Isotherm free energies

FIGURE 1.29 Free energy isotherms for the interaction of hydrophobic surfaces in aqueous solutions of alcohols (1-4) curve 2 shows calculated values of a,2 for solutions of ethanol. (Redrawn from Shchukin, E.D. 

Depending on the physical-chemical nature of the adsorbent solid surface and on the nature of the adsorbing molecules, the r(p), F(c), and the interfacial free energy isotherms may have significantly different shapes. For example, the Tip) isotherm may contain a region corresponding to the adsorption of vapors in narrow capillaries. 

Figure 4 Free-energy isotherms of the liquid and two polymorphic forms Sx and S2 exhibiting (a) enantiotropic and (b) monotropic transformation.

The present discussion is restricted to an introductory demonstration of how, in principle, adsorption data may be employed to determine changes in the solid-gas interfacial free energy. A typical adsorption isotherm (of the physical adsorption type) is shown in Fig. X-1. In this figure, the amount adsorbed per gram of powdered quartz is plotted against P/F, where P is the pressure of the adsorbate vapor and P is the vapor pressure of the pure liquid adsorbate. 

A somewhat subtle point of difficulty is the following. Adsorption isotherms are quite often entirely reversible in that adsorption and desorption curves are identical. On the other hand, the solid will not generally be an equilibrium crystal and, in fact, will often have quite a heterogeneous surface. The quantities ys and ysv are therefore not very well defined as separate quantities. It seems preferable to regard t, which is well defined in the case of reversible adsorption, as simply the change in interfacial free energy and to leave its further identification to treatments accepted as modelistic. 

Some representative plots of entropies of adsorption are shown in Fig. XVII-23, in general, T AS2 is comparable to Ah2, so that the entropy contribution to the free energy of adsorption is important. Notice in Figs. XVII-23 i and b how nearly the entropy plot is a mirror image of the enthalpy plot. As a consequence, the maxima and minima in the separate plots tend to cancel to give a smoothly varying free energy plot, that is, adsorption isotherm. 

Just as one may wish to specify the temperature in a molecular dynamics simulation, so may be desired to maintain the system at a constant pressure. This enables the behavior of the system to be explored as a function of the pressure, enabling one to study phenomer such as the onset of pressure-induced phase transitions. Many experimental measuremen are made under conditions of constant temperature and pressure, and so simulations in tl isothermal-isobaric ensemble are most directly relevant to experimental data. Certai structural rearrangements may be achieved more easily in an isobaric simulation than i a simulation at constant volume. Constant pressure conditions may also be importai when the number of particles in the system changes (as in some of the test particle methoc for calculating free energies and chemical potentials see Section 8.9). 

An early application of the free energy perturbation method was the determination of t] tree energy required to create a cavity in a solvent. Postma, Berendsen and Haak determin the free energy to create a cavity (A = 1) in pure water (A = 0) using isothermal-isobai... 

Umbrella sampling can give free-energy differences, but not absolute free energies Usually done in NPT—isothermal/isobaric ensembles, including a water box... 

Fig. 7. Unfolding (a) isotherm, where the half-maximal unfolding for this protein occurs at 2.6 M denaturant and (b) free energy where in the absence of denaturant, the protein has an extrapolated stability,, of 17.6 kj/mol (4.2 kcal/mol) as shown. To convert to cal, divide by 4.184.

The Van t Hoff isotherm identifies the free energy relationship for bulk chemical reactions. 

The change in total energy or free energy, under isothermal conditions, is then given by... 

An isothermal reaction can proceed spontaneously only if the total Gibbs free energy of the system decreases, i.e. the free energy of the reactants must be greater than the free energy of the products. For a reaction... 

The relationship between the change in free energy AG and the equilibrium constant K is given by the vant Hofif isotherm... 

If a piece of metal, such as silver, is dipping into a solvent, and a positive atomic core is taken from the surface into the solvent, the ion is again surrounded by its electrostatic field but free energy has been lost by the dielectric, and a relatively small amount of work has had to be done. The corresponding potential-energy curve (Fig. 96) is therefore much less steep and has a much shallower minimum than that of Fig. 9a. For large distances d from a plane metal surface this curve is a plot of — c2/4td where t is the dielectric constant of the medium at the temperature considered The curve represents the work done in an isothermal removal of the positive core. 

Our problem is to determine how the changes of total and free energy, AU and A P, or, what are the same, the heat absorption at constant configuration and the maximum work, Qx and At, of an isothermal and reversible process, alter with the temperature of execution of the process. 

Examples.—(1) If a mol of gas expands isothermally and reversibly from volume Vi to volume V2 the diminution of free energy is ... 

If a gas obeys Boyle s law the diminution of potential on isothermal reversible expansion is equal to the diminution of free energy, and both are equal to the maximum work. 

Again, if we consider the initial substances in the state of liquids or solids, these will have a definite vapour pressure, and the free energy changes, i.e., the maximum work of an isothermal reaction between the condensed forms, may be calculated by supposing the requisite amounts drawn off in the form of saturated vapours, these expanded or compressed to the concentrations in the equilibrium box, passed into the latter, and the products then abstracted from the box, expanded to the concentrations of the saturated vapours, and finally condensed on the solids or liquids. Since the changes of volume of the condensed phases are negligibly small, the maximum work is again ... 

We shall now calculate the diminution of free energy which results from the admixture of Ni mols of [1] and N2 mols of [2], both in the liquid state. The simplest method is an application of equation (13) of 52, which states that the work done in the isothermal and reversible execution of a process is equal to the diminution of free energy ... 

Now suppose N2 mols of pure liquid [2] are isothermally and reversibly distilled into Ni mols of pure liquid [1]. The change of free energy for distillation of 8N2 mols of [2] into a mixture over which the partial pressure is p2 is, as we have shown ... 

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