Partition functions molecular properties derivation

The partition function provides the bridge to calculating thermodynamic quantities of interest. Using the molecular partition function and formulas derived in this section, we will be able to calculate the internal energy E, the heat capacity Cp, and the entropy S of a gas from fundamental properties of the molecule, such as its mass, moments of inertia, and vibrational frequencies. Thus, if thermodynamic data are lacking for a species of interest, we usually know, or can estimate, these molecular constants, and we can calculate reasonably accurate thermodynamic quantities. In Section 8.6 we illustrate the practical application of the formulas derived here with a numerical example of the thermodynamic properties for the species CH3. 

Calculation of Thermodynamic Functions from Molecular Properties The calculation methods for thermodynamic functions (entropy S, heat capacities Cp and Cy, enthalpy H, and therefore Gibbs free energy G) for polyatomic systems from molecular and spectroscopic data with statistical methods through calculation of partition functions and its derivative toward temperature are well established and described in reference books such as Herzberg s Molecular Spectra and Molecular Structure [59] or in the earlier work from Mayer and Mayer [7], who showed, probably for the first time in a comprehensive way, that all basic thermochemical properties can be calculated from the partition function Q and the Avagadro s number N. The calculation details are well described by Irikura [60] and are summarized here. Emphasis will be placed on calculations of internal rotations. 

The thermodynamic properties that we have considered so far, such as the internal energy, the pressure and the heat capacity are collectively known as the mechanical properties and can be routinely obtained from a Monte Carlo or molecular dynamics simulation. Other thermodynamic properties are difficult to determine accurately without resorting to special techniques. These are the so-called entropic or thermal properties the free energy, the chemical potential and the entropy itself. The difference between the mechanical emd thermal properties is that the mechanical properties are related to the derivative of the partition function whereas the thermal properties are directly related to the partition function itself. To illustrate the difference between these two classes of properties, let us consider the internal energy, U, and the Fielmholtz free energy, A. These are related to the partition function by ... 

We now have equations for the partition functions for the ideal gas and equations for relating the partition functions to the thermodynamic properties. We are ready to derive the equations for calculating the thermodynamic properties from the molecular parameters. As an example, let us calculate Um - t/o.m for the translational motion of the ideal gas. We start with... 

It is also important to keep in mind the independent (state) variables that were specified in deriving q. That is, the partition function was derived with the number of molecules N, the volume of the system V, and the temperature T specified as the independent variables. Thus, when taking the derivative with respect to temperature, as will be needed later, it is good to keep in mind that q = q(N, V, T). The partition function for the entire system of identical molecules, with independent variables N, V, and T, is denoted by a capital Q. If the molecules are indistinguishable, as would normally be the case when calculating thermochemical properties for a given species, then the system partition function is related to the molecular partition function by... 

Here we note that only a single polarizability or susceptibility exists for any system. The reconstruction from local contributions is in fact an abstraction, the result of which depends on the detail wanted macroscopic with local susceptibilities or microscopic with local polarizabilities and—more importantly—on the partitioning of such properties. However, experimental chemists are used to such procedures from well-chosen series of compounds they derive bond energies as local contributions to heats of formation and ionic radii from crystal structures. Theoretical chemists obtain atomic charges from, e.g., a Mulliken analysis of their wave functions. We are able, following similar reasoning, to construct molecular polarizabilities from atomic ones [38,60], although there is formally no connection between them. In an opposite direction we can decompose a molecular polarizability into a many-center... 

To calculate the static thermodynamic and molecular ordering properties of a system of molecules, the configurational partition function Qc of the system must be derived. Qc does not contain the kinetic energy, intramolecular and intermolecular vibrations, and very small rotations about molecular bonds. Qc does contain terms which deal with significant changes in the shapes of the molecules due to rotations about semiflexible bonds (such as about carbon-carbon bonds in n-alkyl [i.e., (-CH2-)X] sections) in a molecule. For mathematical tractability in deriving Qc, the description of the molecules in continuum space is mapped onto a... 

By means of the equations derived above it should be possible to calculate the heat capacity of a gas at any temperature provided information concerning the partition function is available. The problem is thus reduced to a study of the evaluation of this property of a molecular species. 

Hence, in the light of our both accounts of causality, the molecular dynamics model represents causal processes or chains of events. But is the derivation of a molecule s structure by a molecular dynamics simulation a causal explanation Here the answer is no. The molecular dynamics model alone is not used to explain a causal story elucidating the time evolution of the molecule s conformations. It is used to find the equilibrium conformation situation that comes about a theoretically infinite time interval. The calculation of a molecule s trajectory is only the first step in deriving any observable structural property of this molecule. After a molecular dynamics search we have to screen its trajectory for the energetic minima. We apply the Boltzmann distribution principle to infer the most probable conformation of this molecule.17 It is not a causal principle at work here. This principle is derived from thermodynamics, and hence is statistical. For example, to derive the expression for the Boltzmann distribution, one crucial step is to determine the number of possible realizations there are for each specific distribution of items over a number of energy levels. There is no existing explanation for something like the molecular partition function for a system in thermodynamic equilibrium solely by means of causal processes or causal stories based on considerations on closest possible worlds. 

Thus, reaction rate coefficients can be estimated from the thermochemistry of the transition states, whose molecular properties can be calculated with quantum chemical programs. In calculating reaction rate coefficients, the only negative second derivative of energy with respect to atomic coordinates (called imaginary vibrational frequency ) from the transition state is ignored, so that there are only 37/-7 molecular vibrations in the transition structure (37/ — 6 if linear) and all internal and external symmetry numbers have to be included in the rotational partition functions (then any reaction path degeneracy is usually included automatically). 

The problem is to derive the equation of state and thermodynamic functions of a particular liquid crystal phase from properties of constituting molecules (a form, a polarizability, chirality, etc.). The problem we are going to discuss is one of the most difficult in physics of liquid crystals and the aim of this chapter is very modest just to introduce the reader to the basic ideas of the theory with the help of comprehensive works of the others [2, 5, 19]. To consider the problem quantitatively we need special methods of the statistical physics. In this context, the most useful function is free energy F, which is based microscopically rai the so-called partition function, see below. For the partition function, we need that energy spectrum of a molecular system, which is relevant to the problem imder cmisider-ation. The energy spectrum is related to the entropy of the system and we would like to recall the microscopic sense of the entropy. 

The translational partition function for a molecular confined within volume V is given by Eq. (A27). If we assume this to be the partition function for an ideal gas molecule with v - K/A we derive the correct values for many of the thermodynamic properties with the notable exception of the entropy. The entropy, however, is lower by k per molecule. This discrepancy arises because the gas is not really a system in which the molecules can properly be... 

Because we have derived expressions for each of the partition functions of a molecule, we can evaluate the expressions in equations 18.50, and similarly for each of the other thermodynamic functions, for each part of the overall molecular partition function. These expressions are given in Table 18.5. You should be able to derive most of the expressions in Table 18.5 by simply performing the appropriate derivation given in equations 18.50 and the equivalent for the other thermodynamic properties. (Remember that [(3In q)/aT] = lq) dqldT).)... 

This is the general TST expression for the thermal rate coefficient. It contains a universal frequency factor ksT/h = 6.25 x 10 s at 300 K) which is independent of the nature of the reactants. Specific molecular properties appear in the ratio of the partition functions and in Ael. Equation (3.54) differs therefore from the collision theory rate-coefficient expression [Eq. (3.7)] in that all quantities contained in this equation are at least in principle derivable from molecular properties. 

For the purpose of discussing mesophase formation in polymers it is conveiuent to partition the polymers into two categories and introduce abbreviations that refer to these categories. Polymerized liquid crystals, here abbreviated PLCs, are derived from known, low molecular weight monomer liquid crystals (MLCs) that contain polymerizable functionality (e.g. vinyl units). We designate liquid-crystalline polymers (LCPs) to be semiflexible, linear polymers that are structurally related to conventional engineering thermoplastics, i.e. polymers derived from poly(ester)s, poly(amide)s, poly(imide)s, etc. We will examine the attributes of polymerized liquid crystals first, stressing the similarities between their properties and those of MLCs. 

---