

[edit] Further reading

Goldstein, Martin, and Inge F., 1993. The Refrigerator and the Universe. Harvard Univ. Press. A gentle introduction, a bit less technical than this entry.

[edit] External links

110+ Variations of the 2nd Law.
Mechanical Theory of Heat – Nine Memoirs by Rudolf Clausius [1850-1865] on the 1st and 2nd Laws of Thermodynamics.
Stanford Encyclopedia of Philosophy: "Philosophy of Statistical Mechanics." by Lawrence Sklar.
The evolution of Carnot's principle, by E.T. Jaynes, in G. J. Erickson and C. R. Smith (eds.) Maximum-Entropy and Bayesian Methods in Science and Engineering vol. 1, p. 267 (1988).
The Second Law of Thermodynamics.

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Categories: Laws of thermodynamics | Non-equilibrium thermodynamics | Philosophy of thermal and statistical physics