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Elements of Statistical Thermodynamics
By Leonard K. Nash
Dover Publications, Inc.Copyright © 1974 Leonard K. Nash
All rights reserved.
The Statistical Viewpoint
In every change, however drastic it may appear, we surmise a "something" that remains constant. From the very beginning of the modern era, certain men (e.g., Descartes) have conceived that "something" in terms suggestive of what we would call energy. And energy—or, better, mass-energy-is surely conceived by us as a "something constant" enduring through all change. The energy concept thus gives quantitative expression to our firm conviction that "plus ça change, plus c'est la même chose." But we have too another conviction scarcely less intense: the conviction that the future will not repeat the past, that time unrolls unidirectionally, that the world is getting on. This second conviction finds quantitative expression in the concept of entropy (from Gr. en, in + trope, turning). By always increasing in the direction of spontaneous change, entropy indicates the "turn," or direction, taken by all such change.
From a union of the entropy and energy concepts, little more than a century ago, there was born a notably abstract science with innumerable concrete applications; a science of thermodynamics that combines magnificent generality with unfailing reliability to a degree unrivaled by any other science known to man. Yet, for all its immense power, thermodynamics is a science that fails to reward man's quest for understanding. Yielding impressively accurate predictions of what can happen, thermodynamics affords us little or no insight into the why of those happenings. Thus it permits us to calculate what is the position of equilibrium in the system N2—H2—NH3, for example, but it fails entirely to tell us why that is the equilibrium condition for this specific system.
To be sure, given that certain thermodynamic parameters (the "free energies") are what they are, we readily see that a particular equilibrium condition is entailed. But we can find in thermodynamics no explanation of why the free energies are what they are. And in general, though thermodynamics teaches us to see important relations among the various macroscopic properties of a substance, so that many can be calculated from experimental measurements of a few, thermodynamics is powerless to produce from its own calculations numerical values for the few.
What is it about NH3 that determines the magnitude of the free-energy characteristic of that compound? In principle this question should, we feel, be answerable. But we find scant prospect of any such answer in a classical thermodynamics which, focusing solely on the properties of matter in bulk, eschews all concern with the microcosmic constitution of matter. For consider that we can hope to explain the free energy of some substance only by showing how that particular free energy is entailed by the distinctive values of the atomic and/or molecular parameters of the substance. That is, given a (spectroscopic) determination of such parameters as the length, angle, and flexibility of the bonds in NH3, we must be able to see that the free energy of NH3 could not be other than it is. This will be possible only if we can bridge the gap between the microcosmic realm of atoms and molecules and the macroscopic realm of classical thermodynamics.
Statistical mechanics provides such a bridge, by teaching us how to conceive a thermodynamic system as an assembly of units. More specifically, it demonstrates that the thermodynamic parameters of the system are interpretable in terms of—and are indeed calculable from—the parameters descriptive of such constituent units as atoms and molecules. In a bounded system, the crucial characteristic of these microcosmic units is that their energies are "quantized." That is, where the energies accessible to a macroscopic system form a virtual continuum of possibilities, the energies open to any of its submicroscopic components are limited to a discontinuous set of alternatives associated with integral values of some "quantum number."
Perhaps the most familiar example of what is meant by quantization is presented by the Bohr interpretation of the hydrogen emission spectrum. This spectrum consists of a series of sharp "lines," characterized by particular wavelengths. Each of these lines is supposed to arise in the emission by the hydrogen atom of an energy packet of some particular size. Such an energy packet is emitted when the atom passes from a state of higher energy to one of lower energy. From a study of the sizes of the emitted energy packets, one infers that the atom can exist only in a certain well-defined set of quantum states. The energy (εH) associated with any of these permissible states is given by the equation:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Here h symbolizes Planck's universal constant, m and e respectively represent the mass and charge of the "orbital" electron in the hydrogen atom, and n is a quantum number that can assume any ITL∫ITL value within the range 1 to oo. The possible states of the hydrogen atom, each characterized by some integral value of the quantum number n, are thus linked with the discontinuous set of permissible energies given by the last equation— which expresses the energy-quantization condition for the hydrogen atom. Rather more complicated relations, involving additional quantum numbers, express analogous energy-quantization conditions applicable to other species of gaseous atoms.
Like atoms, molecules also can exist only in particular sets of states characterized by different electronic configurations, with which are associated correspondingly restricted series of permissible energy states. But, unlike atoms, molecules exhibit fully quantized modes of energy storage other than that represented by electronic excitation. For example, when in any given electronic state, a molecule may perform various vibrational motions. A study of molecular spectra indicates that, when the vibration can be approximated as a harmonic oscillation, the only permissible values of the vibrational energy (ευ) are given by the equation
[member of]υ (υ + 1/2)hv.
Here v is a frequency characteristic of the particular vibration involved, and υ is a quantum number that can assume any ITL∫ITL value within the range 0 to oo. The possible vibrational states, each specified by some distinctive integral value of the vibrational quantum number v, are thus linked by the last equation with an evenly spaced set of quantized vibrational energies. The rotational motions of molecules, and the translational motions of both atoms and molecules, are similarly associated with sets of discrete quantum states—to which correspond similarly discontinuous series of permissible rotational and translational energies.
We seek then to view a macroscopic thermodynamic system as an assembly of myriad submicroscopic entities in myriad ever-changing quantum states. This may at first seem a completely hopeless pretension. For how can we possibly hope to give any account of an assembly that, if it contains just one mole of material, contains no less than 6 × 1023 distinct units? Even a three-body problem defies solution in a completely analytical form; yet we face a 6 × 1023-body problem. Actually, just because of the enormous numbers involved, this problem proves unexpectedly tractable when we give it a statistical formulation. From a consideration of assemblies of quantized units, in the next section, we develop three propositions that will prove useful in our statistical analysis. Observe that our concern here is purely mathematical, and that we could instead obtain the desired propositions by considering, say, in how many different ways a number of counters can be distributed over the squares of a gameboard.
MICROSTATES AND CONFIGURATIONS
For simplicity, let us consider first an assembly of identical units, localized in space, with permissible quantum states that are associated with an evenly spaced set of energies. An assembly meeting these specifications might be an array of identical one-dimensional harmonic oscillators occupying various fixed positions in a schematic crystal lattice. We stipulate localization of the oscillators so that, their identity notwithstanding, each will be rendered distinguishable in principle by its unique geometric placement. We stipulate identity of the oscillators so that, in the energy-quantization law ευ = (v + 1/2)hv, the characteristic frequency v will be the same for any among all the oscillators concerned. The quantum states of any such oscillator can then be depicted as shown in Fig. 1. Since all that concerns us is the spacing of these levels, for convenience we have chosen to make our reference zero of energy coincident with the energy of the lowest possible quantum state. That is, for this so-called "ground" state with υ = 0, we now write (ε0 = 0. The energy quantum hv represents the constant margin by which each of the higher ("excited") states surpasses in energy the state immediately below it. To bring any oscillator from its ground state to an excited state characterized by some integral value of v, we need only add v quanta with energy hv.
Let us begin with a very simple assembly of three localized oscillators which share three quanta of energy. In how many ways can these three identical quanta be distributed among the three distinguishable oscillators? The ten possible distributions are indicated in Fig. 2—in which the dots are so placed that the letter markings along the abscissa indicate the particular oscillator concerned, and the number of energy quanta assigned to it can be read from the ordinate. Each of the ten detailed distributions we call a microstate, and it is easy to see that the ten microstates fall in the three groups indicated by Roman numbers. That is, all ten are simply variants of the three basic configurations shown in Fig. 3. In configuration I all three energy quanta are assigned to one oscillator, no quanta to the remaining two oscillators, and three microstates develop from this configuration according to whether the three-quantum packet is assigned to oscillator a or to b or to c. In configuration II two quanta are assigned to some one oscillator, one quantum to a second oscillator, no quanta to the third oscillator; and, as indicated in Fig. 2, there are six distinguishable ways in which such assignments can be made. In configuration III one quantum is assigned to each of the three oscillators, and it is evident that there can be but one microstate associated with this configuration.
How shall we obtain a systematic count of all the microstates associated with any given configuration? To arrive at the requisite formula, return again to configuration II. Observe that we can assign the first (two-quantum) parcel of energy to any one of three oscillators; having done so, we can assign the second (one-quantum) parcel to either of the two remaining oscillators; there then remains but one oscillator to which we assign the third (nil) parcel. The total number of ways in which the assignments can be made is thus 3 · 2 · 1 = 3! (i.e., "three factorial")—which, indeed, duly represents the 6 microstates associated with configuration II. Turning next to configuration I, we have again three choices in assigning the first (three-quantum) parcel, two choices when we assign the second (nil) parcel, and one choice when we assign the third (nil) parcel. But observe that, the last two parcels being the same, the final distribution is independent of the order in which we assign them. Whether, say, we assign the second parcel to oscillator b and the third parcel to oscillator c, or vice versa, the two verbally distinguishable orders result in precisely the same final microstate. That is, 2 · 1 = 2! verbally distinguishable assignments collapse into 1 microstate because the two oscillators wind up in the same (v = 0) quantum level. Hence the total number of microstates associated with configuration I is not 3! but rather 3!/2! = 3. The same kind of shrinkage of possibilities is seen in even more extreme form in configuration III. Here there is triple occupancy of the same (v = 1) quantum level, and the 3! verbally distinguishable assignments collapse into one and the same final microstate. The number of microstates associated with configuration III is then simply 3!/3! = 1.
By extending this style of analysis, we can easily extract a general formula abundantly useful in more difficult cases. Consider an assembly of some substantial number (N) of localized harmonic oscillators. In how many different ways can we distribute among these oscillators the particular set of energy parcels (including nil parcels) characteristic of the configuration in question? We have N choices of the oscillator to which we assign the first parcel, (N – 1) choices in assigning the second, and so on—representing a total of (N)(N – 1)(N – 2) ... (1) = N! distinguishable possibilities if no two of the energy parcels are the same. If, on the other hand, some number (ηα) of the parcels are the same, we can obtain only N!/ ηα! distinct microstates; if ηa of the parcels are of one kind and ηb of some other one kind, we can obtain only N!/(ηa!)(ηb!) microstates, and so on. The general conclusion is now quite clear. Symbolizing by W the total number of microstates associated with any configuration involving N distinguishable units, we can write:
W = N!/(ηa!)(ηb!) ...,
where ηa represents the number of units assigned the same number of energy quanta (and, hence, occupying the same quantum level), ηb represents the number of units occupying some other one quantum level, ...
The last equation can be represented more compactly as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)
where the symbol Π instructs us to make a continuing product (even as the symbol Δ instructs us to make a continuing sum) extended over all terms of the form following the symbol, and each of the ηn terms represents the number of units resident in each of the populated quantum levels. Observe that, though we arrived at equation (1) by considering assemblies of harmonic oscillators, with uniform energy spacing between their quantum levels, the actual argument is wholly independent of the supposition of uniformity. Equation (1) is a general relation, equally applicable to any species of distinguishable unit with any energy spacing between its quantum levels. As indicated below, straightforward. multiplication of the expanded factorials suffices to establish the number of microstates associated with any configuration for which N is small (< 10). For medium-sized values of N (10 to 1000), one can use tabulated values of N! in evaluating W. For very large values of N, we can follow neither of these courses. But, precisely in the limit of large N, an excellent value for N!—or, rather, the natural logarithm of N! which we symbolize as ln N!—is supplied by the simplest form of Stirling's approximation,
ln N! = N ln N - N. (2)
With equation (1) in hand we can make short work of two additional simple examples. Consider that 5 energy quanta are shared among 5 oscillators. The possible configurations, and the number of microstates associated with each of them, are shown in Fig. 4. Note that even a slight increase in the number of units (and quanta) has produced a sharp increase in the total number of microstates = [summation]Wi = 126.
As a last example, consider an assembly in which the number of energy quanta is not equal to the number of units present: suppose that 5 energy quanta are shared among 10 oscillators. The possible configurations of this assembly are easily obtained by adding 5 units to the ground level in each of the configurations shown in Fig. 4—with the results shown in Fig. 5. The calculation of the number of microstates associated with each configuration is given in extenso, to call attention to a simple method we will use repeatedly in handling factorial ratios:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Note that, by doubling the number of units, we have produced close to a ninefold increase in the total number of microstates = [summation]Wi = 2002.
As the number of units increases further, the total number of microstates skyrockets to unimaginable magnitudes. Thus one can calculate that an assembly of 1000 localized harmonic oscillators sharing 1000 energy quanta possesses more than 10600 different microstates. This is an unimaginable magnitude: our entire galaxy contains fewer than 1070 atoms. Even the estimated total number of atoms in the entire universe is as nothing in comparison with 10600. And though we can offer a compact expression for the total number of microstates that can be assumed by 6 × 1023 oscillators sharing an equal number of energy quanta, that number ([equivalent] 101023) is essentially meaningless, inconceivably immense.
Excerpted from Elements of Statistical Thermodynamics by Leonard K. Nash. Copyright © 1974 Leonard K. Nash. Excerpted by permission of Dover Publications, Inc..
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