More Statistical Thermodynamics

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. 

But molecules have other energy states that atoms don t. They have rotational and vibrational energy states that can have an important impact on q. Indeed, we found in our discussion of rotational spectroscopy that molecules occupy excited (that is, / 0) rotational energy levels at normal temperatures This suggests, correctly, that the existence of such energy levels has an impact on q, and correspondingly on the thermodynamic properties of molecular gases. 

Once we establish the complete partition function of a molecular gaseous species, we will consider one additional application of the partition function the chemical change. In the last chapter, a few exercises asked for a determination of the A(something) of a physical process, like the expansion of a monatomic gas. However, in chemistry we are often concerned with the change in the chemical identity of a species—a chemical reaction. It may surprise you to learn that the partition functions of each chemical species in a balanced chemical reaction can be used to determine a characteristic property of that reaction its equilibrium constant. 

Unless otherwise noted, all art on this page is Cengage Learning 2014. 

Recognizing that molecules are an important part of chemistry, we will define a molecular partition function, Q, that is the product of partition functions from various energies of a molecule translational, vibrational, rotational, electronic, and nuclear. 

---

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. 

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. 

Thermodynamic, statistical This discipline tries to compute macroscopic properties of materials from more basic structures of matter. These properties are not necessarily static properties as in conventional mechanics. The problems in statistical thermodynamics fall into two categories. First it involves the study of the structure of phenomenological frameworks and the interrelations among observable macroscopic quantities. The secondary category involves the calculations of the actual values of phenomenology parameters such as viscosity or phase transition temperatures from more microscopic parameters. With this technique, understanding general relations requires only a model specified by fairly broad and abstract conditions. Realistically detailed models are not needed to un-... 

Current use of statistical thermodynamics implies that the adsorption system can be effectively separated into the gas phase and the adsorbed phase, which means that the partition function of motions normal to the surface can be represented with sufficient accuracy by that of oscillators confined to the surface. This becomes less valid, the shorter is the mean adsorption time of adatoms, i.e. the higher is the desorption temperature. Thus, near the end of the desorption experiment, especially with high heating rates, another treatment of equilibria should be used, dealing with the whole system as a single phase, the adsorbent being a boundary. This is the approach of the gas-surface virial expansion of adsorption isotherms (51, 53) or of some more general treatment of this kind. 

For a calculation of d. see R- H. Fowler. Statistical Thermodynamics. Second Edition, Cambridge University Press. 1956. p. 127. In Section 1.5a of Chapter 1 we defined the compressibility and cautioned that this compressibility can be applied rigorously only for gases, liquids, and isotropic solids. For anisotropic solids where the effect of pressure on the volume would not be the same in the three perpendicular directions, more sophisticated relationships are required. Poisson s ratio is the ratio of the strain of the transverse contraction to the strain of the parallel elongation when a rod is stretched by forces applied at the end of the rod in parallel with its axis. 

Chapter 10, the last chapter in this volume, presents the principles and applications of statistical thermodynamics. This chapter, which relates the macroscopic thermodynamic variables to molecular properties, serves as a capstone to the discussion of thermodynamics presented in this volume. It is a most satisfying exercise to calculate the thermodynamic properties of relatively simple gaseous systems where the calculation is often more accurate than the experimental measurement. Useful results can also be obtained for simple atomic solids from the Debye theory. While computer calculations are rapidly approaching the level of sophistication necessary to perform computations of... 

Self-organization seems to be counterintuitive, since the order that is generated challenges the paradigm of increasing disorder based on the second law of thermodynamics. In statistical thermodynamics, entropy is the number of possible microstates for a macroscopic state. Since, in an ordered state, the number of possible microstates is smaller than for a more disordered state, it follows that a self-organized system has a lower entropy. However, the two need not contradict each other it is possible to reduce the entropy in a part of a system while it increases in another. A few of the system s macroscopic degrees of freedom can become more ordered at the expense of microscopic disorder. This is valid even for isolated, closed systems. Eurthermore, in an open system, the entropy production can be transferred to the environment, so that here even the overall entropy in the entire system can be reduced. 

It is not the purpose of chemistry, but rather of statistical thermodynamics, to formulate a theory of the structure of water. Such a theory should be able to calculate the properties of water, especially with regard to their dependence on temperature. So far, no theory has been formulated whose equations do not contain adjustable parameters (up to eight in some theories). These include continuum and mixture theories. The continuum theory is based on the concept of a continuous change of the parameters of the water molecule with temperature. Recently, however, theories based on a model of a mixture have become more popular. It is assumed that liquid water is a mixture of structurally different species with various densities. With increasing temperature, there is a decrease in the number of low-density species, compensated by the usual thermal expansion of liquids, leading to the formation of the well-known maximum on the temperature dependence of the density of water (0.999973 g cm-3 at 3.98°C). 

The more recent statistical thermodynamic work approaches the... 

We will adopt this statement as the working form of the third law of thermodynamics. This statement is the most convenient formulation for making calculations of changes in the Gibbs function or the Planck function. Nevertheless, more elegant formulations have been suggested based on statistical thermodynamic theory [5]. 

A satisfactory explanation for this discrepancy was not available until the development of statistical thermodynamics with its methods of calculating entropies from spectroscopic data and the discovery of the existence of ortho- and parahydrogen. It then was found that the major portion of the deviation observed between Equations (11.24) and (11.25) is from the failure to obtain a tme equilibrium between these two forms of H2 molecules (which differ in their nuclear spins) during thermal measurements at very low temperatures (Fig. 11.4). If true equilibrium were established at all times, more parahydrogen would be formed as the temperature is lowered, and at 0 K, all the hydrogen molecules would be in the... 

We shall treat more compUcated cases, such as systems with a larger number of identical or different sites, and also cases of more than one type of ligand. But the general rules of constructing the canonical PF, and hence the GPF, are the same. The partition functions, either Q or have two important properties that make the tool of statistical thermodynamics so useful. One is that, for macroscopic systems, each of the partition functions is related to a thermodynamic potential. For the particular PFs mentioned above, these are... 

From statistical thermodynamic treatments it is possible to calculate the terms on the right-hand side of Eq. (6.251). This is done by assuming a given model of the species considered, e.g., whether the adsorbed ion is immobilized or nonlocalized on the surface of the electrode (Section 6.8.3), and how many cells it occupies. Then one adds the corresponding entropies according to Eq. (6.251) and finally compares the calculated values of (AS ds)model with the value obtained experimentally, (AS ds)exp. The model that gives the value more similar to that obtained experimentally corresponds to the model closer to reality. 

---