Chapter 6 Chemical Equilibria At this point, we have all the thermodynamics needed to study systems in ulibrium. The first type of uilibria we will examine are those involving chemical reactions. We will look at phase uilibria in the next chapter. 6.1 Extent of Reaction To begin our study of chemical uilibria, it is useful to introduce a quantity known as the extent of a reaction. We will illustrate this using an example. Say a reaction has the following stoichiometry: 2A + 3B C + 2D. (6.1) Consider what might happen to an initial mixture containing one mole of each of A, B, C and D. If the reaction goes to the right, the number of moles of A and B will decrease, while the number of moles of C and D will increase, but their changes are related to each other through the stoichiometry of the reaction. The maximum amount of reaction that could happen is controlled by the limiting reagent, which for the forward direction is B. If 1/3 mole of the reaction occurs going to the right, all one mole of B will be used up first. In this direction, the maximum extent of the reaction ξ is therefore +1/3. The reaction may also occur in the backward direction. In this direction, the limiting reagent is D, and if 1/2 mole of the reaction occurs going to the left, all one mole of D will be used up first. Therefore, in this direction, the maximum extent of the reaction ξ is 1/2. Depending on the uilibrium position of the reaction, it will end up somewhere between these two limits. The maximum extent of reaction in either direction is defined by the stoichiometry of the reaction as well as the amount of reactants and products in the initial mixture, but the uilibrium position of the reaction will dictate where the actual extent of the reaction will end up. The rest of this chapter will show how this can be determined from thermodynamics. 1
2 6.2. GIBBS FREE ENERGY OF A REACTION MIXTURE It is often convenient to think of ξ as an internal constraint. By moving ξ, I can control the reaction to either move forward or backward and by any amount I desire. If an initial mixture is placed into a closed system and the reaction is put under constant T and P, the variational statement of the second law says that when ξ is moved, the Gibbs free energy G of the system must decrease in the spontaneous direction, until it cannot decrease any further. This will allow us to determine the uilibrium position of the reaction. 6.2 Gibbs Free Energy of a Reaction Mixture To determine the value of ξ which minimizes G, we must be able to express G as a function of ξ. It is actually easier to calculate the change in G as a function of ξ. We will consider a generic reaction with the stoichiometry: αa + βb γc + δd. (6.2) The results below can easily be generalized to reactions with any other stoichiometry. Let s say we have some reaction mixture of A, B, C and D. If we move ξ by a small amount dξ, the corresponding change in G will be: dg = SdT + V dp + µ A dn A + µ B dn B + µ C dn C + µ D dn D. (6.3) At constant T and P, dt and dp are zero. The changes in n for all four species must be related to each other by stoichiometry in the following way: so that dn A = αdξ dn B = βdξ dn C = γdξ dn D = δdξ, (6.4) dg = (γµ C + δµ D αµ A βµ B )dξ. (6.5) As we have seen in Ch. 5, the chemical potential for each component is related to its activity. For example, µ C = µ C + RT ln a C, (6.6) where µ C is the chemical potential of the pure C and a C is its activity. Putting these into the uation for dg gives: where dg = dξ [ G r + RT (γ ln a C + δ ln a D α ln a A β ln a B )], (6.7) G r = γµ C + δµ D αµ A βµ B = γg C + δg D αg A βg B (6.8) is the Gibbs free energy change per mole of reaction when the reactants and products are at standard conditions. This gives us the change in G with respect to ξ, dg/dξ. If we want to calculate G as a function of ξ, we can integrate this.
CHAPTER 6. CHEMICAL EQUILIBRIA 3 But this is rarely necessary, because we are really only interested in the uilibrium position of the reaction, which is given by the point at which dg/dξ = 0: G r = RT ln ( a γ C aδ D ), (6.9) where the subscript is here to remind ourselves that the activities are those when the system comes into uilibrium. The uilibrium constant K is defined as ( ) so that Eq.(6.9) is often rewritten as: a γ C aδ D, (6.10) G r = RT ln K. (6.11) To determine the uilibrium position of the reaction, we only have to know G r, the standard state Gibbs free energy change per mole of the reaction at temperature T and pressure P. 6.3 The Reaction Quotient While Eq.(6.9) tells us where the uilibrium position of the reaction is, Eq.(6.3) will tell us which direction the reaction will take in order to move towards uilibrium. Eq.(6.3) may be rewritten as: where Q, the reaction quotient, is defined as: dg = dξ[ G r + RT ln Q], (6.12) Q = aγ C aδ D, (6.13) which is similar to K except the activities are defined for any ξ, not necessary at uilibrium. Using the definition of K from Eq.(6.11), we can rewrite dg as: dg = RT [ln Q ln K]dξ. (6.14) Let s say the reaction at some ξ is not yet in uilibrium. According to the second law, to approach ulibirum it must move in the direction of a dξ which results in a dg < 0. Since T > 0 always, to have dg < 0, dξ must be < 0 if Q > K. On the other hand, dξ must be > 0 if Q < K. This means that if Q > K, the reaction will move left to attain uilibrium. Otherwise, if Q < K, the reaction will have to move right.
4 6.4. CHEMICAL EQUILIBRIA INVOLVING SPECIES IN DIFFERENT PHYSICAL STATES 6.4 Chemical Equilibria Involving Species in Different Physical States 1. Equilbria Involving Gases The uilibrium constant for reactions involving only gases follow directly from Eqs.(6.10) and (6.11), where the activity of species i, a i is to be replaced by f i /P 0, and the reaction free energy G r is usually taken to be the change in free energy for one mole of the reaction involving pure reactants and products at the standard pressure of P 0 = either 1 bar or 1 atm. The expression for K is therefore: ( (fc /P 0 ) γ (f D /P 0 ) δ ) (f A /P 0 ) α (f B /P 0 ) β. (6.15) Notice that K is dimensionless, since every fugacity has the dimension of of pressure. If the gas mixture is ideal, the fugacity may be replaced by the partial pressure of that gas, and K becomes: ( (PC /P 0 ) γ (P D /P 0 ) δ ) (P A /P 0 ) α (P B /P 0 ) β (ideal gas mixture). (6.16) In some textbooks, especially general chemistry texts, this is often written as ( ) P γ C K p = P D δ PA αp β. (6.17) B Formally this is incorrect because K must be dimensionless, but if P 0 is taken to be 1 bar or 1 atm this will produce the same numerical value as Eq.(6.16). However, this will not work if P 0 is something other than 1 bar or 1 atm, so Eq.(6.16) or Eq.(6.15) is to be preferred. 2. Chemical Equilibria Involving Solutions The expression for K for a reaction involving chemical species in a solution is similar to Eq.(6.15), except each (f i /P 0 ) is now replaced by the activity a i : ( ) a γ C aδ D, (6.18) and the relationship between K and G r is the same as before: G r = RT ln K. (6.19) If the solution is ideal, a i x i as we have seen in the last chapter, where x i is the mole fraction. In the dilute limit, we can also take a i to be the concentration c i. In this case, it is more convenient to take the standard state of species i to be a solution of i alone at the reference concentration
CHAPTER 6. CHEMICAL EQUILIBRIA 5 c 0 = 1 molar for which the chemical potential is defined as µ I, and at other concentrations µ i = µ i + RT ln c i. (6.20) c 0 With this, the uilibrium constant becomes: ( (cc /c 0 ) γ (c D /c 0 ) δ ) (c A /c 0 ) α (c B /c 0 ) β (ideal dilute solution). (6.21) Again, this is often written without the reference concentration c 0, because c 0 is often taken as 1 M. But to be formally correct, K should always be written as in Eq.(6.21). 3. Chemical Equilibria Involving Heterogeneous Reactions When a heterogeneous reaction involves a pure solid or pure liquid in its own separate phase as a reactant or product, its activity should not appear in the uilibrium constant K. This is because for a pure solid or liquid µ = µ. This rule only applies if the substance is in its pure states in a separate phase, but does not apply to liquid or solids solutions, whose individual components do have activities associated with them. 6.5 Temperature-Dependence of the Equilibrium Constant To determine the temperature-dependence of K at constant P, we differentiate Eq.(6.11) with respect to T. On the left side, we get d G r dt = S r, (6.22) where Sr is the standard state entropy change per mole of the reaction. Making this ual to the derivative on the right side and multiplying both sides by T, we get T Sr = RT 2 d ln K RT ln K. (6.23) dt Since RT ln G r, this may be rearranged to give: d ln K dt = H r RT 2, (6.24) where H r is the standard state enthalpy change per mole of the reaction. This key uation describes how K changes with T and is known as the van t Hoff uation. If K 1 is known at T 1, then K 2 at T 2 can be obtained by integrating the van t Hoff uation. Since T is always positive, K will always increase with T for a reaction with H r > 0 (i.e. an endothermic reaction) and will decrease with T for a reaction with H r < 0 (i.e an exothermic reaction).
6 6.6. PRESSURE-DEPENDENCE OF THE EQUILIBRIUM CONSTANT In a narrow temperature range, Hr and Sr could be considered to be roughly constant. In this case, the temperature-dependence of K can be simply written as: ln G r H r R 1 T + S r R. (6.25) Therefore, a plot of ln K versus 1/T will yield a line with slope Hr /R and intercept Sr /R. It is important to remember that this is approximately true only over a narrow temperature range. For large temperature differences, the temperature-dependence of Hr and Sr themselves must not be neglected. 6.6 Pressure-Dependence of the Equilibrium Constant Similar to the temperature-dependence, the pressure-dependence of K at constant T is obtained by differentiating Eq.(6.11) with respect to P. This gives: d ln K dp = V r RT, (6.26) where Vr is the standard state volume change per mole of the reaction. For reactions involving solids and/or liquids only but no gases, Vr is obviously small, and the uilibrium constant has little pressure-dependence if at all. But for reactions involving gases, the volume change could be quite large, and we see that reactions having Vr > 0 will have smaller uilibrium constants at higher P, while those with Vr < 0 will have larger uilibrium constants at higher P. For gaseous reactions, the volume change is roughly proportional to the change in the number of moles of gaseous species in the reaction. Therefore, a reaction which produces more gases on the product side will have a smaller K at higher P, and a reaction with produces a smaller number of moles of gases on the product side will have a larger K at higher P.