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Gibbs Free Energy and Chemical Potential
Appendix IV Gibbs Free Energy and Chemical Potential The concept of chemical potential is introduced in Chapter 2 (Section 2.2) and used throughout the rest of the book. In order to not overburden the text with mathematical details, certain points are stated without proof. Here we will derive an expression for the chemical potential, justify the form of the pressure term in the chemical potential, and also provide insight into how the expression for the Gibbs free energy arises.
IV.A. Entropy and Equilibrium A suitable point of departure is to reconsider the condition for equilibrium. The most general statement we can make concerning the attainment of equilibrium by a system is that it occurs when the entropy of the system plus its surroundings is at a maximum. Unfortunately, entropy has proved to be an elusive concept to master and a difficult quantity to measure. Moreover, reference to the surroundingsdthe “rest of the universe” in the somewhat grandiloquent language of physicsdis a nuisance. Consequently, thermodynamicists sought a function that would help describe equilibrium but would depend only on readily measurable parameters of the system under consideration. As we will see, the Gibbs free energy is such a function for most applications in biology. The concept of entropy (S) is really part of our day-to-day observations. We know that an isolated system will spontaneously change in certain waysda system proceeds toward a state that is more random, or less ordered, than the initial one. For instance, neutral solutes will diffuse toward regions where they are less concentrated (Fig. 1-5). In so doing, the system lowers its capacity for further spontaneous change. For all such processes, DS is positive, whereas DS
633
634
Appendix IV
becomes zero and S achieves a maximum at equilibrium. Equilibrium means that no more spontaneous changes will take place; entropy is therefore an index for the capacity for spontaneous change. It would be more convenient in some ways if entropy had been originally defined with the opposite sign. In fact, some authors introduce the quantity negentropy, which equals S and reaches a minimum at equilibrium. In any case, we must ultimately use a precise mathematical definition for entropy, such as dS ¼ dQ/T, where dQ refers to the heat gain or loss in some reversible reaction taking place at temperature T. We can represent the total entropy of the universe, Su, as the entropy of the system under consideration, Ss, plus the entropy of the rest of the universe, Sr. We can express this in symbols as follows: Su ¼ Ss þ Sr or
(IV.1) dSu ¼ dSs þ dSr
An increase in Su accompanies all real processesdthis is the most succinct way of stating the second law of thermodynamics. Su is maximum at equilibrium. The heat absorbed by a system during some process is equal to the heat given up by the rest of the universe. Let us represent the infinitesimal heat exchange of the system by dQs. For an isothermal reaction or change, dQs is simply dQr because the heat must come from the rest of the universe. From the definition of entropy,1 dS ¼ dQ/T, we can obtain the following relationship: dSr ¼
dQr dQs dUs þ PdVs ¼ ¼ T T T
(IV.2)
The last step in Equation IV.2 derives from the principle of the conservation of energy for the case when the only form of work involved is mechanicalda common assumption in stating the first law of thermodynamics. It is thus possible to express dQs as the sum of the change in internal energy (dUs) plus a work term (PdVs). The internal energy (Us) is a function of the state of a system, i.e., its magnitude depends on the characteristics of the system but is independent of how the system got to that state. PVs is also a well-defined variable. However, heat (Qs) is not a function of the state of a system. As we indicated previously, equilibrium occurs when the entropy of the universe is maximum. This means that dSu then equals zero. By substituting Equation IV.2 into the differential form of Equation IV.1, we can express this equilibrium condition solely in terms of system parameters:
1. This definition really applies only to reversible reactions, which we can in principle use to approximate a given change; otherwise, dQ is not uniquely related to dS.
635
IV.B. Gibbs Free Energy
0 ¼ dSs þ
dUs þ PdVs T
or
(IV.3)
TdSS þ dUs þ PdVs ¼ 0 Equation IV.3 suggests that there is some function of the system that has an extremum at equilibrium. In other words, we might be able to find some expression determined by the parameters describing the system whose derivative is zero at equilibrium. If so, the abstract statement that the entropy of the universe is a maximum at equilibrium could then be replaced by a statement referring only to measurable attributes of the systemdeasily measurable ones, we hope. In the 1870s, Josiah Willard Gibbsdperhaps the most brilliant thermodynamicist to datedchose a simple set of terms that turned out to have the very properties for which we are searching. This function is now referred to as the Gibbs free energy and has the symbol G: G ¼ U þ PV TS
(IV.4a)
which, upon differentiating, yields dG ¼ dU þ PdV þ VdP TdS SdT
(IV.4b)
Equation IV.4b indicates that, at constant temperature (dT ¼ 0) and constant pressure (dP ¼ 0), dG is dG ¼ dU þ PdV TdS
at constant T and P
(IV.5)
By comparing Equation IV.5 with the equilibrium condition expressed by Equation IV.3, we see that dG for a system equals zero at equilibrium at constant temperature and pressure. Moreover, G depends only on U, P, V, T, and S of the system. The extremum condition, dG ¼ 0, actually occurs when G reaches a minimum at equilibrium. This useful attribute of the Gibbs free energy is strictly valid only when the overall system is at constant temperature and pressure, conditions that closely approximate those encountered in many biological situations. Thus our criterion for equilibrium shifts from a maximum of the entropy of the universe to a minimum in the Gibbs free energy of the system.
IV.B. Gibbs Free Energy We will now consider how the internal energy, U, changes when material enters or leaves a system. This will help us derive an expression for the Gibbs free energy that is quite useful for biological applications.
636
Appendix IV
The internal energy of a system changes when substances enter or leave it. For convenience, we will consider a system of fixed volume and at the same temperature as the surroundings so that there are no heat exchanges. If dnj moles of species j enter such a system, U increases by mjdnj, where mj is an intensive variable representing the free energy contribution to the system per mole of species j entering or leaving. Work is often expressed as the product of an intensive quantity (such as mj, P, T, E, and h) times an extensive one (dnj, dV, dS, dQ, and dm, respectively); that is, the amount of any kind of work depends on both: some thermodynamic parameter characterizing the internal state of the system and the extent or amount of change for the system. In our current example, the extensive variable describing the amount of change is dnj, and mj represents the contribution to the internal energy of the system per mole of species j. When more than one species crosses the boundary of our system, which P is at constant volume and the same temperature as the surroundings, the term j mj dnj is added to dU, where dnj is positive if the species enters the system and negative if it leaves. In the general case, when we consider all of the ways that the internal energy of a system can change, we can represent dU as follows: X dU ¼ dQ PdV þ mj dnj (IV.6) j
We now return to the development of a useful relation for the Gibbs free energy of a system. When dU as expressed by Equation IV.6 is substituted into dG as given by Equation IV.5, we obtain X mj dnj þ PdV TdS dG ¼ TdS PdV þ X ¼ mj dnj
j
(IV.7)
j
where dQ has been replaced by TdS. Equation IV.7 indicates that the particular form chosen for the Gibbs free energy leads to a very simple expression for dG at constant T and Pdnamely, dG then depends only on mj and dnj. To obtain an expression for G, we must integrate Equation IV.7. To facilitate the integration, we will define a new variable, a, such that dnj is equal to nj da, where nj is the total number of moles of species j present in the final system; that is, nj is a constant describing the final system. The subsequent integration from a ¼ 0 to a ¼ 1 corresponds to building up the system by a simultaneous addition of all of the components in the same proportions that are present in the final system. (The intensive variable mj is also held constant for this integration pathway, i.e., the chemical potential of species j does not depend
637
IV.C. Chemical Potential
on the size of the system.) Using Equation IV.7 and this easy integration pathway, we obtain the following expression for the Gibbs free energy: Z
G ¼ dG ¼
Z X
mj dnj ¼
Z 1X
j
¼
X
j
mj nj
Z 1
0
da ¼
X
mj nj
0 j
mj nj da (IV.8)
j
The well-known relation between G and mj’s in Equation IV.8 can also be obtained by a method that is more elegant mathematically but somewhat involved. In Chapter 6 (Section 6.1) we presented without proof an expression for the Gibbs free energy (Eq. 6.1 is essentially Eq. IV.8) and also noted some of the properties of G. For instance, at constant temperature and pressure, the direction for a spontaneous change is toward a lower Gibbs free energy; minimum G is achieved at equilibrium. Hence, DG is negative for such spontaneous processes. Spontaneous processes can in principle be harnessed to do useful work, where the maximum amount of work possible at constant temperature and pressure is equal to the absolute value of DG (some of the energy is dissipated by inevitable inefficiencies such as frictional losses, so DG represents the maximum work possible). To drive a reaction in the direction opposite to that in which it proceeds spontaneously requires a free energy input of at least DG (cf. Fig. 2-6).
IV.C. Chemical Potential We now P examine the properties of the intensive variable mj. Equation IV.8 (G ¼ j mj nj ) suggests a very useful way of defining mj. In particular, if we keep mj and ni constant, we obtain the following expression: vG vG mj ¼ ¼ (IV.9) vnj mi ;ni vnj T;P;E;h;ni where ni and mi refer to all species other than species j. Because mj can depend on T, P, E (the electrical potential), h (the height in a gravitational field), and ni, the act of keeping mi constant during partial differentiation is the same as that of keeping T, P, E, h, and ni constant, as is indicated in Equation IV.9. Equation IV.9 indicates that the chemical potential of species j is the partial molal Gibbs free energy of a system with respect to that species and that it is obtained when T, P, E, h, and the amount of all other species are held constant. Thus mj corresponds to the intensive contribution of species j to the extensive quantity G, the Gibbs free energy of the system.
638
Appendix IV
In Chapters 2 and 3 we argued that mj depends on T, aj (aj ¼ gjcj; Eq. 2.5), P, E, and h in a solution and that the partial pressure of species j, Pj, is also involved for the chemical potential in a vapor phase. We can summarize the two relations as follows (see Eqs. 2.4 and 2.21): liquid
mj
¼ mj þ RT ln aj þ V j P þ zj FE þ mj gh vapor
mj
¼ mj þ RT ln
Pj þ mj gh Pj
(IV.10a) (IV.10b)
The forms for the gravitational contribution (mjgh) and the electrical one (zjFE) can be easily understood. We showed in Chapter 3 (Section 3.2A) that RT ln aj is the correct form for the concentration term in mj. The reasons for the forms of the pressure terms in a liquid (Vj P) and in a gas [RT ln ðPj =Pj Þ] are not so obvious. Therefore, we will examine the pressure dependence of the chemical potential of species j in some detail.
IV.D. Pressure Dependence of mj To derive the pressure terms in the chemical potentials of solvents, solutes, and gases, we must rely on certain properties of partial derivatives as well as on commonly observed effects of pressure. To begin with, we will differentiate the chemical potential in Equation IV.9 with respect to P: " # vmj v vG ¼ vP vnj T;P;E;h;ni vP T;E;h;ni ;nj T;E;h;ni ;nj
"
# v vG ¼ vnj vP T;E;h;ni ;nj
(IV.11)
T;P;E;h;ni
where we have reversed the order for partial differentiation with respect to P and nj (this is permissible for functions such as G, which have well-defined and continuous first-order partial derivatives). The differential form of Equation IV.4 (dG ¼ dU þ PdV þ VdP TdS SdT) gives us a suitableP form for dG. If we substitute dU given by Equation IV.6 (dU ¼ TdS PdV þ j mj dnj , where dQ is replaced by TdS) into this expression for the derivative of the Gibbs free energy, we can express dG in the following useful form: dG ¼ VdP SdT þ
X mj dnj
(IV.12)
j
Using Equation IV.2 we can readily determine the pressure dependence of the Gibbs free energy as needed in the last bracket of Equation IV.11dnamely,
639
IV.D. Pressure Dependence of mj
ðvG=vPÞT;E;h;ni ;nj is equal to V by Equation IV.12. Next, we have to consider the partial derivative of this V with respect to nj (see the last equality of Eq. IV.11). Equation 2.6 indicates that ðvV=vnj ÞT;P;E;h;ni is Vj, the partial molal volume of species j. Substituting these partial derivatives into Equation IV.11 leads to the following useful expression: vmj ¼ Vj (IV.13) vP T;E;h;ni ;nj Equation IV.13 is of pivotal importance in deriving the form of the pressure term in the chemical potentials of both liquid and vapor phases. Let us first consider an integration of Equation IV.13 appropriate for a liquid. We will make use of the observation that the partial molal volume of a species in a solution does not depend on the pressure to any significant extent. For a solvent, this means that the liquid generally is essentially incompressible. If we integrate Equation IV.13 with respect to P at constant T, E, h, ni, and nj with Vj independent of P, we obtain the following relations: Z Z mliquid j vmj liquid dP ¼ dmj ¼ mj mj vP mj (IV.14) Z Z ¼ V j dP ¼ V j dP ¼ V j P þ “constant” where the definite integral in the top line is taken from the chemical potential of species j in a standard state as the lower R limit up to the general mj in a liquid as the upper limit. The integration of Vj dP leads to our pressure term Vj P plus a constant. Because the integration was performed while holding T, E, h, ni, and nj fixed, the “constant” can depend on all of these variables but not on P. Equation IV.14 indicates that the chemical potential of a liquid contains a pressure term of the form Vj P. The other terms (mj , RT ln aj, zjFE, and mjgh; see Eq. IV.10a) do not depend on pressure, a condition used throughout this text. The experimental observation that gives us this useful form for mj is that Vj is generally not influenced very much by pressure; for example, a liquid is often essentially incompressible. If this should prove invalid under certain situations, Vj P would then not be a suitable term in the chemical potential of species j for expressing the pressure dependence in a solution. Next, we discuss the form of the pressure term in the chemical potential of a gas, where the assumption of incompressibility that we used for a liquid is not valid. Our point of departure is the perfect or ideal gas law: Pj V ¼ nj RT
(IV.15)
where Pj is the partial pressure of species j, and nj is the number of moles of species j in volume V. Thus we will assume that real gases behave like ideal gases, which is generally justified for biological applications. Based on
640
Appendix IV
Equations IV.15 and 2.6 ½Vj ¼ ðvV=vnj ÞT;P;E;h;ni , the partial molal volume Vj for gaseous species j is RT/Pj. We also note that the total pressure P is equal to P j Pj , where the summation is over all gases present (Dalton’s law of partial pressures); hence, dP equals dPj when ni, T, and V are constant. When we integrate Equation IV.13, we thus find that the chemical potential of gaseous species j depends on the logarithm of its partial pressure: Z Z mvapor Z Z j vmj RT vapor dP ¼ dmj ¼ mj mj ¼ V j dP ¼ dPj Pj vP mj (IV.16) ¼ RT ln Pj þ “constant” where the “constant” can depend on T, E, h, and ni but not on nj (or Pj). In particular, the “constant” equals RT ln Pj þ mj gh, where Pj is the saturation partial pressure for species j at atmospheric pressure and some particular temperature. Hence, the chemical potential for species j in the vapor phase vapor (mj ) is vapor
mj
¼ mj þ RT ln Pj RT ln Pj þ mj gh ¼ mj þ RT ln
Pj þ mj gh Pj
(IV.17)
which is essentially the same as Equations 2.21 and IV.10b. We have defined the standard state for gaseous species j, mj , as the chemical potential when the gas phase has a partial pressure for species j (Pj) equal to the saturation partial pressure (Pj ), when we are at atmospheric pressure (P ¼ 0) and the zero level for the gravitational term (h ¼ 0), and for some specified temperature. Many physical chemistry texts ignore the gravitational term (we calculated that it has only a small effect for water vapor; see Chapter 2, Section 2.4C) and define the standard state for the condition when Pj equals 1 atm and species j is the only species present (P ¼ Pj). The chemical potential of such a standard state equals mj RT ln Pj in our symbols. The partial pressure of some species in a vapor phase in equilibrium with a liquid depends slightly on the total pressure in the systemdloosely speaking, when the pressure on the liquid is increased, more molecules are squeezed out of it into the vapor phase. The exact relationship between the pressures involved, which is known as the Gibbs equation, is as follows for water: vðln Pwv Þ V w ¼ vP RT or
(IV.18) vPwv Vw ¼ vP V wv
where the second equality follows from the derivative of a logarithm [v ln u/vx ¼ (1/u) (vu/vx)] and the ideal or perfect gas law [PwvV ¼ nwvRT (Eq. IV.15),
IV.E. Concentration Dependence of mj
641
so vV=vnwv ¼ Vwv ¼ RT=Pwv ]. Because Vw is much less than Vwv , the effect is quite small (e.g., at 20 C and atmospheric pressure, V w ¼ 1:8 105 m3 mol1 and Vwv ¼ 2:4 102 m3 mol1 ). From the first equality in Equation IV.18, we see that Vw dP equals RTd ln Pwv. Hence, if the chemical potential of the liquid phase (mw) increases by Vw dP as an infinitesimal pressure is applied, then an equal increase, RTd ln Pwv, occurs in mwv (see Eq. IV.10b for a definition of mwv), and hence we will still be in equilibrium (mw ¼ mwv). This relation can be integrated to give RT ln Pwv ¼ Vw P þ constant, where the constant is RT ln P0wv and P0wv is the partial pressure of water vapor at standard atmospheric pressure; hence, RT ln Pwv =P0wv is equal to Vw P, a relation used in Chapter 2 (see Section 2.4C). We note that effects of external pressure on Pwv can be of the same order of magnitude as deviations from the ideal gas law for water vapor, both of which are usually neglected in biological applications.
IV.E. Concentration Dependence of mj We will complete our discussion of chemical potential by using Equation IV.17 to obtain the logarithmic term in concentration that is found for mj in a liquid phase. First, it should be pointed out that Equation IV.17 has no concentration term per se for the chemical potential of species j in a gas phase. However, the partial pressure of a species in a gas phase is really analogous to the concentration of a species in a liquid; e.g., PjV ¼ njRT for gaseous species j (Eq. IV.15), and concentration means number/volume and equals nj/V, which equals Pj/RT. Raoult’s law states that at equilibrium the partial pressure of a particular gas above its volatile liquid is proportional to the mole fraction of that solvent in the liquid phase. A similar relation more appropriate for solutes is Henry’s law, which states that Pj in the vapor phase is proportional to the Nj of that solute in the liquid phase. Although the proportionality coefficients in the two vapor relations are different, they both indicate that Pj depends linearly on solution Nj . For dilute solutions, the concentration of species j, cj, is proportional to its mole fraction, Nj (this is true for both solute and solvent). Thus when vapor Pj changes from one equilibrium condition to another, we expect a similar liquid vapor change in csolution because mj is equal to mj at equilibrium. In particular, j vapor Equation IV.17 indicates that mj depends on RT ln ðPj =Pj Þ, and hence the chemical potential of a solvent or solute should contain a term of the form RT ln cj, as in fact it does (see Eqs. 2.4 and IV.10). As we discussed in Chapter 2 (Section 2.2B), we should be concerned about the concentration that is thermodynamically active, aj (aj ¼ gjcj; Eq. 2.5), so the actual term in the chemical potential for a solute or solvent is RT ln aj, not RT ln cj. In Chapter 3 (Section 3.2A), instead of the present argument based on Raoult’s and Henry’s laws, we used a comparison with Fick’s first law to justify the RT ln aj term. Moreover,
642
Appendix IV
the BoyleeVan’t Hoff relation, which was derived assuming the RT ln aj term, has been amply demonstrated experimentally. Consequently, the RT ln aj term in the chemical potential for a solute or solvent can be justified or derived in a number of different ways, all of which depend on agreement with experimental observations.
IV.A. Entropy and Equilibrium A suitable point of departure is to reconsider the condition for equilibrium. The most general statement we can make concerning the attainment of equilibrium by a system is that it occurs when the entropy of the system plus its surroundings is at a maximum. Unfortunately, entropy has proved to be an elusive concept to master and a difficult quantity to measure. Moreover, reference to the surroundingsdthe “rest of the universe” in the somewhat grandiloquent language of physicsdis a nuisance. Consequently, thermodynamicists sought a function that would help describe equilibrium but would depend only on readily measurable parameters of the system under consideration. As we will see, the Gibbs free energy is such a function for most applications in biology. The concept of entropy (S) is really part of our day-to-day observations. We know that an isolated system will spontaneously change in certain waysda system proceeds toward a state that is more random, or less ordered, than the initial one. For instance, neutral solutes will diffuse toward regions where they are less concentrated (Fig. 1-5). In so doing, the system lowers its capacity for further spontaneous change. For all such processes, DS is positive, whereas DS
633
634
Appendix IV
becomes zero and S achieves a maximum at equilibrium. Equilibrium means that no more spontaneous changes will take place; entropy is therefore an index for the capacity for spontaneous change. It would be more convenient in some ways if entropy had been originally defined with the opposite sign. In fact, some authors introduce the quantity negentropy, which equals S and reaches a minimum at equilibrium. In any case, we must ultimately use a precise mathematical definition for entropy, such as dS ¼ dQ/T, where dQ refers to the heat gain or loss in some reversible reaction taking place at temperature T. We can represent the total entropy of the universe, Su, as the entropy of the system under consideration, Ss, plus the entropy of the rest of the universe, Sr. We can express this in symbols as follows: Su ¼ Ss þ Sr or
(IV.1) dSu ¼ dSs þ dSr
An increase in Su accompanies all real processesdthis is the most succinct way of stating the second law of thermodynamics. Su is maximum at equilibrium. The heat absorbed by a system during some process is equal to the heat given up by the rest of the universe. Let us represent the infinitesimal heat exchange of the system by dQs. For an isothermal reaction or change, dQs is simply dQr because the heat must come from the rest of the universe. From the definition of entropy,1 dS ¼ dQ/T, we can obtain the following relationship: dSr ¼
dQr dQs dUs þ PdVs ¼ ¼ T T T
(IV.2)
The last step in Equation IV.2 derives from the principle of the conservation of energy for the case when the only form of work involved is mechanicalda common assumption in stating the first law of thermodynamics. It is thus possible to express dQs as the sum of the change in internal energy (dUs) plus a work term (PdVs). The internal energy (Us) is a function of the state of a system, i.e., its magnitude depends on the characteristics of the system but is independent of how the system got to that state. PVs is also a well-defined variable. However, heat (Qs) is not a function of the state of a system. As we indicated previously, equilibrium occurs when the entropy of the universe is maximum. This means that dSu then equals zero. By substituting Equation IV.2 into the differential form of Equation IV.1, we can express this equilibrium condition solely in terms of system parameters:
1. This definition really applies only to reversible reactions, which we can in principle use to approximate a given change; otherwise, dQ is not uniquely related to dS.
635
IV.B. Gibbs Free Energy
0 ¼ dSs þ
dUs þ PdVs T
or
(IV.3)
TdSS þ dUs þ PdVs ¼ 0 Equation IV.3 suggests that there is some function of the system that has an extremum at equilibrium. In other words, we might be able to find some expression determined by the parameters describing the system whose derivative is zero at equilibrium. If so, the abstract statement that the entropy of the universe is a maximum at equilibrium could then be replaced by a statement referring only to measurable attributes of the systemdeasily measurable ones, we hope. In the 1870s, Josiah Willard Gibbsdperhaps the most brilliant thermodynamicist to datedchose a simple set of terms that turned out to have the very properties for which we are searching. This function is now referred to as the Gibbs free energy and has the symbol G: G ¼ U þ PV TS
(IV.4a)
which, upon differentiating, yields dG ¼ dU þ PdV þ VdP TdS SdT
(IV.4b)
Equation IV.4b indicates that, at constant temperature (dT ¼ 0) and constant pressure (dP ¼ 0), dG is dG ¼ dU þ PdV TdS
at constant T and P
(IV.5)
By comparing Equation IV.5 with the equilibrium condition expressed by Equation IV.3, we see that dG for a system equals zero at equilibrium at constant temperature and pressure. Moreover, G depends only on U, P, V, T, and S of the system. The extremum condition, dG ¼ 0, actually occurs when G reaches a minimum at equilibrium. This useful attribute of the Gibbs free energy is strictly valid only when the overall system is at constant temperature and pressure, conditions that closely approximate those encountered in many biological situations. Thus our criterion for equilibrium shifts from a maximum of the entropy of the universe to a minimum in the Gibbs free energy of the system.
IV.B. Gibbs Free Energy We will now consider how the internal energy, U, changes when material enters or leaves a system. This will help us derive an expression for the Gibbs free energy that is quite useful for biological applications.
636
Appendix IV
The internal energy of a system changes when substances enter or leave it. For convenience, we will consider a system of fixed volume and at the same temperature as the surroundings so that there are no heat exchanges. If dnj moles of species j enter such a system, U increases by mjdnj, where mj is an intensive variable representing the free energy contribution to the system per mole of species j entering or leaving. Work is often expressed as the product of an intensive quantity (such as mj, P, T, E, and h) times an extensive one (dnj, dV, dS, dQ, and dm, respectively); that is, the amount of any kind of work depends on both: some thermodynamic parameter characterizing the internal state of the system and the extent or amount of change for the system. In our current example, the extensive variable describing the amount of change is dnj, and mj represents the contribution to the internal energy of the system per mole of species j. When more than one species crosses the boundary of our system, which P is at constant volume and the same temperature as the surroundings, the term j mj dnj is added to dU, where dnj is positive if the species enters the system and negative if it leaves. In the general case, when we consider all of the ways that the internal energy of a system can change, we can represent dU as follows: X dU ¼ dQ PdV þ mj dnj (IV.6) j
We now return to the development of a useful relation for the Gibbs free energy of a system. When dU as expressed by Equation IV.6 is substituted into dG as given by Equation IV.5, we obtain X mj dnj þ PdV TdS dG ¼ TdS PdV þ X ¼ mj dnj
j
(IV.7)
j
where dQ has been replaced by TdS. Equation IV.7 indicates that the particular form chosen for the Gibbs free energy leads to a very simple expression for dG at constant T and Pdnamely, dG then depends only on mj and dnj. To obtain an expression for G, we must integrate Equation IV.7. To facilitate the integration, we will define a new variable, a, such that dnj is equal to nj da, where nj is the total number of moles of species j present in the final system; that is, nj is a constant describing the final system. The subsequent integration from a ¼ 0 to a ¼ 1 corresponds to building up the system by a simultaneous addition of all of the components in the same proportions that are present in the final system. (The intensive variable mj is also held constant for this integration pathway, i.e., the chemical potential of species j does not depend
637
IV.C. Chemical Potential
on the size of the system.) Using Equation IV.7 and this easy integration pathway, we obtain the following expression for the Gibbs free energy: Z
G ¼ dG ¼
Z X
mj dnj ¼
Z 1X
j
¼
X
j
mj nj
Z 1
0
da ¼
X
mj nj
0 j
mj nj da (IV.8)
j
The well-known relation between G and mj’s in Equation IV.8 can also be obtained by a method that is more elegant mathematically but somewhat involved. In Chapter 6 (Section 6.1) we presented without proof an expression for the Gibbs free energy (Eq. 6.1 is essentially Eq. IV.8) and also noted some of the properties of G. For instance, at constant temperature and pressure, the direction for a spontaneous change is toward a lower Gibbs free energy; minimum G is achieved at equilibrium. Hence, DG is negative for such spontaneous processes. Spontaneous processes can in principle be harnessed to do useful work, where the maximum amount of work possible at constant temperature and pressure is equal to the absolute value of DG (some of the energy is dissipated by inevitable inefficiencies such as frictional losses, so DG represents the maximum work possible). To drive a reaction in the direction opposite to that in which it proceeds spontaneously requires a free energy input of at least DG (cf. Fig. 2-6).
IV.C. Chemical Potential We now P examine the properties of the intensive variable mj. Equation IV.8 (G ¼ j mj nj ) suggests a very useful way of defining mj. In particular, if we keep mj and ni constant, we obtain the following expression: vG vG mj ¼ ¼ (IV.9) vnj mi ;ni vnj T;P;E;h;ni where ni and mi refer to all species other than species j. Because mj can depend on T, P, E (the electrical potential), h (the height in a gravitational field), and ni, the act of keeping mi constant during partial differentiation is the same as that of keeping T, P, E, h, and ni constant, as is indicated in Equation IV.9. Equation IV.9 indicates that the chemical potential of species j is the partial molal Gibbs free energy of a system with respect to that species and that it is obtained when T, P, E, h, and the amount of all other species are held constant. Thus mj corresponds to the intensive contribution of species j to the extensive quantity G, the Gibbs free energy of the system.
638
Appendix IV
In Chapters 2 and 3 we argued that mj depends on T, aj (aj ¼ gjcj; Eq. 2.5), P, E, and h in a solution and that the partial pressure of species j, Pj, is also involved for the chemical potential in a vapor phase. We can summarize the two relations as follows (see Eqs. 2.4 and 2.21): liquid
mj
¼ mj þ RT ln aj þ V j P þ zj FE þ mj gh vapor
mj
¼ mj þ RT ln
Pj þ mj gh Pj
(IV.10a) (IV.10b)
The forms for the gravitational contribution (mjgh) and the electrical one (zjFE) can be easily understood. We showed in Chapter 3 (Section 3.2A) that RT ln aj is the correct form for the concentration term in mj. The reasons for the forms of the pressure terms in a liquid (Vj P) and in a gas [RT ln ðPj =Pj Þ] are not so obvious. Therefore, we will examine the pressure dependence of the chemical potential of species j in some detail.
IV.D. Pressure Dependence of mj To derive the pressure terms in the chemical potentials of solvents, solutes, and gases, we must rely on certain properties of partial derivatives as well as on commonly observed effects of pressure. To begin with, we will differentiate the chemical potential in Equation IV.9 with respect to P: " # vmj v vG ¼ vP vnj T;P;E;h;ni vP T;E;h;ni ;nj T;E;h;ni ;nj
"
# v vG ¼ vnj vP T;E;h;ni ;nj
(IV.11)
T;P;E;h;ni
where we have reversed the order for partial differentiation with respect to P and nj (this is permissible for functions such as G, which have well-defined and continuous first-order partial derivatives). The differential form of Equation IV.4 (dG ¼ dU þ PdV þ VdP TdS SdT) gives us a suitableP form for dG. If we substitute dU given by Equation IV.6 (dU ¼ TdS PdV þ j mj dnj , where dQ is replaced by TdS) into this expression for the derivative of the Gibbs free energy, we can express dG in the following useful form: dG ¼ VdP SdT þ
X mj dnj
(IV.12)
j
Using Equation IV.2 we can readily determine the pressure dependence of the Gibbs free energy as needed in the last bracket of Equation IV.11dnamely,
639
IV.D. Pressure Dependence of mj
ðvG=vPÞT;E;h;ni ;nj is equal to V by Equation IV.12. Next, we have to consider the partial derivative of this V with respect to nj (see the last equality of Eq. IV.11). Equation 2.6 indicates that ðvV=vnj ÞT;P;E;h;ni is Vj, the partial molal volume of species j. Substituting these partial derivatives into Equation IV.11 leads to the following useful expression: vmj ¼ Vj (IV.13) vP T;E;h;ni ;nj Equation IV.13 is of pivotal importance in deriving the form of the pressure term in the chemical potentials of both liquid and vapor phases. Let us first consider an integration of Equation IV.13 appropriate for a liquid. We will make use of the observation that the partial molal volume of a species in a solution does not depend on the pressure to any significant extent. For a solvent, this means that the liquid generally is essentially incompressible. If we integrate Equation IV.13 with respect to P at constant T, E, h, ni, and nj with Vj independent of P, we obtain the following relations: Z Z mliquid j vmj liquid dP ¼ dmj ¼ mj mj vP mj (IV.14) Z Z ¼ V j dP ¼ V j dP ¼ V j P þ “constant” where the definite integral in the top line is taken from the chemical potential of species j in a standard state as the lower R limit up to the general mj in a liquid as the upper limit. The integration of Vj dP leads to our pressure term Vj P plus a constant. Because the integration was performed while holding T, E, h, ni, and nj fixed, the “constant” can depend on all of these variables but not on P. Equation IV.14 indicates that the chemical potential of a liquid contains a pressure term of the form Vj P. The other terms (mj , RT ln aj, zjFE, and mjgh; see Eq. IV.10a) do not depend on pressure, a condition used throughout this text. The experimental observation that gives us this useful form for mj is that Vj is generally not influenced very much by pressure; for example, a liquid is often essentially incompressible. If this should prove invalid under certain situations, Vj P would then not be a suitable term in the chemical potential of species j for expressing the pressure dependence in a solution. Next, we discuss the form of the pressure term in the chemical potential of a gas, where the assumption of incompressibility that we used for a liquid is not valid. Our point of departure is the perfect or ideal gas law: Pj V ¼ nj RT
(IV.15)
where Pj is the partial pressure of species j, and nj is the number of moles of species j in volume V. Thus we will assume that real gases behave like ideal gases, which is generally justified for biological applications. Based on
640
Appendix IV
Equations IV.15 and 2.6 ½Vj ¼ ðvV=vnj ÞT;P;E;h;ni , the partial molal volume Vj for gaseous species j is RT/Pj. We also note that the total pressure P is equal to P j Pj , where the summation is over all gases present (Dalton’s law of partial pressures); hence, dP equals dPj when ni, T, and V are constant. When we integrate Equation IV.13, we thus find that the chemical potential of gaseous species j depends on the logarithm of its partial pressure: Z Z mvapor Z Z j vmj RT vapor dP ¼ dmj ¼ mj mj ¼ V j dP ¼ dPj Pj vP mj (IV.16) ¼ RT ln Pj þ “constant” where the “constant” can depend on T, E, h, and ni but not on nj (or Pj). In particular, the “constant” equals RT ln Pj þ mj gh, where Pj is the saturation partial pressure for species j at atmospheric pressure and some particular temperature. Hence, the chemical potential for species j in the vapor phase vapor (mj ) is vapor
mj
¼ mj þ RT ln Pj RT ln Pj þ mj gh ¼ mj þ RT ln
Pj þ mj gh Pj
(IV.17)
which is essentially the same as Equations 2.21 and IV.10b. We have defined the standard state for gaseous species j, mj , as the chemical potential when the gas phase has a partial pressure for species j (Pj) equal to the saturation partial pressure (Pj ), when we are at atmospheric pressure (P ¼ 0) and the zero level for the gravitational term (h ¼ 0), and for some specified temperature. Many physical chemistry texts ignore the gravitational term (we calculated that it has only a small effect for water vapor; see Chapter 2, Section 2.4C) and define the standard state for the condition when Pj equals 1 atm and species j is the only species present (P ¼ Pj). The chemical potential of such a standard state equals mj RT ln Pj in our symbols. The partial pressure of some species in a vapor phase in equilibrium with a liquid depends slightly on the total pressure in the systemdloosely speaking, when the pressure on the liquid is increased, more molecules are squeezed out of it into the vapor phase. The exact relationship between the pressures involved, which is known as the Gibbs equation, is as follows for water: vðln Pwv Þ V w ¼ vP RT or
(IV.18) vPwv Vw ¼ vP V wv
where the second equality follows from the derivative of a logarithm [v ln u/vx ¼ (1/u) (vu/vx)] and the ideal or perfect gas law [PwvV ¼ nwvRT (Eq. IV.15),
IV.E. Concentration Dependence of mj
641
so vV=vnwv ¼ Vwv ¼ RT=Pwv ]. Because Vw is much less than Vwv , the effect is quite small (e.g., at 20 C and atmospheric pressure, V w ¼ 1:8 105 m3 mol1 and Vwv ¼ 2:4 102 m3 mol1 ). From the first equality in Equation IV.18, we see that Vw dP equals RTd ln Pwv. Hence, if the chemical potential of the liquid phase (mw) increases by Vw dP as an infinitesimal pressure is applied, then an equal increase, RTd ln Pwv, occurs in mwv (see Eq. IV.10b for a definition of mwv), and hence we will still be in equilibrium (mw ¼ mwv). This relation can be integrated to give RT ln Pwv ¼ Vw P þ constant, where the constant is RT ln P0wv and P0wv is the partial pressure of water vapor at standard atmospheric pressure; hence, RT ln Pwv =P0wv is equal to Vw P, a relation used in Chapter 2 (see Section 2.4C). We note that effects of external pressure on Pwv can be of the same order of magnitude as deviations from the ideal gas law for water vapor, both of which are usually neglected in biological applications.
IV.E. Concentration Dependence of mj We will complete our discussion of chemical potential by using Equation IV.17 to obtain the logarithmic term in concentration that is found for mj in a liquid phase. First, it should be pointed out that Equation IV.17 has no concentration term per se for the chemical potential of species j in a gas phase. However, the partial pressure of a species in a gas phase is really analogous to the concentration of a species in a liquid; e.g., PjV ¼ njRT for gaseous species j (Eq. IV.15), and concentration means number/volume and equals nj/V, which equals Pj/RT. Raoult’s law states that at equilibrium the partial pressure of a particular gas above its volatile liquid is proportional to the mole fraction of that solvent in the liquid phase. A similar relation more appropriate for solutes is Henry’s law, which states that Pj in the vapor phase is proportional to the Nj of that solute in the liquid phase. Although the proportionality coefficients in the two vapor relations are different, they both indicate that Pj depends linearly on solution Nj . For dilute solutions, the concentration of species j, cj, is proportional to its mole fraction, Nj (this is true for both solute and solvent). Thus when vapor Pj changes from one equilibrium condition to another, we expect a similar liquid vapor change in csolution because mj is equal to mj at equilibrium. In particular, j vapor Equation IV.17 indicates that mj depends on RT ln ðPj =Pj Þ, and hence the chemical potential of a solvent or solute should contain a term of the form RT ln cj, as in fact it does (see Eqs. 2.4 and IV.10). As we discussed in Chapter 2 (Section 2.2B), we should be concerned about the concentration that is thermodynamically active, aj (aj ¼ gjcj; Eq. 2.5), so the actual term in the chemical potential for a solute or solvent is RT ln aj, not RT ln cj. In Chapter 3 (Section 3.2A), instead of the present argument based on Raoult’s and Henry’s laws, we used a comparison with Fick’s first law to justify the RT ln aj term. Moreover,
642
Appendix IV
the BoyleeVan’t Hoff relation, which was derived assuming the RT ln aj term, has been amply demonstrated experimentally. Consequently, the RT ln aj term in the chemical potential for a solute or solvent can be justified or derived in a number of different ways, all of which depend on agreement with experimental observations.