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Spontaneity and Equilibrium III: A History of Misinformation Lionel M. Raff* Department of Chemistry, Oklahoma State University, Stillwater, Oklahoma 74078, United States ABSTRACT: Necessary and sufficient criteria for reaction spontaneity in a given direction and for spontaneity of finite transformations in single-reaction, closed systems are developed. The criteria are general in that they hold for reactions conducted under either conditions of constant T and p or constant T and V. These results are illustrated using a simple, liquid-to-vapor phase transition as an example. Following this development, the paper investigates the source of the mathematical and logical errors contained in textbooks ca. 1950 to the present that led to the fallacious statements still present in most introductory chemistry textbooks and some more advanced texts that the conditions for spontaneity are ΔG < 0 at constant T and p and ΔA < 0 at constant T and V, whereas the corresponding conditions for equilibrium are ΔG = 0 or ΔA = 0. This investigation shows the principal errors to be (i) incorrect evaluation of definite integrals; (ii) failure to determine whether the results of such integrations produce criteria for spontaneity and equilibrium that are necessary conditions, sufficient conditions, both, or neither; and (iii) incorrect logical arguments related to equilibrium. The question of why such errors have been perpetuated in textbooks for over 50 years is also addressed. Some recommendations for revision of textbooks, for more frequent use of the concept of chemical force in discussing spontaneity and equilibrium, and for revision of the notation and nomenclature related to spontaneity and equilibrium are made. KEYWORDS: Upper-Division Undergraduate, Graduate Education/Research, Chemical Engineering, Inorganic Chemistry, Physical Chemistry, Misconceptions/Discrepant Events, Equilibrium, Nomenclature/Units/Symbols, Thermodynamics

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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 reaction products.1 Specifically, ξ = (ni − nio)/νi, with νi being the stoichiometric coefficient and nio the initial number of moles of product i. It is important to note that G, dG, and (∂G/ ∂ξ)T,p are functions of ξ and dξ. These quantities are not defined for a finite interval over a reaction process but rather vary continuously as the reaction proceeds. Consequently, the issue of spontaneity or nonspontaneity is a question that must be addressed at each point along the reaction path. A similar statement is true for A, dA, and (∂A/∂ξ)T,V. 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. Here, we note for clarity that in this paper reaction spontaneity addresses the question of whether or not the reaction system will move spontaneously toward products or reactants without the necessity to do work. It does not refer to the spontaneity of a finite transformation from State A to State B. Such questions are denoted by the phrase “transformation spontaneity”. 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 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. 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. In a multireaction system with N possible reactions, the criteria for spontaneity and equilibrium are more complex.2 In this case, G and A will depend upon N independent reaction coordinates, ξα (α = 1, 2, ..., N), if either T and p or T and V are held constant. Consequently, ⎡⎛ ∂G ⎞ d G = ⎢⎜ ⎟ dξ1 + ⎢⎣⎝ ∂ξ1 ⎠ T ,p,ξ β

⎤ ⎥ dξα ⎥ = ⎥⎦

namic potential, either G(ξ1, ξ2, ..., ξN) or A(ξ1, ξ2, ..., ξN). When this is the reaction path, the component of dG in the direction of ξα, dGα, is always proportional to the reaction chemical potential associated with reaction α, Δrμα. Consequently, it is the reaction chemical potentials that determine spontaneity, equilibrium, and the most probable reaction path in multireaction systems, not G, A, ΔG, or ΔA. The erroneous criteria given in eqs 1a−1d have pervaded our introductory chemistry textbooks and even some more advanced texts for over half a century in spite of the fact that these criteria cannot be derived. This situation raises several questions: How did such errors occur? What was the nature of the mistakes in the derivations that led to these statements? Why did such mistakes escape the scrutiny of scientists for over 50 years? Why are these errors still in some modern textbooks? If science is to learn from its mistakes, it is first necessary that the source or sources of the errors be determined. Once that is done, the question of what led to the perpetuation of the errors for such an extended period of time needs to be addressed. In this paper, the above questions are investigated by examination of some of the textbooks published from 1950 onward. Without exception, all of these texts have made major contributions to the education of countless numbers of students over the past decades and in the process significantly advanced the understanding of physical chemistry. The objective here is to learn from our mistakes, not to criticize. It is virtually impossible for any author to write a text containing approximately 1000 pages without making some errors. Making mistakes is a ubiquitous human condition.

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

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A PHASE TRANSITION EXAMPLE To illustrate the points made in the following sections, it will be advantageous to have a correctly analyzed reference system available. The criteria for spontaneity and equilibrium stated in eqs 2a−2d hold for processes involving phase transitions as well as for chemical reactions. The simplicity of a phase transition makes such a process an ideal choice. Consider the liquid−vapor phase transformation A(l) → A(g) in a closed, constant volume container at 298.15 K that initially contains noA(l) mol of A(l) and no vapor. At constant T and V, spontaneity and equilibrium are determined by dA or (∂A/∂ξ)T,V. The variation of the Helmholtz free energy with T, V, and composition is given by3,4

β

N

∑ Δr μα dξα α=1

(3)

where the subscript ξβ on the partial derivatives means the derivative with respect to ξα is taken, with all other reaction coordinates held constant. Although the criteria for spontaneity and equilibrium are still dG ≤ 0 or dA ≤ 0, 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. A similar statement about A when T and V are constant is also true. Spontaneity requires that the sum Σα (∂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. With regard to the spontaneity or nonspontaneity of finite transformations of a multireaction system from State A to State B, the situation is the same as that for single-reaction systems: The result that ΔG < 0 for the transformation is a necessary condition for spontaneity, but it is not a sufficient condition. The result that ΔG > 0 for the transformation is a sufficient condition to ensure nonspontaneity, but it is not a necessary condition. 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.2 This path is in the direction described by the negative gradient of the thermody-

dA = − S dT − p dV +

∑ μi dni i

(4)

where the summation runs over all substances in the system. ni is the number of moles of compound i, and μi is the chemical potential for that compound at 298.15 K. At constant T and V, eq 4 becomes dA =

∑ μi dni = μA(l)dnA(l) + μA(g)dnA(g) i

(5)

for the phase transition being considered. It should be noted that eqs 4 and 5 implicitly assume 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. B

dx.doi.org/10.1021/ed500253e | J. Chem. Educ. XXXX, XXX, XXX−XXX

Journal of Chemical Education

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for the vapor pressure and write nA(l) and nA(g) in terms of the reaction coordinate, eq 14 becomes

In order to ensure that positive changes in the reaction coordinate, ξ, correspond to reaction in the forward direction, we define ξ = nA(g) − no A(g)

G = (no A(l) − ξ)μo A(l) + ξ[μo A(g) + RT ln(p /po )]

(6)

= no A(l)μo A(l) + Δr μo ξ + ξRT ln(ξRT /po V )

noA(g)

With no vapor initially present, = 0 and dξ = dnA(g) Conservation of mass requires that nA(l) = noA(l) − ξ so that dξ = dnA(g) = −dnA(l). Therefore, eq 5 becomes dA = [μA(g) − μA(l)]dξ = Δr μdξ

Combination of eqs 13 and 15 yields the Helmholtz free energy for the vaporization reaction A = no A(l)μo A(l) + Δr μo ξ + ξRT ln(ξRT /po V ) − ξRT

(7)

(16)

where Δrμ is defined to be the factor inside the square brace of eq 7. Because this factor is the difference between the chemical potentials of the products and those for the reactants, it is appropriate to call it the reaction chemical potential and denote it with the notation Δrμ. Equation 7 may also be written in the form ⎛ ∂A ⎞ = Δr μ ⎜ ⎟ ⎝ ∂ξ ⎠T , V

Straightforward partial differentiation of eq 16 with respect to ξ at constant T and V yields eq 11, as expected. To provide a numerical example, we take A(l) to be water at 298.15 K so that μοA(l) = −237 130 J mol−1 and μοA(g) = −228 570 J mol−1. Therefore, Δrμo = 8 560 J mol−1. For simplicity, we take V = 24.790 L so that RT/(poV) = 1.0000 mol−1. Finally, we take the initial number of moles of H2O(l) to be 0.05000 mol. It is assumed throughout the analysis that the volume occupied by liquid water is negligible relative to that of the container, V, so that VA(g) may be equated to V. In the case of water, this approximation introduces a maximum error of 0.0036%. Figure 1 shows the Helmholtz free energy for the H2O(l)− H2O(g) system computed using eq 16. The reaction chemical

(8)

For the condensed phase, A(l), the chemical potential is essentially independent of the pressure so we can write μA(l) = μοA(l), where μo denotes the standard chemical potential at the standard pressure, po. The accuracy of this approximation may be assessed by noting that (∂μA(l)/∂p)T = V̅ , where V̅ is the partial molar volume of the liquid. In the case where A(l) is water, the variation of μA(l) for a one bar change of pressure from po, is about 0.00076%. Vapor pressures are typically very small so that the ideal gas equation of state is nearly exact. Under these conditions, the chemical potential for the vapor is given by3,4 μA(g) = μo A(g) + RT ln(p /po )

(15)

(9) o

where p is the pressure of vapor and p is the standard pressure. In most cases, po is taken to be 1 bar. Combination of eqs 7 and 9 with this choice gives dA = [(μo A(g) − μo A(l)) + RT ln(p /po )]dξ = [Δr μo + RT ln(p /po )]dξ

(10)

After p is replaced with its ideal gas equivalent, eqs 8 and 10 become ⎛ ∂A ⎞ = [Δr μo + RT ln(ξRT /po V )] = Δr μ ⎜ ⎟ ⎝ ∂ξ ⎠T , V

Figure 1. Helmholtz free energy, A, as a function of the reaction coordinate for the vaporization of 0.050 mol of water at 298.15 K in a container with V = 24.790 L. The energies are given relative to the chemical potentials of the elements at one bar pressure and 298.15 K being assigned the values of 0.

(11)

and dA = [Δr μo + RT ln(ξRT /po V )]dξ = Δr μdξ

potential for the system obtained from eq 11 is shown in Figure 2. The values of ξ and A at the labeled points in Figure 1 are provided in Table 1. The condition for phase equilibrium is (∂A/∂ξ)T,V = 0. As can be seen, this condition is attained when ξ = ξeq = 0.03165 mol. At this point, the computed equilibrium vapor pressure of water is peq = ξeqRT/V = 0.03165 bar =23.74 Torr, which is in excellent accord with the 23.77 Torr value given in the CRC Handbook of Chemistry and Physics.5 As can be seen in Figure 2, when ξ < ξeq, we have (∂A/∂ξ)T,V < 0. Consequently, the reaction will proceed spontaneously in the direction H2O(l) → H2O(g). When ξ > ξeq, the reaction will proceed spontaneously in the opposite direction. These criteria, which are summarized in Table 2, hold for any single-reaction, closed system. We may also formulate similar, necessary and sufficient criteria for finite transformations to be spontaneous. In order

(12)

respectively. The Helmholtz free energy for the vaporization process is related to the Gibbs free energy by A = G − pV = G − nA(g)RT = G − ξRT

(13)

The Gibbs free energy is given by G=

∑ niμi = nA(l)μA(l) + nA(g)μA(g) i

(14)

because the chemical potentials at any composition give the Gibbs free energies per mole. If we replace the chemical potentials with their equivalent expressions in terms of the standard chemical potentials and the ideal gas law expression C



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Table 3. Necessary and Sufficient Criteria for Finite Transformations in Single-Reaction, Closed Systems Conditionsa

Spontaneity/Nonspontaneity Result

ξI and ξJ ≤ ξeq and ξI < ξJ ξI and ξJ ≥ ξeq and ξI < ξJ ξI < ξeq and ξJ > ξeq ξI > ξeq and ξJ < ξeq

Spontaneous Transformation I → J Spontaneous Transformation J → I Both Transformations I → J and J → I are nonspontaneous Both Transformations I → J and J → I are nonspontaneous

ξI and ξJ are the reaction coordinates for States I and J, respectively, and ξeq denotes the reaction coordinate at equilibrium. When ξ is defined in terms of one of the products, these criteria are general in that they hold for reactions conducted at either constant T and p or constant T and V. a

Figure 2. Reaction chemical potential, (∂A/∂ξ)T,V, as a function of the reaction coordinate for the vaporization of 0.050 mol of water at 298.15 K in a container with V = 24.790 L. The values of A are relative to the chemical potentials of the elements at one bar pressure and 298.15 K being assigned the values of 0.

in their 1955 textbook. When a thermodynamic development is employed in textbooks, the usual procedure is to first introduce the second law and then demonstrate that this law requires the differential δqrev/T to be exact, where δqrev is the heat absorbed by the system in a reversible process and T is the absolute temperature. This differential is denoted by dSsys, where Ssys is the entropy for the system. Some textbooks omit the proof that the second law requires δqrev/T to be exact.7,8 Others provide this proof in varying degrees of detail. For example, see: Levine,3 pp. 78−87; Atkins,9 pp. 129−133; or Raff,4 pp. 157− 164. In most textbooks, the next step is to obtain the Clausius inequality10

Table 1. Reaction Coordinate and A Values at the Labeled Points in Figure 1 Point

ξ (mol)

A (J)

n2 = 0 A B C D E F G

0.000 0.0165 0.05000 0.0190 0.0470 0.03165 0.0250 0.0010

−11 856.5 −11 923.7 −11 923.7 −11 927.6 −11 927.6 −11 934.96 −11 934.4 −11 867.5

dStot = dSsys + dSsur = dSsys − δq/T = δqrev /T − δq/T ≥ 0 (17)

where “tot”, “sys”, and “sur” denote total, system, and surroundings, respectively, and −δq is the heat removed from the surroundings, which is less than δqrev when the process is spontaneous and irreversible. Consequently, the inequality applies for spontaneous, irreversible processes and the equality when the process occurs reversibly under equilibrium conditions. Although the details of the derivations and notation differ from textbook to textbook, the end result is always eq 17. After obtaining eq 17, Maron and Prutton11 integrate this equation over a finite interval from state A to state B to obtain

Table 2. Necessary and Sufficient Criteria for Spontaneous Reaction Directions at a Specific Point ξi along the Reaction Coordinate for Single-Reaction, Closed Systems Conditionsa

Spontaneous Reaction Direction

ξi < ξeq ξi > ξeq

Toward the Products Toward the Reactants

ξeq denotes the reaction coordinate at equilibrium. When ξ is defined in terms of one of the products, these criteria are general in that they hold for reactions conducted at either constant T and p or constant T and V. a

ΔStot =

for a system to spontaneously move from I to J, the reaction must be spontaneous in the direction I → J at every point in the path leading from I to J. Examination of Figure 2 shows that this requires that the reaction coordinates associated with States I and J, ξI and ξJ, must be less than ξeq with ξI < ξJ. For a J → I transformation to occur spontaneously, ξI and ξJ must be greater than ξeq with ξI < ξJ. These criteria are summarized in Table 3. With this simple example available as a reference system, we are now ready to examine the contents of various textbooks to determine what happened that led to the erroneous criteria given in eqs 1a−1d for reaction spontaneity and equilibrium.

∫A

B

dStot =

∫A

B

[dSsys − (δq/T )] ≥ 0

(18)

On the basis of this result, it is then stated that ΔStot = 0 for any reversible, equilibrium process and (19a)

ΔStot > 0 for any spontaneous, irreversible process

(19b)

Immediately following these equations, the text states, Therefore, whether an increase in entropy does or does not occur for processes taking place in isolated systems depends entirely on whether the processes are irreversible or reversible. The integration of a differential inequality, such as eq 17, to obtain an analogous expression in terms of the change of the quantity over a finite interval is not a proper mathematical procedure. Let f(X) be a function of a continuous variable X over some defined range of X. Suppose it is known that at some point X = Xo, we have df(X)/dX |X = Xo ≥ 0. That is, at the point X = Xo, we know that the slope of f(X) is either positive or zero. If both sides of this inequality are integrated over some finite

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HISTORY Physical chemistry textbooks published before the mid-1950s often addressed equilibrium from a kinetic point of view in which forward and reverse rates are equated at equilibrium. For example, this is the approach employed by Daniels and Alberty6 D



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interval from X1 to X2 that includes the point Xo, it is not generally correct to write x2

∫x1

d f (X ) dX = dX

x2

∫x1

df (X ) = Δf = f (X 2) − f (X1) ≥ 0.

With no information about the value of df(X) at any point other than Xo, it is obvious that nothing can be safely said about the value of the integral. It could be positive, negative, or zero. Only if it is known that df(X) is positive over the entire interval can we be certain that the integral will be positive. In the general case, the integral will not be zero unless df(X) is known to be zero over the entire interval. Consequently, it is clear that eq 19a is incorrect. Equation 19b will be correct only for the condition stated, that is, the process is known to be spontaneous. What is not specifically stated in Maron and Prutton’s text8 or in other physical chemistry texts is that to be certain that eq 19b will hold, it must be known that the transformation from State A to State B is spontaneous. That is, ΔStot > 0 is a necessary condition for the transformation from State A to State B to be spontaneous. Equation 19b does not say, and the derivations in Maron and Prutton’s text or any other, do not prove that if we find ΔStot > 0 for the transformation of a system from State A to State B, that transformation must be spontaneous. The word “entirely” in the sentence quoted above suggests that ΔStot > 0 is both a necessary and a sufficient condition for spontaneity. This is incorrect. If the transformation from State A to State B is known to be spontaneous, eq 17 ensures that dStot will be positive at every point along the path leading from A to B. If it were not, State B would be unattainable. Consequently, ΔStot will be greater than zero, and ΔStot > 0 is a necessary condition for a transformation to be spontaneous. However, imagine a path leading from State A to State B for which dStot is positive over a portion of the path and negative over the remainder. Under this condition, the transformation from A to B will not be spontaneous, but ΔStot might still be positive. In other words, finding ΔStot > 0 is not a suf f icient condition to ensure that the transformation will be spontaneous. The opposite conclusion is true for a nonspontaneous process. If we find ΔStot < 0 for the transformation from A → B, this is sufficient to ensure that the transformation will not be spontaneous, but this is not a necessary condition for nonspontaneity. The truth of the above statements may be easily seen by considering a very simple example of entropy. Consider two identical boxes and n particles. The entropy associated with a given random distribution of these particles between the two boxes is provided by the Boltzmann expression for S, ⎛ n! ⎞ S = kb ln(W ) = kb ln⎜ ⎟ ⎝ n1! n2! ⎠

Figure 3. S/kb for a system of 30 particles distributed between two identical boxes as a function of the number of particles distributed to box 2.

expect that a random process will spontaneously result in an even distribution of the particles between the two boxes, that is, ξeq = 15 particles in each box. Because Table 3 shows the transformation ξI = 0 → ξJ = 15 to be spontaneous, we must have dStot > 0 at all points in the interval. As Figure 3 shows, this is the case. With dStot > 0 at all points in the interval, we are assured that ΔStot = ∫ 15 0 dStot > 0, that is, ΔStot > 0 is a necessary condition for spontaneity. However, the transformation ξI = 0 → ξJ = 20 is not spontaneous because dStot is negative over the latter portion of the interval from ξI = 15 → ξJ = 20. This is the result obtained using Table 3. Nevertheless, as seen in Figure 3, we still have ΔStot > 0 for this process. Clearly, finding ΔStot > 0 is not sufficient to ensure a spontaneous process. On the other hand, any transformation for which ΔStot < 0 will be nonspontaneous. An example is the transformation ξI = 14 → ξJ = 18. Thus, ΔStot < 0 is a sufficient condition to ensure nonspontaneity. However, it is not a necessary one. For example, the transformation ξ I = 20 → ξ J = 13 is nonspontaneous, but ΔStot > 0 for this process. Note that the criteria given in Table 3 correctly predict the result in each case. Incorrect evaluation of definite integrals combined with the failure of textbooks to address questions of whether the resulting criteria for spontaneity and nonspontaneity are necessary or sufficient conditions lies at the heart of the misunderstandings that have been associated with these topics for over half a century. When A and G are introduced to treat questions of spontaneity and equilibrium under conditions of constant T and V or constant T and p, respectively, the errors made in stating the criteria for spontaneity in terms of ΔStot propagate to the corresponding expressions containing ΔA and ΔG. The usual derivation starts with eq 17 and then replaces δq with dU − δwn if T and V are constant and with dH − δwn if T and p and constant, where δwn is the differential nonpressure·volume work done on the system by the surroundings in the process. These replacements yield

(20)

where W is the thermodynamic probability for the distribution, kb is the Boltzmann constant, and n1 and n2 are the number of particles distributed to boxes 1 and 2, respectively. Figure 3 shows the value of S/kb as a function of the reaction coordinate, ξ = n2, for the case in which n = 30. Although this is too small a number for the Boltzmann expression to yield a highly accurate value for the entropy, the results are sufficiently accurate to illustrate the above principles. The distribution with n1 = 30 and ξ = n2 = 0 corresponds to perfect order for which eq 20 gives S = 0, as expected. We E

dS −

dU − δwn ≥ 0 for constant T and V and T

(21)

dS −

dH − δwn ≥ 0 for constant T and p T

(22)



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mol of liquid water at 298.15 K will not spontaneously form the same amount of water vapor with a pressure of 37.50 Torr. That is, finding ΔA < 0 is not a sufficient condition to ensure a spontaneous process. However, the criteria in Table 3 correctly predict this transformation to be nonspontaneous. The last sentence of Maron and Prutton’s statements,

where the inequality holds if the process is spontaneous and the equality if the process occurs reversibly at equilibrium. If we multiply these equations by −T, we obtain dU − δwn − T dS = dA − δwn ≤ 0 for constant T and V and

(23)

A positive sign for ΔG signifies, therefore, that the reaction in the given direction is not spontaneous is correct. For constant T and V processes, the equivalent statement is A positive sign for ΔA signifies, therefore, that the reaction in the given direction is not spontaneous. The reverse of the previous process, Point B (ξ = 0.050 mol) → H2O(l) (ξ = 0), has ΔA = 67.2 J, and it is correct that 0.050 mol of water vapor at 298.15 K with V = 24.790 L will not spontaneously condense to 0.50 mol of liquid water. However, although ΔA > 0 at constant T and V is a sufficient condition to ensure nonspontaneity of the process, it is not a necessary condition. The transformation of 0.050 mol H2O(g) to 0.025 mol H2O(g) and 0.025 mol H2O(l) (Point B → Point F) has ΔA = −10.7 J, but the transformation is nonspontaneous as predicted by the criteria in Table 3. Condensation will progress from Point B to Point E in Figure 1 and then stop. Thus, ΔA > 0 is not a necessary condition for a nonspontaneous process. Similar misstatements about the criteria for spontaneity and nonspontaneity are found in many physical chemistry textbooks. Hamill, Williams, and MacKay12 state on page 93 that at constant T and V, permitted processes have ΔAV,T < 0 and forbidden processes have ΔAV,T > 0 with no discussion of whether these are necessary or sufficient conditions. After correctly deriving the criteria for spontaneity and nonspontaneity in terms of dG and dA, Alberty and Silbey13 state on page 105 that, The same relations may be applied to finite changes as well as infinitesimal changes, replacing the d’s by Δ’s thereby arriving at the erroneous eqs 1a and 1c. In effect, this statement is equivalent to an improper definite integration, as previously discussed. On pages 180−182, Castellan14 correctly obtains eqs 2a and 2c as the conditions for spontaneity. The text then states that by integration of these equations, we have ΔA < 0 and ΔG < 0, eqs 1a and 1c. The argument that just because dA or dG are negative at some point, the integral ΔG = ∫ BA dG must also be negative is clearly incorrect. Conversely, even if the integral yielding ΔG is negative, dG does not have to be negative everywhere, which may prevent the process from reaching the final state. Cogent examples of this fact have previously been provided in ref 1. There, it is shown that the transformation of two moles of NO2(g) to one mole of N2O4(g), 2 NO2(g) → N2O4(g), at 298.15 K, has ΔA < 0 at constant T and V and ΔG < 0 at constant T and p, but the transformation is not spontaneous.1 On pages 169 and 170 of his text, Moore15 essentially repeats the statements made by Maron and Prutton11 about work being released by the system if ΔG < 0 for the reaction but not otherwise. From this statement, which is incorrect, it is concluded that a transformation is spontaneous if ΔG < 0 and nonspontaneous if ΔG > 0. Necessary and sufficient conditions are never considered. In a more recent textbook, Engel and Reid16 summarize spontaneity conditions at constant T and p on page 119 with the statements,

dH − δwn − T dS = dG − δwn ≤ 0 for constant T and p (24)

When there is no non-pV work done, δwn = 0, and eqs 23 and 24 become dA ≤ 0 for constant T and V and

(25)

dG ≤ 0 for constant T and p

(26)

The criteria expressed by eqs 23−26 are derived correctly in most physical chemistry textbooks. However, as previously shown, problems arise when an attempt is made to convert the differential expressions contained in eqs 25 and 26 to corresponding equations involving ΔG and ΔA by integrating both sides of the differential inequality over a finite interval as shown in eqs 27 and 28

∫A

B

dA = ΔA ≤ 0

(27)

and

∫A

B

dG = ΔG ≤ 0

(28)

Many textbooks do not actually write eqs 27 and 28, but rather simply state that by integrating eqs 25 and 26, eqs 27 and 28 are obtained. The fact that one has no idea what the sign of these integrals may be is generally ignored. Only if the differentials in eqs 27 and 28 are known to be negative over the entire interval can it be correctly stated that the integrals are negative. Once again, questions related to whether the criteria obtained by such an improper integration are necessary or sufficient conditions are not addressed. For example, on pages 196−197, Maron and Prutton11 state, The sign of the free energy change of a process is very important. When the driving tendency of a reaction is from left to right, energy is emitted on reaction, and the sign of ΔF is negative. A minus sign denotes, therefore, that the reaction tends to proceed spontaneously...A positive sign for ΔF signifies, therefore, that the reaction in the given direction is not spontaneous. (Note: the text uses the letter F instead of G for the Gibbs free energy.) It is clear that the above statement implicitly assumes that eq 28 is a sufficient condition to ensure that the transformation from State A to State B will be spontaneous. Examination of Figure 1 clearly shows that eq 27 is a necessary condition for the transformation A → B to be spontaneous, but it is not a sufficient one. Experimental measurement shows that 0.05 mol H2O(l) in a 24.790 L container at 298.15 K will spontaneously form a mixture of 0.03165 mol H2O(g) at a pressure of 23.74 Torr and 0.01835 mol H2O(l). Therefore, it is certain that ΔA will be negative for this process. Table 1 shows that ΔA for the process is −78.46 J, negative as expected. However, the transformation of 0.05 mol H2O(l) (ξ = 0) → Point B (ξ = 0.050 mol) has ΔA = −67.2 J, but the transformation is not spontaneous. An aliquot of 0.05 F



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are constant. For example, on page 315, Maron and Prutton8 state, Finally, when the system is in equilibrium, there is no tendency to proceed in either direction, no work can be done by the system, and hence ΔF = 0. (F is used instead of G in this text.) Using this logic, we can easily “prove” that Point F in Figure 1 is the point of equilibrium. Because we have ΔA(F →B) = +10.7 J and ΔA(F →G) = +66.9 J, reaction cannot proceed in either direction starting at Point F. Consequently, Point F must be the equilibrium point. As a final point, we address the question why the erroneous criteria contained in eqs 1a−1d have persisted for over 50 years and even now are still present in almost all introductory textbooks18−24 and even some more advanced textbooks, as noted in the previous discussion. Because it is much more difficult to determine why something has not happened than it is to detect logical or mathematical errors, the following represents no more than the author’s opinion. Perhaps part of the reason for the perpetuation of eqs 1a−1d for such an extended period lies in the subtlety of the errors plus the fact that the spontaneity criteria are partially correct. Another reason that may be even more important is the fact that once an error finds its way into the research literature and textbooks and is repeated in a nearly uncountable number of lectures, it tends to become an ingrained part of the science. In this event, the misinformation is difficult to dispel. In addition to the above, the notation and the nomenclature surrounding the subject contribute to the problem. In his 1964 textbook, Castellan14 carefully and correctly derives the equations for dG at constant T and p that control spontaneity and equilibrium. His final equation for the reaction υAA + υBB → υCC + υDD was

For macroscopic changes at constant p and T in which no nonexpansion work is possible, the condition for spontaneity is ΔG < 0 where ΔG = ΔH − TΔS and For all other cases, the relative magnitudes of ΔH and TΔS determine if the chemical transformation is spontaneous. Clearly, these statements assume, without proof, that finding ΔG < 0 is a sufficient condition to guarantee spontaneity. Similar statements to the effect that at constant T and V, the result ΔA < 0 ensures spontaneity of the transformation follow the above statements. After deriving eqs 2a and 2c in terms of dA and dG, Mortimer17 states, The criteria for finite processes are completely analogous to those for infinitesimal processes. This is followed by the claim that at constant T and p, ΔG ≤ 0 is “the most useful criterion for the spontaneity of chemical reactions”. Once again, these statements involve an improper analysis of the results of integration over a finite interval, as previously discussed. The statements that the criteria for equilibrium to exist are those given by eqs 1b and 1d are more egregiously incorrect than those for spontaneity, eqs 1a, 1c, and 19a. At least the criteria for spontaneity are partially correct. ΔStot > 0, ΔG < 0, and ΔA < 0 are necessary criteria under appropriate conditions for a transformation to be spontaneous. They are just not sufficient conditions. Likewise, ΔStot < 0, ΔG > 0, and ΔA > 0 are sufficient conditions to ensure that a transformation is nonspontaneous. They are just not necessary conditions. However, ΔStot = 0, ΔG = 0, and ΔA = 0 are never the appropriate criteria for equilibrium. Equilibrium occurs at a single point in both single-reaction and multiple-reaction systems, not over a finite interval.1,2 The result ΔStot = 0, ΔG = 0, or ΔA = 0 for some finite transformation does not denote a state of equilibrium nor does such a result provide any information about the location of the equilibrium point.1,2 In fact, the very concept of equilibrium existing over any finite interval makes no logical sense. The results shown in Figure 1 for the phase transition of water, as well as the examples given in refs 1 and 2, make this point very clear. Figure 1 and Table 1 demonstrate that there exist an infinite number of finite intervals for which ΔA = 0. For example, the intervals A → B and C → D shown by the dashed lines in the figure have ΔA = ∫ BA dA = ∫ DC dA = 0. Neither of these results have anything to do with the equilibrium point E for the liquid−vapor equilibrium. In multiple-reaction systems the number of finite intervals for which ΔA or ΔG is zero is a much higher-order of infinity, none of which have anything to do with the location of the single point of equilibrium.2 Physical chemistry textbooks have usually obtained eqs 1b, 1d, and 19a as the thermodynamic criteria for equilibrium by two different procedures. The first of these involves the mistaken notion that both sides of the differential equalities in eqs 17, 25, and 26 can be integrated to obtain eqs 1b, 1d, and 19a. This point has already been addressed. The second “derivation” relies on faulty logic. Using the erroneous assumptions that at constant T and p, ΔG < 0 is a necessary and sufficient condition for spontaneity and ΔG > 0 is a necessary and sufficient condition for nonspontaneity, it is argued that at the equilibrium point, there is no tendency for reaction to proceed in either direction, so we cannot have either ΔG < 0 or ΔG > 0. Therefore, it follows that we must have ΔG = 0. Similar statements are made in terms of A when T and V

⎛ ∂G ⎞ ⎜ ⎟ = υCμC + υDμD − υA μA − υBμB ⎝ ∂ξ ⎠T , p

(29)

This led Castellan to correctly write the criteria for spontaneity and equilibrium as ⎛ ∂G ⎞ ⎜ ⎟ < 0 for spontaneous reaction in the direction of ⎝ ∂ξ ⎠T , p products,

(30a)

⎛ ∂G ⎞ ⎜ ⎟ > 0 for spontaneous reaction in the direction of ⎝ ∂ξ ⎠T , p reactants, and

⎛ ∂G ⎞ ⎜ ⎟ = 0 for equilibrium ⎝ ∂ξ ⎠T , p

(30b)

(30c)

Immediately following this derivation on page 209, Castellan stated, The derivative in eq (11−29) [here, eq 29] has the form of an increase of free energy, ΔG, since it is the sum of the free energies of the products of the reaction less the sum of the free energies of the reactants. Consequently, we will write ΔG for (∂ G/∂ ξ)T,p and call ΔG the “reaction free energy”. This nomenclature and notation are ill-judged. The quantity on the right-hand side of eq 29 is not a free energy nor does it have the units of energy. It is the rate of change of the Gibbs G



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free energy with respect to ξ with units of energy mol−1. The notation ΔG does not suggest a derivative, a rate of change, or even a quantity that is a function of ξ. ΔG has a fixed value for any specified interval. It does not vary with the reaction coordinate whereas (∂G/∂ξ)T,p is a very strong function of ξ as is evident from Figure 2. When (∂G/∂ξ)T,p is replaced with ΔG, the immediate result is the conversion of correct criteria contained in eqs (2a−2d) to the incorrect expressions of eqs (1a−1d). The nomenclature makes matters worse. Today, the common notation is ΔrG, but the same objections apply to this notation. This type of notation and nomenclature contributes in some measure to the continuation of the misinformation surrounding spontaneity and equilibrium.

quantity to which (∂G/∂ξ)T,p is equal involves the difference between the sum of product chemical potentials and reactant chemical potentials. Consequently, either Δμ or Δrμ is an obvious choice. This notation has the right units. It correctly represents a difference in rates of change of G instead of a finite difference in G itself. The quantity Δμ or Δrμ is a function of the reaction coordinate, ξ, rather than a fixed variation of the Gibbs free energy for some finite interval. It is also recommended that the name of the quantity currently labeled ΔrG be changed from “free energy of reaction” to “reaction chemical potential”. The reasons are the same as those for recommending the notation change. These changes would greatly facilitate and speed the removal of the erroneous statements concerning spontaneity and equilibrium that currently pervade chemistry textbooks.

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SUMMARY AND RECOMMENDATIONS The analyses and examples provided in this paper, as well as those contained in refs 1 and 2, show clearly that chemical potentials and the associated chemical forces determine spontaneity, nonspontaneity, the thermodynamically favored reaction path in multiple-reaction systems, and equilibrium, not S, G, A, ΔStot, ΔG, or ΔA. Consequently, statements to the contrary contained in many introductory texts should be removed as soon as possible and not appear in new textbooks. Clearly, more advanced textbooks on physical chemistry should avoid improper integration methods and bogus qualitative arguments in an effort to convert differential equalities and inequalities to analogous expressions involving ΔS, ΔG, or ΔA over a finite interval. Using the methods employed in this paper or by an analysis of the work produced by processes known to be spontaneous, it is possible to rigorously show that for such processes, we must have ΔStot,> 0 for an isolated system, ΔG < 0 at constant T and p, and ΔA < 0 at constant T and V. When this analysis is presented in a physical chemistry textbook, it should always be followed by an appropriate discussion and proof that these criteria are necessary conditions for spontaneity, but they are not sufficient conditions. If the opposite inequalities are presented for nonspontaneous processes, the presentation should include a proof that the inequalities, while sufficient to show nonspontaneity, are not necessary conditions. For simplicity in introductory textbooks, it should be emphasized that thermodynamics provides iron-clad criteria, ΔStot < 0 and ΔG > 0 under appropriate conditions, that guarantee that a transformation is nonspontaneous. To obtain criteria that are both necessary and sufficient for spontaneity at constant T and p or at constant T and V, Tables 2 and 3 should be used. Recommendations have already been advanced in refs 1 and 2 for using the chemical force to introduce the student to the concepts of spontaneity and equilibrium. In a single-reaction system, the chemical force is Fξ = −(∂G/∂ξ)T,p or Fξ = −(∂A/ ∂ξ)T,V. In ref 2, it was also recommended that physical chemistry texts should expand the coverage of spontaneity and equilibrium to include multireaction systems. 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. Such expansion would provide an excellent opportunity to demonstrate how the Boltzmann entropy expression influences not only equilibrium constants but also reaction pathways in nonequilibrium systems. Finally, it is recommended that the notation currently in use for (∂ G/∂ ξ)T,p, ΔrG, be changed to either Δμ or Δrμ. The

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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 Mark Rockley for reading a draft of the manuscript and providing cogent suggestions to increase the clarity of the presentation. I also thank Paul V. Fleming, Senior Graphic Designer, Oklahoma State University, for preparing the abstract graphic.

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REFERENCES

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