Free energy units

Substituting in equation 11 the known experimental parameters for phenol dissociation (AG, = 13.8 kcalmol" calculated from the ground-state equilibrium constant, pX, = 10.0), AGt((PhO ) — (PhOH)) of the phenolate/phenol system is about —76 kcalmoH, which is about 10% less than the accepted value for the electrostatic solvation energy of the chloride anion in water, AGe(Cr) = —85 kcalmol". These simple considerations imply that the AGt((PhO ) — (PhOH)) contribution to the overall free energy of solvation is largely electrostatic, and that relatively small differences in the gas-phase proton affinity of the base and in specific solvent-solute interactions of the photoacid and the base determine the relatively narrow (in free-energy units) acidity scale in aqueous solution. It... 

Check Note that in parts (a) and (c), we made the free energy units (kJ) consistent with the units in R (J). For significant figures in addition and subtraction, we retain one digit to the right of the decimal place in part (c). 

Displacing the boundary by A/ increases the film area by 2dAl and requires that work equal to F Al is expended (Figure 1.1). Per unit of the newly formed surface, this work is F All2dAl = F 2d=a. Consequently, the meaning of a now is specific surface free energy (units of work per unit surface area). In a thermodynamic sense, the term free means that this (equilibrium) process takes place at constant temperature and volume. The identity of those two notions, that is, of the surface tension and the surface free energy, is true only for common (unassociated) liquids. For solids, the situation is more complex [1-3]. 

It may be readily shown [28,31] that the generalized entry coefficient is related to the interaction free energy/unit area, W(h), of the pseudoemulsion film where... 

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 >... 

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... 

The interfacial free energy per unit area is given by the adhesion force Fo/lrR", estimate the Hamaker constant responsible for the adhesion force in the crossed-cylinder geometry illustrated in the inset to Fig. VI-6. 

In Chapter III, surface free energy and surface stress were treated as equivalent, and both were discussed in terms of the energy to form unit additional surface. It is now desirable to consider an independent, more mechanical definition of surface stress. If a surface is cut by a plane normal to it, then, in order that the atoms on either side of the cut remain in equilibrium, it will be necessary to apply some external force to them. The total such force per unit length is the surface stress, and half the sum of the two surface stresses along mutually perpendicular cuts is equal to the surface tension. (Similarly, one-third of the sum of the three principal stresses in the body of a liquid is equal to its hydrostatic pressure.) In the case of a liquid or isotropic solid the two surface stresses are equal, but for a nonisotropic solid or crystal, this will not be true. In such a case the partial surface stresses or stretching tensions may be denoted as Ti and T2-... 

Referring to Section V-2, the double-layer system associated with a surface whose potential is some value j/o requires for its formation a free energy per unit area or a t of... 

Figure A2.5.9. (Ap), the Helmholtz free energy per unit volume in reduced units, of a van der Waals fluid as a fiinction of the reduced density p for several constant temperaPires above and below the critical temperaPire. As in the previous figures the llill curves (including the tangent two-phase tie-lines) represent stable siPiations, the dashed parts of the smooth curve are metastable extensions, and the dotted curves are unstable regions. See text for details.

The central quantity of interest in homogeneous nucleation is the nucleation rate J, which gives the number of droplets nucleated per unit volume per unit time for a given supersaturation. The free energy barrier is the dommant factor in detenuining J J depends on it exponentially. Thus, a small difference in the different model predictions for the barrier can lead to orders of magnitude differences in J. Similarly, experimental measurements of J are sensitive to the purity of the sample and to experimental conditions such as temperature. In modem field theories, J has a general fonu... 

A way of looking at the points raised in the previous section is to compare energy distributions in two systems whose free energies we wish to relate. In particular, consider measuring, in a simulation of system 0, the fiinction Pq(AE), i.e., the probability density per unit AE of configurations for which and differ by the... 

In this relationship. S is alkane solubility, A is the cavity surface area and a is the hydrophobic free energy per unit area. Extensive fitting of this equation [24] yields a value of 88 kJ mol A for the proportionality constant a. This value corresponds to an unfavourable free energy of about 3.6 kJ mol for the transfer of a CH2 group to aqueous solution. 

When the e.m.f. of a cell is measured by balancing it against an external voltage, so that no current flows, the maximum e.m.f. is obtained since the cell is at equilibrium. The maximum work obtainable from the cell is then nFE J, where n is the number of electrons transferred, F is the Faraday unit and E is the maximum cell e.m.f. We saw in Chapter 3 that the maximum amount of work obtainable from a reaction is given by the free energy change, i.e. - AG. Hence... 

The expressions appearing in the exponents are the free energy change of the NA-water system per unit mole in the U A and A—>B conformational transitions. The terms AF p, introduced to take into account the... 

The expressions for the changes in the free energy of the hydration shell of the NA (per NA units mole) in the U A and A— B conformational transitions should be written in the form ... 

Equilibrium constants for protein-small molecule association usually are easily measured with good accuracy it is normal for standard free energies to be known to within 0.5 kcal/mol. Standard conditions define temperature, pressure and unit concentration of each of the three reacting species. It is to be expected that the standard free energy difference depends on temperature, pressure and solvent composition AA°a also depends on an arbitrary choice of standard unit concentrations. 

The confinement term is unique because it alone causes a dependence of the binding free energy on the choice of unit concentration in the standard state the volume available per ligand molecule in the free state, and hence the compression factor, depend on the unit concentration. 

Jayaram, B., Beveridge, D. L. A simple method to estimate free energy from molecular simulation Renormalization on the unit interval. J. Phys. Chem. 94 (1990) 7288-7293... 

Table 3. Conformational free energy simulation of linear DPDPE. Changes in free energy and its components. Units kcal/mol...

Mesoscale simulations model a material as a collection of units, called beads. Each bead might represent a substructure, molecule, monomer, micelle, micro-crystalline domain, solid particle, or an arbitrary region of a fluid. Multiple beads might be connected, typically by a harmonic potential, in order to model a polymer. A simulation is then conducted in which there is an interaction potential between beads and sometimes dynamical equations of motion. This is very hard to do with extremely large molecular dynamics calculations because they would have to be very accurate to correctly reflect the small free energy differences between microstates. There are algorithms for determining an appropriate bead size from molecular dynamics and Monte Carlo simulations. 

As in the qualitative discussion above, let 7 be the Gibbs free energy per unit area of the interface between the crystal and the surrounding hquid. This is undoubtedly different for the edges of the plate than for its faces, but we... 

Use Eq. (4.14), the results in Fig. 4.5, and the data in Table 4.1 to estimate a value for 7 for polyethylene. Figure 4.10 shows the unit cell of polyethylene Fig. 4.10b shows the equivalent of two chains emerging from an area 0.740 by 0.493 nm. On the basis of the calculated value of 7 and the characteristics of the unit cell, estimate the free energy of the fold surface per mole of repeat units. 

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