Equilibrium constant standard free energy

Free Energy Change and Equilibrium Constant Standard free energy change and equilibrium constant... 

Through all these calculations of the effect of pH and metal ions on the ATP hydrolysis equilibrium, we have assumed standard conditions with respect to concentrations of all species except for protons. The levels of ATP, ADP, and other high-energy metabolites never even begin to approach the standard state of 1 M. In most cells, the concentrations of these species are more typically 1 to 5 mM or even less. Earlier, we described the effect of concentration on equilibrium constants and free energies in the form of Equation (3.12). For the present case, we can rewrite this as... 

The need for entropy values is bypassed when the van t Hoff equation (d In K/dT) =AH/RT2 is used. This can be integrated, either assuming AH is temperature-independent, or by incorporating a specific heat-temperature variation. This is the so-called second law method which contrasts with the third law method. In the latter method, the standard enthalpy is obtained from each equilibrium constant using free-energy functions of all the species present, for example... 

Finally, the relationship between equilibrium constant and free energy change in the standard state on the one hand, and rate constant and energy of... 

Numerous applications of standard electrode potentials have been made in various aspects of electrochemistry and analytical chemistry, as well as in thermodynamics. Some of these applications will be considered here, and others will be mentioned later. Just as standard potentials which cannot be determined directly can be calculated from equilibrium constant and free energy data, so the procedure can be reversed and electrode potentials used for the evaluation, for example, of equilibrium constants which do not permit of direct experimental study. Some of the results are of analjrtical interest, as may be shown by the following illustration. Stannous salts have been employed for the reduction of ferric ions to ferrous ions in acid solution, and it is of interest to know how far this process goes toward completion. Although the solutions undoubtedly contain complex ions, particularly those involving tin, the reaction may be represented, approximately, by... 

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

There can be many reasons why a reaction does not proceed to equilibrium. The standard free energy, then, does not define the amount of work that will be obtained from a reaction it is the maximum energy available under defined conditions. When the standard free energy is known, it can be used to determine the equilibrium constant of a reaction. This, it must be remembered, measures the extent to which a reaction may proceed, but it does not indicate the speed of a reaction or even that a reaction will occur at all. AF is related indirectly to the relative rates of enzyme-catalyzed reactions by the Haldane relationship (p. 12), but the absolute rate of reaction is determined by the amount of enzyme and substrate in a given system. 

To proceed fiirther, to evaluate the standard free energy AG , we need infonnation (experimental or theoretical) about the particular reaction. One source of infonnation is the equilibrium constant for a chemical reaction involving gases. Previous sections have shown how the chemical potential for a species in a gaseous mixture or in a dilute solution (and the corresponding activities) can be defined and measured. Thus, if one can detennine (by some kind of analysis)... 

The value of the standard free energy AG depends on the choice of reference state, as does the equilibrium constant. Thus it would be safer to write the equilibrium constant K for a gaseous reaction as... 

Equilibria in Solution The stability of a protein-ligand complex in solution is measured in terms of the equilibrium constant and the standard free energy of association based on it. For association of species P and L in solution to form a complex PL, i.e., for... 

Equilibrium constants for protein-small molecule association usually are easily measured with good accuracy it is normal for standard free energies to be known to within 0.5 kcal/mol. Standard conditions define temperature, pressure and unit concentration of each of the three reacting species. It is to be expected that the standard free energy difference depends on temperature, pressure and solvent composition AA°a also depends on an arbitrary choice of standard unit concentrations. 

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... 

One of the fundamental equations of thermo dynamics concerns systems at equilibrium and relates the equilibrium constant K to the dif ference in standard free energy (A6°) between the products and the reactants... 

The distribution coefficient is an equilibrium constant and, therefore, is subject to the usual thermodynamic treatment of equilibrium systems. By expressing the distribution coefficient in terms of the standard free energy of solute exchange between the phases, the nature of the distribution can be understood and the influence of temperature on the coefficient revealed. However, the distribution of a solute between two phases can also be considered at the molecular level. It is clear that if a solute is distributed more extensively in one phase than the other, then the interactive forces that occur between the solute molecules and the molecules of that phase will be greater than the complementary forces between the solute molecules and those of the other phase. Thus, distribution can be considered to be as a result of differential molecular forces and the magnitude and nature of those intermolecular forces will determine the magnitude of the respective distribution coefficients. Both these explanations of solute distribution will be considered in this chapter, but the classical thermodynamic explanation of distribution will be treated first. 

Equation (5-43) has the practical advantage over Eq. (5-40) that the partition functions in (5-40) are difficult or impossible to evaluate, whereas the presence of the equilibrium constant in (5-43) permits us to introduce the well-developed ideas of thermodynamics into the kinetic problem. We define the quantities AG, A//, and A5 as, respectively, the standard free energy of activation, enthalpy of activation, and entropy of activation from thermodynamics we now can write... 

At equilibrium, the ratio of concentrations is an equilibrium constant, so we can write the standard free energy change for the process as... 

The standard free-energy change for a reaction M + X - MX is also related to the equilibrium constant for the corresponding reaction... 

Because the equilibrium constant, Keq, and the standard free-energy change, AG°, both measure whether a reaction is favored, they are mathematical related by the equation... 

The equation just written is generally applicable to any system. The equilibrium constant may be the K referred to in our discussion of gaseous equilibrium (Chapter 12), or any of the solution equilibrium constants (Rw Ra, Rj, K, . . . ) discussed in subsequent chapters. Notice that AG° is the standard free energy change (gases at 1 atm, species in solution at 1M). That is why, in the expression for K, gases enter as their partial pressures in atmospheres and ions or molecules in solution as their molarities. 

One of the most important characteristics of a cell is its voltage, which is a measure of reaction spontaneity. Cell voltages depend on the nature of the half-reactions occurring at the electrodes (Section 18.2) and on the concentrations of species involved (Section 18.4). From the voltage measured at standard concentrations, it is possible to calculate the standard free energy change and the equilibrium constant (Section 18.3) of the reaction involved. 

As pointed out previously, the value of the standard cell voltage, E°, is a measure of the spontaneity of a cell reaction. In Chapter 17, we showed that the standard free energy change, AG°, is a general criterion for reaction spontaneity. As you might suppose, these two quantities have a simple relation to one another and to the equilibrium constant, K, for the cell reaction. 

In the introductory chapter we stated that the formation of chemical compounds with the metal ion in a variety of formal oxidation states is a characteristic of transition metals. We also saw in Chapter 8 how we may quantify the thermodynamic stability of a coordination compound in terms of the stability constant K. It is convenient to be able to assess the relative ease by which a metal is transformed from one oxidation state to another, and you will recall that the standard electrode potential, E , is a convenient measure of this. Remember that the standard free energy change for a reaction, AG , is related both to the equilibrium constant (Eq. 9.1)... 

Enzymes accelerate reaction rates by lowering the activation barrier AGp. While they may undergo transient modification during the process of catalysis, enzymes emerge unchanged at the completion of the reaction. The presence of an enzyme therefore has no effect on AG for the overall reaction, which is a function solely of the initial and final states of the reactants. Equation (25) shows the relationship between the equilibrium constant for a reaction and the standard free energy change for that reaction ... 

The standard free energy change can be calculated from the equilibrium constant... 

If two reactions differ in maximum work by a certain amount 8wm (= -SAG ), it follows from the Brpnsted relation [when taking into account the Arrhenius equation and the known relation between the equilibrium constant and the Gibbs standard free energy of reaction, A m = exp(-AGm/J r)] that their activation energies will differ by a fraction of this work, with the opposite sign ... 

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... 

In all metallurgical processing, heterogeneous reactions and the approach to equilibrium between two or more phases are of great importance. Much of the information on equilibrium is contained in the equilibrium constant which, as mentioned earlier, is related to the standard free energy change by the equation... 

It is instructive to consider the rationale underlying the various linear free energy correlations and to indicate in qualitative fashion how substituents may influence reaction rates. The relation between an equilibrium constant and the standard free energy change accompanying a reaction is given by... 

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