Thennodynamics

Many phenomena in solid-state physics can be understood by resort to energy band calculations. Conductivity trends, photoemission spectra, and optical properties can all be understood by examining the quantum states or energy bands of solids. In addition, electronic structure methods can be used to extract a wide variety of properties such as structural energies, mechanical properties and thennodynamic properties. 

Themiodynamics is a powerful tool in physics, chemistry and engineering and, by extension, to substantially all other sciences. However, its power is narrow, since it says nothing whatsoever about time-dependent phenomena. It can demonstrate that certain processes are impossible, but it caimot predict whether thennodynamically allowed processes will actually take place. 

It is important to recognize that thennodynamic laws are generalizations of experimental observations on systems of macroscopic size for such bulk systems the equations are exact (at least within the limits of the best experimental precision). The validity and applicability of the relations are independent of the correchiess of any model of molecular behaviour adduced to explain them. Moreover, the usefiilness of thennodynamic relations depends cmcially on measurability, unless an experimenter can keep the constraints on a system and its surroundings under control, the measurements may be worthless. 

First, a few definitions a system is any region of space, any amount of material for which the boundaries are clearly specified. At least for thennodynamic purposes it must be of macroscopic size and have a topological integrity. It may not be only part of the matter in a given region, e.g. all the sucrose in an aqueous solution. A system could consist of two non-contiguous parts, but such a specification would rarely be usefLil. 

To define the thennodynamic state of a system one must specify fhe values of a minimum number of variables, enough to reproduce the system with all its macroscopic properties. If special forces (surface effecls, external fields—electric, magnetic, gravitational, etc) are absent, or if the bulk properties are insensitive to these forces, e.g. the weak terrestrial magnetic field, it ordinarily suffices—for a one-component system—to specify fliree variables, e.g. fhe femperature T, the pressure p and the number of moles n, or an equivalent set. For example, if the volume of a surface layer is negligible in comparison with the total volume, surface effects usually contribute negligibly to bulk thennodynamic properties. 

For example, the definition of a system as 10.0 g FI2O at 10.0°C at an applied pressure p= 1.00 atm is sufficient to specify that the water is liquid and that its other properties (energy, density, refractive index, even non-thennodynamic properties like the coefficients of viscosity and themial condnctivify) are uniquely fixed. 

Redlich [3] has criticized the so-called zeroth law on the grounds that the argument applies equally well for the introduction of any generalized force, mechanical (pressure), electrical (voltage), or otherwise. The difference seems to be that the physical nature of these other forces has already been clearly defined or postulated (at least in the conventional development of physics) while in classical thennodynamics, especially in the Bom-Caratheodory approach, the existence of temperature has to be inferred from experiment. 

The work depends on the detailed path, so Dn is an inexact differential as symbolized by the capitalization. (There is no established convention about tliis symbolism some books—and all mathematicians—use the same symbol for all differentials some use 6 for an inexact differential others use a bar tln-ough the d still others—as in this article—use D.) The difference between an exact and an inexact differential is crucial in thennodynamics. In general, the integral of a differential depends on the path taken from the initial to the final state. Flowever, for some special but important cases, the integral is independent of the path then and only then can one write... 

As we shall see, because of the limitations that the second law of thennodynamics imposes, it may be impossible to find any adiabatic paths from a particular state A to another state B because In this... 

The surfaces in which the paths satisfying the condition = 0 must lie are, thus, surfaces of constant entropy they do not intersect and can be arranged in an order of increasing or decreasmg numerical value of the constant. S. One half of the second law of thennodynamics, namely that for reversible changes, is now established. 

Equation (A2.1.53) is frequently called the Clausius-Clapeyronequation, although this name is sometimes applied to equation (A2.1.52). Apparently Clapeyron first proposed equation (A2.1.52) in 1834, but it was derived properly from thennodynamics decades later by Clausius, who also obtained tlie approximate equation (A2.1.53).)... 

This new quantity Sv p, the negative of which De Bonder (1920) has called the affinity and given the symbol of a script A, is obviously the important thennodynamic fiinction for chemical equilibrium ... 

For a free energy of fonnation, the preferred standard state of the element should be the thennodynamically stable (lowest chemical potential) fonn of it e.g. at room temperature, graphite for carbon, the orthorhombic crystal for sulfiir. 

The usefid thennodynamic fiinctions (e.g. G, H, S, C, etc) are all state fiinctions, so their values in any particular state are independent of the path by which the state is reached. Consequently, one can combine (by... 

The validity of equation (A2.1.70) has sometimes been questioned when enthalpies of reaction detennined from calorimetric experiments fail to agree with those detennined from the temperature dependence of the equilibrium constant. The thennodynamic equation is rigorously correct, so doubters should instead examine the experunental uncertainties and whether the two methods actually relate to exactly the same reaction. 

For those who are familiar with the statistical mechanical interpretation of entropy, which asserts that at 0 K substances are nonnally restricted to a single quantum state, and hence have zero entropy, it should be pointed out that the conventional thennodynamic zero of entropy is not quite that, since most elements and compounds are mixtures of isotopic species that in principle should separate at 0 K, but of course do not. The thennodynamic entropies reported in tables ignore the entropy of isotopic mixing, and m some cases ignore other complications as well, e.g. ortho- and para-hydrogen. 

In the Lewis and Gibson statement of the third law, the notion of a perfect crystalline substance , while understandable, strays far from the macroscopic logic of classical thennodynamics and some scientists have been reluctant to place this statement in the same category as the first and second laws of thennodynamics. Fowler and Guggenheim (1939), noting drat the first and second laws both state universal limitations on processes that are experunentally possible, have pointed out that the principle of the unattainability of absolute zero, first enunciated by Nemst (1912) expresses a similar universal limitation ... 

Consider how the change of a system from a thennodynamic state a to a thennodynamic state (3 could decrease the temperature. (The change in state a —> f3 could be a chemical reaction, a phase transition, or just a change of volume, pressure, magnetic field, etc). Initially assume that a and (3 are always in complete internal equilibrium, i.e. neither has been cooled so rapidly that any disorder is frozen in. Then the Nemst heat... 

The principle of tire unattainability of absolute zero in no way limits one s ingenuity in trying to obtain lower and lower thennodynamic temperatures. The third law, in its statistical interpretation, essentially asserts that the ground quantum level of a system is ultimately non-degenerate, that some energy difference As must exist between states, so that at equilibrium at 0 K the system is certainly in that non-degenerate ground state with zero entropy. However, the As may be very small and temperatures of the order of As/Zr (where k is the Boltzmaim constant, the gas constant per molecule) may be obtainable. 

The microcanonical ensemble is a set of systems each having the same number of molecules N, the same volume V and the same energy U. In such an ensemble of isolated systems, any allowed quantum state is equally probable. In classical thennodynamics at equilibrium at constant n (or equivalently, N), V, and U, it is the entropy S that is a maximum. For the microcanonical ensemble, the entropy is directly related to the number of allowed quantum states C1(N,V,U) ... 

The grand canonical ensemble is a set of systems each with the same volume V, the same temperature T and the same chemical potential p (or if there is more than one substance present, the same set of p. s). This corresponds to a set of systems separated by diathennic and penneable walls and allowed to equilibrate. In classical thennodynamics, the appropriate fimction for fixed p, V, and Tis the productpV(see equation (A2.1.3 7)1 and statistical mechanics relates pV directly to the grand canonical partition function... 

Oyy/Ais of the order of hT, as is Since a macroscopic system described by themiodynamics probably has at least about 10 molecules, the uncertainty, i.e. the typical fluctuation, of a measured thennodynamic quantity must be of the order of 10 times that quantity, orders of magnitude below the precision of any current experimental measurement. Consequently we may describe thennodynamic laws and equations as exact . 

Wlien = N/2, the value of g is decreased by a factor of e from its maximum atm = 0. Thus the fractional widtii of the distribution is AOr/A i M/jV)7 For A 10 the fractional width is of the order of 10 It is the sharply peaked behaviour of the degeneracy fiinctions that leads to the prediction that the thennodynamic properties of macroscopic systems are well defined. 

In equilibrium statistical mechanics, one is concerned with the thennodynamic and other macroscopic properties of matter. The aim is to derive these properties from the laws of molecular dynamics and thus create a link between microscopic molecular motion and thennodynamic behaviour. A typical macroscopic system is composed of a large number A of molecules occupying a volume V which is large compared to that occupied by a molecule ... 

A statistical ensemble can be viewed as a description of how an experiment is repeated. In order to describe a macroscopic system in equilibrium, its thennodynamic state needs to be specified first. From this, one can infer the macroscopic constraints on the system, i.e. which macroscopic (thennodynamic) quantities are held fixed. One can also deduce, from this, what are the corresponding microscopic variables which will be constants of motion. A macroscopic system held in a specific thennodynamic equilibrium state is typically consistent with a very large number (classically infinite) of microstates. Each of the repeated experimental measurements on such a system, under ideal... 

If this condition is not satisfied, there is no unique way of calculating the observed value of ff, and the validity of the statistical mechanics should be questioned. In all physical examples, the mean square fluctuations are of the order of 1/Wand vanish in the thennodynamic limit. 

Snch a generalization is consistent with the Second Law of Thennodynamics, since the //theorem and the generalized definition of entropy together lead to the conchision that the entropy of an isolated non-eqnilibrium system increases monotonically, as it approaches equilibrium. 

Since Tis positive for systems in thennodynamic equilibrium,. S and hence log S should both be monotonically increasing fiinctions of E. This is the case as discussed above. 

Between two systems there can be a variety of interactions. Thennodynamic equilibrium of a system implies themial, chemical and mechanical equilibria. It is therefore logical to consider, in sequence, the following interactions between two systems thermal contact, which enables the two systems to share energy material contact, which enables exchange of particles between them and pressure transmitting contact, which allows an exchange of volume between the two systems. In each of the cases, the combined composite system is... 

One can trivially obtain the other thennodynamic potentials U, H and G from the above. It is also interesting to note that the internal energy U and the heat capacity Cy can be obtained directly from the partition fiinction. Since V) = 11 exp(-p , ), one has... 

This behaviour is characteristic of thennodynamic fluctuations. This behaviour also implies the equivalence of various ensembles in the thermodynamic limit. Specifically, as A —> oo tire energy fluctuations vanish, the partition of energy between the system and the reservoir becomes uniquely defined and the thennodynamic properties m microcanonical and canonical ensembles become identical. 

If //is 00 (very large) or T is zero, tire system is in the lowest possible and a non-degenerate energy state and U = -N xH. If eitiier // or (3 is zero, then U= 0, corresponding to an equal number of spins up and down. There is a synnnetry between the positive and negative values of Pp//, but negative p values do not correspond to thennodynamic equilibrium states. The heat capacity is... 

Once the partition function is evaluated, the contributions of the internal motion to thennodynamics can be evaluated. depends only on T, and has no effect on the pressure. Its effect on the heat capacity can be... 

The constant of integration is zero at zero temperature all the modes go to the unique non-degenerate ground state corresponding to the zero point energy. For this state S log(g) = log(l) = 0, a confmnation of the Third Law of Thennodynamics for the photon gas. 

In many experiments the sample is in thennodynamic equilibrium, held at constant temperature and pressure, and various properties are measured. For such experiments, the T-P ensemble is the appropriate description. In this case the system has fixed and shares energy and volume with the reservoir E = E + E" and V=V + V", i.e. the system... 

The coimection between the grand canonical ensemble and thennodynamics of fixed (V, T, p) systems is provided by the identification... 

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