Ways of Expressing Equilibrium Constants

To use equilibrium constants, we must express them in terms of the reactant and product concentrations. Our oidy guidance is the law of mass action [Equation (15.2)]. However, because the concentrations of the reactants and products can be expressed in different nnits and becanse the reacting species are not always in the same phase, there may be more than one way to express the equilibrium constant for the same reaction. To begin with, we will consider reactions in which the reactants and products are in the same phase. 

The term homogeneous equilibrium applies to reactions in which all reacting species are in the same phase. An example of homogeneous gas-phase equilibrimn is the dissociation of N2O4. The equilibrium constant, as given in Equation (15.1), is 

Note that the subscript in denotes that the concentrations of the reacting species are expressed in moles per hter. The concentrations of reactants and products in gaseous reactions can also be ejqtressed in terms of their partial pressures. From Equation (5.8) we see that at constant terrqrerature the pressme P of a gas is directly related to the concentration in moles per hter of the gas that is, P = (n/V)RT. Thus, for the equiUbrium process 

In general, is not equal to Kp, because the partial pressmes of reactants and products are not equal to their concentrations expressed in moles per liter. A simple relationship between Kp and K can be derived as follows. Let us consider this equilibrium in the gas phase  

Substituting these relations into the expression for Kp, we obtain 

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The standard-state electrochemical potential, E°, provides an alternative way of expressing the equilibrium constant for a redox reaction. Since a reaction at equilibrium has a AG of zero, the electrochemical potential, E, also must be zero. Substituting into equation 6.24 and rearranging shows that... 

As the LiF example illustrates, the most direct way to determine the value of an equilibrium constant is to mix substances that can undergo a chemical reaction, wait until the system reaches equilibrium, and measure the concentrations of the species present once equilibrium is established. Although the calculation of an equilibrium constant requires knowledge of the equilibrium concentrations of all species whose concentrations appear in the equilibrium constant expression, stoichiometric analysis often can be used to deduce the concentration of one... 

We have said that AG ° is a way of expressing the equilibrium constant for a reaction. For reaction (1) above,... 

The value of the equilibrium constant for a reaction makes it possible to judge the extent of reaction, predict the direction of reaction, and calculate equilibrium concentrations (or partial pressures) from initial concentrations (or partial pressures). The farther the reaction proceeds toward completion, the larger the value of Kc. The direction of a reaction not at equilibrium depends on the relative values of Kc and the reaction quotient Qc, which is defined in the same way as Kc except that the concentrations in the equilibrium constant expression are not necessarily equilibrium concentrations. If Qc Kcr net reaction goes from left to right to attain equilibrium if Qc > Kc/ net reaction goes from right to left if Qc = Kc/ the system is at equilibrium. 

The Law of Chemical Equilibrium is based on the constancy of the equilibrium constant. This means that if one disturbs the equilibrium, for example by adding more reactant molecules, there will be an increase in the number of product molecules in order to maintain the producl/reactant ratio unchanged and thus preserving the numerical value of the equilibrium constant. The Le Chatelier Principle expresses this as follows If an external stress is applied to a system in equilibrium, the system reacts in such a way as to partially relieve the stress. In our present experiment, we demonstrate the Le Chatelier Principle in two manners (a) disturbing the equilibrium by changing the concentration of a product or reactant (b) changing the temperature. 

The three columns in the table are three different ways of expressing the same information. However, just looking at the percentages column, it is not immediately obvious to see how much more of the equatorial conformer there is—after all, the percentages of equatorial conformer for methyl, ethyl, isopropyl, f-butyl, and phenyl-cyclohexanes are all 95% or more. Looking at the equilibrium constants gives a much clearer picture... 

A more general way of expressing solubilities is through the vapor—liquid equilibrium constant m defined by... 

Chapters 3-5 have described the calculation of various transformed thermodynamic properties of biochemical reactants and reactions from standard thermodynamic properties of species, but they have not discussed how these species properties were determined. Of course, some species properties came directly out of the National Bureau of Standard Tables (1) and CODATA Tables (2). One way to calculate standard thermodynamic properties of species not in the tables of chemical thermodynamic properties is to express the apparent equilibrium constant K in terms of the equilibrium constant K of a reference chemical reaction, that is a reference reaction written in terms of species, and binding polynomials of reactants, as described in Chapter 2. In order to do this the piiTs of the reactants in the pH range of interest must be known, and if metal ions are bound, the dissociation constants of the metal ion complexes must also be known. For the hydrolysis of adenosine triphosphate to adenosine diphosphate, the apparent equilibrium constant is given by... 

The equilibrium constant expression can be useful in another way. Knowing the equilibrium constant expression, a chemist can calculate the equilibrium concentration of any substance involved in a reaction if the concentrations of all other reactants and products are known. 

We learn to interpret the magnitude of an equilibrium constant and how its value depends on the way the corresponding chemical equation is expressed. 

As you can see, K2 is the square root of Ki. Thus, dividing by two in the stoichiometry leads to a division by two in the exponents of the equilibrium expression. It s clear that we must look carefully at the way a chemical equation is written when we consider the value of the equilibrium constant. 

Section 15.4 If the concentrations of all species in an equilibrium are known, the equilibrium-constant expression can be used to calculate the value of the equilibrium constant. The changes in the concentrations of reactants and products on the way to achieving equilibrium are governed by the stoichiometry of the reaction. 

The theory can be developed in more detail by expressing dw/ST in terms of the equilibrium constant and the hH of reaction. In this way it can be shown that Cp passes through a maximum value at a certain temperature— as is intuitively obvious. At low temperature there is very little NOa and at high temperature there is very little NfOi. Hence the contribution to c, due to the shift in the equilibrium per unit rise in temperature, reaches a maximum at some intermediate temperature. 

We have seen how statistical thermodynamics can be applied to systems composed of particles that are more than just a single atom. By applying the partition function concept to electronic, nuclear, vibrational, and rotational energy levels, we were able to determine expressions for the thermodynamic properties of molecules in the gas phase. We were also able to see how statistical thermodynamics applies to chemical reactions, and we found that the concept of an equilibrium constant presents itself in a natural way. Finally, we saw how some statistical thermodynamics is applied to solid systems. Two similar applications of statistical thermodynamics to crystals were presented. Of the two, Einstein s might be easier to follow and introduced some new concepts (like the law of corresponding states), but Debye s agrees better with experimental data. 

Because both metal and metal oxide are pure condensed phases, and their activities are constant and defined to be unity, the oxygen activity at the phase boundary is fixed by the temperature according to the phase law or, expressed in a different way, its temperature dependence is given by that of the equilibrium constant. 

This equilibrium constant is dimensionless, as are all equilibrium constants expressed in terms of activities, because the equilibrium constant is a ratio of products of activities to various powers (+ and —), and activities are all dimensionless. That was one of the reasons for defining the activity, to make all equilibrium constants dimensionless. However, the numerical value of the activity depends on the choice of standard states. If we change standard states we will change the computed numerical value of the equilibrium constant, but in a way that will not change the computed concentrations at equilibrium (if we pay careful attention to standard states ). As we will see later, we often use modified equilibrium constants that have built-in dimensions however, the basic equilibrium constant, which is defined by Eqs. 12.14and 12.15,is always dimensionless. 

Other conventions for treating equiUbrium exist and, in fact, a rigorous thermodynamic treatment differs in important ways. Eor reactions in the gas phase, partial pressures of components are related to molar concentrations, and an equilibrium constant i, expressed directiy in terms of pressures, is convenient. If the ideal gas law appHes, the partial pressure is related to the molar concentration by a factor of RT, the gas constant times temperature, raised to the power of the reaction coefficients. 

Given the foregoing assumptions, it is a simple matter to construct an expression for the transition state theory rate constant as the probability of (1) reaching the transition state dividing surface and (2) having a momenrnm along the reaction coordinate directed from reactant to product. Stated another way, is the equilibrium flux of reactant states across... 

The equilibrium equation for the first step is shown in Scheme 5-2. Introducing the equilibrium constant Kw of water (Kw = [H+][0H ]/[H20] leads to the equation shown in Scheme 5-3. ATW can be combined with the constant K (defined by Kx = K[KW) to give the equation of Scheme 5-4. In the same way, the second step can be expressed as in Scheme 5-5. 

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