Thermodynamics, statistical

By now we should be convinced that thermodynamics is a science of immense power. But it also has serious limitations. Our fifty million equations predict what — but they tell us nothing about why or how. For example, we can predict for water, the change in melting temperature with pressure, and the change of vapor fugacity with temperature or determine the point of equilibrium in a chemical reaction but we cannot use thermodynamic arguments to understand why we end up at a particular equilibrium condition. 

The reason is that classical thermodynamics tells us nothing about the atomic or molecular state of a system. We use thermodynamic results to infer molecular properties, but the evidence is circumstantial. For example, we can infer why a (hydrocarbon + alkanol) mixture shows large positive deviations from ideal solution behavior, in terms of the breaking of hydrogen bonds during mixing, but our description cannot be backed up by thermodynamic equations that involve molecular parameters. 

Statistical thermodynamics provides the relationships that we need in order to bridge this gap between the macro and the micro. Our most important application will involve the calculation of the thermodynamic properties of the ideal gas, but we will also apply the techniques to solids. The procedure will involve calculating U — Uo, the internal energy above zero Kelvin, from the energy of the individual molecules. Enthalpy differences and heat capacities are then easily calculated from the internal energy. Boltzmann s equation 

In this appendix, we will use statistical thermodynamics to relate the equilibrium constant of a chemical reaction to the microscopic properties of the molecules involved. Our strategy will be to define functions for the microscopic properties of molecules and then to connect these to the change in the free energy of a reaction. The relationship between this property and the equilibrium constant is well known. 

In addition to the relation between the energy of the system and the molecular partition functions, we must also consider the relationship between the entropy, S, and the number of configurations, which the molecules can take in this system. If we consider N molecules, the first of these can be arranged in N different ways, the second in (A—1) ways and we can continue in the same fashion until the Ath molecule, which can only be arranged in one way. Thus, the total number of arrangements is 

Using Stirling s approximation, that In jc w jc In x—jc, we obtain lnW =NlnN-N- 

The quantitative relationship between the statistical entropy and the molecular disorder is given by the Boltzmann relationship 

Each particular system within the ensemble can occupy the energy states Eq, E, E2, Ej, and the total energy of the state j of the system is given by 

This section is meant only as a compilation of some of the important results of statistical thermodynamics. No proof will be given.The fundamental quantity which relates the mechanical properties of molecules to the thermodynamic properties of a dilute gas is the molecular partition function q T) 

SO that knowledge of the partition functions determines the equilibrium constant. 

Another important result is that statistical thermodynamics allows calculation of the fraction of molecules,/, in a particular energy level. The expression is 

If this is summed over all energy levels it reduces to 

A fuller discussion can be found in texts on this subject, e.g., T. L. Hill, Introduction to Statistical Thermodynamics (Reading, Mass. Addison-Wesley, 1960), Chapters 4, 8, and 9. 

The goal here is to provide a systematic, if streamlined, derivation of the quantities of interest using statistical thermodynamics. These concepts are outside the range of topics usually considered in mechanical engineering or chemical engineering treatments of fluid flow. However, the results are essential for understanding and estimating the thermodynamic properties that are needed. 

The material covered in this chapter is self-contained, and is derived from well-known relationships such as Newton s second law and the ideal gas law. Some quantum mechanical results and the statistical thermodynamics definition of entropy are given without rigorous derivation. The end result will be a number of practical formulas that can be used to calculate thermodynamic properties of interest. 

Phenomenological thermodynamics describes changes in energies, temperatures, volumes, etc. Unless additional assumptions are made, however, it cannot give any information about the molecular phenomena that lie behind these processes. Statistical thermodynamics attempts to obtain such information through the use of probability functions. 

N is the number of segments in the chain and / is the segment length. Inserting equation (11-21) into the Boltzmann equation s, = JtlnQ we obtain the following for the entropy of a chain  

The constant serves to maintain dimensionality, and is, therefore, purely a factor of convenience (in the subsequent mathematical treatment it is canceled out). It is then assumed, in fact, that the coordinates of each individual segment change in the same proportion as the external coordinates of the test body. Thus if the external coordinates are extended by an elongation ratio a, the coordinate in the x direction of the i segment should be a times as large after elongation as the initial unextended value (x,- q), etc.  

Equation (11-24), together with equation (11-23) for the change in entropy of a segment becomes, for extension. 

The total change in entropy must be additive. With chains of equal length it then follows that 

The previous section is concerned mostly with an isolated molecule and its properties. When we have an ensemble of molecules in equilibrium with a particular volume and temperature, we find that the molecules are distributed among different energies and we are interested mostly in their average properties, which is the concern of statistical mechanics. 

If each molecule in an ensemble of molecules can exist in various quantized states with energies E, E2, E-i. then the probability pj that a molecule will be found in the state with energy Ej is given by the Boltzmann distribution  

Since the sum of all the probabilities must equal one, the normalization constant for the probabilities is l/ 2, so that the absolute value of the probability is 

Many thermodynamic functions can be derived from the partition function of the canonical ensemble by a weighted average, or by differentiation of the partition function. For instance, the average energy of the ensemble can be given by a weighted 

Let us consider a system of M ideal monatomic gas molecules in a cubic box kept at a constant temperature T. For a very dilute gas, where the molecules do not interact with one another, the quantum mechanical solution is a number of electronic wave functions with three quantum numbers nx, riy, and for the translational energies in three dimensions. The energy of a molecule for a set of quanmm numbers, the observed average energy, and the heat capacity at constant volume are given by 

The previous summary provides the basic relationships, derived from the first and second laws, used for the manipulation of available experimental data. However, statistical thermodynamics is then required to develop expressions for the thermodynamic properties in terms of the fluctuating quantities of interest here. First, we will use statistical thermodynamics to provide the characteristic thermodynamic potentials in terms of the appropriate partition function, which will involve a sum over the microscopic states available to the systan. Second, we will provide relevant expressions for the fluctuations nnder one set of variables, which can then be used to rationalize the thermodynamic properties of a system characterized by a different set of variables. 

There are four main ensembles in statistical thermodynamics for which the independent variables are NVE (microcanonical), NVT (canonical), NpT (Gibbs or isothermal isobaric), and VT (grand canonical). The characteristic fnnetions provided in Equations 1.2 and 1.3 can be expressed in terms of a series of partition functions such that (Hill 1956) 

Using Equation 1.4 and Equation 1.28 it can be shown that the volume and enthalpy in the Gibbs ensemble are given by 

Furthermore, using the above expressions in Equation 1.5 through Equation 1.7, and Equation 1.29, and then evaluating the derivatives leads to expressions for the compressibility, thermal expansion, and heat capacity. The results expressed in terms of fluctuations in the isothermal-isobaric ensemble are 

It is important to note that the above formulas represent fluctuations (8X=X - (X)) in the properties of the whole system, that is, bulk fluctuations. They are useful expressions but provide no information concerning fluctuations in the local vicinity of atoms or molecules. These latter quantities will prove to be most useful and informative. One can also derive expressions for partial molar quantities by taking appropriate first (to give the chemical potential) and second (to give partial molar volume and enthalpy) derivatives of the expressions presented in Equation 1.28. However, these do not typically lead to useful simple formulas that can be applied directly to theory or simulation. For instance, while it is straightforward to calculate the compressibility, thermal expansion, and heat capacity from simulation, the determination of chemical potentials is much more involved (especially for large molecules and high densities). 

So the whole set of equations can be solved in one stroke. Applications for this technology are the Brunauer - Emmet - Teller adsorption isotherm, various polymerization reactions and depolymerization reactions, and counting problems in statistical thermodynamics. 

Basically, it is believed that the properties of a macroscopic system can be derived from the properties and interactions of the microscopic particles that constitute the macroscopic system by a mechanical theory. 

Actually, the number of particles is too large that this can be achieved by strict application of the laws known in mechanics. Therefore, a statistical method has been developed. For example, the pressure of a gas is expressed as the average force per unit area exerted by its particles when they collide with the container walls. 

Thus far we have explored the field of classical thermodynamics. As mentioned previously, this field describes large systems consisting of billions of molecules. The understanding that we gain from thermodynamics allows us to predict whether or not a reaction will occur, the amount of heat that will be generated, the equilibrium position of the reaction, and ways to drive a reaction to produce higher yields. This otherwise powerful tool does not allow us to accurately describe events at a molecular scale. It is at the molecular scale that we can explore mechanisms and reaction rates. Events at the molecular scale are defined by what occurs at the atomic and subatomic scale. What we need is a way to connect these different scales into a cohesive picture so that we can describe everything about a system. The field that connects the atomic and molecular descriptions of matter with thermodynamics isknown as statistical thermodynamics. 

Statistical thermodynamics is based on a statistical interpretation of how atoms and molecules behave. This statistical nature arises because we have so many atoms and molecules in systems and because matter is intrinsically defined based on probabilities, which is the crux of all quantum mechanics. Rather than delve into the great details of statistical thermodynamics, which would far exceed the scope of this text, we will present its foundations only. 

The understanding of the distribution of energies in a system provides an important tool in describing many processes important in polymer science. For example, the rates of reactions, crystallization and degradation rely on energy distributions. 

The names and symbols given here are in agreement with those recommended by IUPAP [4] and by ISO [5.i]. 

In order to dissolve crystalline polymers, it is necessary to consider the Gibbs energy of fusion. This additional energy expenditure is not taken into account in the concept of the solubility parameter. Crystalline polymers therefore often dissolve only above their melting temperatures and in solvents with roughly the same solubility parameter. Unbranched, highly crystalline poly(ethylene) (62 = 8.0) only dissolves in decane (61 = 7.8) at temperatures close to the melting point of 135°C. 

The crystallinity of polymers is also responsible for the curious effect where a polymer at constant temperature first dissolves in a solvent and later, at the same temperature, precipitates out again. In these cases, the original polymer is of low crystallinity and therefore dissolves well. On dissolution, the chains become mobile. A crystalline polymer-solvent equilibrium is rapidly achieved with precipitation of polymer of higher crystallinity than the original material. 

As described in the previous chapter, the orientational distribution function corresponding to the single-molecule potential has the form 

There is one such self-consistency equation for each term L included in the potential Fi, Eq. [13]. In each of these equations one of the Pl appears on the left-hand side, and all the included  

The internal energy, entropy, and free energy are obtained in exactly the same way as computed in the simpler theory of the previous chapter  

Just as in the case of the simpler theory, the free energy is found to include additional terms beyond the usually expected InZj term. The form is correct, however, and arises because we have approximated the action of the pair potential 12 by the temperature-dependent single-molecule potential V. Note that setting the partial derivatives F/ d Pj T to zero regains the required self-consistency equations, Eq. [17]. Furthermore, testing Eqs. [18] and [20], we see that they do satisfy the required thermodynamic identity, E = (dpF/dp). 

The main physics of the problem, the character of the intermolecular interactions, is contained in the parameters Ui oi the one molecule potential Vi, In spite of their central importance in the theory, however, not all that much is known about their magnitude or volume dependence. Several attempts to estimate their properties have been made, but very little in the way of definitive results has been obtained. What is usually done in practice is to treat the Z/x, as simple volume-dependent functions with constants determined by fitting the theory to experimental results. 

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R. H. Fowler and E. A. Guggenheim, Statistical Thermodynamics, Cambridge University Press, London, 1939. 

S. A. Sairan, Statistical Thermodynamics of Surfaces, Interfaces and Membranes, Addison-Wesley, Reading, MA, 1994. 

S. Ross and I. D. Morrison, Colloidal Systems and Interfaces, Wiley, New York, 1988. S. A. Saffan, Statistical Thermodynamics of Surfaces, Interfaces and Membranes, Addison-Wesley, Reading, MA, 1994. 

Statistical Thermodynamics of Adsorbates. First, from a thermodynamic or statistical mechanical point of view, the internal energy and entropy of a molecule should be different in the adsorbed state from that in the gaseous state. This is quite apart from the energy of the adsorption bond itself or the entropy associated with confining a molecule to the interfacial region. It is clear, for example, that the adsorbed molecule may lose part or all of its freedom to rotate. 

The preceding derivation, being based on a definite mechanical picture, is easy to follow intuitively kinetic derivations of an equilibrium relationship suffer from a common disadvantage, namely, that they usually assume more than is necessary. It is quite possible to obtain the Langmuir equation (as well as other adsorption isotherm equations) from examination of the statistical thermodynamics of the two states involved. 

Thus from an adsorption isotherm and its temperature variation, one can calculate either the differential or the integral entropy of adsorption as a function of surface coverage. The former probably has the greater direct physical meaning, but the latter is the quantity usually first obtained in a statistical thermodynamic adsorption model. 

In general, it seems more reasonable to suppose that in chemisorption specific sites are involved and that therefore definite potential barriers to lateral motion should be present. The adsorption should therefore obey the statistical thermodynamics of a localized state. On the other hand, the kinetics of adsorption and of catalytic processes will depend greatly on the frequency and nature of such surface jumps as do occur. A film can be fairly mobile in this kinetic sense and yet not be expected to show any significant deviation from the configurational entropy of a localized state. 

Mason E A and Spurling T H 1969 The VIrlal Equation of State (Oxford Pergamon) McQuarrie D A 1973 Statistical Thermodynamics (Mill Valley, CA University Science Books)... 

A2.1.7.4 THIRD STATEMENT (SIMPLE LIMITS AND STATISTICAL THERMODYNAMICS)... 

By the standard methods of statistical thermodynamics it is possible to derive for certain entropy changes general formulas that cannot be derived from the zeroth, first, and second laws of classical thermodynamics. In particular one can obtain formulae for entropy changes in highly di.sperse systems, for those in very cold systems, and for those associated, with the mixing ofvery similar substances. 

Chandler D 1987 Introduction to Modern Statistical Mechanics (Oxford Oxford University Press) Hill T L 1960 Introduction to Statistical Thermodynamics (Reading, MA Addison-Wesley)... 

Ben-Naim A 1992 Statistical Thermodynamics for Chemists and Biologists (London Plenum)... 

It is curious that he never conuuented on the failure to fit the analytic theory even though that treatment—with the quadratic fonn of the coexistence curve—was presented in great detail in it Statistical Thermodynamics (Fowler and Guggenlieim, 1939). The paper does not discuss any of the other critical exponents, except to fit the vanishing of the surface tension a at the critical point to an equation... 

Keizer J 1987 Statistical Thermodynamics of Nonequilibrium Processes (New York Springer)... 

Lavenda B H 1985 Nonequilibrium Statistical Thermodynamics (New York Wley) oh 3... 

Lavenda B FI 1985 Nonequilibrium. Statistical Thermodynamics (New York Wiley)... 

Gilson et al., 1997] Gilson, M., Given, J., Bush, B., and McCammon, J. The statistical-thermodynamic basis for computation of binding affinities A critical review. Biophys. J. 72 (1997) 1047-1069... 

Statistical thermodynamics tells us that Cv is made up of four parts, translational, rotational, vibrational, and electronic. Generally, the last part is zero over the range 0 to 298 K and the first two parts sum to 5/2 R, where R is the gas constant. This leaves us only the vibrational part to worry about. The vibrational contr ibution to the heat capacity is... 

From the third law of thermodynamics, the entiopy 5 = 0 at 0 K makes it possible to calculate S at any temperature from statistical thermodynamics within the hamionic oscillator approximation (Maczek, 1998). From this, A5 of formation can be found, leading to A/G and the equilibrium constant of any reaction at 298 K for which the algebraic sum of AyG for all of the constituents is known. A detailed knowledge of A5, which we already have, leads to /Gq at any temperature. Variation in pressure on a reacting system can also be handled by classical thermodynamic methods. 

Irikura, K. K. Essential Statistical Thermodynamics in Computational Thermochemistry, In Irikura, K. K. Erurip, D. J. Eds., 1998. Computational Thermochemistry. American Chemical Society, Washington, DC. 

K (66.46 e.u.) with the spectroscopic value calculated from experimental data (66.41 0.009 e.u.) (295, 289) indicates that the crystal is an ordered form at 0°K. Thermodynamic functions of thiazole were also determined by statistical thermodynamics from vibrational spectra (297, 298). 

The results of a comparison between values of n estimated by the DRK and BET methods present a con. used picture. In a number of investigations linear DRK plots have been obtained over restricted ranges of the isotherm, and in some cases reasonable agreement has been reported between the DRK and BET values. Kiselev and his co-workers have pointed out, however, that since the DR and the DRK equations do not reduce to Henry s Law n = const x p) as n - 0, they are not readily susceptible of statistical-thermodynamic treatment. Moreover, it is not easy to see how exactly the same form of equation can apply to two quite diverse processes involving entirely diiferent mechanisms. We are obliged to conclude that the significance of the DRK plot is obscure, and its validity for surface area estimation very doubtful. 

It is not particularly difficult to introduce thermodynamic concepts into a discussion of elasticity. We shall not explore all of the implications of this development, but shall proceed only to the point of establishing the connection between elasticity and entropy. Then we shall go from phenomenological thermodynamics to statistical thermodynamics in pursuit of a molecular model to describe the elastic response of cross-linked networks. 

In another sense the title is too restrictive, implying that only pure, phenomenological thermodynamics are discussed herein. Actually, this is far from true. Both thermodynamics and statistical thermodynamics comprise the contents of the chapter, with the second making the larger contribution. But the term statistical is omitted from the title, as it is too intimidating. 

The treatment of heat capacity in physical chemistry provides an excellent and familiar example of the relationship between pure and statistical thermodynamics. Heat capacity is defined experimentally and is measured by determining the heat required to change the temperature of a sample in, say,... 

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