Using equation 7.3, U can also be expressed in terms of the free energy F [Pg.327]

The basic problem of statistical mechanics is to evaluate the sum-over-states in equation 7.2 and obtain Z and F as functions of T and any other variables (such as external magnetic fields) that might appear in %. Any thermodynamic observable of interest can then be obtained in a straightforward manner from equation 7.5. In practice, however, the sum-over-states often turns out to be prohibitively difficult to evaluate. Instead, the physical system is usually replaced with a simpler model system and/or some simplifying approximations are made so that the sum-over-states can be evaluated directly. [Pg.327]

Helmholtz free energy, represented by the symbol A, is defined as [Pg.240]

In this expression U is the internal energy, T is the absolute temperature and S is the entropy. [Pg.240]

Gibbs free energy, represented by the symbol G (the symbol F is also sometimes used in place of symbol G), is defined as [Pg.240]

Keeping in view the expression for G, one obtains, in the differential form, [Pg.240]

It has been seen earlier that by combining the first and the second laws, the following relationship can be arrived at [Pg.240]

A= Work function, Helmholtz Free Energy °C = Degrees Centigrade (temperature) [Pg.8]

C = Heat Capacity at constant pressure C = Heat Capacity at constant volume d = Differential [Pg.8]

E = EMF, or Electromotive Force, or Cell Potential (In the context of galvanic cells) also see below [Pg.8]

W = Number of Micro-states (In the context of Statistical Thermodynamics) [Pg.9]

An ideal gas system assisting in simulating a real process in a reversible manner. 37 [Pg.10]

Nearly 10 years after Zwanzig published his perturbation method, Benjamin Widom [6] formulated the potential distribution theorem (PDF). He further suggested an elegant application of PDF to estimate the excess chemical potential -i.e., the chemical potential of a system in excess of that of an ideal, noninteracting system at the same density - on the basis of the random insertion of a test particle. In essence, the particle insertion method proposed by Widom may be viewed as a special case of the perturbative theory, in which the addition of a single particle is handled as a one-step perturbation of the liquid. [Pg.3]

Finding the best estimate of the free energy difference between two canonical ensembles on the same configurational space, for which finite samples are available, is a nontrivial problem. Charles Bennett [11] addressed this problem by developing the acceptance ratio estimator, which corresponds to the minimum statistical [Pg.3]

3 Early Successes and Failures of Free Energy Calculations [Pg.4]

As we have already pointed out, the theoretical basis of free energy calculations were laid a long time ago [1,4,5], but, quite understandably, had to wait for sufficient computational capabilities to be applied to molecular systems of interest to the chemist, the physicist, and the biologist. In the meantime, these calculations were the domain of analytical theories. The most useful in practice were perturbation theories of dense liquids. In the Barker-Henderson theory [13], the reference state was chosen to be a hard-sphere fluid. The subsequent Weeks-Chandler-Andersen theory [14] differed from the Barker-Henderson approach by dividing the intermolecular potential such that its unperturbed and perturbed parts were associated with repulsive and attractive forces, respectively. This division yields slower variation of the perturbation term with intermolecular separation and, consequently, faster convergence of the perturbation series than the division employed by Barker and Henderson. [Pg.4]

Analytical perturbation theories led to a host of important, nontrivial predictions, which were subsequently probed by and confirmed in numerical simulations. The elegant theory devised by Lawrence Pratt and David Chandler [15] to explain the hydrophobic effect constitutes a noteworthy example of such predictions. [Pg.4]

Click Coached Problems for a self-study module on putting Aff and AS together. [Pg.458]

A spontaneous process is capable of producing useful work. [Pg.458]

The sign of the free energy change can be used to determine the spontaneity of a reaction carried out at constant temperature and pressure. [Pg.458]

If AG is positive, the reaction will not take place spontaneously. Instead, the reverse reaction will be spontaneous. [Pg.458]

Role of Solvent in Ligand Binding in the Active Site of [Pg.1]

Future Prospects for Binding Free Energy Studies on [Pg.1]

Howard Hughes Medical Institute, Department of Chemistry and Biochemistry, and Department of Pharmacology, University of California, San Diego, La Jolla, CA 92093 [Pg.3]

Calculation of the standard free energy of binding itself can be viewed as a special case of the above, in which one of the pair of ligands contains no atoms.10 Some care is required to be sure that such calculations yield answers that actually correspond to the desired standard state.11 12 Unfortunately, many calculations of free energies of binding have not made appropriate contact with a standard state, so that results in the literature must be interpreted with caution. [Pg.4]

It was noted above that a continuum treatment of the solvent can be helpful, although representing certain solvent molecules explicitly may be necessary. The expressions for handling the free energy contributions in such hybrid models have been derived by Gilson et al.11 [Pg.5]

As hydrophobic surfaces contact each other, the ordered water molecules that occupied the surfaces are liberated to go about their normal business. The increased entropy (disorder) of the water is favorable and drives (causes) the association of the hydrophobic surfaces. [Pg.10]

The surface area of a hydrophobic molecule determines how unfavorable the interaction between the molecule and water will be. The big- [Pg.10]

These are very short-range interactions between atoms that occur when atoms are packed very closely to each other. [Pg.11]

When the hydrophobic effect brings atoms very close together, van der Waals interactions and London dispersion forces, which work only over very short distances, come into play. This brings things even closer together and squeezes out the holes. The bottom line is a very compact, hydrophobic core in a protein with few holes. [Pg.11]

Hydrogen bonding means sharing a hydrogen atom between one atom that has a hydrogen atom (donor) and another atom that has a lone pair of electrons (acceptor) [Pg.11]

At the triple point, the free energies of each phase are equal [Pg.149]

The total free energy of the system is then made up of the molar free energy times the total number of moles of the liquid plus G, the surface free energy per unit area, times the total surface area. Thus [Pg.48]

Gy = molar free energy of the liquid G = molar free energy of the gas [Pg.149]

Gg = partial free energy of component i in the gas phase at temperature T and pressure P [kJ/kmol] [Pg.151]

G = Gibbs molar free energy S = molar entropy F = Helmholtz free molar energy H = molar enthalpy U = molar internal energy [Pg.148]

Gibbs function (see Gibbs free energy, free energy) gram [Pg.7]

The total surface energy generally is larger than the surface free energy. It is frequently the more informative of the two quantities, or at least it is more easily related to molecular models. [Pg.49]

CALCULATE RESIDUAL CONTRIBUTION TO EXCESS FREE ENERGY lAO DO 141 l =lfN [Pg.312]

Gibbs-Helmholtz equation This equation relates the heats and free energy changes which occur during a chemical reaction. For a reaction carried out at constant pressure [Pg.190]

The relations which permit us to express equilibria utilize the Gibbs free energy, to which we will give the symbol G and which will be called simply free energy for the rest of this chapter. This thermodynamic quantity is expressed as a function of enthalpy and entropy. This is not to be confused with the Helmholtz free energy which we will note sF (L" j (j, > ) [Pg.148]

AP such that the work against this pressure difference APAvr dr is just equal to the decrease in surface free energy. Thus [Pg.5]

A quantitative theory of rate processes has been developed on the assumption that the activated state has a characteristic enthalpy, entropy and free energy the concentration of activated molecules may thus be calculated using statistical mechanical methods. Whilst the theory gives a very plausible treatment of very many rate processes, it suffers from the difficulty of calculating the thermodynamic properties of the transition state. [Pg.402]

The determination of equilibria is done theoretically via the calculation of free energies. In practice, the concept of fugacity is used for which the unit of measurement is the bar. The equation linking the fugacity to the free energy is written as follows > [Pg.149]

This is exact—see Problem 11-8. Notice that Eq. 11-14 is exactly what one would write, assuming the meniscus to be hanging from the wall of the capillary and its weight to be supported by the vertical component of the surface tension, 7 cos 6, multiplied by the circumference of the capillary cross section, 2ar. Thus, once again, the mathematical identity of the concepts of surface tension and surface free energy is observed. [Pg.13]

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