Isothermal change, free energy

JtXln K equilibrium isotherm, Standard free energy change is directly linked to equilibrium constant in activities... 

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. 

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). 

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

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

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. 

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 ... 

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 ... 

Here, the a s refer to the activities in the chosen arbitrary state. The concept of activity is presented separately in a later section. For the present, the activity of a species in a system may just be considered to be a function of its concentration in the system, and when the species is in a pure form (or in its standard state), its activity is taken to be unity. The activities ac, aD, aA, aB given above correspond to the actual conditions of the reaction, and these may or may not correspond to the state of equilibrium. Two special situations can be considered. In the first, the arbitrary states are taken to correspond to those for the system at equilibrium. Q would then become identical to the equilibrium constant K and, according to the Van t Hoff isotherm, AG would then be zero. In the second situation, all the reactants and the products are considered to be present as pure species or in their standard states, and aA, aB, ac, and aD are all equal to 1. Then (7=1 and the free energy change is given by... 

The Van t Hoff isotherm establishes the relationship between the standard free energy change and the equilibrium constant. It is of interest to know how the equilibrium constant of a reaction varies with temperature. The Varft Hoff isochore allows one to calculate the effect of temperature on the equilibrium constant. It can be readily obtained by combining the Gibbs-Helmholtz equation with the Varft Hoffisotherm. The relationship that is obtained is... 

From the Van t Hoff isotherm it follows that for the oxidation reaction considered, the standard free energy change at temperature T is given by... 

The standard Gibbs free energy change for a reaction refers to the process wherein the reaction proceeds isothermally, starting with stoichiometric quantities of reactants each in its standard state of unit activity and ending with products each at unit activity. In general it is nonzero and given by... 

Poisoning is caused by chemisorption of compounds in the process stream these compounds block or modify active sites on the catalyst. The poison may cause changes in the surface morphology of the catalyst, either by surface reconstruction or surface relaxation, or may modify the bond between the metal catalyst and the support. The toxicity of a poison (P) depends upon the enthalpy of adsorption for the poison, and the free energy for the adsorption process, which controls the equilibrium constant for chemisorption of the poison (KP). The fraction of sites blocked by a reversibly adsorbed poison (0P) can be calculated using a Langmuir isotherm (equation 8.4-23a) ... 

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