As early as 1799 C. Berthollet proposed the idea that chemical reactions are reversible. noting the ion exchange reaction:
2NaCl + Ca(CO3)2 ´ Na2CO3 + CaCl2
Later, in the mid 1800's Berthelot and Gilles showed that concentration of reactants has an effect on the concentration of products in chemical reactions. Shortly afterward, Guldberg and Waage showed that an equilibrium is reached in chemical reactions, and equilibrium can be approached from either direction. In 1877 van't Hoff quantified the expression for the equilibrium of a chemical reaction and showed that it is a function of the concentration of the various species involved, and that the concentrations appear as powers corresponding to the stoichiometric number in the balanced chemical equation.
The relationship between concentration and thermodynamic force (or lack thereof) in a chemical reaction or simple transformation is one of the most important in all of thermochemistry. It is important to recognize that this driving force results from maximizing entropy and therefore, minimizing Gibbs energy and naturally leads to the concept of equilibrium. The relationship between equilibrium conditions and concentration results from the definition of Gibbs energy or chemical potential,
II. Equilibrium Expression
In order to derive a generalized expression for the equilibrium of a chemical reaction we must start with a balanced equation (conservation of mass). For example,
2A + 3B´ A2B3
For any balanced chemical reaction we can write the following expression,
where Ai=atom type, ni=stoichiometric # (positive integer for products, negative integer for reactants). Because an equilibrium condition can occur partway through completion of the reaction in either direction, a measure of the extent of reaction must be introduced. Moreover, since when a chemical reaction occurs, the changes in the amounts of species involved in the reaction are proportional to their stoichiometric numbers, ni, in the balanced chemical equation, we can write:
where ni° is the amount of species i at time zero, and ni is the amount of species i at some later time, andx is the extent of reaction . The differential change in ni is simply given by:
Since the change in Gibbs energy is our convenient thermodynamic property to probe (or define) equilibrium at constant T and P (conditions appropriate in the lab), we consider the total differential of G for an open system:
Substituting the expression for dni in terms of extent of reaction,x, we have,
The quantity represents the change in Gibbs energy per change in the extent of reaction by 1 mol, and is given a more simplified symbol,DrG called the reaction Gibbs energy.
As a reaction takes place, the Gibbs energy of the system continues to decrease until it reaches a minimum value. At equilibrium (constant T and P), the Gibbs energy is at the minimum, and orDrG is equal to zero, thus,
Recall, the chemical potential of species (i) is related to the activity ai, of the species at constant T and P by,
Substituting this into the expression forDrG we have,
The quotient of activities to the power of the stoichiometric number is given a special symbol Q, and is called the reaction quotient. Thus,
At equilibriumDrG = 0, and Q is written as K to symbolize equilibrium and is referred to as the equilibrium constant.
B. Equilibrium Expression for Gas Reactions
In the case of reactions involving only gases, we can rewrite the equilibrium expression in terms of partial pressure (ideal gases) or fugacity (real gases). Thus,
Substituting these into the general expression for the equilibrium constant,
Note: The value of K can only be interpreted if the chemical equation is balanced, and the standard state of each species is specified.
C. Thermodynamics of Gaseous Reactions (Dependence of G onx)
We have already discussed the thermodynamics of mixing gases, and discovered that althoughDhmix and DVmix are zero for ideal gases, DGmix is not zero,and results from the systems tendency to maximize entropy (i.e. a mixture of gases has more possible states than each gas individually). Naturally, in gaseous reactions the relative proportion of reactant gas and product gas change as a function of extent of reaction, x. Therefore, DGmix must be a contributing term in the Gibbs energy of the system for reactions involving more than one gas. We can illustrate this point by considering a simple reaction to convert A(g) to B(g),
Starting with 1 mole of A, the amounts at some time later are,
and at any value ofx the Gibbs energy of the mixture is,
G = (1 x)mA + x mB
Recall, the chemical potential for an ideal gas mixture at constant T
Since we get,
where P = total pressure atx. Substituting these into the expression for G above,
where the last term is the contribution due to ĘGmix,
Note: We can generalize the above analysis to say that no chemical reaction involving only gases can go to completion!!
III. Determination of Equilibrium Constants
We have derived the correct expression for the equilibrium constant for a general system as well as for a gaseous system, and related it to the change in Gibbs energy with extent of reaction. We have yet to discuss how to actually determine the equilibrium constant, there are several different ways:
A. Determination of Concentrations of Reactants and Products at Equilibrium
Given that K is directly related to the amounts (pressures for gaseous rxn.) of reactants and products at equilibrium, a direct way to determine K is to simply determine the quantities of each species at equilibrium. There are many different ways to measure the relative amounts of species in a mixture (i.e. spectroscopy, refractive index, density, etc.) Often however, only the extent of reaction can be determined, not the individual amounts of each species. Determination of the density of a partially dissociated gas represents a simple example of a method which provides the extent of reaction at equilibrium. In order to use the extent of reaction in calculating the equilibrium constant we must derive the expression for the equilibrium constant in terms of the equilibrium extent of reaction and the total pressure. Consider the simple reaction,
1. A (g)
If we start with moles of A and the equilibrium extent of reaction is given byxeq, then at equilibrium we have,
xeq ´ 2 xeq
We can just as easily divide these quantities by the initial number of moles of A, , to obtain a dimensionless extent of reactionx'eq. = xeq/ which leads to values at equilibrium:
(equilibrium amounts) 1 x'eq ´ 2x'eq
We can relate the amounts of each species at equilibrium to the partial pressures through mole fractions;
(equilibrium mole fraction)
The equilibrium constant in terms of partial pressures is,
Since Pi = yiP, we can substitute,
Note: Like the equilibrium constant in terms of partial pressure, the equilibrium constant in terms ofx is dependent on the chemical reaction and its stoichiometry.
B. Determination ofDrG°:
At equilibrium, the complete equilibrium expression is given by,
whereDrG° is the standard reaction Gibbs energy. Therefore, the equilibrium constant, K, can also be determined directly from DrG°. There are several ways to determine DrG° for given chemical reaction including the use of Statistical Mechanics (Chem. 3308). DrG°can also be obtained from the thermodynamic expression,
in which the standard reaction enthalpies and entropies can be obtained calorimetrically at various temperatures. Instead of tabulating many different values forDrG°at different temperature, standard Gibbs energies of formation, DfG°, are used to reference Gibbs energies of various species to their respective elements. The standard reaction Gibbs energy, DrG°, can then be determined from the following expression, . Thus,
This is directly analogous to the way enthalpies of formations were used to determine . Like of elements in their standard states, are also zero for elements. Note: in the above equation, are not zero.
IV. The Effects on Equilibrium due to Changes in Independent Variables
Just as was done for other thermodynamic parameters, it is useful to probe the effects of changes in certain independent variables (i.e. temperature, pressure, initial composition, etc.) on the equilibrium, and hence equilibrium constant, of chemical reactions.
A. LeChatelier Principle
We can probe the effects of temperature and pressure in a qualitative sense by considering how the quantity
If we assume this quantity is a function of T, P, andx, we can write the full expression in terms of a total differential,
At equilibrium , and and since G is a minimum must be > 0. Therefore,
At constant temperature and pressure respectively,
These expressions are simply qualitative statements of LeChatelier Principle which states: "If the external constraints under which an equilibrium is established are changed, the equilibrium will shift in such a way as to moderate the effect on the change."
B. Changes in the Equilibrium Constant Expression - although the previous discussion is qualitatively useful, it would be more useful to have exact expressions for the equilibrium constant as a function of certain variables, especially for gaseous reactions.
1) Effects of pressure - the equilibrium constant in terms of partial pressures is given by,
Rewriting we have,
2) Effects of Temperature - the equilibrium constant in terms of temperature can be easily derived by starting from the Gibbs-Helmholtz equation introduced in the previous discussion of Gibbs energy. Recall,
The equilibrium constant can be written as an explicit function of temperature by integrating the above expression,
IfDrH° is independent of temperature we have,
We can derive a similar expression for the equilibrium constant as a function of temperature by starting from the fundamental expression obtained earlier, namely,
SubstitutingDrG° =DrH° TDrS°,
Again, ifDrH° is assumed to be independent of temperature then DrS° is independent of temperature, and the above expression is exact (moreover, it must be identical to the one derived above). In such a case, a plot of lnK versus should yield a straight line with slope and intercept .
IfDrH° and DrS° are not independent of temperature the above expressions get much more complicated!!
3) Equilibrium expression in terms of concentrations - since kinetic rate equations are written in terms of concentrations of each species, it would be worthwhile to have an expression for the equilibrium constant in terms of concentration. We will use Kc for concentration and Kp for partial pressures. Recall,
For an ideal gas therefore,
In order to keep both terms dimensionless we simply introduce a standard concentration by multiplying by