Thermodynamic Gibbs free energy

That is, the equilibrium constant for a reaction is equal to the ratio of the rate constants for the forward and reverse elementary reactions that contribute to the overall reaction. We can now see in kinetic terms rather than thermodynamic (Gibbs free energy) terms when to expect a large equilibrium constant K 1 (and products are favored) when k for the forward direction is much larger than k for the reverse direction. In this case, the fast forward reaction builds up a high concentration of products before reaching equilibrium (Fig. 13.21). In contrast, K 1 (and reactants are favored) when k is much smaller than k. Now the reverse reaction destroys the products rapidly, and so their concentrations are very low. 

The standard entropy difference between the reactant(s) of a reaction and the activated complex of the transition state, at the same temperature and pressure. Entropy of activation is symbolized by either A5 or and is equal to (A// - AG )IT where A// is the enthalpy of activation, AG is the Gibbs free energy of activation, and T is the absolute temperature (provided that all rate constants other than first-order are expressed in temperature-independent concentration units such as molarity). Technically, this quantity is the entropy of activation at constant pressure, and from this value, the entropy of activation at constant volume can be deduced. See Transition-State Theory (Thermodynamics) Gibbs Free Energy of Activation Enthalpy of Activation Volume of Activation Entropy and Enthalpy of Activation (Enzymatic)... 

The relation between the non-stoichiometry and the equilibrium oxygen pressure mentioned in Section 1.1 can be deduced from the phase rule. For the purpose of the derivation of the phase rule, we shall review fundamental thermodynamics. Gibbs free energy G is defined by the relation... 

The American convention would assign a positive value to E° for the Zn Zn2+(aq) half cell written as an oxidation, but a negative sign if written as a reduction. It is seen that the European convention refers to the invariant electrostatic potential of the electrode with respect to the SHE, whereas the American convention relates to the thermodynamic Gibbs free energy which is sensitive to the direction of the cell reaction. 

The key expression here is the Nernst equation that, under ideal conditions, relates the electrical potential (E) of a system to the standard thermodynamic Gibbs free energy (AG°) of the process and the concentrations (strictly activities) of the... 

The first and second conditions ensure the existence of the thermodynamic Gibbs free energy function or, using the mathematical term, the convex Lyapunov function for kinetic equations. The Lyapunov function is a strictly positive function with a nonpositive derivative. The one exception to this definition is that at the equilibrium point, the Lyapunov function equals zero. In physicochemical sciences, the Gibbs free energy is an extremely important Lyapunov function for understanding the stability of equilibria. 

In contrast, stability is an intrinsic property of a reactive intermediate. We define a structure as stable or stabilized if it is thermodynamically (Gibbs free energy) more stable than some reference structure, as discussed above. Flere, our focus is upon electronic stability rather than sterics, because for most reactive intermediates their electronics dominates their reactivity. For example, the benzyl cation is stabilized, because it is thermodynamically more stable than its reference, the methyl cation. However, under typical conditions the benzyl cation is not expected to be persistent. Fundamentally, stability is a thermodynamic notion, while persistence is a kinetic one. Stability is intrinsic to a structure, while persistence is very much context sensitive. We will do our best to keep these distinctions clear in the following sections. In the chemical literature, however, such precision in terminology is not always maintained. 

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, > )... 

Thermodynamics shows that equilibrium constants can be related to Gibbs free energies, AG, by Eq. (3). 

A quantity of great importance in chemical thermodynamics is the Gibbs free energy G. The latter is defined in terms of enthalpy H as... 

Figure 4.3 Behavior of thermodynamic variables at T for an idealized phase transition (a) Gibbs free energy and (b) entropy and volume.

Next we consider how to evaluate the factor 6p. We recognize that there is a local variation in the Gibbs free energy associated with a fluctuation in density, and examine how this value of G can be related to the value at equilibrium, Gq. We shall use the subscript 0 to indicate the equilibrium value of free energy and other thermodynamic quantities. For small deviations from the equilibrium value, G can be expanded about Gq in terms of a Taylor series ... 

Physical Equilibria and Solvent Selection. In order for two separate Hquid phases to exist in equiHbrium, there must be a considerable degree of thermodynamically nonideal behavior. If the Gibbs free energy, G, of a mixture of two solutions exceeds the energies of the initial solutions, mixing does not occur and the system remains in two phases. Eor the binary system containing only components A and B, the condition (22) for the formation of two phases is... 

P rtl IMol r Properties. The properties of individual components in a mixture or solution play an important role in solution thermodynamics. These properties, which represent molar derivatives of such extensive quantities as Gibbs free energy and entropy, are called partial molar properties. For example, in a Hquid mixture of ethanol and water, the partial molar volume of ethanol and the partial molar volume of water have values that are, in general, quite different from the volumes of pure ethanol and pure water at the same temperature and pressure (21). If the mixture is an ideal solution, the partial molar volume of a component in solution is the same as the molar volume of the pure material at the same temperature and pressure. 

Themodynamic State Functions In thermodynamics, the state functions include the internal energy, U enthalpy, H and Helmholtz and Gibbs free energies, A and G, respectively, defined as follows ... 

Equilibrium combustion product compositions and properties may be readily calculated using thermochemical computer codes which minimize the Gibbs free energy and use thermodynamic databases... 

Now, equations are given to identify whether the reactions are thermodynamically favorable. The method uses Gibbs free energy of formation for the reactants and products. 

Since the change in Gibbs free energy for the reaction is highly negative, the thermodynamics for the reaction at these temperatures are favorable (reaction promising). 

Thermodynamic data Data associated with the aspects of a reaction that are based on the thermodynamic laws of energy, such as Gibbs free energy, and the enthalpy (heat) of reaction. 

An important question for chemists, and particularly for biochemists, is, Will the reaction proceed in the direction written J. Willard Gibbs, one of the founders of thermodynamics, realized that the answer to this question lay in a comparison of the enthalpy change and the entropy change for a reaction at a given temperature. The Gibbs free energy, G, is defined as... 

Other thermodynamical functions, such as the enthalpy H, the entropy S and Gibbs free energy G, may be constructed from these relations. 

As noted above, it is very difficult to calculate entropic quantities with any reasonable accmacy within a finite simulation time. It is, however, possible to calculate differences in such quantities. Of special importance is the Gibbs free energy, as it is the natoal thermodynamical quantity under normal experimental conditions (constant temperature and pressme. Table 16.1), but we will illustrate the principle with the Helmholtz free energy instead. As indicated in eq. (16.1) the fundamental problem is the same. There are two commonly used methods for calculating differences in free energy Thermodynamic Perturbation and Thermodynamic Integration. 

This expression shows that the maximum possible useful work (i.e., reversible work) that can be obtained from any process occurring at constant temperature and pressure is a function of the initial and final states only and is independent of the path. The combination of properties U + PV - TS or H - TS occurs so frequently in thermodynamic analysis that it is given a special name and symbol, F, the free energy (sometimes called the Gibbs Free Energy). Using this definition, Equation 2-143 is written... 

Crystalline non-polar polymers and amorphous solvents Most polymers of regular structure will crystallise if cooled below a certain temperature, i.e. the melting point T. This is in accordance with the thermodynamic law that a process will only occur if there is a decrease in Gibbs free energy (-AF) in going from one state to another. Such a decrease occurs on crystallisation as the molecules pack regularly. 

Other thermodynamic functions described above in that the change in free energy AG is determined solely by the initial and final states of the system. The maximum work, or maximum available energy, defined in terms of the Gibbs free energy G, which is now called the free enthalpy, is... 

The importance of the Gibbs free energy and the chemical potential is very great in chemical thermodynamics. Any thermodynamic discussion of chemical equilibria involves the properties of these quantities. It is therefore worthwhile considering the derivation of equation 20.180 in some detail, since it forms a prime link between the thermodynamics of a reaction (AG and AG ) and its chemistry. 

In most applications, thermodynamics is concerned with five fundamental properties of matter volume (V), pressure (/ ), temperature (T), internal energy (U) and entropy (5). In addition, three derived properties that are combinations of the fundamental properties are commonly encountered. The derived properties are enthalpy (//). Helmholtz free energy (A) and Gibbs free energy ) ... 

In addition to the fundamental variables p, V, T, U, and S that we have described so far, three other thermodynamic variables are commonly encountered enthalpy Helmholtz free energy and Gibbs free energy. They are extensive variables that do not represent fundamental properties of the... 

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