The Third Law of Thermodynamics

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Also known as Nernst=s heat theorem, it was devised by Walther Hermann Nernst (1864 Jan 25 - 1941 Nov 18), a German chemist who presented it on 1905 Dec 23 at a meeting of the Königliche Gesellschaft der Wissenschaften zu Göttingen. Nernst stated it as, AThe entropy change in a reaction between pure substances approaches zero at T=0K.@ Physicists broadened it to state, AThe entropy of any physical system vanishes in the state for which the absolute temperature equals zero.@ They then restated it in terms of its chief consequence, AThere exists no thermodynamic system whose operation can reduce the temperature of any body to absolute zero in a finite number of steps.@ Now we want to add that theorem to the Map of Physics in an appropriately deductive manner. To that end let=s first quickly review the fundamental laws of thermodynamics that come before it.

The Zeroth Law

This one sounds like Euclid=s first common notion (Things equal to the same thing are equal to each other): if two physical systems are in thermodynamic equilibrium with a third system, then they are in equilibrium with each other. We take equilibrium to mean that if two systems are in equilibrium with each other, then those systems will not change if we connect them to each other in a way that would allow change. For example, if we connect two gas-filled containers with a thin tube and no gas flows through the tube, then the containers are in equilibrium, which means that the pressures in them are equal. If we bring two heated bodies into physical contact with each other and no heat flows from one to the other, then the bodies are in equilibrium with each other, which means that their temperatures are equal. Thus, we can determine that two systems are in equilibrium if we find that their accessible intensive parameters (pressure, temperature, chemical potential, etc.) are equal, one system=s to the other=s.

The First Law

This one necessitates conservation of energy. We usually express it by saying that no thermodynamic process can create energy ex nihilo or destroy energy, but can only transform energy or transfer it. In thermodynamic systems the various forms in which energy comes manifest consist of the intensive parameters taken conjointly with their associated extensive parameters. Thus, pressure multiplied by volume corresponds to the energy of compressing a gas; temperature multiplied by entropy corresponds to the energy of heating a material; and so on.

Fundamentally, energy denumerates a relationship between two bodies. Kinetic energy measures relative motion between bodies and potential energy measures force exerted between bodies. If we have a system in which all of the component particles are subject to the minimum forces possible in the system and all motions of the particles have ceased, then that system occupies a state of zero thermodynamic energy; that is, except for the energy bound up in the rest masses of the particles, the system contains zero energy. The system cannot go into a state of lower energy because negative energy cannot exist: to do so would require the existence of negative mass or imaginary velocity, neither of which we admit into physics as part of a correct description of Reality.

The Second Law

In its simplest form the law of entropy states that heat cannot, of itself, go from a colder to a hotter body. What constitutes hotter and colder involves the concept of temperature and its relationship with entropy. The state in which a system contains zero energy gives us a natural state of zero hotness. Because we sense the hotness of bodies through touch, we associate hotness with the form of energy that bodies exchange through physical contact (randomized kinetic energy) and call that form of energy heat. We denote the degree of a body=s hotness with the word temperature.

The temperature of a body has a value at every point in the body. We know that fact stands true to Reality because the body must have a degree of hotness everywhere something (even itself) can touch it and because the hotness of a body can differ from place to place on the body (think of a block of iron touching fire on one side and ice on its opposite side). Thus we know that temperature is an intensive property of thermodynamic systems. We associate temperature with heat, so we infer the existence of an extensive property of the system, denote it by S, and call it entropy. Just as pressure and volume combine to do work,

(Eq=n 1)

so temperature and entropy combine in heat,

(Eq=n 2)

If we have two systems at different temperatures, then an increment of heat dQ going from the hotter body to the colder body will diminish the entropy of the hotter body less than it will increase the entropy of the colder body, in accordance with

(Eq=n 3)

Thus the overall entropy of the system increases. If we isolate this combined system from all external thermodynamic processes, then heat will flow from the hotter body to the colder body until both bodies have the same temperature. In that final state the system will have its maximum possible entropy. So Clausius= version of the second law has as its chief consequence the statement that any thermodynamic system left to itself will evolve toward the state that maximizes its entropy.

The Third Law

Heat flows naturally from hotter to colder, but we can move heat from a colder body to a hotter body if we do work upon the system containing those bodies. To achieve that end we use a refrigerator, a mechanism that draws heat out of a cold body at that body=s temperature, does work upon the part of the system into which the heat went to add energy to that part of the system and thereby raise its temperature, and then expels the heat into a hot body. That is, the refrigerator acts as a heat pump, drawing heat Qc from the cold body at the temperature Tc, adding work W, and expelling heat Qh=Qc+W into the hot body at the temperature Th. The second law requires that, at a minimum, the entropy of the system must change by zero (though in real systems ΔS>0): that statement defines an ideal refrigerator. Using an ideal refrigerator, we have, in light of Equation 3,

(Eq=n 4)

That equation tells us that as Tc approaches absolute zero the work W must go toward infinity. Of course, nothing can do an infinite amount of work, but we can imagine breaking the process up into smaller steps. We still can=t do an infinite amount of work, but now we can say that no thermodynamic process can reduce the temperature of any body to absolute zero in a finite number of steps, which restates the popular form of Nernst=s heat theorem.

How does that relate to the way in which Nernst stated his theorem? In 1907 he stated the theorem as, AAny entropy changes in an isothermal reversible process approach zero as the temperature approaches zero.@ We can use that statement to guide us from the popular form of Nernst=s theorem to his original formulation.

The ideal refrigerator works through a reverse Carnot cycle, which operates in four discrete steps:

1. We put an expandable container filled with pressurized Ćtherium (an imaginary gas that will never, under any circumstances, condense into a liquid) into contact with the colder body and allow the gas to expand isothermally (that is, the process occurs at the single temperature Tc), drawing heat out of the colder body;

2. We disconnect the container from the colder body and compress the gas adiabatically (isentropically). The work that we do upon the gas turns into heat to keep the entropy of the gas unchanged as the temperature increases;

3. We put the container into contact with the hotter body and compress the gas isothermally, driving heat into the hotter body;

4. We disconnect the container from the hotter body and allow the gas to expand adiabatically, the work done by the gas reducing the heat in the gas to keep its entropy unchanged as the temperature decreases; and

5. We repeat the cycle.

If we draw a description of the gas on a graph that plots pressure against volume, then the isotherms, the lines representing processes that occur at an unchanging temperature, comprise a set of non-intersecting curves that descend from the graph=s upper left, curve away from the origin of the graph, and extend to the lower right. The adiabats, the lines representing processes that occur without changing the gas= s entropy, comprise a set of non-intersecting curves that descend more steeply and cross the isotherms. Where two adiabats cross two isotherms the four line segments defined by the intersections form a figure that represents a Carnot cycle: the area enclosed by the figure represents the amount of work that the system must do in order to pull a certain amount of heat out of a body at the temperature of the lower isotherm and drive it into a body at the temperature of the upper isotherm. As we go to ever colder isotherms (as the colder body=s temperature goes down) the relative curvatures of the isotherms and of the adiabats must change in a way that makes that work increase in accordance with Equation 4. If the lower isotherm represents processes occurring at T=0, then the work that the system does in the Carnot cycle must go to infinity. That fact necessitates that the adiabat on the left side of the diagram, the one representing the lower of the two entropies at which the system operates, must coincide with the isotherm at T=0. That adiabat can never cross another isotherm, because if it did, it could represent part of a process that could drive a body to absolute zero with a finite amount of work. Thus, any isothermal process carried out at T=0 (represented by motion of a point on the isotherm) will not change the entropy of the system that carries it out, which corresponds to what Nernst said.

Further, the adiabat that coincides with the isotherm at T=0 represents the minimum possible entropy that the system can have: any lower entropy would correspond to negative temperatures, which cannot exist. The definition of entropy, encoded in Equation 3, determines the value of a system=s entropy to within an arbitrary additive constant. Having the freedom to decide that constant=s value, we set it by stating that at T=0 all systems have zero entropy. We can then calculate a system=s entropy at any temperature by starting with S=0 at T=0 and integrating up the temperature scale. Thus we have Nernst=s heat theorem, the third law of thermodynamics, in full.

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