Free Energy, and the Equilibrium Constant

The condition for equilibrium is determined from the combined form of the first and second laws of thermodynamics that is, 

It is apparent from the definition of enthalpy H and the introduction of the concept of the Gibbs free energy G 

Recall that P and T are intensive properties that are independent of the size of mass of the system, whereas E, El, G, and S (as well as V and n) are extensive properties that increase in proportion to mass or size. By writing the general relation for the total derivative of G with respect to the variables in Eq. (1.16), one obtains 

The concept of the chemical potential is introduced here because this property plays an important role in reacting systems. In this context, one may consider that a reaction moves in the direction of decreasing chemical potential, reaching equilibrium only when the potential of the reactants equals that of the products [3], 

from Eq. (1.16) the criterion for equilibrium for combustion products of a chemical system at constant T and P is 

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Changes in free energy and the equilibrium constants for Reactions 1, 2, 3, and 4 are quite sensitive to temperature (Figures 2 and 3). These equilibrium constants were used to calculate the composition of the exit gas from the methanator by solving the coupled equilibrium relationships of Reactions 1 and 2 and mass conservation relationships by a Newton-Raphson technique it was assumed that carbon was not formed. Features of the computer program used were as follows (a) any pressure and temperature may be specified (b) an inert gas may be present (c) after... 

C19-0039. Write a paragraph explaining the linkages among cell potential, free energy, and the equilibrium constant. 

The thermodynamic heats of adsorption (AH) were calculated using equation 3, which is derived as follows from the relationship between free energy and the equilibrium constant ... 

We can now utilize some of the statistical mechanics relationships derived in Chapter 8 to find expressions for the free energy and the equilibrium constant in term of the molecular partition functions. From the definition of the free energy (Eq. 9.1) the expression for the enthalpy of an ideal gas (Eq. 8.121), and recalling that Ho = Eq (for an ideal gas), we obtain... 

Equation 10 allows us to write the relation between reaction free energy and the equilibrium constant at two temperatures, T, and T2 ... 

We can now derive a relationship between free energy and the equilibrium constant. At equilibrium, AG for a reaction is zero and the reaction quotient Q equals the equilibrium constant K. Substituting AG = 0 and Q = K into the equation... 

TABLE 10.7 Qualitative Relationship Between the Change in Standard Free Energy and the Equilibrium Constant for a Given Reaction... 

Keeping in mind the relationship between standard free energy and the equilibrium constant, it is very easy to modify the Langmuir isotherm to take into account the variation of AG with 0. To do this,... 

The relation between the standard free energy and the equilibrium constant of a reaction can be readily derived. This equation is important because it displays the energetic relation between products and reactants in terms of their concentrations. At equilibrium, A G = 0. Equation 1 then becomes... 

This expression (2.3) follows on from another linear relation seen in thermodynamics linking the variation in free energy and the equilibrium constant K, (AG = —RT InK), for a homologous family of compounds in which each term differs from the preceding one by a supplementary CH2 unit. Since K= kj3 therefore fp = Ktfjf/fS, then log will increase as In K for the homologous family In fp = In K + (In fM//3). 

Before we discuss redox titration curves based on reduction-oxidation potentials, we need to learn how to calculate equilibrium constants for redox reactions from the half-reaction potentials. The reaction equilibrium constant is used in calculating equilibrium concentrations at the equivalence point, in order to calculate the equivalence point potential. Recall from Chapter 12 that since a cell voltage is zero at reaction equilibrium, the difference between the two half-reaction potentials is zero (or the two potentials are equal), and the Nemst equations for the halfreactions can be equated. When the equations are combined, the log term is that of the equilibrium constant expression for the reaction (see Equation 12.20), and a numerical value can be calculated for the equilibrium constant. This is a consequence of the relationship between the free energy and the equilibrium constant of a reaction. Recall from Equation 6.10 that AG° = —RT In K. Since AG° = —nFE° for the reaction, then... 

In Chapter 20, we discussed the relationship of useful work, free energy, and the equilibrium constant. In this section, we examine this relationship in the context of electrochemical cells and see the effect of concentration on cell potential. 

The above calculation used free energies and the equilibrium constants for specific chemical reactions. For this reason the calculations apply only to the system, species, and reactions considered. It is possible to generalize free energy based equilibrium calculations so that they can be used with systems and species chosen arbitrarily, in a manner analogous to the program EQBRM described above. 

FREE ENERGY AND THE EQUILIBRIUM CONSTANT Finally, we consider how the standard free-energy change for a chemical reaction can be used to calculate the equilibrium constant for the reaction. 

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