Statistical thermodynamics expression

Buckingham (1962) has given the (classical) statistical thermodynamic expression for as... 

To express the result in terms of the statistical thermodynamic expression, we use (GmT-—instead of //mo)- Hence, we have the following... 

It is useful to be able to express the pressure in terms of the partition function. (The result will be used subsequently in the statistical thermodynamic expression derived for the enthalpy.)... 

Transition-state theory is based on the assumption of chemical equilibrium between the reactants and an activated complex, which will only be true in the limit of high pressure. At high pressure there are many collisions available to equilibrate the populations of reactants and the reactive intermediate species, namely, the activated complex. When this assumption is true, CTST uses rigorous statistical thermodynamic expressions derived in Chapter 8 to calculate the rate expression. This theory thus has the correct limiting high-pressure behavior. However, it cannot account for the complex pressure dependence of unimolecular and bimolecular (chemical activation) reactions discussed in Sections 10.4 and 10.5. 

The thermophysical properties necessary for the growth of tetrahedral bonded films could be estimated with a thermal statistical model. These properties include the thermodynamic sensible properties, such as chemical potential /t, Gibbs free energy G, enthalpy H, heat capacity Cp, and entropy S. Such a model could use statistical thermodynamic expressions allowing for translational, rotational, and vibrational motions of the atom. 

In order to assign the Raman bands and determine the absolute oceupancies of O2 and N2 molecules in the small and large cavities, we use the statistical thermodynamic expression derived by van der Waals and Platteeuw. " Let us consider an equilibrium state of the ice-hydrate-gas system. Then, the difference between the chemical potential of water molecules in ice, //h 0), and that in a hypothetical empty lattice of structure II hydrate, ju (h ), is given by... 

According to the principle of microscopic reversibility ki/k.i can be equated to the equilibrium constant for Reaction I, which in turn can be expressed in terms of the electron affinity through the statistical thermodynamic expression for an ideal gas (24). 

In order to get expressions for Gibbs energy G and the Helmholtz energy A, we will need an expression for the entropy, S. The statistical thermodynamic approach for S is somewhat different. Rather than derive a statistical thermodynamic expression for S (which can be done but will not be given here ), we present Ludwig Boltzmanns 1877 seminal contribution relating entropy S and the distribution of particles in an ensemble il ... 

Chapter 17 introduced some of the basic concepts that led to the development of a statistical approach to energy and entropy. This is statistical thermodynamics. By the end of the chapter, equations were applied to monatomic gases, and thermodynamic state functions—mostly entropy—were calculated whose values were very close to experimental values. Also, in some of the exercises you were asked to derive some simple expressions that were also derived from phenomenological thermodynamics. For example, we know from early chapters in this book that the equation AS = i In (V2/V1) is applicable for an isothermal change in volume of an ideal gas. We can also get this expression using the Sackur-Tetrode statistical thermodynamic expression for S. These correspondences are just two examples where phenomenological and statistical thermodynamics are consistent with each other. That is, they ultimately make the same predictions about the state functions of a system, and how they change with a process. 

The frequency distribution function in equation 18.66 can be substituted into the statistical thermodynamic expressions for the various state functions, and various thermodynamic properties determined for crystals. We are interested in the expression for the heat capacity. It is (omitting the details of the derivation) ... 

One could also try to take the reverse route, starting from equation (35) (which implies ((y)) = I, not important for the present discussion). Then one derives Jfcoo(T) from detailed balance and the statistical thermodynamical expression for the equilibrium constant. Finally, one might obtain specific rate constants k(E) by an inverse Laplace transformation (operator in equation 73), with = (kT) ... 

This expression has a formal character and has to be complemented with a prescription for its evaluation. A priori, we can vary the values of the fields independently at each point in space and then we deal with uncountably many degrees of freedom in the system, in contrast with the usual statistical thermodynamics as seen above. Another difference with the standard statistical mechanics is that the effective Hamiltonian has to be created from the basic phenomena that we want to investigate. However, a description in terms of fields seems quite natural since the average of fields gives us the actual distributions of particles at the interface, which are precisely the quantities that we want to calculate. In a field-theoretical approach we are closer to the problem under consideration than in the standard approach and then we may expect that a simple Hamiltonian is sufficient to retain the main features of the charged interface. A priori, we have no insurance that it... 

This equation of state applies to all substances under all conditions of p, and T. All of the virial coefficients B, C,. .. are zero for a perfect gas. For other materials, the virial coefficients are finite and they give information about molecular interactions. The virial coefficients are temperature-dependent. Theoretical expressions for the virial coefficients can be found from the methods of statistical thermodynamic s. 

One can write for Eq. (7-49) an expression for the equilibrium constant. Statistical thermodynamics allows its formulation in terms of partition functions ... 

On expressing Kj in terms of statistical thermodynamics, an approximation can be adopted that the partition function of the middle oxidation level is an average of functions for the oxidized and reduced forms i.e., the preexponential... 

Flory-Huggins model for polymer solutions, based on statistical thermodynamics, is often used for illustrating the behavior of polymer blends [6,7]. The expression for the free energy change... 

By applying the machinery of statistical thermodynamics we have derived expressions for the adsorption, reaction, and desorption of molecules on and from a surface. The rate constants can in each case be described as a ratio between partition functions of the transition state and the reactants. Below, we summarize the most important results for elementary surface reactions. In principle, all the important constants involved (prefactors and activation energies) can be calculated from the partitions functions. These are, however, not easily obtainable and, where possible, experimentally determined values are used. 

There are three approaches that may be used in deriving mathematical expressions for an adsorption isotherm. The first utilizes kinetic expressions for the rates of adsorption and desorption. At equilibrium these two rates must be equal. A second approach involves the use of statistical thermodynamics to obtain a pseudo equilibrium constant for the process in terms of the partition functions of vacant sites, adsorbed molecules, and gas phase molecules. A third approach using classical thermodynamics is also possible. Because it provides a useful physical picture of the molecular processes involved, we will adopt the kinetic approach in our derivations. 

Occasionally alternative expressions of the PDT (9.5) have been proposed [23-25], These alternatives arise from consideration of statistical thermodynamic manipulations associated with a particular ensemble, and the distinguishing features of those alternative formulae are relics of the particular ensemble considered. On the other hand, relics specific to an ensemble are not evident in the PDT formula (9.5). These alternative formulae should give the same result in the thermodynamic limit. 

A rams, D. S., and J. M. Prausnitz, "Statistical Thermodynamics of Liquid Mixtures A New Expression for the Excess Gibbs Energy of Partly or Completely Miscible Systems," AIChE J., 1975, 21, 116. 

Fig. 6. All paths leading from the initial to the final points in time t contribute an interfering amplitude to the path sum describing the resultant probability amplitude for the quantum propagation. In this double slit free particle case, two paths of constant speed are local functional stationary points of the action, and these two dominant paths provide the basis for a (semiclassical) classification of subsets of paths which contribute to the path integral. In the statistical thermodynamic path expression, the path sum is equal to the off-diagonal electronic thermal density matrix...

From classic thermodynamics alone, it is impossible to predict numeric values for heat capacities these quantities are determined experimentally from calorimetric measurements. With the aid of statistical thermodynamics, however, it is possible to calculate heat capacities from spectroscopic data instead of from direct calorimetric measurements. Even with spectroscopic information, however, it is convenient to replace the complex statistical thermodynamic equations that describe the dependence of heat capacity on temperature with empirical equations of simple form [15]. Many expressions have been used for the molar heat capacity, and they have been discussed in detail by Frenkel et al. [4]. We will use the expression... 

Data for a large number of organic compounds can be found in E. S. Domalski, W. H. Evans, and E. D. Hearing, Heat capacities and entropies in the condensed phase, J. Phys. Chem. Ref. Data, Supplement No. 1, 13 (1984). It is impossible to predict values of heat capacities for solids by purely thermodynamic reasoning. However, the problem of the solid state has received much consideration in statistical thermodynamics, and several important expressions for the heat capacity have been derived. For our purposes, it will be sufficient to consider only the Debye equation and, in particular, its limiting form at very low temperamres ... 

Abrams, D.S. and Prausnitz, J.M., Statistical thermodynamics of liquid mixtures a new expression for the excess Gibbs energy of partly or completely miscible systems, A. I. Chem. E. /., 21 (1975) 116-128. 

Although the statistical mechanical theories such as those described above yield exact analytic expressions for various quantities characterizing the conformation of an interrupted helix, those expressions are so complicated that it is of both theoretical and practical value to simplify them, with the imposition of suitable restrictions on parameters, to forms that are amenable to straightforward computations and also, hopefully, to direct comparisons with observed data. Various attempts have been made, and they are summarized in Poland-Scheraga s book (10). Though not available at the time this book was published, the approximations worked out by Okita et al. (13) are of great practical use for their wide applicability and simplicity. Their method is described below in some detail, because it has been consistently used in our statistical-thermodynamic analyses of helix-coil transition phenomena. 

The Hamaker constants of nonpolar fluids and polymeric liquids can be obtained using an expression similar to Equation (67) in combination with the corresponding state theory of thermodynamics and an expression for interfacial energy based on statistical thermodynamics (Croucher 1981). This leads to a simple, but reasonably accurate and useful, relation for Hamaker constants for nonpolar fluids and polymeric liquids. We present in this section the basic details and an illustration of the use of the equation derived by Croucher. 

We shall now discuss the phase transition from the viewpoint of statistical thermodynamics. " The total free energy G can be expressed as a function of N (total number of cation sites = total number of anion sites), (total number of anions), (number of cations on the A sites), Ag, A,-, and Aq as G = G(A,Ax,Aa,Ab,Ac,Ad) (1.234)... 

Another assumption of CTST is that the activated complex C can be treated as a distinct chemical species. Thus one can use standard statistical mechanics expressions to derive its thermodynamic properties. 

R. H. Fowler and E. A. Guggenheim [Statistical Thermodynamics (Cambridge University Press, Cambridge, 1939)] criticized this statement as well as similar statements (to be quoted below) which imply that the entropy of perfect crystalline substances is zero. According to Fowler and Guggenheim, the only valid third-law inference is the unattainability of absolute zero, as expressed in the following statement ... 

Statistical thermodynamics can provide explicit expressions for the phenomenological Gibbs energy functions discussed in the previous section. The statistical theory of point defects has been well covered in the literature [A. R. Allnatt, A. B. Lidiard (1993)]. Therefore, we introduce its basic framework essentially for completeness, for a better atomic understanding of the driving forces in kinetic theory, and also in order to point out the subtleties arising from the constraints due to the structural conditions of crystallography. 

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