Entropy derivation from partition functions

However, sufficiently accurate measurements of gaseous heat capacities down to the boiling point have usually not been made, and so the entropy contribution of the gaseous state, assuming ideal behavior, is derived from heat capacities obtained from partition functions as described in Chapter VI. If the heat capacity can be expressed as a function of the temperature over the required range, the entropy change can be derived analytically, as described in 20a. 

Expressions similar to those given above may be derived easily from partition functions in other ensembles.The choice of ensemble is very important in calculations of hydration entropy, enthalpy, and heat capacity, as discussed below. Many other quantities, including all free energies, are ensemble invariant, with the choice of ensemble affecting only system size dependence. For simplicity, the discussion here is therefore limited to the canonical ensemble except in such cases where a true ensemble dependence exists. 

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 ... 

This expression accounts for the configurational entropy of an ideal binary mixture with identical molecular sizes, but not for that of a polymer solution, since polymer chains are large and flexible. For that case, more contributions arise from the chain conformational entropy, first considered by Meyer [19] and then derived by Huggins [20] and Flory [21]. In analogy with a nonreversing random walk on a lattice, the conformational contribution of polymer chains to the partition function is given by... 

In practice, it proves more convenient to work within a convention where we define tire ground state for each energy component to have an energy of zero. Thus, we view 1/eiec as the internal energy that must be added to I/q, which already includes Eeiec (see Eq. (10.1)), as the result of additional available electronic levels. One obvious simplification deriving from this convention is that the electronic partition function for the case just described is simply eiec = 1, Inspection of Eq. (10.5) then reveals that the electronic component of the entropy will be zero (In of 1 is zero, and the constant 1 obviously has no temperature dependence, so both terms involving eiec are individually zero). 

The superscript o indicates that a standard state is being referred to (see below).) Conventionally, however, the last two terms in the final line of Eq. (10.9), i.e., those deriving from Stirling s approximation, are typically assigned to the translational partition function as well. As they have no temperature dependence, this has no impact on f/transl however, the entropy of translation becomes... 

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. 

The derivatives of the partition functions with respect to temperature are needed to calculate entropies from eq. (4). For one degree of translational, rotational, or internal rotational freedom, treated classically, the derivative is given by the expression... 

It is seen from the foregoing results, c.g., equations (24.11) and (24.12), that by combining statistical mechanics with the Boltzmann-Planck equation it is possible to derive a relationship between the molar entropy of any gas, assuming it to behave ideally, and the partition function of the given species. Since the partition function and its temperature coefficient may be regarded as known, from the discussion in Chapter VI, the problem of calculating entropies may be regarded as solved, in principle. In order to illustrate the procedure a number of cases will be considered. 

For accurate results, should be the combined rotational and vibrational partition function derived from the actual energy levels of the molecule as obtained from spectroscopic measurements ( 16k). For most purposes, at ordinary temperatures, very little error results from the separation of Qi into the product of two independent factors, viz., Qr and Q., representing the rotational and vibrational partition functions, respectively. Because equation (24.18) involves Q< in logarithmic terms only, it follows that an expression of the same form can be used to give the separate rotational and vibrational entropies. Thus, if is replaced by Qr, the result is Sr, the rotational contribution to the entropy, and similarly the vibrational contribution , is obtained by using Q, for Qi in equation (24.18). The sum of Sr and Sv derived in this manner represents Si, which added to St, as given by equation (24.14), etc., yields the total entropy. 

The free energy of the system also includes entropic contributions arising from the internal fluctuations, which are expected to be different for the separate species and for the liganded complex. These can be estimated from normal-mode analyses by standard techniques,136,164 or by quasi-harmonic calculations that introduce approximate corrections for anharmonic effects 140,141 such approaches have been described in Chapt. IV.F. From the vibrational frequencies, the harmonic contribution to the thermodynamic properties can be calculated by using the multimode harmonic oscillator partition function and its derivatives. The expressions for the Helmholtz free energy, A, the energy, E, the heat capacity at constant volume, C , and the entropy are (without the zero-point correction)164... 

The progress of a chemical reaction comprises variation of the positions of the atomic nuclei along a multidimensional potential hyperface. The path with minimum energy with respect to the other degrees of freedom is called the reaction coordinate. If the potential along this coordinate exhibits a maximum, it is called the transition state and its difference to the initial state is denoted as activation energy E. From the partition functions of the initial and transition states, the activation entropy AS is derived. Within the framework of transition state theory (TST) [1], the rate constant for the reaction is then given by... 

A simple way (actually the only way) to determine this low temperature amorphous phase is to use a theory that correctly predicts the behavior of the liquid and extend it to low temperatures. One obviously should use the most realistic existing equilibrium theory to obtain the low temperature phase. The predictions are the two equations of state, S-V-T and P-V-T which are each derived from the Helmholtz free energy F which is in turn obtained from the partition function (F=-kTLnQ). In obtaining the S-V-T equation of state it is discovered that the configurational entropy Sc defined as the total entropy minus the vibrational entropy, approaches zero at a finite temperature (4), This vanishing of Sc is taken as the thermodynamic criterion of glass formation (5,6). 

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. 

In 1956, Hory introduced the semi-flexibility into the classical lattice statistical thermodynamic theory of polymer solutions (Flory 1956). From the classical lattice statistics of flexible polymers, we have derived the total number of ways to arrange polymer chains in a lattice space, as given by (8.15). The first two terms on the right-hand side of that equation are the combinational entropy between polymers and solvent molecules, and the last three terms belong to polymer conformational entropy. Thus the contribution of polymer conformation in the total partition function is... 

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. 

In the meantime, Scheutjens and Fleer [35] had formulated their now classical numerical self-consistent field (lattice) (SF) fheory for equilibrium adsorption of polymers, which provides a very detailed partition function from which strucfural and fhermodynamic information can be derived. The key idea of a self-consisfenf field approach is that, on the one hand, the interactions between molecules are treated as if they constitute an external field on an individual molecule (fhereby defining potential energies), and that each molecule finds its optimum distribution of conformations and locations as a Boltzmann distribution in that field. On the other hand, the distribution found has to be such that it will indeed produce the field, hence the term self-consistent. An iterative scheme is employed to reach consistency, and as soon as the solution is found, energy and entropy of fhe system are known as well. 

The energy and entropy become a sum of translational, vibrational, or rotational contributions, because they depend logarithmically on the partition function. Because of (4.55) the vibrational partition function has been defined with respect to Uq. This can be understood from the following derivation ... 

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