Chemical potential from partition function

Chapters 13 through 16 focus on the chemical potential, the quantity that describes the tendency for molecules to move from one place to another, or from one phase to another. It is the basis for treating boiling and freezing, for describing the partitioning of solute molecules between phases, and for treating chemical reaction equilibria and kinetics. To treat the gas phase in these processes, the key relationship that we will need is p (p), the chemical potential as a function of pressure. 

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 is our principal result for the rate of desorption from an adsorbate that remains in quasi-equihbrium throughout desorption. Noteworthy is the clear separation into a dynamic factor, the sticking coefficient S 6, T), and a thermodynamic factor involving single-particle partition functions and the chemical potential of the adsorbate. The sticking coefficient is a measure of the efficiency of energy transfer in adsorption. Since energy supply from the... 

Other thermodynamic quantities such as chemical potential and entropy also follow directly from the partition function, as we demonstrate later on. However, to illustrate what a partition function means, we will first discuss two relatively simple but instructive examples. 

By using Eq. (35) we find the chemical potential directly from the partition function ... 

Thus, given sufEcient detailed knowledge of the internal energy levels of the molecules participating in a reaction, we can calculate the relevant partition functions, and then the equilibrium constant from Eq. (67). This approach is applicable in general Determine the partition function, then estimate the chemical potentials of the reacting species, and the equilibrium constant can be determined. A few examples will illustrate this approach. 

The molecule reaches the transition state and from there it desorbs into the gas phase. To evaluate the rate constant we use the same procedure as above Write down the partition functions for the participating species, equalize the chemical potentials, and find an expression for the number of molecules in the transition state. Since it is much more practical to do this in terms of coverages we immediately obtain ... 

For t vo systems in chemical equilibrium we can calculate the equilibrium constant from the ratio of partition functions by requiring the chemical potentials of the t vo systems to be equal. 

Other thermodynamic functions may be derived from the partition function Q, or from the expression for the osmotic pressure. The chemical potential of the solvent in the solution (not to be confused with the excess chemical potential (mi —within a region of uniform segment expectancy, or density) is given, of course, by ... 

Computing thermodynamic properties is the most important validation of simulations of solutions and biophysical materials. The potential distribution theorem (PDT) presents a partition function to be evaluated for the excess chemical potential of a molecular component which is part of a general thermodynamic system. The excess chemical potential of a component a is that part of the chemical potential of Gibbs which would vanish if the intermolecular interactions were to vanish. Therefore, it is just the part of that chemical potential that is interesting for consideration of a complex solution from a molecular basis. Since the excess chemical potential is measurable, it also serves the purpose of validating molecular simulations. 

To make contact with atomic theories of the binding of interstitial hydrogen in silicon, and to extrapolate the solubility to lower temperatures, some thermodynamic analysis of these data is needed a convenient procedure is that of Johnson, etal. (1986). As we have seen in Section II. l,Eqs. (2) et seq., the equilibrium concentration of any interstitial species is determined by the concentration of possible sites for this species, the vibrational partition function for each occupied site, and the difference between the chemical potential p, of the hydrogen and the ground state energy E0 on this type of site. In equilibrium with external H2 gas, /x is accurately known from thermochemical tables for the latter. A convenient source is the... 

Here /i j3 is the chemical potential of the ideal gas at the standard pressure. It will be seen subsequently that qi for an ideal gas depends linearly on the volume V, so fif is a function only of the temperature. It does of course depend on the distribution of energy levels of the ideal gas molecules. The form of Equation 4.59 for the chemical potential of an ideal gas component is the same as that previously derived from thermodynamics (Equation 4.47). The present approach shows how to calculate m through the evaluation of the molecular partition function. Furthermore, the... 

The Langmuir isotherm can be derived from a statistical mechanical point of view. Thus, for the reaction M + Agas Aads, equilibrium is established when the chemical potential on both phases is the same, i.e., pgas = p,ads. The partition function for the adsorbed molecules as a system is given by... 

The statistical thermodynamic approach to the derivation of an adsorption isotherm goes as follows. First, suitable partition functions describing the bulk and surface phases are devised. The bulk phase is usually assumed to be that of an ideal gas. From the surface phase, the equation of state of the two-dimensional matter may be determined if desired, although this quantity ceases to be essential. The relationships just given are used to evaluate the chemical potential of the adsorbate in both the bulk and the surface. Equating the surface and bulk chemical potentials provides the equilibrium isotherm. 

A partitioning function for a system of rigid rod-like particles with partial orientation around an axis is derived from the use of a modified lattice model. The free energy of mixing is shown as a function of the mole numbers, the axis ratio of the solute particles and a disorientation parameter this function passes through a minimum with increase in the disorientation parameter. The chemical potentials display discontinuities at the concentration at which the minimum appears and then separation into an isotropic phase and a somewhat more concentrated anisotropic phase arises. The critical concentration, v, is given in the form 13) ... 

In the context of van der Waals theory, a and b are positive parameters characterizing, respectively, the magnitude of the attractive and repulsive (excluded volume) intermolecular interactions. Use this partition function to derive an expression for the excess chemical potential of a distinguished molecule (the solute) in its pure fluid. Note that specific terms in this expression can be related to contributions from either the attractive or excluded-volume interactions. Use the Tpp data given in Table 3.3 for liquid n-heptane along its saturation curve to evaluate the influence of these separate contributions on test-particle insertions of a single n-heptane molecule in liquid n-heptane as a function of density. In light of your results, comment on the statement made in the discussion above that the use of the potential distribution theorem to evaluate pff depends on primarily local interactions between the solute and the solvent. 

Once you have the chemical potential for the van der Waals model of the previous exercise, find the equation of state by integrating the Gibbs-Duhem relation. Compare your result with the equation of state obtained from the approximate partition function using... 

To transform from the canonical to the grandcanonical ensemble with respect to guest molecules using the chemical potential of the guest species, the grand partition function, E, is written as... 

In order to evaluate the thermodynamic functions of the process (5), it is necessary to know the interaction energy, equilibrium geometry and frequencies of the normal vibration modes of the bases and base pairs involved in equilibrium process. Interaction energies and geometries are evaluated using empirical potential or quantum chemically (see next section), and normal vibrational frequencies are determined by a Wilson FG analysis implemented in respective codes. Partition functions, computed from AMBER 4.1, HF/6-31G and MP2/6-31G (0.25) constants (see next section), are evaluated widiin the rigid rotor-harmonic oscillator-ideal gas approximations (RR-HO-IG). We have collected evidence [26] that the use of RR-HO-IG approximations yields reliable thermodynamic characteristics (comparable to experimental data) for ionic and moderately strong H-bonded complexes. We are, therefore,... 

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