Equilibria and free energy

To understand the relationship between free energy and equilibrium, let s consider the following simple hypothetical reaction  

Note that Ca and Cb are defined here as the total free energies of A and B and are dependent on the moles of A and B as well as the pressures of A and B. 

Total free energy of A = Ga Total free energy of B = Gb 

Suppose that for the experiment just described, the plot of free energy ver-sns the mole fraction of A reacted is defined as shown in Fig. 10.15(a). In this experiment minimum free energy is reached when 75% of A has been changed to B. At this point the pressure of A is 0.25 times the original pressure, or 

For the reaction A(g) B(g) the pressure is constant during the reaction, since the same number of gas molecules is always present. 

Having calculated the standai d values AyW and S foi the participants in a chemical reaction, the obvious next step is to calculate the standard Gibbs free energy change of reaction A G and the equilibrium constant from 

P)(]uation (3-34) makes clear a difficulty that will bedevil us throughout computational chemistry Although the accuracy of compi.itatknial chemistry is extremely high, the demands placed on our results ntay he even higher. In the present case, the equilibrium constant is dependent on the exponential of the standard free energy 

Because of the severe demands placed on us for aceuraey if we tre to calculate an equilibrium constant, let us choose a simple reaction, the isomerization of but-2-eiic, 

Using MMd. calculate A H and. V leading to ATT and t his reaction has been the subject of computational studies (Kar, Len/ and Vaughan, 1994) and experimental studies by Akimoto et al, (Akimoto, Sprung, and Pitts. 1972) and by Kapej n et al, (Kapeijn, van der Steen, and Mol, 198.V), Quantum mechanical systems, including the quantum harmonic oscillator, will be treated in more detail in later chapters. 

Minimal input lilcs fur cyclupentciie and cyclopciitanc can be constructed from a pentag(.)n drawn un grapli paper the way minimal ethylene was drawn (Fig. 4-4). F(.)r more complicated molecules, however, the draw function of PCMODEL or some similar file constnicting program becomes less a convenience and more a necessity, 

In Fig. 17.7(a) the ball will roll to point B. This diagram is analogous to a phase change. For example, at 25°C ice will spontaneously change completely to liquid water, because the latter has the lowest free energy. In this case liquid water is the only choice. There is no intermediate mixture of ice and water with lower free energy. 

The situation is different for a chemical reaction system, as illustrated in Fig. 17.7(b). In Fig. 17.7(b) the ball will not get to point B because there is a lower potential energy at point C. Like the ball, a chemical system will seek the lowest possible free energy, which, for reasons we will discuss below, is the equilibrium position. 

Therefore, although the value of AG for a given reaction system tells us whether the products or reactants are favored under a given set of conditions, it does not mean that the system will proceed to pure products (if AG is negative) or remain at pure reactants (if AG is positive). Instead, the system will spontaneously go to the equilibrium position, the lowest possible free energy available to it. hi the next section we will see that the value of AG° for a particular reaction teUs us exactly where this position will be. 

When the components of a given chemical reaction are mixed, they will proceed, rapidly or slowly depending on the kinetics of the process, to the equilibrium position. In Chapter 13 we defined the equilibrium position as the point at which the forward and reverse reaction rates are equal. In this chapter we look at equilibrium from a thermodynamic point of view, and we find that the equilibrium point occurs at the lowest value of free energy available to the reaction system. As it turns out, the two definitions give the same equilibrium state, which must be the case for both the kinetic and thermodynamic models to be valid. 

The free energy profile for A(g) B(g) in a system containing 1.0 mol (A plus B) at Ptotal = 2.0 atm. Each point on the curve corresponds to the total free energy of the system for a given combination of A and B. 

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See the box entitled Enthalpy Free Energy and Equilibrium Constant accompanying this section for a discussion of these relationships... 

For biochemical reactions in which hydrogen ions (H ) are consumed or produced, the usual definition of the standard state is awkward. Standard state for the ion is 1 M, which corresponds to pH 0. At this pH, nearly all enzymes would be denatured, and biological reactions could not occur. It makes more sense to use free energies and equilibrium constants determined at pH 7. Biochemists have thus adopted a modified standard state, designated with prime ( ) symbols, as in AG°, AH°, and so on. For values determined... 

The units of AG are joules (or kilojoules), with a value that depends not only on E, but also on the amount n (in moles) of electrons transferred in the reaction. Thus, in reaction A, n = 2 mol. As in the discussion of the relation between Gibbs free energy and equilibrium constants (Section 9.3), we shall sometimes need to use this relation in its molar form, with n interpreted as a pure number (its value with the unit mol struck out). Then we write... 

Relation between standard reaction Gibbs free energy and equilibrium constant van t Hoff equation ... 

The second key equation having to do with equilibrium relates to the difference in energy between reactants and products. The particular form of energy important in this relationship is free energy, G. The difference in free energy between product and reactant states is AG = Gp Q y j - G, The relationship between free energy and equilibrium is... 

The importance of interactions amongst point defects, at even fairly low defect concentrations, was recognized several years ago. Although one has to take into account the actual defect structure and modifications of short-range order to be able to describe the properties of solids fully, it has been found useful to represent all the processes involved in the intrinsic defect equilibria in a crystal (with a low concentration of defects), as well as its equilibrium with its external environment, by a set of coupled quasichemical reactions. These equilibrium reactions are then handled by the law of mass action. The free energy and equilibrium constants for each process can be obtained if we know the enthalpies and entropies of the reactions from theory or... 

Conceptualization of the Ion Exchange Sorptive Reactions , Free Energies, and Equilibrium Constants... 

Enthalpy, Free Energy, and Equilibrium Constant CHAPTER 4... 

We mentioned that biochemists usually define the standard state of protons as 10-7 m and report values of free energy and equilibrium constants for solutions at pH 7. These values are designated by a prime and written as AG°, AG and K q. Unprimed symbols are used to designate values based on a standard state of 1 m for protons (pH 0). For a reaction that releases one proton, the relationship between K eq and K q is K eq = 107/feq. In evaluating the standard free energies AG° and AG°, it is critical to use the equilibrium constants K eq and Keq, respectively, because these can be very different quantities. 

By using the relationship 2.7 between free energy and equilibrium, Equation 2.6 can be rewritten as Equation 2.8, which in turn simplifies to Equation 2.9. 

Chapter 17 Thermodynamics Entropy Free Energy and Equilibrium... 

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