Partitioning chemical potential

The partitioning chemical potentials, Ui, and the dispersion force coefficients, Bi, are treated as adjustable parameters, although the latter can in principle be estimated from spectroscopic data. These two models are quite different in terms of their details, but also in terms of principle. The dispersion-force model treats the ion-lipid interface as a mathematical discontinuity and modifies the ion-lipid wall PMF, while the partitioning model promotes an alternative, Swiss-cheese-like picture of the ion-lipid interface. It is worthwhile to mention here that this penetration of anions within the lipid interface was observed in recent computer simulation work." ... 

The first important piece of information from this model system is that the interaction of anions with lipids is not of the local chemical binding type. The fact that the other two models are successful is gratifying but not in itself surprising. Both partitioning chemical potentials and dispersion-force constants are roughly proportional to the ionic volume, so they provide a good representation of ionic size. The fitted U and B parameters are presented in Table 2 and their relation to the ionic volumes in solution is presented in Fig. 9. 

The grand canonical ensemble is a set of systems each with the same volume V, the same temperature T and the same chemical potential p (or if there is more than one substance present, the same set of p. s). This corresponds to a set of systems separated by diathennic and penneable walls and allowed to equilibrate. In classical thennodynamics, the appropriate fimction for fixed p, V, and Tis the productpV(see equation (A2.1.3 7)1 and statistical mechanics relates pV directly to the grand canonical partition function... 

For both first-order and continuous phase transitions, finite size shifts the transition and rounds it in some way. The shift for first-order transitions arises, crudely, because the chemical potential, like most other properties, has a finite-size correction p(A)-p(oo) C (l/A). An approximate expression for this was derived by Siepmann et al [134]. Therefore, the line of intersection of two chemical potential surfaces Pj(T,P) and pjj T,P) will shift, in general, by an amount 0 IN). The rounding is expected because the partition fiinction only has singularities (and hence produces discontinuous or divergent properties) in tlie limit i—>oo otherwise, it is analytic, so for finite Vthe discontinuities must be smoothed out in some way. The shift for continuous transitions arises because the transition happens when L for the finite system, but when i oo m the infinite system. The rounding happens for the same reason as it does for first-order phase transitions whatever the nature of the divergence in thennodynamic properties (described, typically, by critical exponents) it will be limited by the finite size of the system. 

At equilibrium, in order to achieve equality of chemical potentials, not only tire colloid but also tire polymer concentrations in tire different phases are different. We focus here on a theory tliat allows for tliis polymer partitioning [99]. Predictions for two polymer/colloid size ratios are shown in figure C2.6.10. A liquid phase is predicted to occur only when tire range of attractions is not too small compared to tire particle size, 5/a > 0.3. Under tliese conditions a phase behaviour is obtained tliat is similar to tliat of simple liquids, such as argon. Because of tire polymer partitioning, however, tliere is a tliree-phase triangle (ratlier tlian a triple point). For smaller polymer (narrower attractions), tire gas-liquid transition becomes metastable witli respect to tire fluid-crystal transition. These predictions were confinned experimentally [100]. The phase boundaries were predicted semi-quantitatively. 

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 ... 

Here Zint is the intramolecular partition function accounting for rotations and vibrations. However, in equilibrium, the chemical potential in the gas phase is equal to that in the adsorbate, fi, so that we can write the desorption rate in (I) as... 

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. 

Hint Write the partition functions for the atoms occupying step and terrace sites and equal their chemical potentials. 

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. 

The first term on the right is the formula for the chemical potential of component a at density pa = na/V in an ideal gas, as would be the case if interactions between molecules were negligible, fee is Boltzmann s constant, and V is the volume of the solution. The other parameters in that ideal contribution are properties of the isolated molecule of type a, and depend on the thermodynamic state only through T. Specifically, V/A is the translational contribution to the partition function of single a molecule at temperature T in a volume V... 

The chemical potential pa on the left is the full chemical potential including ideal and excess parts. In this chapter we will scale the chemical potentials by (3 and often refer to this unitless quantity as the chemical potential. 3/ia yields the absolute activity. The first term on the right is the ideal-gas chemical potential, where pa is the number density, Aa is the de Broglie wavelength, and q 1 is the internal (neglecting translations) partition function for a single molecule without interactions with any other molecules. 

As mentioned above, there are multiple ways to derive the PDT for the chemical potential. Here we utilize the older method in the canonical ensemble which says that 3/j,0 is just minus the logarithm of the ratio of two partition functions, one for the system with the distinguished atom or molecule present, and the other for the system with no solute. Using (11.7) we obtain [9, 48,49]... 

We now use a trick to partition this exact expression for the chemical potential into classical and quantum correction parts [29]. To do this we multiply and divide inside the logarithm of the excess term by the classical average... 

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