Partition Functions in Solution

Partition Functions in Solution For molecules in the gas phase the partition functions can be evaluated in a straightforward manner from eqs. (5)-(9). Unfortunately 

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Theoretical calculations are less fundamental and rigorous for solution reactions. This is a consequence of the difficulty of calculating partition functions in solution. The main focus for solution reactions has been on the thermodynamic formulation of transition state theory. 

The effect of the solvent on the rate constant is considered in terms of non-ideality, charge on reactants, relative permittivity and change in solvation pattern of the solvent. Because of the difficulty of assessing partition functions in solution, the thermodynamic formulation is used. A simplified version is given here. 

In the molecular-motion contribution, the molecular partition function is the product of translational, rotational, vibrational, and electronic partition functions. If the molecule in solution is assumed to have the entire volume of the solution available to it, the ratio of gas-phase and solution-phase translational partition functions equals one. Likewise, the electronic partition function ratio will be one. It is unclear what one should use for the rotational partition function in solution, but if this is assumed to have the same form as that in the gas phase, the rotational partition function ratio (which involves the moments of inertia) will be very close to one, since structural changes from gas to solution are slight. Significant contributions to the vibrational partition function are made only by the low-frequency vibrational normal modes, and these modes sometimes show substantial changes in frequency on going from the gas phase to solution. If a vibrational calculation is done in the gas phase and in solutitm, one can calculate AG°oiv m, but the most common procedure is to omit it, assuming that its contribution is negligible. 

Using the formalism of statistical mechanics, Giddings et al. [135] investigated the effects of molecular shape and pore shape on the equilibrium distribution of solutes in pores. The equilibrium partition coefficient is defined as the ratio of the partition function in the pore... 

From a series of transformations of Equation 1 we obtain a new partition function (T) whose independent variables are temperature, pressure, solvent mole number, and the chemical potentials of the solutes (components 2 and 3). These transformations consist of first creating a partition function with pressure rather than volume as an independent variable, and then using this result to create yet another partition function in which we have switched independent variables from solute mole numbers to solute chemical potentials. These operations are analogous to the Legendre transforms commonly employed in thermodynamics. 

In 1957 Hill introduced a binary solution theory based on an analysis of the semigrand partition function in which the pressure P, temperature T, and number of solvent particles, Nj, are held fixed (10, 11). In this section, we extend his derivation to a four-component system containing solvent (component 1), two polymers (components 2 and 3) and protein (component 4). The objective of the calculation is to derive expressions for the chemical potentials of all components. Later, by equating the chemical potentials of each component in each phase, we will determine the composition of each phase and hence the protein partition coefficient which is defined to be the ratio of protein compositions in the top and bottom phases. 

The chemical potential of the solute D in the liquid l is obtained from the partition functions in equations (7.187) and (7.188) ... 

A fundamental assumption that underlies the polymer solution theory in this chapter is the factorization of the chain partition function in Equations (31.4) and (31.7) into a product of two terms. One term, which depends on factors... 

The analytical solution of this equation towards finding the explicit form of the atomic partition function in molecule involves a procedure for solving the Fredholm integral equations. However, such equation can be formally solved, by considering it as a infinite linear set of algebraic equations, and after matrix-formalization one can resuming it as following (Rychlewski Parr, 1986 Li Parr, 1986)... 

Because the transition state geometry optimized in solution and the solution-path reacton path may be very different from the gas-phase saddle point and the gas-phase reaction path, it is better to follow the reaction path given by the steepest-descents-path computed from the potential of mean force. This approach is called the equilibrium solvation path (ESP) approximation. In the ESP method, one also substitutes W for V in computing the partition functions. In the ESP approximation, the solvent coordinates are not involved in the definition of the generalized-transition-state dividing surface, and hence, they are not involved in the definition of the reaction coordinate, which is normal to that surface. One says physically that the solvent does not participate in the reaction coordinate. That is the hallmark of equilibrium solvation. 

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

Unlike the trivial solution x = 0, the instanton, as well as the solution x(t) = x, is not the minimum of the action S[x(t)], but a saddle point, because there is at least one direction in the space of functions x(t), i.e. towards the absolute minimum x(t) = 0, in which the action decreases. Hence if we were to try to use the approximation of steepest descents in the path integral (3.13), we would get divergences from these two saddle points. This is not surprising, because the partition function corresponding to the unbounded Hamiltonian does diverge. 

The last quantity that we discuss is the mean repulsive force / exerted on the wall. For a single chain this is defined taking the derivative of the logarithm of the chain partition function with respect to the position of the wall (in the —z direction). In the case of a semi-infinite system exposed to a dilute solution of polymer chains at polymer density one can equate the pressure on the wall to the pressure in the bulk which is simply given by the ideal gas law The conclusion then is that [74]... 

The reader who is less familiar with the theory of grand partition functions may directly proceed to Eqs. 12a and 13. The physical basis of these formulas and the significance of the quantities CK% will then become apparent in the subsequent paragraph is the vapor pressure (or fugacity) of solute K and y i is the probability of finding a K molecule in a cavity of type i. 

The theory introduced by Lennard-Jones and Devonshire13 17 for the study of liquids provides a powerful method for the quantitative evaluation of the partition function of a solute molecule within its cavity.51 Because the application of this method to the present problem has been described in detail,62 we shall restrict ourselves to its most essential features. 

When we deal with the subject of reactions in solution, which has been our primary emphasis, we are not concerned with ab initio calculations of Ki, since in general the partition functions are unavailable for the participants. 

The notation K is used to emphasize that the scale for K% is based on activities, not simply concentrations. In the gas phase one has available partition functions that can be used to calculate Kt. In solution, however, partition functions are not available. The... 

Rikvold and Stell [319,320,365] have developed an expression for the partition coefficient in a random two-phase medium made up of spherical particles. They found the partition coefficient to be essentially an exponential function of the solute radius, which is in qualitative agreement with the Ogston theory. 

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 ... 

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