Thermodynamic Properties from Statistical Thermodynamics

Now that we have established the complete form of our partition function, how can we determine thermodynamic properties from it We will start with energy. The total energy of the ensemble is given by equation 17.10  

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Consider briefly the derivative of equation 17.32 with respect to temperature  

According to the rules of calculus, the left side of the above equation is (a In q/dT). Moving the kT term to the left side, we have 

The right side of this equation is most of the right side of equation 17.33. Substituting  

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This volume also contains four appendices. The appendices give the mathematical foundation for the thermodynamic derivations (Appendix 1), describe the ITS-90 temperature scale (Appendix 2), describe equations of state for gases (Appendix 3), and summarize the relationships and data needed for calculating thermodynamic properties from statistical mechanics (Appendix 4). We believe that they will prove useful to students and practicing scientists alike. 

Statistical mechanics relates macroscopic or bulk properties to the properties of the smaller units which make up the macroscopic system. The smaller units are usually atoms and molecules, though they need not be. In favorable cases we can reliably calculate bulk properties from statistical models. When the discipline is restricted to equilibrium properties, it is referred to as statistical thermodynamics. The relation between thermod3mamics and statistical thermodynamics touches upon the philosophical issues of inter-theoretic reduction. We should examine the chemical applications of statistical thermodjmamics before we discuss the nature of the relation between the two subjects. 

The drawback of the statistical approach is that it depends on a model, and models are bound to oversimplify. Nevertheless, we can learn a great deal from the attempt to evaluate thermodynamic properties from molecular models, even if the effort falls short of quantitative success. 

The most common states of a pure substance are solid, liquid, or gas (vapor), state property See state function. state symbol A symbol (abbreviation) denoting the state of a species. Examples s (solid) I (liquid) g (gas) aq (aqueous solution), statistical entropy The entropy calculated from statistical thermodynamics S = k In W. statistical thermodynamics The interpretation of the laws of thermodynamics in terms of the behavior of large numbers of atoms and molecules, steady-state approximation The assumption that the net rate of formation of reaction intermediates is 0. Stefan-Boltzmann law The total intensity of radiation emitted by a heated black body is proportional to the fourth power of the absolute temperature, stereoisomers Isomers in which atoms have the same partners arranged differently in space, stereoregular polymer A polymer in which each unit or pair of repeating units has the same relative orientation, steric factor (P) An empirical factor that takes into account the steric requirement of a reaction, steric requirement A constraint on an elementary reaction in which the successful collision of two molecules depends on their relative orientation. 

In classical molecular dynamics, on the other hand, particles move according to the laws of classical mechanics over a PES that has been empirically parameterized. By means of their kinetic energy they can overcome energetic barriers and visit a much more extended portion of phase space. Tools from statistical mechanics can, moreover, be used to determine thermodynamic (e.g. relative free energies) and dynamic properties of the system from its temporal evolution. The quality of the results is, however, limited to the accuracy and reliability of the (empirically) parameterized PES. 

A method for the estimation of thermodynamic properties of the transition state and other unstable species involves analyzing parts of the molecule and assigning separate properties to functional groups (Benson, 1976). Another approach stemming from statistical mechanics is outlined in the next section. 

Exploiting the principles of statistical mechanics, atomistic simulations allow for the calculation of macroscopically measurable properties from microscopic interactions. Structural quantities (such as intra- and intermolecular distances) as well as thermodynamic quantities (such as heat capacities) can be obtained. If the statistical sampling is carried out using the technique of molecular dynamics, then dynamic quantities (such as transport coefficients) can be calculated. Since electronic properties are beyond the scope of the method, the atomistic simulation approach is primarily applicable to the thermodynamics half of the standard physical chemistry curriculum. 

We will delay a more detailed discussion of ensemble thermodynamics until Chapter 10 indeed, in this chapter we will make use of ensembles designed to render the operative equations as transparent as possible without much discussion of extensions to other ensembles. The point to be re-emphasized here is that the vast majority of experimental techniques measure molecular properties as averages - either time averages or ensemble averages or, most typically, both. Thus, we seek computational techniques capable of accurately reproducing these aspects of molecular behavior. In this chapter, we will consider Monte Carlo (MC) and molecular dynamics (MD) techniques for the simulation of real systems. Prior to discussing the details of computational algorithms, however, we need to briefly review some basic concepts from statistical mechanics. 

Consider the equilibrium between reactants A, B,.. . and products M, N,.. . in the gas phase as well as in solution in a given solvent S (Figure 2.2). The equilibrium constant in the gas phase, K%, depends on the properties of the reactants and the products. In very favourable cases it can be estimated from statistical thermodynamics via the relevant partition functions, but for the present purposes it is regarded as given. The problem is to estimate the magnitude of the equilibrium constant in the solution, A , and how it changes to KP-, when solvent Su is substituted for solvent St. 

Statistical mechanics is often thought of as a way to predict the thermodynamic properties of molecules from their microscopic properties, but statistical mechnics is more than that because it provides a complementary way of looking at thermodynamics. The transformed Gibbs energy G for a biochemical reaction system at specified pH is given by... 

The main objective of statistical mechanics is to calculate macroscopic (thermodynamic) properties from a knowledge of microscopic information like quantum mechanical energy levels. The purpose of the present appendix is merely to present a selection1 of the results that are most relevant in the context of reaction dynamics. 

Gibbs also wrote an influential book on statistical mechanics, which developed a molecular theory of thermodynamic properties from first principles, with a treatment that was general enough to accommodate quantum mechanics, a theory that had not even been imagined yet. He championed the now standard use of vector notation over William Rowan Hamilton s quaternion algebra and wrote several seminal papers on electromagnetism in the 1880s that supported Maxwell s theory, see also Equilibrium Maxwell, James Clerk Thermodynamics. 

In addition to a general knowledge of laboratory techniques, creative research work requires the ability to apply two different kinds of theory. Many an experimental method is based on a special phenomenological theory of its own this must be well rmderstood in order to design the experiment properly and in order to calculate the desired physical property from the observed raw data. Once the desired result has been obtained, it is necessary to understand its significance and its interrelationship with other known facts. This requires a sound knowledge of the fundamental theories of physical chemistry (e g., thermodynamics, statistical mechanics, qrrantrrm mechanics, and kinetics). Cortsider-able emphasis has been placed on both kirrds of theory in this book. 

In a similar manner, many of the classical thermodynamic relationships and properties can be derived by statistical techniques. Thus, the macro thermodynamic properties are derivable from statistical concepts. 

In addition, many thermodynamic properties can be estimated taking clues from the molecular properties and applying concepts from statistical thermodynamics. 

In order to measure temperatures above 4000°K., it is often convenient to consider the properties of the molecules, atoms, and charged particles in the system. Since the laws of thermodynamics apply only to systems that are macroscopic, it is desirable to find another definition of temperature which can be based on measured microscopic properties of these particles. One definition, supplied from statistical mechanics is temperature is the measure of broadness of a certain kind of distribution (F6). 

Two important objectives of statistical mechanics are (1) to verify the laws of thermodynamics from a molecular viewpoint and (2) to make possible the calculation of thermodynamic properties from the molecular structure of the material composing the system. Since a thorough discussion of the foundations, postulates, and formal development of statistical mechanics is beyond the scope of this summary, we shall dispose of objective (1) by merely stating that for all cases in which statistical mechanics has successfully been developed, the laws quoted in the preceding section have been found to be valid. Furthermore, in discussing objective (2), we shall merely quote results the reader is referred to the literature [3-7] for amplification. 

Statistical thermodynamics of monolayers is the obvious pendant of phenomenological classical thermodynamics, discussed in the previous section. In this approach some model assumptions are made on the properties and interactions of molecules, moving from a molecular picture to macroscopic properties, using statistical strategies, the foundations of which were laid down in chapter 1.3. Basically two approaches are open ... 

Ideal-gas tables of thermodynamic properties derived from statistical mechanics are based on the thermodynamic temperatures (as well as on the values of the physical constants used) and are hence independent of any practical temperature scale. The enthalpy of formation, Gibbs energy of formation, and logarithm of the equilibrium constant might depend on temperature-adjusted data. 

In this section, we review some of the important formal results in the statistical mechanics of interaction site fluids. These results provide the basis for many of the approximate theories that will be described in Section III, and the calculation of correlation functions to describe the microscopic structure of fluids. We begin with a short review of the theory of the pair correlation function based upon cluster expansions. Although this material is featured in a number of other review articles, we have chosen to include a short account here so that the present article can be reasonably self-contained. Cluster expansion techniques have played an important part in the development of theories of interaction site fluids, and in order to fully grasp the significance of these developments, it is necessary to make contact with the results derived earlier for simple fluids. We will first describe the general cluster expansion theory for fluids, which is directly applicable to rigid nonspherical molecules by a simple addition of orientational coordinates. Next we will focus on the site-site correlation functions and describe the interaction site cluster expansion. After this, we review the calculation of thermodynamic properties from the correlation functions, and then we consider the calculation of the dielectric constant and the Kirkwood orientational correlation parameters. 

In equilibrium statistical mechanics, one is concerned with the thermodynamic 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 thermodynamic behaviour. A typical macroscopic system is composed of a large number N of molecules occupying a volume V which is large compared to that occupied by a molecule ... 

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