Statistical thermodynamics internal energy

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. 

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... 

The freely-jointed chain considered previously has no internal restraint, and hence, its internal energy is zero regardless of its present configuration. The entropy (S) is not constant, however, since the number of available configurations decreases with the chain end separation distance. The variation which follows from chain length change by a small amount (dr) at constant temperature (T) is given by the Boltzmann rule of statistical thermodynamics ... 

Here an additional distinction is to be made between thermodynamic averages of a conformational observable such as the internal energy, which converges well if potential minima are correctly sampled, and statistical properties such as free energies, which depend on the entire partition function. 

The most important new concept to come from thermodynamics is entropy. Like volume, internal energy and mole number it is an extensive property of a system and together with these, and other variables it defines an elegant self-consistent theory. However, there is one important difference entropy is the only one of the extensive thermodynamic functions that has no obvious physical interpretation. It is only through statistical integration of the mechanical behaviour of microsystems that a property of the average macrosystem, that resembles the entropy function, emerges. 

The use of statistical calculations of configuration integrals to determine thermodynamic adsorption characteristics of zeolites dates back to the late 1970s (49). Kiselev and Du (22) reported calculations based on atom-atom potentials for Ar, Kr, and Xe sorbed in NaX, NaY, and KX zeolites. Then-calculations, which included an electrostatic contribution, predicted changes in internal energy in excellent agreement with those determined experimentally. The largest deviation between calculated and experimental values, for any of the sorbates in any of the hosts, was a little over 1 kJ/mol. 

In order to introduce basic equations and quantities, a preliminary survey is made in Section II of the statistical mechanics foundations of the structural theories of fluids. In particular, the definitions of the structural functions and their relationships with thermodynamic quantities, as the internal energy, the pressure, and the isothermal compressibility, are briefly recalled together with the exact equations that relate them to the interparticular potential. We take advantage of the survey of these quantities to introduce what is a natural constraint, namely, the thermodynamic consistency. 

In Section 2.1, we remarked that classical thermodynamics does not offer us a means of determining absolute values of thermodynamic state functions. Fortunately, first-principles (FP), or ab initio, methods based on the density-functional theory (DFT) provide a way of calculating thermodynamic properties at 0 K, where one can normally neglect zero-point vibrations. At finite temperatures, vibrational contributions must be added to the zero-kelvin DFT results. To understand how ab initio thermodynamics (not to be confused with the term computational thermochemistry used in Section 2.1) is possible, we first need to discuss the statistical mechanical interpretation of absolute internal energy, so that we can relate it to concepts from ab initio methods. 

The Entropy and Irreversible Processes.—Unlike the internal energy and the first law of thermodynamics, the entropy and the second law are relatively unfamiliar. Like them, however, their best interpretation comes from the atomic point of view, as carried out in statistical mechanics. For this reason, we shall start with a qualitative description of the nature of the entropy, rather than with quantitative definitions and methods of measurement. 

Thermodynamic principles arise from a statistical treatment of matter by studying different idealized ensembles of particles that represent different thermodynamic systems. The first ensemble that we study is that of an isolated system a collection of N particles confined to a volume V, with total internal energy E. A system of this sort is referred to as an NVE system or ensemble, as N, V, and E are the three thermodynamic variables that are held constant. N, V, and E are extensive variables. That is, their values are proportional to the size of the system. If we combine NVE subsystems into a larger system, then the total N, V, and E are computed as the sums of N, V, and E of the subsystems. Temperature, pressure, and chemical potential are intensive variables, for which values do not depend on the size of the system. 

It should be noted that it is assumed that the intermolecular forces do not affect the internal degrees of freedom so that is independent of whether these forces are present or not. When they are absent (Zf = 0), the integral Z collapses to and equation (2.2.31) becomes the same as equation (2.2.23). The important task of the statistical thermodynamics of imperfect gases and liquids is to evaluate Z. This subject is discussed in detail later in this chapter. However, the nature of the intermolecular forces which give rise to the potential energy U is considered next. 

Temperature (T), pressure (P), partial specific Gibbs function (G), and entropy (S) for a cell in a nonequilibrium state depend upon the internal energy (U) and the mass units (m,) of the molecular species V exactly in the same manner as in an equilibrium situation. In other words each cell of a system in nonequilibrium is treated exactly in the same manner (statistically as well as thermodynamically) as if equilibrium exists in each of these cells. 

In Chapter 4 we considered that the total energy of a system is given by relativity theory as E,. = rru - and that the internal energy of thermodynamics is some unspecified smaller quantity such that = U + constant. In statistical mechanics the energy... 

Formulate an expression for the (a) partition function Z, (b) internal energy U, and (c) specific heat at constant pressure Cp for 1 mol of chloromethane (CH3CI) based on statistical thermodynamics. 

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