CHAPTER Chemical Equilibrium 4 ow we arrive at the point where real chemistry begins. Chemical thermo- Thermodynamic background dynamics is used to predict whether a mixture of reactants has a sponta- 4.1 The reaction Gibbs energy neous tendency to change into products, to predict the composition of the ⌬ N 4.2 The variation of rG with reaction mixture at equilibrium, and to predict how that composition will be mod- composition ified by changing the conditions. In biology, life is the avoidance of equilibrium, 4.3 Reactions at equilibrium and the attainment of equilibrium is death, but knowing whether equilibrium lies CASE STUDY 4.1: Binding of in favor of reactants or products under certain conditions is a good indication of oxygen to myoglobin and the feasibility of a biochemical reaction. Indeed, the material we cover in this chap- hemoglobin ter is of crucial importance for understanding the mechanisms of oxygen transport 4.4 The standard reaction in blood, metabolism, and all the processes going on inside organisms. Gibbs energy There is one word of warning that is essential to remember: thermodynamics is silent about the rates of reaction. All it can do is to identify whether a particular re- The response of equilibria to the conditions action mixture has a tendency to form products; it cannot say whether that ten- dency will ever be realized. We explore what determines the rates of chemical re- 4.5 The presence of a catalyst actions in Chapters 6 through 8. 4.6 The effect of temperature Coupled reactions in Thermodynamic background bioenergetics 4.7 The function of adenosine The thermodynamic criterion for spontaneous change at constant temperature and triphosphate pressure is ⌬G Ͻ 0. The principal idea behind this chapter, therefore, is that, at CASE STUDY 4.2: The constant temperature and pressure, a reaction mixture tends to adjust its composition un- biosynthesis of proteins til its Gibbs energy is a minimum. If the Gibbs energy of a mixture varies as shown 4.8 The oxidation of glucose in Fig. 4.1a, very little of the reactants convert into products before G has reached its minimum value, and the reaction “does not go.” If G varies as shown in Proton transfer equilibria Fig. 4.1c, then a high proportion of products must form before G reaches its min- 4.9 Brønsted-Lowry theory imum and the reaction “goes.” In many cases, the equilibrium mixture contains al- 4.10 Protonation and most no reactants or almost no products. Many reactions have a Gibbs energy that deprotonation varies as shown in Fig. 4.1b, and at equilibrium the reaction mixture contains sub- 4.11 Polyprotic acids stantial amounts of both reactants and products. CASE STUDY 4.3: The fractional composition of a 4.1 The reaction Gibbs energy solution of lysine 4.12 Amphiprotic systems To explore metabolic processes, we need a measure of the driving power of a 4.13 Buffer solutions chemical reaction, and to understand the chemical composition of cells, we need CASE STUDY 4.4: Buffer action to know what those compositions would be if the reactions taking place in them in blood had reached equilibrium. Exercises To keep our ideas in focus, we consider two important processes. One is the iso- merism of glucose-6-phosphate (1, G6P) to fructose-6-phosphate (2, F6P), which is an early step in the anaerobic breakdown of glucose (Section 4.8): G6P(aq) 0ˆˆˆ9 F6P(aq) (A) 151 152 Chapter 4 • Chemical Equilibrium 2– OPO 3 2– OPO 3 OH O H H O G OH H H HO HO OH H OH OH (a) (b) (c) H OH H Gibbs energy, Gibbs energy, 1 Glucose-6-phosphate 2 Fructose-6-phosphate The second is the binding of O2(g) to the protein hemoglobin, Hb, in blood (Case Pure Pure study 4.1): reactants products Fig. 4.1 The variation of Hb(aq) ϩ 4 O2(g) 0ˆˆˆ9 Hb(O2)4(aq) (B) Gibbs energy of a reaction mixture with progress of the These two reactions are specific examples of a general reaction of the form reaction, pure reactants on the left and pure products on the ϩ ˆˆ9 ϩ right. (a) This reaction “does a A b B 0ˆˆ c C d D (C) not go”: the minimum in the Gibbs energy occurs very close with arbitrary physical states. to the reactants. (b) This First, consider reaction A. Suppose that in a short interval while the reaction reaction reaches equilibrium is in progress, the amount of G6P changes infinitesimally by Ϫdn. As a result of with approximately equal this change in amount, the contribution of G6P to the total Gibbs energy of the amounts of reactants and system changes by Ϫ␮G6Pdn, where ␮G6P is the chemical potential (the partial mo- products present in the lar Gibbs energy) of G6P in the reaction mixture. In the same interval, the amount mixture. (c) This reaction goes of F6P changes by ϩdn, so its contribution to the total Gibbs energy changes by almost to completion, as the ϩ␮ ␮ minimum in Gibbs energy lies F6Pdn, where F6P is the chemical potential of F6P. The change in Gibbs en- very close to pure products. ergy of the system is dG ϭ ␮F6Pdn Ϫ ␮G6Pdn On dividing through by dn, we obtain the reaction Gibbs energy, ⌬rG: dG ᎏᎏ ϭ ␮ Ϫ ␮ ϭ⌬G (4.1a) dn F6P G6P r There are two ways to interpret ⌬rG. First, it is the difference of the chemical potentials of the products and reactants at the composition of the reaction mixture. Second, we can think of ⌬rG as the derivative of G with respect to n, or the slope of the graph of G plotted against the changing composition of the system (Fig. 4.2). The binding of oxygen to hemoglobin provides a slightly more complicated ex- ample. If the amount of Hb changes by Ϫdn, then from the reaction stoichiome- try we know that the change in the amount of O2 will be Ϫ4dn and the change in the amount of Hb(O2)4 will be ϩdn. Each change contributes to the change in the total Gibbs energy of the mixture, and the overall change is ⌬ ϭ ␮ ϫ Ϫ ␮ ϫ Ϫ ␮ ϫ G Hb(O2)4 dn Hb dn O2 4dn ϭ ␮ Ϫ ␮ Ϫ ␮ ( Hb(O2)4 Hb 4 O2)dn Thermodynamic background 153 where the ␮J are the chemical potentials of the species in the reaction mixture. In this case, therefore, the reaction Gibbs energy is dG ⌬ G ϭ ᎏᎏ ϭ ␮ Ϫ (␮ ϩ 4␮ ) (4.1b) r dn Hb(O2)4 Hb O2 G Note that each chemical potential is multiplied by the corresponding stoichiomet- ric coefficient and that reactants are subtracted from products. For the general ∆G ∆ G = reaction C, ∆G r ∆n Gibbs energy, Gibbs energy, ⌬ G ϭ (c␮ ϩ d␮ ) Ϫ (a␮ ϩ b␮ ) (4.1c) r C D A B ∆n The chemical potential of a substance depends on the composition of the mix- Pure Pure ture in which it is present and is high when its concentration or partial pressure is reactants products high. Therefore, ⌬rG changes as the composition changes (Fig. 4.3). Remember Composition that ⌬rG is the slope of G plotted against composition. We see that ⌬rG Ͻ 0 and the slope of G is negative (down from left to right) when the mixture is rich in the Fig. 4.2 The variation of Gibbs energy with progress of reactants A and B because ␮A and ␮B are then high. Conversely, ⌬rG Ͼ 0 and the slope of G is positive (up from left to right) when the mixture is rich in the prod- reaction showing how the reaction Gibbs energy, ⌬rG, is ucts C and D because ␮C and ␮D are then high. At compositions corresponding to ⌬ Ͻ ⌬ Ͼ related to the slope of the rG 0 the reaction tends to form more products; where rG 0, the reverse re- curve at a given composition. action is spontaneous, and the products tend to decompose into reactants. Where When ⌬G and ⌬n are both ⌬ ϭ rG 0 (at the minimum of the graph where the derivative is zero), the reaction infinitesimal, the slope is has no tendency to form either products or reactants. In other words, the reaction written dG/dn. is at equilibrium. That is, the criterion for chemical equilibrium at constant temperature and pressure is ⌬rG ϭ 0 (4.2) 4.2 The variation of ⌬rG with composition The reactants and products in a biological cell are rarely at equilibrium, so we need to know how the reaction Gibbs energy depends on their concentrations. ∆rG < 0 G Spontaneous ∆rG > 0 Gibbs energy, Gibbs energy, Spontaneous Fig. 4.3 At the minimum of the curve, ⌬ ϭ Equilibrium corresponding to equilibrium, rG 0. To the ∆ G = 0 r left of the minimum, ⌬rG Ͻ 0, and the forward Pure Pure reaction is spontaneous. To the right of the reactants products minimum, ⌬rG Ͼ 0, and the reverse reaction is Composition spontaneous. 154 Chapter 4 • Chemical Equilibrium Our starting point is the general expression for the composition dependence of the chemical potential derived in Section 3.11: ␮J ϭ ␮J ϩ RT ln aJ (4.3) where aJ is the activity of the species J. When we are dealing with systems that may be treated as ideal, which will be the case in this chapter, we use the identifica- tions given in Table 3.3: For solutes in an ideal solution, aJ ϭ [J]/c , the molar concentration of J relative to the standard value c ϭ 1 mol LϪ1.