Second Law of Thermodynamics

From SkepticWiki

1 Definition
2 The Clausius statement implies the Kelvin-Planck statement
3 The Kelvin-Planck statement implies the Clausius inequality
4 Interlude: defining entropy
5 The Clausius inequality implies increase of entropy in isolated systems
6 Increase of entropy for isolated systems implies the generalized increase of entropy law
7 The generalized increase of entropy principle implies the Clausius statement
8 Statistical Mechanics and the Second Law
9 Misconceptions
10 Quotations
- 10.1 Thermodynamics and the Arrow of Time
11 References
12 Related Articles

[edit] Definition

There are many ways of formulating the Second Law of Thermodynamics. Some of the more popular are:

The Clausius statement: There is no mechanism which operates in a cycle the sole effect of which is to transfer heat from a body at a lower temperature to a body at a higher temperature.

We shall call a machine which could carry out such a cycle a "magic fridge". A diagram of such a device is given to the right. You will notice that the really crucial difference between a magic fridge and an ordinary fridge is that the magic fridge has no power supply.

You might be tempted to wonder if a fridge powered by an internal battery would constitute a magic fridge. It would not, because in operating, such a fridge would run down its battery, so it would not be operating in a cycle.

Note that there is nothing about the existence of a magic fridge which is forbidden by the First Law of Thermodynamics. Heat goes into the fridge, heat goes out, no energy is destroyed. So we are in genuine need of a second law to say that this can't be done.

The Kelvin-Planck statement: There is no mechanism which operates in a cycle, which is connected to a single thermal reservoir, and which produces a positive amount of net work.

We shall call such an impossible device a "magic heat engine": one is shown in the diagram to the left. Such a device is also referred to as a "perpetual motion machine of the second kind". This is perhaps a misnomer, because in real life there is no such thing as a thermal reservoir - if such a magic heat engine existed, it would eventually tap all the heat made available to it and then stop working.

Again, note that the existence of a magic heat engine is not an impossibility according to the First Law - heat is going in, work is coming out, and all the First Law has to say about it is that the two quantities must be equivalent.

While the First Law already indicates that it is impossible to get more work out of an engine than the heat used, the Second Law states that there will always be some waste, and you will always get less work than the equivalent amount of heat. In practice, this means that for a heat engine to run continuously, it must not only draw heat from a reservoir but also discharge some waste heat.

The Clausius inequality: For any cyclic process, the integral over the cycle of δQ / T is less than or equal to zero (where as usual Q is the net flow of heat into the system and T is the temperature.)

The Clausius inequality may be seen as a stepping stone to more useful forms of the Second Law, such as the next statement.

The entropy of an isolated system cannot decrease.

The concept of entropy is rather a subtle one: it will be defined below. It is sometimes explained as a measure of the "disorder" of a system - this is not quite accurate: indeed, entropy isn't similar to anything except more entropy, and the concept cannot really be grasped except by seeing the equations that define it.

The total entropy of a system and its surroundings cannot decrease.

As there are no truly isolated systems in nature, except, arguably, the Universe itself, it is convenient in practice to use this generalized principle.

These five statements do not look equivalent, or even related, at first glance. We shall prove their equivalence by showing that the Clausius statement implies the Kelvin-Planck statement; that the Kelvin-Planck statement implies the Clausius inequality; that the Clausius inequality implies the increase of entropy for isolated systems; that the increase of entropy for isolated systems implies the generalized increase of entropy principle; and that the generalized increase of entropy principle implies the Clausius statement.

[edit] The Clausius statement implies the Kelvin-Planck statement

The proof is by the contrapositive. Suppose that we violated the Kelvin-Planck statement and constructed a magic heat engine. Then we could take its work output and use it to power a (non-magical) fridge, which draws heat from a cooler thermal reservoir and discharges this heat into the thermal reservoir that the magic heat engine runs off, as shown in the diagram to the right. Then this combined system of the fridge and the magic heat engine is a magic fridge, since it moves heat from a lower temperature to a higher temperature while having no other effect on its environment - in particular, without having an external power source.

So if we had a magic heat engine, we could have a magic fridge. Therefore, the statement that you can't have a magic fridge implies that you can't have a magic heat engine.

[edit] The Kelvin-Planck statement implies the Clausius inequality

We now wish to prove that the statement that you can't have a magic heat engine implies the Clausius inequality - that for any closed, stationary system operating in a cycle

$\oint \delta Q / T \leq 0$

where Q is as usual the heat transferred to the system and T is the temperature at the boundary of the system.

We shall prove this with the assistance of the piece of imaginary equipment shown in the diagram below. We choose any closed, stationary system (hereafter referred to as "the chosen system"). We imagine that the chosen system is attached to a reversible system which is connected to a thermal reservoir, as shown on the diagram.

By convention, the diagram shows heat flowing in to the chosen system and work coming out of it. We emphasize, however, that this is not a restriction on what the chosen system is allowed to do: it merely means that work input or heat output will have negative sign. (See the article on the First Law of Thermodynamics for more information.)

It is important to note that the inequality we're going to derive relates only to the boundary temperature and heat flow of the chosen system, and is therefore true whether or not the chosen system is connected to a reversible system which is connected to a thermal reservoir. The reversible system and the thermal reservoir are, so to speak, intellectual scaffolding, which will be discarded after we have produced our result.

So, consider the diagram. The combined system of the chosen system and the reversible system to which it's attached is, like everything else, subject to the First Law of Thermodynamics, which we give for the combined system in its differential form: δQ_r - δW = dE.

Now, as the reversible system is reversible, then if we are using a Thermodynamic Temperature Scale (eg degrees kelvin) then by the definition of a thermodynamic temperature scale we have δQ_r / δQ = T_r / T.

We can rearrange this as δQ_r = T_r δQ / T and substitute this into the First Law equation above, giving us: T_r δQ / T - δW = dE

For convenience, we shall assume that the reversible system performs an whole number of cycles in the same time that the chosen system takes to execute one. Now, let's integrate the equation above over one cycle of the chosen system.

$\oint T_r \delta Q / T - \oint \delta W = \oint dE$

Of course, $\oint \delta W$ is simply W. To evaluate $\oint dE$ , recall that over a cycle, the combined system can neither gain nor lose energy - or it wouldn't be a cycle - and so $\oint dE = 0$ . This gives us

$\oint T_r \delta Q / T - W = 0$

which we shall rearrange to give

$\oint T_r \delta Q / T = W$

Now, here's the clever part. There is no such thing as a magic heat engine. That's the Kelvin-Planck statement of the Second Law, and the whole thing we're basing our argument on.

Now, look at the diagram given above, and observe that the combined system is connected to only one thermal reservoir. If the net work output W was greater than zero, the combined system would be a magic heat engine. Therefore, the net work output W of the combined system is less than or equal to zero, and so

$\oint T_r \delta Q / T \leq 0$

Finally, we note that T_r is a constant (being the temperature of a thermal reservoir) and positive, since we are using a thermodynamic temperature scale. So we can divide both sides of the inequality through by T_r, giving us

$\oint \delta Q / T \leq 0$

as required.

[edit] Interlude: defining entropy

We are now almost in a position to define entropy, but a few preliminary remarks are required.

For a internally reversible system, the integral $\oint \delta Q / T$ is not merely less than or equal to zero; it is exactly equal to zero.

We demonstrate this as follows. The system is reversible, so imagine what happens when we run it in reverse. Run in reverse, heat flowing into the system becomes heat flowing out of the system, and vice versa. This means that for the system run backwards, the expression to be integrated is the same except that the sign of δQ is reversed. Hence, the result of the integral for the system run backwards is the same as for the system run forwards, but with the sign reversed. If the integral for the system run forwards sums to x, then for the system run backwards the integral sums to minus x. But as the Clausius inequality holds for the system whether it's run backwards or forwards, this means that we have x ≤ 0 and -x ≤ 0. This, is only possible if x = 0.

Hence for a internally reversible system

$\oint \delta Q / T = 0$

This has interesting consequences. Take any two states (call them state 1 and state 2) and any two internally reversible processes between them (call them process A and process B). Then the integral $\int \delta Q / T$ is the same when summed over process A or process B - that is:

$\int_1^2 (\delta Q / T)_A = \int_1^2 (\delta Q / T)_B$

Why so? Well, consider that if you followed process A, and then followed process B backwards (which you can do because it is a reversible process), then this would get you back where you started from, so it would be a cycle, and, since it's the composition of two reversible processes, it would be a reversible cycle. So the equality deduced above for reversible cycles applies, and we may write

$\int_1^2 (\delta Q / T)_A + \int_2^1 (\delta Q / T)_{B-backwards} = 0$

Now, following B backwards reverses the direction of heat flow, and hence reverses the sign of δQ; hence the result of the integral following B backwards is the same as the integral following B forwards, but with the sign reversed, i.e.

$\int_2^1 (\delta Q / T)_{B-backwards} = - \int_1^2 (\delta Q / T)_{B}$

We substitute this into the equation above and the result follows.

This allows us to define the change in entropy associated with a process. Suppose we have any process (reversible or irreversible) between two states (state 1 and state 2). Then the change in entropy between state 1 and state 2 (written as S₂ - S₁ or simply as ΔS ) is given by the value that the integral of δQ / T would have if you went between those two states by an internally reversible process. That is:

$\Delta S = \int_1^2 (\delta Q / T)_{int.rev.}$

This expression is well-defined because we have shown that the integral comes to the same thing whichever internally reversible process you follow between the two states.

Notice that this expression defines the change in entropy between two states. The formula given does not allow us to say absolutely how much entropy is associated with a given state, but only the difference in entropy between two states. The Third Law of Thermodynamics allows us to fix an absolute standard of entropy - that's what it's for. However, in most practical cases, this is unimportant: the change in entropy between two states is usually what concerns engineers and scientists.

Now, you might find the definition of entropy slightly odd. It might seem more tolerable if it was the integral of δQ / T associated with the actual process taking place. But instead, it's the integral of what δQ / T would be if you got between the two states by an internally reversible process. It is not, at first, clear that this is something that we would particularly want to know. And entropy seems like such an abstract thing. Temperature can be measured with a thermometer, volume with a ruler, pressure with a manometer - but entropy, it seems, is measured by a theoretical calculation concerning a process which cannot actually take place (since there are no truly reversible thermodynamic processes in nature).

Indeed, the only excuse for the concept of entropy is that the science of thermodynamics would be impossible without it.

[edit] The Clausius inequality implies increase of entropy in isolated systems

Suppose we have an actual process between two states. The increase in entropy is given by the integral of δQ / T for any reversible process between these two states. Now, if we followed the actual path, and then followed some such internally reversible process backwards, that would be a cycle, and so the Clausius inequality applies to it. The integral over the cycle is the sum of the integrals over the two paths, so we have

$\int_1^2 (\delta Q / T)_{actual} + \int_2^1 (\delta Q / T)_{int.rev.-backwards} \leq 0$

Once more we make use of the fact that reversing the direction of an integral reverses the sign of the result. So we can rewrite the above expression as

$\int_1^2 (\delta Q / T)_{actual} - \int_1^2 (\delta Q / T)_{int.rev.} \leq 0$

By definition of entropy, $\int_1^2 (\delta Q / T)_{int.rev.}$ is the change in entropy associated with the actual process. Hence we can rewrite the inequality above as as

$\Delta S \geq \int_1^2 (\delta Q / T)_{actual}$

Now, consider the case where the process is completely isolated. In that case, there is (by definition) no flow of energy in or out of the system, which means that δQ is always equal to zero. Applying that fact to the equation given above, we find that for isolated systems

$\Delta S_{isolated} \geq 0$

Or, in English: the entropy of an isolated system never decreases.

[edit] Increase of entropy for isolated systems implies the generalized increase of entropy law

The principle given above is an interesting thing to know, but we do not often have to deal with isolated systems. To generalise the law for non-isolated systems, we simply observe that we can choose to see a system and its surroundings as one big thermodynamic system, and by drawing the boundaries of our system big enough we can effectively treat the system as isolated. (The Universe, for example, is considered to be an isolated system by most scientists).

Hence we may write

$\Delta S_{sys} + \sum \Delta S_{surr} \geq 0$

Or, in English: the combined entropy of a system and its surroundings never decreases.

[edit] The generalized increase of entropy principle implies the Clausius statement

We are not quite home and dry, because in order to show that our various statements of the second law are truly equivalent, it is not sufficient to show that they can all be derived from the law against magic fridges - they must also imply it.

So one last proof is required: we must show that the generalized increase of entropy principle implies that there are no magic fridges.

This is simple enough. Consider the fridge in the diagram opposite. In one cycle, it draws heat in a quantity of Q_C from a thermal reservoir at a temperature of T_C and discharges heat in the quantity Q_H into a thermal reservoir at a temperature of T_H . We may take the system and its surroundings to be the fridge and the two thermal reservoirs.

Let's calculate the change in entropy associated with one cycle of the fridge. The fridge itself will undergo neither an increase nor decrease in entropy, because it is operating on a cycle. The hot thermal reservoir has Q_H heat flow in at a temperature of T_H, and the cooler thermal reservoir has Q_C of heat flow out at a temperature of T_C .

So the total change in entropy is simply Q_H / T_H - Q_C / T_C. The generalized increase in entropy principle tells us that this must be greater than or equal to zero: Q_H / T_H - Q_C / T_C ≥ 0. We can rearrange this as Q_H ≥ Q_C T_H / T_C

Now T_H is greater than T_C, so T_H / T_C > 1, so putting this together with the inequality above, this means that Q_H > Q_C. Or, in English, more heat is coming out of the fridge than is going in. By the First Law of Thermodynamics, this is not possible unless someone is putting energy into the fridge: the quantity W_in is positive. So the fridge is not magic.

Indeed, the relation derived above tells us just how non-magical a refrigerator must be. If you wish to move a joule of heat from a body with a temperature of T_C (in degrees kelvin) to a body with a temperature of T_H, this will cost you at least T_H / T_C - 1 joules of energy, and there is nothing you can do to improve on that - to do so would be as remarkable, if less spectacular, than exceeding the speed of light.

[edit] Statistical Mechanics and the Second Law

Classical thermodynamics deals with variables of state, which are properties of bulk collections of matter. One such variable of state is temperature. Statistical mechanics looks beyond the bulk properties to the particles making up the substance. Temperature is thus a measurement of the average kinetic energy of the individually moving molecules in the system. In other words, classical thermodynamics looks at the macroscopic properties of a system, while statistical thermodynamics relates them to the microscopic properties.

This provides a solid mathematical foundation to define entropy. It is the natural logarithm of the number of ways that the particles can be rearranged while preserving the same macroscopic state relative to the total number of ways of rearranging the particles. Highly ordered states have fewer rearrangments that macroscopically appear the same, and thus have lower entropy. It is, in practice, usually impossible to count the number of rearrangements in a material, so entropy is usually expressed in relative terms between two systems or with respect to a zero point, given by the Third Law of Thermodynamics.

The unification of the macroscopic and the microscopic worlds through statistical thermodynamics was one of the great unifications in science. In the 20th century, thermodynamics has been unified with information theory and computing, most notably by Claude Shannon. In information theoretical systems, it is possible to count the number of rearrangements, so the expression of entropy is slightly different. Nevertheless, it is directly related to thermodynamic entropy. Richard Feynman in the Feynman Lectures on Computation described a little car that could be driven by the information on a tape. Entropy plays a strong role in cryptography and the generation of random sequences of numbers.

[edit] Misconceptions

[edit] Disorder

Entropy is often described as a measurement of the disorder of a system. The concept of "order" is a metaphor, and like all metaphors, it can lead to misunderstanding as well as understanding. Our common-sense notions of order can fail. We may, for example, compare a crystalline solid with an amorphous solid of the same material and conclude that the crystal is more orderly and therefore must have lower entropy. In some cases, depending on the material, the crystalline solid may have higher entropy than the amorphous solid.

[edit] Application

The Second Law of Thermodynamics applies clearly only within the domain of classical thermodynamics, which describes certain macroscopic properties of systems. As a system is viewed at a finer level, or as the number of states becomes smaller, the Second Law becomes weaker and weaker.

To state categorically that the entropy of a system can never spontaneously decrease is wrong. Consider of system of 100 coins, where the macroscopic variable of state is the number of heads versus the number of tails. The coins are flipped randomly. The entropy of the system is easy to calculate, and the state of 50 heads and 50 tails has the highest entropy, and we may state, from the Second Law, that the system will tend to go to that state. However, states with, say, 47 heads and 53 tails will also be common.

These fluctuations do not matter much, because the system of coins will still tend to the macroscopic state of 50 heads and 50 tails, give or take a few. The systems of classical thermodynamics also work this way. The variables of state are such that small fluctuations tend to smooth out and tend not to affect the variables of state much.

However, while the principles of the Second Law are universally sound, a great deal of care must be taken in interpreting its implications in different kinds of systems. Particular care must be taken in systems that work in accordance with chaos theory and are thus extremely sensitive to small changes.

There are many important examples of such systems:

Weather: A small local fluctuation consistent with the Second Law can produce dramatic results, given time.
Computers: The fluctuation of a single bit in memory can sometimes cause the whole system to crash.
Biology: A single point mutation or replication error can cause huge changes in the resulting creature.

[edit] Evolution

It is sometimes said that Evolution by Natural Selection violates the Second Law Of Thermodynamics. At first glance, this seems plausible. After all, human beings and the rest of life seem complex and highly ordered, and therefore it would seem that the the tendency toward disorder suggested by the Second Law would conclude it.

This conclusion is, in every possible way, completely wrong.

Recall that the Second Law can be stated as "it is impossible to build a magic fridge". To say that evolution would violate the second law is therefore logically equaivalent to claiming that if the theory of evolution was correct, it would be possible to build a magic fridge. But the theory of evolution implies no such thing.

Every single chemical and biological reaction in the entire history of life has worked in accordance with the Second Law. This obviously includes the processes to which evolution is ascribed, such as mutation and natural selection. It also includes all other processes which produce an increase of complexity, such as the development from embryo to adult.

Creatures are not closed systems. They eat, excrete, use sunlight, and/or feed off the chemicals of thermal vents. Nor is the Earth an isolated system. It recieves light from this sun, without which most of evolution would not have been possible. A tally of the entropy of the entire solar system would show that it has increased during the evolution of life.

Local variations in entropy are not precluded by the Second Law. If they were, refrigerators would be impossible. Rather, refrigerators are enabled by the properties of nature that the Second Law describes. The Second Law puts a limit on their efficiency, and so it puts a limit on the efficiency of evolution by natural selection. Yet the efficiency of evolution by natural selection is nowhere near that of a refrigerator.

For further information, see the main article on the creationist claim that Evolution Violates The Second Law of Thermodynamics.

[edit] Quotations

[edit] Thermodynamics and the Arrow of Time

by Jacob Bronowski

It is often said that the progression from simple to complex runs counter to the normal statistics of chance that are formalized in the Second Law of Thermodynamics. Strictly speaking, we could avoid this criticism simply by insisting that the Second Law does not apply to living systems in the environment in which we find them. For the Second Law applies only when there is no overall flow of energy into or out of a system, whereas all living systems are sustained by a net inflow of energy.

But though this reply has a formal finality, in my view it evades the underlying question that is being asked. True, life could not have evolved in the absence of a steady stream of energy from the sun -- a kind of energy wind on the earth. But if there were no more to the mechanism of molecular evolution than this, we should still be at a loss to understand how more and more complex molecules cam to establish themselves. All the energy wind can do, in itself, is to increase the range and frequency of variation around the average state: that is, to stimulate the formation of more complex molecular arrangements. But most of these variant arrangements fall back to the norm almost at once, by the usual thermodynamic processes of degradation; so that it remains to be explained why they do not all do so, and how instead some complex arrangements establish themselves, and become the base for further complexity in their turn.

It is therefore relevant to discuss the Second Law, which is usually interpreted to mean that all constituent parts of a system must fall progressively to their simplest states. But this interpretation quite misunderstands the character of statistical laws in general in nonequilibrium states. The Second Law describes the final equilibrium state of a system; if we are to apply it, as here, to stable states which are far from equilibrium, we must interpret it and formulate it differently. In these conditions, the Second Law of Thermodynamics becomes a physical law only if there is added to it the condition that there are no preferred states of configurations.

In itself, the Second Law merely enumerates all the configurations which a system could take up, and it remarks that the largest number in this count are average or featureless. Therefore, if there are no preferred configurations (that is, no hidden stabilities in the system on the way to equilibrium), we must expect that any special feature we find is exceptional and temporary, and will revert to the average in the long run. This is a true theorem in combinatorial arithmetic, and (like other statistical laws) a fair guess at the behavior of long runs. But it tells us little about the natural world which, in the years since the Second Law seemed exciting, has turned out to be full of preferred configurations and hidden stabilities, even at the most basic and inanimate level of atomic structure.

The Second Law describes the statistics of a system around equilibrium whose configurations are all equal, and it makes the obvious remark that chance can only make such a system fluctuate around its average. There are no stable states in such a system, and there is therefore no stratum that can establish itself; the system stays around its average only by a principle of indifference, because numerically the most configurations are bunched around the average.

But if there are hidden relations in the system on the way to equilibrium which cause some configurations to be stable, the statistics are changed. The preferred configurations may be unimaginably rare; nevertheless, they present another level around which the system can bunch, and there is now a countercurrent or tug-of-war within the system between this level and the average. Since the average has no inherent stability, the preferred stable configuration will capture members of the system often enough to change the distribution; and, in the end, the system will be established at this level as a new average. In this way, local systems of a fair size can climb up from one level of stability to the next, even though the configuration at the higher level is rare. When the higher level becomes the new average, the climb is repeated to the next higher level of stability; and so on up the level of strata.

When there are hidden levels of stability, one above another, as there are in our universe, it follows that the direction of time is given by the evolutionary process that climbs them one by one. Indeed, if this were not so, it would have been impossible to conceive how the features that we remark could have arisen. We should have to posit a miraculous beginning to time at which the features (and we among them) were created ready-made, and left to fall apart ever since into a tohubohu of individual particles.

Time in the large, open time, takes its direction from the evolutionary processes which mark and scale it. So it is pointless to ask why evolution has a fixed direction in time, and to draw conclusions from the speculation. It is evolution, physical and biological, that gives time its direction; and no mystical explanation is required when there is nothing to explain. The progression from simple to complex, the building up of stratified stability, is the necessary character of evolution from which time takes its direction. And it is not a forward direction in the sense of a thrust towards the future, a headed arrow. What evolution does is to give the arrow of time a barb which stops it from running backwards; and once it has this barb, the chance play of errors will take it forward of itself.

Taken from the book "Evolution Extended" edited by Connie Barlow

[edit] References

Evolution Extended, ISBN 0262522063

[edit] Related Articles

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