Spontaneity and Equilibrium II: Multireaction Systems - Journal of

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Spontaneity and Equilibrium II: Multireaction Systems Lionel M. Raff* Department of Chemistry, Oklahoma State University, Stillwater, Oklahoma 74078, United States ABSTRACT: The thermodynamic criteria for spontaneity and equilibrium in multireaction systems are developed and discussed. When N reactions are occurring simultaneously, it is shown that G and A will depend upon N independent reaction coordinates, ξα (α = 1,2, ..., N), in addition to T and p for G or T and V for A. The general criteria for spontaneity and equilibrium are the same as those for a single-reaction system, dG ≤ 0 (T and p constant) or dA ≤ 0 (T and V constant). However, dG and dA are now the sum of N terms. It is shown that this sum has the form dG = dA = ∑Nα=1 Δrμα dξα, where Δrμα is the reaction chemical potential for reaction α, (∂G/∂ξα)T,p,ξj. Consequently, the result that dG < 0 at a particular composition provides no information about which of the N reactions are proceeding spontaneously and which are nonspontaneous. At most compositions, there exist an infinite number of both spontaneous and nonspontaneous pathways for the system. Equilibrium in a multireaction system exists at a single composition point, as is the case for a single-reaction system. Determination of this point requires the simultaneous solution of N algebraic equations, which are often nonlinear. Although there exists an infinite number of possible, spontaneous paths, the thermodynamically most probable path is shown to be along the negative gradient of the thermodynamic potential. This path corresponds to that for the maximum chemical force. It is also shown to correspond to the statistically most-probable reaction path. The analysis demonstrates that the direction cosines for the negative gradient vector at each composition point are given by the reaction chemical potentials. The equations yielding the gradient vector comprise a set of N, coupled, first-order differential equations involving the reaction chemical potentials whose solution depends upon the initial state of the system. The calculation is, therefore, analogous to the computation of a trajectory in a molecular dynamics study. The analysis clearly shows that the reaction chemical potentials determine spontaneity, equilibrium, and the thermodynamically expected reaction path, not G, A, ΔG, or ΔA. The principles resulting from the analysis are illustrated by application to a simple reaction system involving unimolecular isomerization along two pathways. KEYWORDS: Upper-Division Undergraduate, Graduate Education/Research, Chemical Engineering, Physical Chemistry, Misconceptions/Discrepant Events, Equilibrium, Thermodynamics reaction products. Specifically, ξ = ni/νi with νi being the stoichiometric coefficient of product i. Because (∂A/∂ξ)T,V = (∂G/∂ξ)T,p = Δrμ, where Δrμ is the difference between the chemical potentials of the reaction products weighted by their stoichiometric coefficients and the same quantity for the reactants, it is seen that the chemical potentials play the central role in determining spontaneity and equilibrium. It is further shown1 that the quantities ΔA and ΔG for a finite interval yield very little information concerning reaction spontaneity at specific points along the interval and no information about the equilibrium point. Specifically, if ΔA < 0 (constant T and V) or ΔG < 0 (constant T and p), no information about reaction spontaneity at the final point or at any other arbitrary point in the interval is available. The reaction is known to be spontaneous in an underdetermined region that includes the initial point of the interval. If ΔA > 0 (constant T and V) or ΔG > 0 (constant T and p), no information about reaction spontaneity at the initial point or at any other arbitrary point in the interval is available. The reaction is known to be nonspontaneous in an underdetermined region that includes the end point of the interval.

I

n a previous publication in this Journal,1 it has been shown that in systems involving a single chemical reaction, the fundamental criteria for chemical reactions to be spontaneous in a given direction are often incorrectly stated as ΔG < 0 or ΔA < 0, where G and A are the Gibbs and Helmholtz free energies, respectively. Similarly, the criteria for equilibrium are also misstated as being ΔG = 0 or ΔA = 0. The correct criteria are shown to be dA < 0 or (∂A /∂ξ)T , V < 0 for reaction spontaneity

(1a)

and dA = 0 or (∂A /∂ξ)T , V = 0 for equilibrium when dT = 0 (1b)

and dV = 0

dG < 0 or (∂G /∂ξ)T , p < 0 for reaction spontaneity

(1c)

and dG = 0 or (∂G /∂ξ)T , p = 0 for equilibrium when dT = 0 and dp = 0

(1d)

where ξ is a reaction coordinate for the process in question that is defined in terms of the number of moles of one of the © XXXX American Chemical Society and Division of Chemical Education, Inc.

A

dx.doi.org/10.1021/ed4008205 | J. Chem. Educ. XXXX, XXX, XXX−XXX

Journal of Chemical Education

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corresponding stoichiometric coefficient. However, if we desire to have positive differential changes of ξα correspond to reaction in the forward direction, the ξα should be defined in terms of a product molecule in each reaction. Consequently, we take the change in the number of moles of the first product α molecule P 1α (g), nj=1 , in each reaction to define the corresponding reaction coordinate. That is

With respect to the spontaneity of the transformation of a system from State A to State B, the result that ΔA < 0 (constant T and V) or ΔG < 0 (constant T and p) for the transformation yields no information about the spontaneity or nonspontaneity of the process. The result ΔA > 0 (constant T and V) or ΔG > 0 (constant T and p) guarantees that the transformation is nonspontaneous. These are sufficient conditions to ensure nonspontaneity, but they are not necessary conditions. If the transformation is known to be spontaneous, this information ensures that ΔA < 0 (constant T and V) or ΔG < 0 (constant T and p). That is, ΔA < 0 or ΔG < 0 is a necessary condition for spontaneity, but it is not a sufficient condition. The spontaneity and equilibrium criteria expressed by eqs 1a−1d hold only for systems involving a single reaction. For such systems, G at constant T and p or A at constant T and V is always a function of a single reaction coordinate. Consequently, dG or dA is a function of a single variable which uniquely determines whether the reaction is spontaneous, nonspontaneous, or at equilibrium. If at ξ = ξo with T and p constant, we have (∂G/∂ξ)T,p |ξo < 0, the reaction will be spontaneous in the direction of ξ > 0. If we have (∂G/∂ξ)T,p |ξo = 0, the reaction will be at the point of equilibrium. The corresponding equations hold for A at constant T and V. When several reactions are occurring simultaneously, the situation is much more complex. The fact that such systems are rarely, if ever, treated in modern physical chemistry textbooks is a reflection of this complexity. In this paper, we present an analysis of spontaneity and equilibrium in multireaction systems. It is shown that at most points in the composition space of the system there exist an infinite number of spontaneous and nonspontaneous reaction pathways. However, it is argued that there is one thermodynamically most-favored reaction pathway that is uniquely determined by the chemical potentials of the components of the system. Equilibrium is more complex in multireaction systems, but like single-reaction systems, it exists at a single point in the multidimensional composition hyperspace of the system. Following the development of the general equations in the next section, the important points are illustrated by two simple numerical examples that involve a single gaseous reactant undergoing unimolecular isomerizations along two different pathways.

Nmax =

νiα R iα(g) →

m(α)

∑j=1

dniα = −(νiα /νjα= 1) dnjα= 1 = −νiα dξα

(4)

(5a)

For the products, dnjα = (νjα /νjα= 1) dnjα= 1 = νjα dξα

(5b)

Consequently, we expect that the Gibbs and Helmholtz free energies will each be dependent upon the N reaction coordinates. That is, we expect to have G = G(T,p,ξ1,ξ2, ..., ξN) and A = A(T,V,ξ1,ξ2, ..., ξN). Although the ξα are independent, their defined ranges will be dependent upon the initial composition of the reaction system, the reaction stoichiometries, and upon mass conservation and limiting reagent considerations. In general, each reaction coordinate will be defined between upper and lower limits, ξ+α and ξ−α , respectively. It should be noted that in writing G = G(T,p,ξ1,ξ2, ..., ξN) and A = A(T,V,ξ1,ξ2, ..., ξN), we are implicitly assuming that the equilibration rates of the thermal energy and pressure are fast compared to the reaction rates so that T, p, and the thermodynamic potentials may be defined. Because translational energy equilibration is generally very fast, this will usually be true. The quantities of importance in the determination of spontaneity and equilibrium are dG and dA. These quantities are derived in almost any modern textbook of physical chemistry.2−4 They are given by dA = − S dT − p dV +

νjα Pαj (g) for α = 1, 2, ..., N

∑ μi dni i

(6)

and

(2)

ναi

N

∑α= 1 [k(α) + m(α)]

However, for a closed system in which no material change with external sources occurs, the number of independent variables will be much less than Nmax because the stoichiometry of each reaction allows the change in the number of moles all compounds involved in reaction α to be expressed in terms of dξα. That is, for the reactants

GENERAL TREATMENT Consider a system involving N simultaneous gaseous reactions. We shall employ a Greek index α to denote the reaction where 1 ≤ α ≤ N. The number of reactants, R(α)(g), in reaction α is k(α). The number of products, P(α)(g), in reaction α is m(α). In this notation with a summation index i for reactants and j for products, the system of reactions can be written in the form k(α)

(3)

α where noα j=1 is the initial number of moles of P1 (g) in reaction α. In the last portion of eq 3, the subscript “j = 1” has been omitted to simplify the notation. The total number of compounds in the system may be very large. If no compound appears in more than one reaction, this number will be a maximum, Nmax, where

■

∑i= 1

nα − noα να

ξα ≡ (njα= 1 − njo=α1)/νjα= 1 =

ναj

where the and are the stoichiometric coefficients in reaction α. With N reactions present, N different reaction coordinates are required to specify a point in the N-dimensional reaction hyperspace. These coordinates are denoted by ξα (α = 1, 2, ..., N). In principle, ξα may be defined in terms of the number of moles of any reactant Rαi (g) or product Pαj (g) and the

dG = − S dT − V dp +

∑ μi dni i

(7)

where μi is the partial molar Gibbs free energy of component i, (∂G/∂ni)T,p,nj. The summations in eqs 6 and 7 range over all the components of the system. The quantity μi is a function of both T and p. It is commonly termed the chemical potential for B



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component i. For simplicity in this paper, we shall assume that the reactions are being conducted at a pressure below 10 bar where the ideal gas equation of state is usually sufficiently accurate. For such ideal systems, the dependence of μi(T,p) on pressure, p, for a pure gaseous compound is μi (T , p) = μi o (T ) + RT ln(p /ps )

At constant T and p, G will be dependent upon N reaction coordinates. That is G = G(ξ1,ξ2, ..., ξN). Consequently, the total differential dG will be ⎛ ∂G ⎞ dG = ⎜ ⎟ dξ1 + ⎝ ∂ξ1 ⎠T , p , ξ j

(8)

dGα = dξα[∑

ν αμ α j=1 j j

(9)

−

k(α)

∑

ν αμ α ] j=1 j i

dG = Δr μ1 dξ1 + Δr μ2 dξ2 + ... + Δr μN dξN =

(16a)

≡ dξαΔr μα

dA = 0 for equilibrium when dT = 0 and dV = 0

m (α ) j=1

νjα ln pjα −

k (α )

∑i = 1

(17a)

and dG = 0 for equilibrium when dT = 0 and dp = 0

νiα ln piα }]

where m(α)

νjαμjoα −

k(α)

∑i= 1

νiαμioα

(12)

oα In eq 12, the μoα j and μi are the standard chemical potentials for products and reactants, respectively, in reaction α. Equations 10−12 show that

⎛ ∂G ⎞ m(α) = [Δr μαo + RT { ∑ j = 1 νjα ln pjα ⎜ ⎟ ⎝ ∂ξα ⎠T , p , ξ j

−

k(α)

∑i= 1

νiα ln piα }] = Δr μα

(17b)

Here, we note for clarity that in this paper reaction spontaneity addresses the question of whether or not the multireaction system will move along a specified path emanating from a specific point in the composition space of the N-reaction system given by (ξ1, ξ2, ..., ξN) without the necessity to do work. It does not refer to the spontaneity of a finite transformation from State A to State B of the multireaction system. Such questions are denoted by the phrase “transformation spontaneity”. The criteria in eqs (16a,b) and (17a,b) are general. They hold regardless of the complexity of the system. The problem presented by multireaction systems is shown by eqs 14 and 15. In this case, the sign of dG is dependent upon the sum of N different terms. Consequently, the result that dG < 0 at a particular composition provides no information about which of the N reactions are proceeding spontaneously in the direction of products and which are nonspontaneous in that direction. All we know for certain is that some of the reactions are proceeding spontaneously (dξ > 0) at that particular composition but others may have dξ < 0. The numerical examples provided in the next section demonstrate that there are, in fact, an infinite number of spontaneous reaction directions at most system compositions as well as infinite number of directions that are nonspontaneous because they have dG > 0. Similar conclusions are obtained from dA when T and V are constant. Consequently, the sign of dG or dA at a specific composition does not determine unequivocally which reactions will proceed spontaneously and which will not. The conditions for equilibrium expressed by eqs 16b and 17b still hold in a multireaction system. However, determination of the composition of the system at equilibrium is much more complex than is the case when only a single reaction is involved. Because the dξα’s in eq 14 are arbitrary differential changes, in

(11)

∑j=1

(16b)

dG < 0 for reaction spontaneity dT = 0 and dp = 0

μαi

dGα = dξα[Δr μαo + RT {∑

(15)

dA < 0 for reaction spontaneity when dT = 0 and dV = 0

where we have replaced (T,p) with to simplify the notation and defined Δrμα to be the factor inside the square brace. Because this factor is the difference between the sum of the chemical potentials for the products and that for the reactants in reaction α, it is appropriate to call it the “reaction chemical potential” for reaction α and denote it with the notation Δrμα. Most texts term this quantity the “free energy of reaction” and denote it with the notation ΔrGα, or for a singlereaction system, ΔrG. These choices introduce numerous problems. Δrμα is not a free energy nor does it have the units of energy. It is the rate of change of the Gibbs free energy with respect to ξα. The notation ΔrGα does not suggest a derivative or a rate of change. When this notation is employed, it is always necessary to point out ΔrGα does not represent a change in the Gibbs free energy over some finite interval but rather the difference in chemical potentials at a specific point in the composition space of the system. This fact is obvious when the notation Δrμα is employed. These problems have been discussed in detail by Quilez.5 Insertion of eq 9 into 10 with ps equal to 1 bar yields

Δr μαo =

N

∑α = 1 Δr μα dξα

The criteria for spontaneity, nonspontaneity, and equilibrium derived in ref 1 and in virtually all physical chemistry textbooks2−4 are

(10)

μαi

j

where all partial derivatives appearing in eq 14 are taken with T, p, and all other reaction coordinates held constant. Similar equations hold when T and V are constant. We simply replace G with A. Equation 13 allows us to write eq 14 in the form

where pi is the partial pressure of gas i in the mixture. In most cases, ps is taken to be 1 bar. In this case, eq 9 can be written as μi(T,p) = μio(T) + RT ln(pi), where it is understood that the pressure is its magnitude expressed in bars because the units have canceled in the ratio pi/ps. At constant T and p, combination of eqs 5a, 5b, and 7 gives dG for reaction α m(α)

j

(14)

where ps is the standard pressure and μio(T) is the standard chemical potential.1−4 As the notation implies, μio(T) is a function of T alone. If the entropy of mixing for a system of ideal gases is included, it is simple to show1−4 that for such a mixture, μi(T,p) is given by μi (T , p) = μi o (T ) + RT ln(pi /ps )

⎛ ∂G ⎞ ⎛ ∂G ⎞ dξ2 + ... + ⎜ dξN ⎟ ⎜ ⎟ ⎝ ∂ξ2 ⎠T , p , ξ ⎝ ∂ξN ⎠T , p , ξ

(13) C



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constant-energy contour surface for the Gibbs free energy. The gradient vector will be normal to this contour surface at ξo, and its direction and magnitude give the maximum rate of increase of G(ξ1,ξ2, ..., ξN).7 Hence, the maximum rate of decrease of G and the greatest chemical force will occur along the path determined by the direction of the negative gradient vector at each point along the path. We expect this to be the thermodynamically most favored reaction path. The gradient of the N + 1 dimensional hypersurface for G = G(ξ1,ξ2, ..., ξN), ∇⃗ G, is given by

order to have dG = 0 at equilibrium, the coefficients of the dξα must all vanish. That is, we must have at equilibrium ⎛ ∂G ⎞ m(α) α ν ln pjα − = [Δr μαo + RT {∑ ⎜ ⎟ j=1 j ⎝ ∂ξα ⎠T , p , ξ

k(α)

∑i = 1

νiα ln piα }]eq

j

for α = 1, 2, ..., N

= (Δr μα )eq = 0

(18)

Equation 18 represents a set of N algebraic equations whose solution is the equilibrium values of the ξα. In the most complex, general case, these equations will be coupled, nonlinear equations whose solution must be obtained by a suitable numerical method. In the special case in which no reactant or product appears in more than one reaction, G will have separable form, that is, G(T,p,ξ1,ξ2, ..., ξN) = G(T,p,ξ1) + G(T,p,ξ2) + ... + G(T,p,ξN). In this case, the set of N equations will be uncoupled allowing them to be solved separately. Using eq 18, we may define a temperature-dependent equilibrium constant, Kαp (T), for each reaction. It has the same form as in the single-reaction system. That is,

⎛ ∂G ⎞ ⎛ ∂G ⎞ ⎛ ∂G ⎞ ϖN ϖ2 + ... + ⎜ ϖ1 + ⎜ ∇⃗G = ⎜ ⎟ ⎟ ⎟ ⎝ ∂ξ1 ⎠T , p , ξ ⎝ ∂ξ2 ⎠T , p , ξ ⎝ ∂ξN ⎠T , p , ξ j

= Δr μ1ϖ1 + Δr μ2 ϖ2 + ... + Δr μN ϖN

α

α

(19)

N

= [∑

α=1

[(Δr μα )]2 ]1/2

(23)

The reaction path that follows the negative gradient of the thermodynamic potential at each point may be determined in a straightforward manner. Consider an infinitesimal change in G, dG, produced by the reaction of dn moles starting at composition point ξo = (ξ1,ξ2, ..., ξN)o, where dn = dξ1 + dξ2 + ... + dξN. The change in the individual reaction coordinates will depend upon the path or direction of change of dG. We ⎯→ ⎯ denote the magnitude and direction of dG by dG. If the change occurs in the direction of the negative gradient that corresponds to the direction of maximum chemical force, then eqs 22 and 23 show that ⎯→ ⎯ ∇⃗G dG = − g ⃗ dn = − dn C = −C −1[Δr μ1ϖ1 + Δr μ2 ϖ2 + ... + Δr μN ϖN ] dn

(24)

⎯→ ⎯ The component of dG along the α reaction coordinate, dGα, is ⎯→ ⎯ given by dG·ϖα. Using the orthogonality of the ϖi, we obtain dGα = −C −1Δr μα dn

(20)

where G functions as the thermodynamic potential if T and p are constant. If T and V are constant, A replaces G in eq 20. In the absence of other influencing factors, the chemical system will always move in the direction of Fξ, and with the right engineering design, the force may be employed to produce useful work, for example in a galvanic cell.6 In a multireaction system, eq 20 becomes

Fξ = −∇⃗G

(22)

1/2 ⎡ ⎡ ⎤2 ⎤ ⎛ ⎞ ⃗ ⎢ ⎥ N ⎢ ∂G ∇G ⎥ , with C = ⎢∑ ⎢⎜ g⃗ = ⎟ ⎥⎥ α = 1 ⎝ ∂ξ ⎠ C α T ,p,ξ ⎦ ⎥ ⎢ ⎣ j ⎣ ⎦

where the partial pressures are those at the equilibrium point. Although there exist an infinite number of spontaneous reaction paths at most system compositions, the various spontaneous paths are not all equally probable. In fact, many are highly improbable. In the absence of other influencing factors, such as reaction rates, the thermodynamically most probable reaction path will be in the direction of maximum chemical force,1 which is the direction that maximizes the rate at which the thermodynamic potential, G or A, decreases. This direction will be given by the negative gradient of the potential that is generally written as either −grad G or −∇⃗ G. When a system possesses a potential energy, Φ, due to the relative position of the atoms or molecules, any change of position of atom i, dqi, either produces or requires a force. This force is given by Fi = −(∂Φ/∂qi)qj. In the absence of other influencing factors, the system will always move in the direction of F⃗, and with the right engineering design, the force may be employed to produce useful work. Similarly, when a singlereaction system possesses potential energy due to chemical composition, any change of composition, dξ, either requires or produces a force whose functional form is the same as that for energy due to position. Thus, the chemical force, Fξ, is given by1 ⎛ ∂G ⎞ Fξ = −⎜ ⎟ ⎝ ∂ξ ⎠T , p

j

where the ϖα are unit displacement vectors in the direction of the ξα axes in a plot of the hypersurface for G = G(ξ1,ξ2, ..., ξN).7 It is important to note that the components of the gradient vector are determined by the reaction chemical potentials associated with the N reaction coordinates. A unit vector in the direction of the gradient, g,⃗ is given by

K pα(T ) = exp[−Δr μαo (T )/RT ] k(α) ν eq = Π mj =(α1)[(p)ν ]eq j / Πi = 1 [(p) ]i

j

for α = 1, 2, ..., N and ξα− ≤ ξα ≤ ξα+

(25)

This result shows that the −C−1Δrμα values can be considered as a set of direction cosines that yield the various reaction ⎯→ ⎯ components that produce dG. Equations 18 and 24 show that at the point of equilibrium, the components of the chemical force in the direction of the unit vectors ϖα are zero for all values of α, as expected. Equation 25 represents an initial-value problem involving a set of N coupled, first-order, differential equations whose variables are the N reaction coordinates along the gradient vector starting at the initial point ξo. The problem of

(21)

In general, the gradient vector of G will be tangential to G at the point at which it is evaluated, ξo = (ξ1, ξ2, ..., ξN)o. If, at this point, the value of G is G(ξo), then G = G(ξo) defines a D



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= nipo/noA, because V = noART/po. This permits dG to be expressed in terms of the two reaction coordinates. If the standard pressure is taken to be 1 bar, the result is

determining the thermodynamically most probable reaction path from configuration point ξo is, therefore, analogous to the computation of a molecular dynamics trajectory starting from some initial state.8,9 Both calculations involve the solution of a set of coupled, first-order, differential equations. Both solutions require the repeated computation of the forces from partial derivatives of the appropriate potential. Consequently, the various numerical methods for solving this problem that have been developed in many molecular dynamics studies can be employed in the solution of eq 25 that yields the thermodynamically most probable reaction path. The previous analysis demonstrates the central importance of the chemical potentials in determining the thermodynamically most probable reaction path and equilibrium in multireaction systems. In the next section, two simple, numerical examples are presented that illustrate all of these principles.

dG(ξ1 , ξ2) = [Δr μ1o + RT ln(po ξ1/nAo ) − RT ln{po (nAo − ξ1 − ξ2)/nAo }] dξ1 + [Δr μ2o + RT ln(po ξ2/nAo ) − RT ln{po (nAo − ξ1 − ξ2)/nAo }] dξ2

The two equilibrium constants for this system can be easily obtained using eq 18. At equilibrium, we must have [∂G(ξ1,ξ2)/ ∂ξ1]T,p,ξ2 and [∂G(ξ1,ξ2)/∂ξ2]T,p,ξ1 equal to zero. These requirements with eq 29 yield

■

⎤ ⎡ ξ1 K1 = exp[−Δr μ1o /RT ] = ⎢ o ⎥ ⎣ nA − ξ1 − ξ2 ⎦eq

A SIMPLE QUANTITATIVE EXAMPLE In this section, we consider the simultaneous isomerization of a hypothetical compound A(g) along two pathways yielding products B(g) and C(g). That is, the reactions of interest are Reaction 1:

A(g) → B(g)

Reaction 2:

A(g) → C(g)

⎤ ⎡ ξ2 K 2 = exp[−Δr μ2o /RT ] = ⎢ o ⎥ ⎣ nA − ξ1 − ξ2 ⎦eq

(ξ1)eq = [nAo K 2/(1 + K1 + K 2)]

G(ξ1 , ξ2) =

+ RT[(nAo − ξ1 − ξ2) ln{po(nAo − ξ1 − ξ2)/nAo } + ξ1 ln(poξ1/nAo ) + ξ2 ln(poξ2/nAo )] for ps = 1 bar and ξ1 + ξ2 ≤ nAo

(34)

Straightforward partial differentiation of eq 34 may easily be shown to yield eq 29. To provide numerical examples that may be used to illustrate spontaneity and equilibrium in a multireaction system, values of V, noA, and the standard chemical potentials for A, B, and C must be specified. For simplicity, we take noA = 1 mol and V = 24.790 L, which yields po = RT/V = 1 bar. This choice for noA makes the lower and upper limits for both reaction coordinates 0 and 1, respectively. For the standard chemical potentials, we consider two cases. Case 1 is a totally symmetric system in which μoA = μoB = μoC = −10,000 J mol−1. Case 2 is an unsymmetrical system with μoA = −10,000 J mol−1, μoB = −12,000 J mol−1, and μoC = −14,000 J mol−1. Once the standard chemical potentials and noA are fixed, eqs 30 and 31 determine the equilibrium constants and the

(27)

+ [Δr μ2o + RT ln(pC /ps ) − RT ln(pA /ps )] dξ2 (28)

where =μB−μ = μ C − μ A, and the pi are the partial pressures. The partial pressures are given by pi = niRT/V o

(33)

G(ξ1 , ξ2) = nAo μAo + Δr μ1o ξ1 + Δr μ2o ξ2

dG = [Δr μ1o + RT ln(pB /ps ) − RT ln(pA /ps )] dξ1

o A, Δrμ2

∑i μi ni = μA nA + μB nB + μCnC

Combination of the stoichiometry and eq 9 with eq 33 yields, after rearrangement,

Equation 10 shows that dG1 is the differential change in G associated with reaction Reaction 1 while dG2 is that for Reaction 2. When the chemical potentials in eq 27 are replaced with eq 9, the result is

o

(32)

We now need the appropriate expression for G(ξ1,ξ2). This may be obtained by indefinite integration of dG1 and dG2. However, it is easier to use the fact that the chemical potentials at any composition give the Gibbs free energy per mole so that we have

We consider the usual chemical reaction case in which the initial state is pure A(g) with noA denoting the initial number of moles of A(g) and noB = noC = 0. Finally, we focus solely upon the thermodynamic constraints on the system and ignore kinetic factors. Following eq 3, the two required reaction coordinates, ξ1 and ξ2, are defined to be ξ1 = nB and ξ2 = nC. The reaction stoichiometry and initial conditions require that nA = noA − ξ1 − ξ2 so that dnA = −dξ1 − dξ2. Combination of these conditions with eq 26 gives

o

and

(ξ2)eq = [nAo K1/(1 + K1 + K 2)]

(26)

Δrμo1

(31)

Although eqs 30 and 31 are coupled, they are easily solved to yield the equilibrium values of the two reaction coordinates. The results are

∑i μi dni = μA dnA + μB dnB + μC dnC

dG = (μB ‐μA ) dξ1 + (μC − μA ) dξ2 ≡ dG1 + dG2

(30)

and

The analysis assumes that the pressure is sufficiently low that we may employ the ideal-gas equation of state. We also assume that translational energy equilibration is fast relative to the isomerization rates so that T, p, G, and A may be defined. This very simple multireaction system allows the principles derived in the previous section to be clearly illustrated with a minimum of complexity. We assume the reactions are conducted in a closed vessel of volume V at a fixed temperature T = 298.15 K. Because there is no change in the total number of moles during the reaction, p will also be constant at the value po, and we will have dG = dA =

(29)

o

E



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Table 1. Equilibrium Constants and the Values of ξ1, ξ2, nA, and the Gibbs Free Energy at Equilibrium for the Symmetric (Case 1) and Nonsymmetric (Case 2) Reaction Systems

a

System

K1

K2

(ξ1)eq (mol)

(ξ2)eq (mol)

(nA)eq (mol)

G(ξ1,ξ2)eq (J)a

Case 1 Case 2

1.000 00 2.240 70

1.000 00 5.020 73

0.333 333 0.271 224

0.333 333 0.607 731

0.333 333 0.121 045

−12723.4 −15234.6

The Gibbs free energy at equilibrium, G(ξ1,ξ2)eq, is computed using eq 34.

Figure 1. (A) Gibbs free energy contours (in kJ) for the reactions A(g) → B(g) and A(g) → C(g) conducted at constant temperature and pressure with T = 298.15 K and a pressure of 1 bar. In this case, the standard chemical potentials for A(g), B(g), and C(g) are all −10 kJ mol−1. The initial state of the system with nA = 1 mol, nB = nC = 0 mol corresponds to the origin of figure where G(ξ1,ξ2) has a value of −10.0 kJ. The contour line values relative to the chemical potentials of the elements at 298.15 K and 1 bar pressure are given in kilojoules (kJ). Point D is the equilibrium point → ⎯ → ⎯ at ξ1 = ξ2 = 1/3 mol. The Gibbs free energy at this point is −12.7234 kJ. The vectors labeled S1 and S2 correspond to paths of steepest descent and → ⎯ maximum chemical force. (B) An enlargement of panel A showing the path S1 in greater dethail.

contrast, Path 2 moves both reactions in the forward direction toward the B(g) and C(g) products. Obviously, there are an infinite number of paths originating at point K which move the system toward a lower energy contour and are, therefore, spontaneous. The same situation exists for nonspontaneous paths originating at point K or any other composition point. This situation does not exist in a single-reaction system because there is only one reaction coordinate, and hence, only one possible spontaneous or nonspontaneous path. With respect to the spontaneity of a transformation of the reaction system from State 1 to State 2, ΔG < 0 is not a sufficient condition to guarantee that such a transformation will be spontaneous. For example, the transformation from point G to point H in Figure 1A has ΔG = −0.7 kJ, but the transformation is not spontaneous. The system in State G will spontaneously move to the equilibrium point D but go no further because dG > 0 at all points from D to H. Thus, ΔG < 0 is a necessary condition for a transformation to be spontaneous, but it is not a sufficient condition. The same is true for a singlereaction system.1 On the other hand, the reverse transformation from point H to point G has ΔG > 0. This result is sufficient to guarantee that the transformation will be nonspontaneous, but, as the previous result shows, the condition that ΔG > 0 is not necessary for a transformation to be nonspontaneous. This is also true for a single-reaction system.1

equilibrium values of the reaction coordinates. These results are summarized in Table 1. With only two reaction coordinates, the easiest and most convenient way to visualize the thermodynamic potential G(ξ1,ξ2) is by means of a planar contour map in which the loci of points with a given value of G(ξ1,ξ2) is plotted as a function of ξ1 and ξ2. Figure 1 shows the G(ξ1,ξ2) contour map for the symmetric system. The Gibbs free energies in kJ associated with each contour are shown in the figure. With all the standard chemical potentials equal, we will have K1 = K2 = 1, and the number of moles of A, B, and C at equilibrium will be the same. This expectation is reflected by the results shown in Table 1 and in Figure 1. Furthermore, we would expect to see a contour map for G(ξ1,ξ2) that has C2v symmetry about the 45° line of the map. Such symmetry is clearly present in both figures. In the previous section, it was noted that there are generally an infinite number of spontaneous reaction directions or paths originating from any point in the (ξ1,ξ2) composition space of the system. For example, consider point K on the −12.3 kJ contour in Figure 1. Any reaction path that moves the system toward a lower-energy contour will have dG < 0 and, therefore, be a spontaneous path. Two such spontaneous paths are shown in Figure 1 where they are labeled “1” and “2”. Path 1 moves reaction 2, A(g) → C(g), in the forward direction whereas Reaction 1, A(g) → B(g), moves in the reverse direction. In F



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is 5.0 × 10−12. When n is on the order of 1023, the ratio is zero. The probability that the reaction system will deviate from the path of maximum chemical force is essentially zero. This is the same logic that led Boltzmann to postulate that S = kB ln Wmax, where kB is the Boltzmann constant. Although there are essentially an infinite number of distributions that may occur, only the most probable distribution need be considered. In the same spirit, although there are an infinite number of spontaneous reaction paths, only the thermodynamically most probable path need be considered. If the initial state of the system is noA = 0.50 mol, noB = 0.50 mol, and noC = 0 mol, numerical solution of eq 25 shows that → ⎯ → ⎯ the reaction will follow path S2 . As expected, S2 is perpendicular to every contour curve. Because the initial amount of B(g) exceeds its equilibrium value of 1/3 mol while the amount of C(g) is zero, it might be anticipated that the initial direction of reaction will involve dξ1 < 0 with B(g) reacting to form A(g) while dξ2 > 0 so that Reaction 2 moves in the forward direction toward C(g). However, an inspection of Figure 1 shows that initially dξ1 = 0. The amount of B(g) remains constant while A(g) begins to form C(g). Only when ξ2 ≈ 0.1, do we see dξ1 becoming significantly negative. Equations 27 and 29 make it clear why this happens. At the point (ξ1= 0.5, ξ2 = 0), (∂G/∂ξ1) T,p,ξ2 = Δrμ1 = 0. Consequently, there is no chemical force acting to move B(g) → A(g). Even in a simple system such as this, chemical intuition often fails to predict the correct reaction path. The potential contour map for Case 2 of this reaction system is shown in Figure 2. Although the entropy considerations for the system are same as those for Case 1, the partial molar enthalpies for A(g), B(g), and C(g) are now different. C(g) is now 2.0 kJ mol−1 more stable than B(g) which is 2.0 kJ mol−1 more stable than A(g). The thermodynamically favored reaction path will, therefore, strike a balance between entropy effects that tend to move the system along the path with ξ1 = ξ2, which is shown as the dashed line in Figure 2, and enthalpy effects, which, in the absence of entropy, would move the system along a path coinciding with the ξ2 axis with ξ1 = 0 at all points. The actual path is obtained quantitatively from the solution of eq 25. It is → ⎯ shown in Figure 2 as path S3. As can be seen, entropy effects play the dominant role in the initial stages of reaction. In the latter half of the reaction, enthalpy considerations become the critical factor. The thermodynamically most probable reaction path from an initial state with noA = 0.285 mol, noB = 0.715 mol, and noC = 0 → ⎯ mol is shown in Figure 2 as path S4 . As was the situation for Case 1, initially, we have dξ1 = −C−1Δrμ1 dn = 0. Because the amount of B(g) present is initially about 2.6 times its equilibrium amount, such a result is not what chemical intuition might suggest.

In Reference1 it was shown that there exist an infinite number of finite transformations of a one-reaction system for which ΔG = 0. None of these have anything to do with equilibrium and none serve to locate the single point of equilibrium. In a multireaction system, the number of finite transformations for which ΔG = 0 is a much higher order of infinity and none of these results are related to equilibrium. For example, the transformation from State E to State F in Figure 1A has ΔG = 0, as would any other transformation whose initial and final states lie on the same contour curve. In the singlereaction system, while finding that ΔG = 0 for a finite transformation does not permit us to determine the equilibrium point, it does guarantee that this point lies somewhere in the finite interval leading from the initial to final state. As can be seen from Figure 1A, this is no longer true in a multireaction system. The equilibrium point, D, does not necessarily lie on the path leading from State E to State F. Although there exist an infinite number of spontaneous paths starting at composition point ξo, not all are equally probable. The path corresponding to the negative of the gradient, which is the path with maximum chemical force, will be the thermodynamically most favored path. Consider the present system initially at composition point (ξ1 = 0, ξ2= 0) which is the origin of Figure 1. At this point, we have one mole of A(g) at a pressure of one bar so that G(0,0) = −10.0 kJ. As Figure 1 shows, all reaction paths emanating from this point are spontaneous. Of the infinitude of possible spontaneous paths, the thermodynamically preferred path is along the negative gradient of G(ξ1,ξ2) given by the solution of eq 25. Numerical solution of this equation with dn = 10−3 mol → ⎯ yields the path labeled S1 in Figure 1. This path is lies along the 45° diagonal of the figures with ξ1 = ξ2 at all points. As can be → ⎯ seen, S1 is perpendicular to all the constant-energy contours of G, as expected. In the absence of other factors, such as kinetic considerations, the number of reacting systems deviating from → ⎯ path S1 will be negligibly small. Thus, systems reaching point I will virtually never change course by moving from point I to point J and from there to the equilibrium point D even though such a pathway is spontaneous at all points. This is the case because the chemical force along the path leading from point I → ⎯ to point J is much less than that along path S1. We reach the same conclusion by a consideration of the statistical probabilities associated with State J compared to that → ⎯ of the corresponding point along path S1. Because the standard chemical potentials of A(g), B(g), and C(g) are identical for Case 1, the reaction process starting with one mole of A(g) essentially consists of distributing a portion ξ1 + ξ2 of the 6.022 × 1023 A(g) molecules among two identical states, B(g) and C(g). The thermodynamic probability, W, associated with a given distribution of n molecules among the two states is W (nA , nB) =

n! nA !nB!

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(35)

BROADER IMPACT AND RECOMMENDATIONS

The foregoing analysis and numerical illustrations for multireaction systems combined with the corresponding analysis and examples previously presented for single-reaction systems1 emphasize the central importance of the chemical potentials and the associated chemical forces in the determination of spontaneity, the thermodynamically favored reaction path, and equilibrium in chemical reaction systems. This importance is the result of the fact that the chemical potentials and the

where nA and nB are the number of molecules distributed to → ⎯ states A and B, respectively, and n = nA + nB. Along path S1, we have ξ1 = nA = ξ2 = nB, which is the distribution that maximizes W(nA, nB). Now consider a composition point that does not lie → ⎯ on path S1, say ξ1 = 0.3 and ξ2 = 0.1. For n = 500, the ratio of the thermodynamic probabilities W(0.3n, 0.1n)/W(0.2n, 0.2n) G



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(∂G/∂ξ)T,p. In addition, the treatment of spontaneity and equilibrium should be expanded to include multireaction systems. This could be accomplished in a few pages of text. Such expansion would allow the students to see how the chemical potentials and the associated chemical forces combine to determine the thermodynamically, most-favored reaction pathway from any starting point. Figures such as 1 and 2 in this paper would also provide a very clear and quantitative visual illustration of the interplay of entropy and enthalpy effects in determining the expected reaction pathway. The opportunity to show how the Boltzmann entropy expression, S = kB ln(Wmax), influences not only equilibrium constants but also reaction pathways in nonequilibrium systems should not be lost.

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SUMMARY

The criteria for equilibrium and spontaneity are dG = 0 and dG < 0, respectively, if T and p are constant. At constant T and V, the same criteria hold for A. If there is only one possible reaction, the result (∂G/∂ξ)T,p or (∂A/∂ξ)T,V < 0 ensures that the reaction will be spontaneous in the direction of increasing ξ. If these derivatives are zero, the system will be at equilibrium. Because (∂G/∂ξ)T,p or (∂A/∂ξ)T,V is always equal to Δrμ, a single reaction chemical potential determines both equilibrium and spontaneity. In a multireaction system with N possible reactions, G and A will depend upon N independent variables if either T and p or T and V are held constant. In this case, the criteria for spontaneity and equilibrium are still dG ≤ 0 or dA ≤ 0. However, it is no longer true that (∂G/∂ξα)T,p,ξβ < 0 guarantees that reaction α will proceed spontaneously in the direction of increasing ξα, where ξα is the reaction coordinate for reaction α. A similar statement about A when T and V are constant is also true. Spontaneity requires that the sum ∑α (∂G/∂ξα)T,p,ξβ dξα < 0. Consequently, in most situations, there will be an infinite number of spontaneous and nonspontaneous paths or directions. The equilibrium condition that dG or dA be zero requires that we have (∂G/∂ξα)T,p,ξβ = Δrμα = 0 for 1 ≤ α ≤ N. The solution of this set of N algebraic equations often requires numerical methods. Although there are generally an infinite number of spontaneous paths, they are not all equally probable. The thermodynamically most probable reaction path will be the one following the path of largest chemical force. This path is along the negative gradient of the thermodynamic potential, either G(ξ1, ξ2, ..., ξN) or A(ξ1, ξ2, ..., ξN). The component of −∇⃗ G along ξα, dGα, is always proportional to the reaction chemical potential associated with reaction α, Δrμα. Therefore, the reaction chemical potentials determine the thermodynamically favored reaction path. Consequently, it is the reaction chemical potentials that determine spontaneity, equilibrium, and reaction path in multireaction systems, not G, A, ΔG, or ΔA. The equations that must be solved to obtain the negative gradient of the thermodynamic potential comprise a set of N, first-order, coupled differential equations whose solution is dependent upon the initial state of the system. Consequently, the problem of finding the thermodynamically most-favored pathway is computationally analogous to the calculation of a molecular dynamics trajectory.

Figure 2. Gibbs free energy contours for the reactions A(g) → B(g) and A(g) → C(g) conducted at constant temperature and pressure with T = 298.15 K and a pressure of 1 bar. In this case, the standard chemical potentials are −10.0, −12.0, and −14.0 kJ mol−1 for A(g), B(g), and C(g), respectively. The initial state of the system with nA = 1 mol, nB = nC = 0 mol corresponds to the origin of the figure where G(ξ1,ξ2) has a value of −10.0 kJ. The contour line values relative to the chemical potentials of the elements at 298.15 K and 1 bar pressure are given in kilojoules (kJ). Point Eq is the equilibrium point defined in Table 1. The Gibbs free energy at this point is −15.2346 kJ. The → ⎯ → ⎯ vectors labeled S3 and S4 correspond to paths of steepest descent and maximum chemical force.

chemical forces are directly related to the rate of change of the thermodynamic potential with respect to the reaction coordinate. This is not surprising because the force associated with any process is always related to the rate of change of the appropriate potential energy with respect to the process coordinate. Because high school and college students are generally aware of the fact that a system always responds spontaneously in the direction of the force, if chemical spontaneity is presented in introductory courses from the point of view of a system responding to a chemical force, understanding the concepts of thermodynamic spontaneity will become much more intuitively obvious. Similarly, chemical equilibrium, like all other equilibria, is a result of the complete absence of force. That is, chemical equilibrium occurs when the chemical force is zero along all possible reaction paths. Introductory textbooks should be modified to present reaction spontaneity and equilibrium in this context rather than using nonintuitive presentations that assert incorrectly that processes for which ΔA < 0 (constant T and V) or ΔG < 0 (constant T and p) are necessarily spontaneous and a reaction is at equilibrium whenever ΔA = 0 or ΔG = 0. Introductory textbooks and courses in physical chemistry generally limit their treatment of spontaneity and equilibrium to single-reaction systems. In most cases, the criteria for spontaneity and equilibrium in these systems are correctly and clearly derived to be those given in eqs 1a−1d. It would be useful if these discussions explicitly pointed out the direct relationship between the chemical force and (∂A/∂ξ)T,V or H



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AUTHOR INFORMATION

Corresponding Author

*E-mail: lionel.raff@okstate.edu. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS I am deeply indebted to Dr. Mark Rockley for reading several drafts of the manuscript and providing cogent suggestions to increase the clarity of the presentation and correct errors in the initial drafts. I also thank Mr. Paul V. Fleming, Senior Graphic Designer, Oklahoma State University, for preparing the abstract graphic.

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REFERENCES

(1) Raff, L. M. Spontaneity and Equilibrium: Why “ΔG < 0 Denotes a Spontaneous Process” and “ΔG = 0 Means the System Is at Equilibrium” Are Incorrect. J. Chem. Educ. 2014, 91, 386−395. (2) Levine, I. N. Physical Chemistry, 6th ed.,; McGraw-Hill Higher Education: New York, NY, 2009; pp 109−139. (3) Engel, T.; Reid, P. Physical Chemistry; Pearson, Benjamin Cummings: Upper Saddle River, NJ, 2006; pp 113−148. (4) Raff, L. M. Principles of Physical Chemistry; Prentice Hall: Upper Saddle River, NJ, 2001; pp 144−199. (5) Quilez, J. First-Year University Chemistry Textbooks’ Misrepresentation of Gibbs Energy. J. Chem. Educ. 2012, 89 (1), 87−93. (6) See, for example, Gislason, E. A.; Craig, N. C. Criteria for Spontaneous Processes Derived from the Global Point of View. J. Chem. Educ. 2013, 90, 584−590. (7) Margenau, H. and Murphy, G. M. The Mathematics of Physics and Chemistry; D. Van Nostrand Co., Inc.: Princeton, NJ, 1956; pp 150− 151. (8) Raff, L. M.; Thompson, D. L.. The Classical Trajectory Approach to Reactive Scattering. In Theory of Chemical Reaction Dynamics; Baer, M., Ed.; CRC Press, Inc.: Boca Raton, FL, 1985; Vol. III, pp 1−122, and references therein. (9) Raff, L. M., Komanduri, R.; Hagan, M.; Bukkapatnam, S. T. S. Neural Networks in Chemical Reaction Dynamics; Oxford University Press: New York City, NY, 2012 and references therein.

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Spontaneity and Equilibrium II: Multireaction Systems - Journal of

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