Research: Science and Education edited by
Advanced Chemistry Classroom and Laboratory
Joseph J. BelBruno
Effect of a Perturbation on the Chemical Equilibrium: Comparison with Le Châtelier’s Principle
Dartmouth College Hanover, NH 03755
Emilio Martínez Torres Departament of Physical Chemistry, Escuela de Magisterio, University of Castilla-La Mancha, Ronda de Calatrava 3, 13003 Ciudad Real, Spain;
[email protected] An important topic in teaching chemistry is the qualitative prediction of the direction of shift in a chemical equilibrium when it is subjected to a perturbation. In elementary chemistry courses such predictions are obtained using Le Châtelier’s principle (LCP), which may be expressed as follows (1): If a chemical system is subjected to a perturbation, the equilibrium will be shifted such as to partially undo this perturbation. However, as it has been pointed out by some authors, this principle is so ambiguous that it often gives contradictory predictions (1–4). Perhaps the best example to illustrate the limitations of LCP is the prediction of the direction of shift in the ammo2NH3(g), upon nia synthesis equilibrium, N2(g) + 3H2(g) addition of nitrogen at constant pressure and temperature when the mole fraction of nitrogen in the equilibrium mixture exceeds 0.5 (1, 5–7). On the one hand, LCP predicts that the addition of nitrogen shifts the equilibrium to the right in order to remove the excess of nitrogen, but, on the other hand, LCP predicts that equilibrium shifts to the left in order to decrease the mole fraction of nitrogen.1 LCP cannot solve this contradiction. For the above reasons the use of LCP has fueled much discussion (8–16). The limitations of LCP can be avoided by using alternative approaches described in the literature (1, 4, 5, 17, 18). In this article we make a thermodynamic analysis of the evolution of a system at chemical equilibrium when it is subjected to an infinitesimal perturbation. The main result of this analysis is an inequality relating the change in the perturbed variable and the change that the equilibrium shift produces in its conjugated variable. This inequality predicts correctly the effect of any infinitesimal or finite perturbation and has allowed us to derive general conclusions about the direction of shift in terms of extensive and intensive variables. Thermodynamic Background The internal energy change of a system in an infinitesimal process is given by dU = T d S − P dV +
∑ µk dnk k
(1)
In chemical open systems, the change, dnk, in the amount of substance k can be caused by adding from outside (dnk´) and by chemical reactions within the system (dnk˝): (2)
dnk = dnk′ + dnk′ ′
If we suppose that only one reaction can take place between
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the substances that compose the system, the quantity dnk⬙ can be written as function of the extent of reaction ξ as2 dnk′ ′ = νk dξ
(3)
where νk is the stoichiometric coefficient3 of the substance k. By substituting eq 3 into eq 2 we obtain dnk = dnk′ + νk dξ
(4)
Then eq 1 becomes dU = T dS − P dV +
∑ µk dnk′ − A dξ k
(5)
where A = − ∑ νk µk k
(6)
is a state variable called affinity, which was introduced by de Donder in the 1920s (20). An isolated system is characterized by the conditions dU = 0, dV = 0, and dnk⬘ = 0, (k = 1, 2, ...) (21). Then, for such a system eq 5 gives: T dS = A dξ
(7)
The conditions of spontaneity and equilibrium for isolated systems are given by dS ≥ 0 (22), where inequality refers to spontaneous processes and equality refers to equilibrium. Since the temperature is always positive we get A dξ ≥ 0
(8)
which is known as de Donder inequality. Therefore for a spontaneous reaction Adξ > 0. Hence if A > 0 then dξ > 0; that is, the reaction goes spontaneously in the forward direction. On the contrary, if A < 0 the reaction goes in the backward direction. If A = 0 the reaction is at equilibrium. Consequently, affinity is regarded as a force that drives the chemical reaction towards equilibrium (20). Although the above conditions have been obtained for isolated systems, they are valid for any system (23). For example, for a closed system at constant temperature and pressure eq 5 can be written as d(U − TS + PV ) = ᎑A dξ, which gives dG = ᎑A dξ , where G is the Gibbs free energy. The conditions of spontaneity and equilibrium for such a system are given by dG ≤ 0 (22) that, according to the above, coincides with Adξ ≥ 0. Hence, we see that the conditions of spontaneity and equilibrium expressed in terms of affinity are more general than in terms of G or any other thermodynamic function such as entropy, internal energy, enthalpy, or Helmholtz
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free energy. The equilibrium condition A = 0 means that at equilibrium one of such functions reaches an extremum (maximum or minimum); this function is the entropy if the system is isolated, the Gibbs free energy if the system is at constant T and P, and so forth.4 Thus, although the use of affinity is not widely spread,5 we prefer to use conditions of spontaneity and equilibrium in terms of affinity to keep our treatment as general as possible.
is not at chemical equilibrium and the affinity is different from zero and is given by
A = 0 + δA =
∑ Yj dXj
dZ =
δA = −
− A dξ
(9)
j
∂ Z ∂ ξ
∂ ∂ Z ∂ ξ ∂ X i
=
{X j}
{
ξ, X j ≠ i
}
∂ A ∂ Xi
∂ Yi ∂ ξ
=
{
ξ , X j ≠ i
}
δX i
(11)
∂ Yi ∂ ξ
{X j}
δX i
(12)
d eYi =
∂ Yi ∂ ξ
dξ
(13)
{X j }
Since the step 2 → 3 is spontaneous de Donder’s inequality must be satisfied. Then, from eqs 8 and 11 it follows that:
{
ξ, X j ≠ i
} {X j}
we obtain the following Maxwell relation
−
}
The second step (step 2 → 3) occurs from the state 2 to a new equilibrium state 3 keeping constant the variables of the set {Xj }. During this step the chemical reaction proceeds spontaneously to a new equilibrium position and ξ changes by a quantity dξ, and affinity takes again the zero value. The change in Yi during the step 2 → 3 is given by (the symbol de will be used to identify changes in the step 2 → 3):
Table 1 shows the pairs of variables Xj and Yj for the most common thermodynamic potentials. From the equality of the second cross derivatives
∂ ∂ X i
{
ξ , X j ≠ i
Bearing in mind eq 10, eq 11 becomes
Deduction of a Rule for Predicting the Direction of Shift Postulate a system at chemical equilibrium whose state is specified by the variables X1, X2, ..., ξ, and let Z be the thermodynamic potential (e.g., U, H, F, G ) that is a function of this set of variables. The differential of Z is
∂ A ∂ X i
(14)
δA d ξ > 0 From eq 12, 13, and 14 it follows that
(10)
{ X j}
where {Xj } represents the set of variables {Xj : j = 1, 2, ...} and {Xj≠i } represents the set {Xj } with exception of Xi. Suppose the system is in an initial state of chemical equilibrium (state 1). Since the system is at equilibrium the affinity is zero. Let us perturb the system by making an infinitesimal change δ Xi in the variable Xi keeping constant the variables belonging to the set {Xj≠i }. The evolution of the system can be decomposed into two consecutive steps, which are schematically represented in Figure 1. The first of such steps (step 1 → 2) occurs from the state 1 to a state 2 keeping constant ξ and the variables belonging to the set {Xj≠i }. The changes in Xi and Yi during such a step are δ Xi and δYi, respectively (the symbol δ will be used to identify changes in the step 1 → 2). Since ξ is constant, no chemical reaction takes place and the system behaves during this step as if it were chemically inert. The system in state 2
d eYi δX i < 0
({ X j ≠ i }
constant )
(15)
Taking into account that deYi is the change in Yi caused by the equilibrium shift, eq 15 can be expressed in words as follows: R1: An increase (decrease) of Xi, keeping constant the set of variables {Xj≠i }, causes the equilibrium position to shift in the direction that Yi decreases (increases).
We should not substitute R1 by the apparently equivalent expression: “an increase (decrease) of a variable causes the equilibrium position to shift in the direction that decreases (increases) its conjugated variable”, which is in general untrue since it does not distinguishes between Xi and Yi. Rule R1 has been obtained for infinitesimal perturbation. Suppose a finite perturbation given by a finite change ∆ Xi. We can decompose ∆ Xi into infinitely many infinitesimal changes δ Xi, so that to every δ Xi corresponds a change de Yi satisfying eq 15. Since every δ Xi has the same sign as ∆ Xi, each de Yi must have opposite sign from ∆ Xi. Thus we
Table 1. Thermodynamic Potentials Z
Variable
U
H
F
Yj
T ᎑P µk
T V µk
᎑S ᎑P µk
᎑S V
µk
Xj
S
S P nk´
T V n k´
T
nk´
V nk´
G
P
NOTE: The Helmholtz free energy is usually represented by the symbol A, but because this is used here for the affinity, the symbol F is used.
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Figure 1. Evolution of a system at chemical equilibrium subjected to a perturbation.
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Figure 2. Changes in volume associated with an increase of pressure at constant temperature.
Figure 3. Changes in pressure associated with an increase of volume at constant temperature.
conclude that ∆ Xi ∆Yi ≤ 0, where ∆Yi is the sum of the infinitely many changes de Yi. This shows that R1 is also valid for finite perturbations.
here xk = (nk 兾n) is the mole fraction, n being the total number of moles. The differentiation of eq 19 at constant temperature and pressure and its substitution into eq 18 leads to
First, let us consider a change of pressure in a closed system at constant temperature. According to Table 1, Xi = P, Yi = V, {Xj≠i } = {T, nk⬘(k = 1, 2, ...)}, and Z = G. Then eq 15 gives (16) d eV δP < 0 As a conclusion, an isothermal increase (decrease) of pressure in a closed system shifts the equilibrium in the direction that decreases (increases) the volume. Figure 2 shows in a P–V diagram the evolution of a system subjected to an increase of pressure at constant temperature and its decomposition into the steps 1 → 2 and 2 → 3 together with the changes of pressure and volume. Now let us consider a change of volume at constant temperature. In this case, Xi = V, Yi = ᎑P, {Xj≠i } = {T, nk⬘(k = 1, 2, ...)}, and Z = F. Then we get: (17)
d e P δV > 0
That is, an isothermal increase (decrease) of volume causes the equilibrium position to shift in the direction that increases (decreases) the pressure. Figure 3 shows the evolution of a system subjected to an increase of volume at constant temperature. Finally we consider the addition of a substance at constant temperature and pressure. The substitution of Xi = nk⬘ and Yi = µk into eq 15 leads to (18)
d e µk δnk′ < 0
Hence, addition of a substance shifts the equilibrium in the direction that decreases its chemical potential. In order to apply eq 18 to particular cases it is necessary to write the chemical potential as function of the extent of reaction. This can easily be accomplished for ideal systems, where the expression of the chemical potential of k is (22) (19) µ = µ* (T, P ) + RT ln x k
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(20)
d e xk δnk′ < 0
Application to Particular Cases
k
•
According to the above, the addition of a substance at constant pressure and temperature shifts the equilibrium in the direction that reduces its mole fraction. Note that this conclusion solves the contradiction produced by LCP in the example of ammonia synthesis equilibrium. In the following it is necessary to make use of the differential calculus, hence it will be understood that the changes are infinitesimal. To obtain the expression for de xk we should bear in mind that it is the change in xk caused by the equilibrium shift (step 2 → 3), then de ni = νi dξ for each i. Thus, by differentiating xk = (nk 兾n) we obtain
( νk
d e xk =
− xk ∆ν) dξ n
(T
and P constant ) (21)
where ∆ν = ∑ i νi. The substitution of eq 21 into eq 20 leads to
(∆ν xk
− νk ) dξ δnk′ > 0
(T
and P constant ) (22)
If we consider only the addition of substance (δ nk⬘ > 0), eq 22 becomes
( ∆ν xk
− ν k ) dξ > 0
(T
and P constant )
(23)
That is, if ( ∆ν xk − νk ) > 0 then dξ > 0; that is, the addition of substance k shifts the equilibrium in the forward direction. On the other hand, if ( ∆ν xk − νk ) < 0 then dξ < 0; that is, the addition of substance k shifts the equilibrium in the backward direction. For an inert substance νk = 0, then eq 23 becomes ∆ν dξ > 0. So the addition of an inert substance shifts the equilibrium in the forward direction if ∆ν > 0 and in the backward direction if ∆ν < 0. From the above it is easy to show that the equilibrium shifts in the direction that reduces the amount of k if (νk 兾∆ν) < 0 or (νk 兾∆ν) > xk. In exchange, the equilibrium shifts in the direction that increases the amount of k if 0 < (νk 兾∆ν) < xk.
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In the example of ammonia synthesis reaction, N2(g) + 3 H2(g) 2NH3(g), we have (νN2兾∆ν) = 0.5, (νH2兾∆ν) = 1.5, and (νNH3兾∆ν) = ᎑1. Then the addition of nitrogen at constant temperature and pressure shifts the equilibrium to the right when xN2 < 0.5 and to the left when xN2 > 0.5, whereas the addition of hydrogen or ammonia always shifts the equilibrium to the right or to the left, respectively. Rule in Terms of Extensive and Intensive Variables The relationship between the changes δ Xi and δYi is ∂ Yi ∂ Xi
δYi =
{
ξ, X j ≠ i
δX i
(24)
}
to enhance such decrease. For this reason the equilibrium will shift in the direction that decreases the volume (Figure 2). Notes 1. It can be easily shown that such a shift decreases the mole fraction of nitrogen. 2. The extent of reaction is usually defined by the expression ξ = (nk − nk0 )兾νk (19). In our case it is given by ξ = ∆nk⬙兾νk , where ∆nk⬙ is the number of moles of k produced by the chemical reaction from the initial instant. 3. The stoichiometric coefficients are taken to be positive for products and negative for reactants. 4. Affinity can be expressed by any of the following derivatives:
The stability criteria of thermodynamics require that (21) ∂ Yi ∂ X i
> 0,
{
ξ, X j ≠ i
}
(Xi
extensive)
A = −
(25a)
∂ S ∂ ξ
= − U, V, {nk′ }
= −
∂ Yi ∂ X i
< 0,
{
ξ, X j ≠ i
}
(Xi
intensive)
(25b)
If Xi is extensive, from eqs 15, 24, and 25a follows that δYi and de Yi have opposite signs; that is, the change of Yi caused by the equilibrium shift tends to compensate the change δYi. In exchange, if Xi is intensive, from eqs 15, 24, and 25b it follows that δYi and de Yi have the same sign; that is, the change of Yi caused by the equilibrium shift tends to enhance the change δYi. This can be expressed in a mathematical way as follows
d eYi δYi < 0
(Xi
extensive and { X j ≠ i } constant) (26a)
d eYi δYi > 0
(Xi
intensive and { X j ≠ i } constant ) (26b)
or in words:
S,V, {nk′ }
= − T, V, {nk′ }
∂ H ∂ ξ ∂ G ∂ ξ
S , P, {nk′ }
T, P, {nk′ }
Literature cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Because equations 26a and 26b together are equivalent to eq 15, R1 is equivalent to R2. Note that if Xi is extensive R2 is in agreement with LCP, but it is not when Xi is intensive. As an example, let us apply R2 to predict the equilibrium shift caused by an increase of volume at constant temperature. According to Table 1 Xi = V and Yi = ᎑P, then eq 25a gives [∂(᎑P )兾(∂V )] ξ,T,nj > 0; that is, an increase of volume at constant temperature causes a decrease of pressure. The volume is extensive, then R2 predicts that the equilibrium shift tends to compensate such decrease, so the equilibrium will shift in the direction that increases the pressure (see Figure 3). The results of eqs 25a and 25b are so intuitive that we do not need to use them explicitly. This means that R2 can be applied without using Table 1. This is an advantage over R1, for which Table 1 is indispensable. To illustrate this, let us consider an increase of pressure at constant temperature. This change causes a decrease of volume (we know this intuitively), then, according to R2, the equilibrium shift tends
12. 13. 14. 15. 16. 17. 18. 19.
•
∂ F ∂ ξ
= −
where F is the Helmholtz free energy. 5. The main reason is that affinity was not employed in the classical textbook of thermodynamics by Lewis and Randall (24).
R2: If we change an extensive (intensive) variable the equilibrium shifts in the direction that tends to reduce (increase) the change in the corresponding intensive (extensive) variable.
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∂ U ∂ ξ
20.
21. 22. 23. 24.
De Heer, J. J. Chem. Educ. 1957, 34, 375–380. De Heer, J. J. Chem. Educ. 1958, 35, 133–136. Treptow, R. S. J. Chem. Educ. 1980, 57, 417–420. Gold, J.; Gold, V. Chem. Brit. 1984, 20, 802–806. Katz, L. J. Chem. Educ. 1961, 38, 375–377. Levine, I. N. Physical Chemistry, 4th ed.; McGraw-Hill: New York, 1995. Lacy, J. E. J. Chem. Educ. 2005, 82, 1192–1193. Allsop, R. T.; George, N. H. Educ. Chem. 1984, March, 54– 56. Mellon, E. K. J. Chem. Educ. 1979, 56, 380–381. Bodner, G. M. J. Chem. Educ. 1980, 57, 117–119. Mazo, R. M.; Barnhard, R. J. Chem. Educ. 1972, 49, 639– 640. Fernández-Prini, R. J. Chem. Educ. 1982, 59, 550–553. Brice, L. K. J. Chem. Educ. 1983, 60, 387–389. Helfferich, F. G. J. Chem. Educ. 1985, 62, 305–308. Campbell, J. A. J. Chem. Educ. 1985, 62, 231–232. Xianmin, Z. J. Chem. Educ. 1989, 5, 401–402. Kemp, H. R. J. Chem. Educ. 1987, 64, 482–484. Canagaratna, S. G. J. Chem. Educ. 2003, 80, 1211–1219. Dumon, A.; Lichanot, A.; Poquet, E. J. Chem. Educ. 1993, 70, 29–30. Kondepudi, D.; Prigogine, I. Modern Thermodynamics—From Heat Engines to Dissipative Structures; John Wiley & Sons: New York, 1998; pp 108–113. Callen, H. B. Thermodynamics; Wiley: New York, 1960. Atkins, P. W. Physical Chemistry, 6th ed.; Oxford University Press: Oxford, 1998. Prigogine, I.; Defay, R. Chemical Thermodynamics; Longmans: London, 1954. Lewis, G.; Randall, M. Thermodynamics, 2nd ed.; McGrawHill: New York, 1961.
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