Approaches to the Treatment of Equilibrium Perturbations - Journal of

Research: Science and Education

Approaches to the Treatment of Equilibrium Perturbations Sebastian G. Canagaratna Department of Chemistry, Ohio Northern University, Ada, OH 45810; [email protected]

The subject of perturbations of an equilibrium system and the direction in which it shifts as it moves towards a new equilibrium position is an important aspect of general and physical chemistry courses. Textbooks base their treatment of equilibrium perturbations largely on Le Châtelier’s principle. The latter is a principle of moderation. de Heer (1) has been a severe critic of the use of Le Châtelier’s principle and has discussed cases where the principle of moderation fails. Treptow (2) emphasized the need for careful phrasing in the statement of the principle to remove ambiguity, and examined five different formulations. In general, perturbations involving intensive variables are subject to moderation while perturbations involving extensive variables are not. However cases are known where moderation fails even for intensive variables (3). The problem of moderation can be become even more complicated when there are several reactions (4). Katz (5) suggested the comparison of the equilibrium constant K with the reaction quotient Q as a method of completely avoiding the use of Le Châtelier’s principle. We begin by reviewing the common methods used in textbooks to deal with predicting the direction of a reaction shift induced by a perturbation. The merits and demerits of these methods are discussed. A rigorous and quantitative treatment of the problem is lacking even in undergraduate physical chemistry texts. Such a treatment would clearly show what restrictions should be placed on our conclusions. In this article the problem is treated directly in a unified way, the question of moderation being avoided altogether. The necessary equations can be derived from thermodynamic relations that should be understandable by junior-level students. We can take advantage of the fact that the treatment of thermodynamics in quantity and presentation has changed considerably both in introductory and physical chemistry texts. Deficiencies in the Le Châtelier Formulation Most textbooks state the Le Châtelier principle in a form similar to the following: If a stress is applied to a system in chemical equilibrium, the system shifts in the direction that tends to relieve that stress.

As de Heer points out, when the temperature of a system is changed and the system allowed to relax at this new temperature, no change in the system can undo the change in temperature and the principle is not valid and makes no sense. The Le Châtelier principle is the only context where the word “stress” appears to be used in introductory chemistry. It is vague, since there is no unique way of deciding what the stress is. When the volume is changed, is the stress the change in volume or the change in pressure or either? Or is the stress the change in some other property? A more important deficiency in the usual treatment in the texts is that there is general failure to make clear under what conditions the perturbed system is relaxing to a new

equilibrium. For example, if the volume is increased, the system can relax at constant pressure or at constant volume or under some other conditions. If the increase in volume is considered as the stress and the system allowed to relax at constant pressure, the volume continues to increase: the stress is not moderated when the relaxation is at constant T and p. However, if the change in pressure is considered to be the stress and the relaxation is at constant T and p, the pressure increases and the stress is moderated (6). Prigogine also discusses the more difficult problem of the system relaxing at constant temperature under adiabatic conditions. Many texts do not specify clearly whether addition or removal of substances is at constant volume or at constant pressure. There is moderation of addition or removal of substances at constant volume, but as we shall show later, there may not be moderation when the addition or removal is at constant pressure. The addition of an indifferent substance is also subject to the same restriction: at constant volume the equilibrium is not perturbed, but again as we will show later, addition at constant pressure could perturb the system. The oversimplified attempt to prove that moderation occurs often produces strange descriptions, as when a gas is said to respond to a reduction in volume by reducing its own volume. This type of explanation can be very misleading. de Heer believed that the usefulness of the Le Châtelier principle was due to a specific but rather arbitrary interpretation of a vague formulation and a careful restriction of the examples considered. The usual interpretations “work” but they are not the only reasonable interpretations. For example, in exothermic reactions taking place at constant temperature, energy escapes from the system. Consequently, if energy is added to the system, it would seem reasonable to assume that moderation would consist of the system decreasing its energy by releasing energy to the surroundings: that is, the reaction takes place in the forward direction—the wrong answer! This supports de Heer’s charge of “a specific but rather arbitrary interpretation”. In order to overcome the problem of vagueness and arbitrariness, most textbooks give subsidiary rules for the separate cases. Thus, Atkins and Jones (7) give the following rules: (a) When a reactant is added to a reaction mixture at equilibrium, the reaction tends to form products; when a reactant is removed, more reactant tends to form; when a product is added, the reaction tends to forms reactants; when a product is removed, more product is formed. (b) Compression of a reaction mixture at equilibrium tends to drive the reactions in the direction that reduces the number of gas-phase molecules; increasing the pressure by introducing an inert gas has no effect on the equilibrium composition. (c) Raising the temperature of an exothermic reaction favors the formation of reactants; raising the temperature of an endothermic reaction favors the formation of products.

JChemEd.chem.wisc.edu • Vol. 80 No. 10 October 2003 • Journal of Chemical Education

1211


Comparing K with Q A way to avoid the use of Le Châtelier’s principle is by comparing the equilibrium constant K with the reaction quotient Q, a method recommended by Katz. This approach is followed by some introductory texts, to supplement and “explain” Le Châtelier’s principle. The method is free of ambiguity and can be used instead of Le Châtelier’s principle. To apply this method, subsidiary rules predicting how Q and K change should be given. Most of the subsidiary rules follow from the definition of Q in terms of concentrations or pressures. In a closed system, as the reaction goes forward under any conditions, Q increases; when the reaction goes backward Q decreases. (i)

At constant pressure, the slope of the change of K with T has the sign of the standard enthalpy of reaction. At constant temperature, the slope of K with p has the sign of the negative of the standard volume of reaction. (For ideal gas reactions, K depends on T only.)

(ii)

When T is changed at constant volume, the concentrations do not change: Qc, the reaction quotient expressed in concentration, ni 兾V, is constant. When T is changed at constant pressure, the partial pressures (= xi ptot) do not change: Qp is constant.

(iii)

When the total pressure is changed by a factor f, each partial pressure is changed by the same factor f. The new value of Qp will be equal to the old value multiplied by a factor to which each product gas molecule in the balanced equation will contribute a factor f in the numerator, while each reactant gas molecule will contribute a factor f in the denominator: Qp will change by a factor f νg, where νg is the increase in number of gas molecules in the balanced equation as we go from left to right: if νg is positive, increasing the total pressure (f > 1) will increase Qp and therefore Qc, while if νg is negative, Qp and Qc will decrease. The effect of changing V can be deduced using the fact that p is inversely proportional to V. When an indifferent substance is added at constant volume, the partial pressures are not changed, so that both Qp and Qc are constant. When it is added at constant pressure, the volume is increased so that the partial pressures changes by a factor f < 1 and Qp will change by a factor f νg. Addition of a reactant at constant volume increases the corresponding concentration or partial pressure term in the denominator of Q and therefore decreases it. Similarly, addition of a product increases the corresponding concentration or partial pressure terms in the numerator of Q and therefore increases it. Addition of reactant or product at constant pressure is rather more complicated and unsuitable for discussion at an introductory level; a statement of the final results will have to suffice. The rate of change of ln Q with ni is related to νi [(1兾xi ) − (νg 兾νi)], where νi is the coefficient of the added species in the balanced equation with sign, + for product and − for reactant. We can use this to predict how Q is changed by the perturbation.

1212

When the system is in equilibrium K = Q. A perturbation will change K or Q or both. During the relaxation K will be constant, so that Q will have to move towards K. Thus if a perturbation produces the condition Q < K, then Q has to increase to become equal to K; this is accomplished by a forward shift of the equilibrium, since a forward shift increases Q. Similarly, if the perturbation in a closed system produces the condition Q > K, then Q has to decrease to reach the value of K; this is accomplished by a shift in the backward direction. This comparison can be presented visually, as shown in Figure 1. This deals with a perturbation of the temperature at constant pressure for an ideal system. Qp remains constant but Kp changes with temperature. The necessary information here is that the slope of K with temperature has the same sign as the enthalpy of reaction. Students should be warned that the K–Q method can be applied in most cases only to ideal systems. There are two reasons for this. If Q has to be evaluated, we rarely have data to evaluate relative activities. Also, we assume that Q does not change with T. This is true only for an ideal system. Because we are dealing with ideal systems, we assume that only K changes with T. This leads to the incorrect result that the enthalpy of reaction involved is the standard enthalpy of reaction, whereas, as we will show later, the quantity of interest is the enthalpy of reaction in the equilibrium system. In applying this method, do we compare Kc with Qc or Kp with Qp? It would be convenient to have only K or Q change, not both. Thus when considering the effect of a change of T at constant volume, the comparison of Qc with Kc is easier to apply since only Kc changes. Similarly, when T is changed at constant pressure, we make use of the fact that Kp changes while Qp remains constant. Similarly, for the addition or removal of substances at constant volume the comparison of Qc with Kc is easier to apply.

K'

Equilibrium Constant

Thus in practice students are probably not deducing the direction of shift by using the principle of moderation, but are instead using subsidiary rules for the separate cases.

K

Q Q moves towards K' : reaction shifts backward

K

K'

T

T'

Temperature Figure 1. Temperature perturbation. ∆rH is negative and the slope of the graph is negative.

Journal of Chemical Education • Vol. 80 No. 10 October 2003 • JChemEd.chem.wisc.edu


We now give a simple illustration of the failure of the Le Châtelier prediction. Consider a vessel with a variable volume containing N2, H2, and NH3 in equilibrium at 500 K. The initial volume is 1 L and the total pressure is 4.145 atm. At this temperature Kp for the equilibrium N2(g) + 3H2(g) 2NH3(g) is 0.036. The equilibrium amounts of N2, H2, and NH3 are 0.075 mol, 0.020 mol, and 0.00603 mol, respectively. Now let us add 0.005 mol of N2 to this vessel and allow the system to relax at constant pressure and temperature. The Le Châtelier principle would predict that the reaction will shift forward. The mole fraction of N2 in the mixture is 0.7424, and νN2 = ᎑1 and νg = ᎑2. The rule given above tells us that, since ᎑1[(1兾0.7424) − (᎑2)兾(᎑1)] is positive, Q will increase. However, to reach a new equilibrium Q must decrease, and the reaction will move backward. Calculation shows that in the new equilibrium position the amounts of N2, H 2, and NH3 are 0.08003 mol, 0.02009 mol, and 0.00597 mol, respectively: the reaction has indeed moved backward. Other examples where the Le Châtelier principle fails (for a certain range of composition, not always!) are: CO in CH4(g) + H2O(g) N2 in N2(g) + 2H2(g) CS2 in CH4(g) + 2H2S(g)

CO(g) + 3H2(g) N2H4(g) CS2(g) + 4H2(g)

The Kinetic Approach A kinetic approach gives a satisfactory qualitative explanation of the direction of shift in most cases. If, for example, a perturbation increases the forward rate relative to the backward rate, then by the time a new equilibrium is established the system would have shifted in the forward direction. Though it is unlikely that the balanced equation represents elementary steps in both directions, we may safely assume that near equilibrium the increase in concentration or pressure of a reactant increases the forward rate and that the increase in concentration of a product increases the backward direction. The equation K = kf兾kb expressing the equilibrium constant as a ratio of the forward and backward rate constants and the equation d(ln K )兾dT = ∆H ⬚兾RT 2 are given in most general chemistry texts. These can be used to show that for endothermic reactions the forward rate is increased more than the backward rate, while the reverse holds for an exothermic reaction. The General Approach When a reaction takes place in a system, the changes in the amounts of the various species are a measure of the “amount of reaction” that has taken place. As the reaction proceeds in the forward direction the amounts of products increase and the amounts of reactants decrease. Taking the reaction N2(g) + 3H2(g) 2NH3(g) as an example, if ∆nN2 = ᎑0.02 mol, we see that ∆nH2 = ᎑3 × 0.02 mol and ∆nNH3 = 2 × 0.02 mol. Though the ∆n’s vary with nature of the species, they are directly proportional to one another. This allows us to describe the amounts of all the species in terms of just one quantity, ξ, the extent of reaction, if we introduce the stoichiometric coefficient νi of species i as the signed coefficient of i in the balanced equation, + for products and −

for reactants. Thus νN2 = ᎑1, νH2= ᎑3 and νNH3 = +2. ν = ∑i νi for this reaction = (᎑1) + (᎑3) + 2 = ᎑2. The relation between ni and ξ is given by

ni = n 0i + νi ξ

(1)

where n0i is the initial value of ni and ξ = 0 at time t = 0. Equation 1 defines the extent of reaction ξ. The extent of reaction ξ may be regarded as numerically equal to the number of times a change equivalent to the balanced equation has taken place. When ξ changes by 1 mol, the amount of N2 decreases by 1 mol, the amount of H2 decreases by 3 mol, and the amount of NH3 increases by 2 mol. We describe this loosely as “the balanced equation has taken place once”. From eq 1, dni = νi dξ

(2)

Stoichiometric coefficients and the extent of reaction are being increasingly used in physical chemistry texts (8). I have recently shared my experience with the use of ξ in introductory courses in the pages of this Journal (9), especially in the calculation of limiting reagents. Though I have preferred to use ξ in the derivations and in the statement of the rules, it is quite possible to avoid it altogether and use, for example, the amount of one of the products—when ξ increases, the amount of any product will increase linearly with it. For given initial amounts, the value of ξ at equilibrium, ξeq, will depend in general on T, p, and composition. What is of interest to us is how ξeq varies with the parameters that determine it. Increase of ξeq corresponds to a shift of the equilibrium in the forward direction. The rules for prediction involve the “reaction quantities”, for example, the enthalpy of reaction. This is numerically equal to the change in enthalpy when the balanced equation takes place once and is denoted here by hrn. It is the same as ∆ rH. The mathematical definition is h rn = (∂H兾∂ξ)T,p. Similarly, the volume of reaction, vrn, is numerically equal to the change in volume when the balanced equation takes place once. Rules for Predicting the Direction of Shift Before deriving the necessary equations from thermodynamic relations we give the conclusions in the form of a few rules. The rules can be reformulated, if desired, in terms of the direction of shift. These should be compared with the subsidiary rules associated with the treatment of the Le Châtelier principle. The possible equilibrium positions of a system will lie on the graph of ξeq as a function of the variables that determine it. In particular, if only one variable is changed, a graph of ξeq versus this variable is all that we need to know how the system will shift when perturbed. For small perturbations, we can linearize the curve and replace it with the slope: the slope of ξeq with respect to the variables on which it depends will play an important part in the formulation of the rules for predicting the direction of shift. Rule I: Change of T at constant pressure or volume. The slope of the graph of ξeq versus T has the same sign as the corresponding heat of reaction in the equilibrium system.


1213


Rule III: Addition of species i only at constant T and V. If the system is an ideal gas mixture, the slope of the graph of ξeq versus ni has the sign of ᎑νi , a sign opposite to the stoichiometric coefficient of i, νi . In particular, (i) addition of indifferent species does not perturb the reaction, (ii) addition of reactant makes the reaction go forward, and (iii) addition of product makes the reaction go backward. Rule IV: Addition of species i only at constant T and p. This is the most difficult of the rules, because the result depends on the composition. If the system is ideal (ideal gas mixture or ideal solution), the slope of the graph of ξeq versus ni has the sign of the quantity ᎑(νi 兾xi ) + ν = ᎑ νi [(1兾xi ) − (ν兾νi )], where ν is the sum of the stoichiometric coefficients of the reactants and products (the increase in the number of molecules as we go from the left to right in the balanced equation), and xi the mole fraction of i; for a gas reaction ν would be represented by νg. Thus the difference from the result for constant volume is the term in brackets. This is positive when ν and νi have opposite signs or when they have the same sign but ν is numerically smaller than νi . For this case the Le Châtelier prediction will be valid; in all other cases, for certain ranges of composition it will fail. In particular, for gas reactions, since vrn = νg RT兾p, the sign of vrn has the sign of νg . νg will be positive if the number of gas molecules increases as the reaction goes forward. For the addition of an indifferent species to a reaction mixture, the slope has the sign of vrn or νg .

We consider an example of how these rules are applied. According to the rule for temperature perturbation at constant pressure the slope of ξeq with T has the same sign as the molar enthalpy of reaction. Suppose the molar enthalpy is equal to ᎑200 kJ/mol. The graph of ξeq with T will therefore have a negative slope, as shown in Figure 2. It is represented near the original equilibrium position O by the line PQ with a negative slope. If the temperature is increased, that is, the temperature shifts to the right, the system will move to the right along PQ to the new equilibrium position O´. ξeq will thus decrease; that is, the reverse reaction will take place. Similarly, a decrease in T will increase ξeq and shift the equilibrium in the forward direction. The application of the rules is simple and straightforward. The rules are direct, unambiguous, and accurate. A knowledge of the sign of the slope will enable students to quickly sketch the graph near equilibrium and to be able to make predictions with little chance of error.

A closed system can be described in terms of three variables, for example, T, p, and ξ or T, V, and ξ. The composition of the reaction mixture is completely described in terms of the original amounts and ξ. In a closed system with one reaction dni = νi dξ according to eq 2, so that the condition of equilibrium ∑i µi dni = 0 can also be written as (∑iνi µi )dξ = 0 whence ∑iνi µi = 0. The quantity ᎑∑iνi µi is referred to as the affinity of the reaction, A.1 Thus the affinity of the reaction N2(g) + 3H2(g) 2NH3(g) is ᎑2µNH3 + 1µN2 + 3µH2. If we flip the equation and write it as 2NH3(g) N2(g) + 3H2(g), the stoichiometric coefficients have changed sign and the affinity of this reaction is the negative of the previous value. Thus for any given balanced equation we can talk of the affinity of the forward and backward reactions such that Aforward = ᎑Abackward. The symbol A stands for Aforward, where the forward reaction is the direction represented by the given balanced equation. This means that when the affinity of the forward reaction is negative, the affinity of the backward reaction is positive. The direction of reaction is determined by A (6; p 40). At equilibrium, A = 0 under any conditions whatsoever. When A is positive, the reaction takes place in the direction of ξ (i.e., the forward direction) while if A is negative the backward reaction will take place. The affinity A may thus be regarded as the driving force of the reaction, since the reaction always takes place in the direction in which A is positive. Figure 3 shows schematically a sketch of A versus ξ near equilibrium. We see from the graph that ∂A兾∂ξ is negative; the variables being held constant can be any selection of the appropriate variables. This is indeed the great advantage in treating the problem in terms of the affinity of reaction because statements relating to it do not have to be qualified by the conditions under which the reaction is taking place. By contrast, if we are dealing with the Gibbs free energy G = H − TS we have to restrict ourselves to constant temperature and

P

␰eq

Rule II: Change of p at constant T. The slope of the graph of ξeq versus p has the sign of ᎑vrn, a sign opposite to that of the volume of reaction in the equilibrium system. Changes in V can also be dealt with by this rule.

O

O'

Review of Basic Relations We make use of the fact that the system is in equilibrium to begin with and returns to equilibrium after the perturbation. The condition dG = ∑i µi dni = 0 for equilibrium (where µi is the partial molar Gibbs free energy of i) for constant T and p should be familiar to students. Most physical chemistry texts deal with the fact that ∑i µi dni = 0 is the condition for equilibrium under other conditions as well (8; Chapter 4 ).

1214

Q

Temperature Figure 2. ξ eq vs T. ∆rH is negative and the slope of the graph is negative.



pressure conditions. The proofs given here are in terms of affinity in order to keep the treatment as general as possible. This can be avoided by using, for example, the Gibbs free energy or the Helmholtz free energy. At constant T and p, the free energy of reaction ∆rG = ∑iνi µi so that under these conditions A = ᎑∆rG. The usual rule that for nonequilibrium ∆rG is negative corresponds to the fact that the affinity A is positive. As reaction proceeds, A will decrease until it reaches a value of 0 at equilibrium. The basic equations relating the affinity to the more familiar Gibbs free energy G and the Helmholtz free energy A = U − TS are dG = −SdT + Vdp − Ad ξ

(3)

dA = −S dT − p dV − A dξ

(4)

∂ A ∂ T ∂ A ∂ p

= V, ξ

= − T, ξ

∂ A ∂ V

∂ V ∂ ξ

= T, ξ

∂ S ∂ ξ

(8)

T, V

= − v rn

(9)

T, p

∂ p ∂ ξ

(10)

T, V

hrn, srn, and vrn are the enthalpy of reaction, the entropy of reaction and volume of reaction respectively. Since A = 0 at equilibrium it follows from eqs 5 and 6 that at equilibrium (11)

hrn = Tsrn

From the definition of A we see, for example, that the first of these equations is the same as dG = ᎑SdT + Vdp + ∑i µi dni, which is the usual form in which it is found in the texts; for closed systems, both forms are equivalent. From these equations we can obtain some useful relations, for example,

∂ U ∂ ξ

∂ S ∂ ξ

= T T, V

(12)

T, V

We also have ∂ G A = − ∂ ξ

=

∂ A ∂ ξ

= − hrn + T s rn

T, V

= p, ξ

+T T, V

∂ S ∂ ξ

∂ S ∂ ξ

= srn T, p

T, V

(6)

(7)

A

O'

0

O

O''

␰ Figure 3. A versus ξ. Nonequilibrium points, O´ and O˝, move towards O. The slope is negative.

(

A = −∑ νi µ i = −∑ νi µ°i + RT ln ai i

∂ U ∂ ξ

=

∂ A ∂ T

(5)

T, p

i

= RT ln

)

K Q

(13) (14)

where ai is the relative activity of species i, K is the equilibrium constant, and Q the reaction quotient. Since ∂A兾∂ξ is negative irrespective of what variables are being held constant, it follows from the above equation that at constant temperature ∂(ln Q )兾∂ξ is positive under any conditions. Thus as the reaction goes forward the reaction Q increases under all conditions; this is true for both ideal and nonideal systems. The rules given earlier deal with graphs of ξeq versus the variable of interest. It is also possible to develop a visual method of predicting qualitatively how the equilibrium shifts in terms of graphs of the affinity A versus ξ. Consider an exothermic reaction. At equilibrium srn = hrn兾T is negative. From eq 7 we see that for a given ξ, A will decrease with temperature. Thus if AB is the graph of A versus ξ for a temperature T, the graph CD for a higher temperature T+ will be below AB and for a lower temperature T− the graph EF will be above AB; this is shown in Figure 4. Thus raising the temperature from T at equilibrium to a temperature T+ moves the system from point O to point O´ on CD. Now the system will move on CD toward the point where A = 0; that is, the reaction will shift backward. Similarly, if the temperature is lowered, the affinity increases and the system shifts to point O˝ on EF. Now the nonequilibrium system will move towards A = 0 along EF; that is, it will move to the right. In a quantitative approach it is preferable to derive an equation for the slope of ξeq with the perturbed variable (6; p 270). For a closed system, the affinity of the reaction may be regarded as a function of any three quantities, for example, T, p, and ξ or T, V, and ξ for given initial amounts. Starting with A = A(T,p,ξ), A = 0 at equilibrium, so that ξeq = f (T, p).


1215


∂ y ∂ x

=

−

z

∂ z ∂ x ∂ z ∂ y

E A O'' C

A

If we perturb the system that is initially at equilibrium at temperature T and pressure p by increasing the temperature at constant pressure to T + dT, and allow the system to come to a new position of equilibrium, the change in ξ has occurred at constant p and constant A(= 0). We therefore need to find (∂ξ兾∂T )A=0,p. In a similar way, if we change the pressure from p to p + dp and allow the system to come to a new equilibrium at the new pressure, we need to find (∂ξ兾∂p)A=0,T. We treat these two cases separately. The proofs will start by using an identity from the calculus

O

0

F

O'

B D

y

T− T T+

(15) x

Figure 4. A versus ξ for different temperatures. The reaction is exothermic. A decreases with T. At higher temperature, T+, O moves down to O+

Temperature Perturbation of a Closed System We have

∂ ξ ∂ T

= − A = 0, p

= −

= −

∂ A ∂ T

ξ, p

∂ A ∂ ξ

T, p

srn ∂ A ∂ ξ T, p hrn ∂ A T ∂ ξ

volume instead of the pressure is kept constant. We obtain

(from eqn 15)

∂ ξ ∂ T

(from eqn 7)

A = 0, V

(16)

(from eqn 11) = −

T, p

The differential coefficients are evaluated for the equilibrium state. Here hrn is the molar enthalpy of reaction in the equilibrium state, not the standard state. Since A decreases with ξ, it follows from eq 16 that the sign of the change of ξ with T at equilibrium is the same as that is hrn at equilibrium. For endothermic reactions, an increase of temperature will increase ξ so that the equilibrium will shift in the forward direction, and for an exothermic reaction an increase of temperature will make the equilibrium shift in the backward direction. The variation of solubility with temperature was the subject of controversy in the pages of this Journal, (10–13). For the case of solubility, ξ in eq 16 would be measured by the amount of solute that has dissolved; that is, solubility should be expressed in terms of the amount of solute dissolved in a fixed amount of solvent. Thus the rule will be valid if solubility is expressed as mass per cent of solute or molality or as a mole ratio. It may not be valid if the solubility is expressed as a molar concentration, since the volume of solution also changes with temperature. The proof is easily modified if the

1216

= −

= −

∂ A ∂ T

ξ, V

∂ A ∂ ξ

T, V

∂ S ∂ ξ

T, V

∂ A ∂ ξ

T, V

∂ U ∂ ξ ∂ A T ∂ ξ

T, V

(from eqn 15)

(from eqn 8) (17)

(from eqn 12)

T, p

The quantity (∂U兾∂ξ)T,V is the heat of reaction at constant volume and is related to the internal energy of reaction urn = (∂U兾∂ξ)T,p by ∂ U ∂ ξ

= urn + T, V

∂ U ∂ p

T, ξ

∂ p ∂ ξ

T, V

(18)

For ideal gas reactions, since U depends only on temperature, (∂U兾∂ξ)T,V = urn. Consequently, for ideal gas reactions, the direction of shift for perturbations at constant volume depends on the sign of the internal energy of reaction, not



the enthalpy of reaction. These can have opposite signs since hrn = urn + RTνg, where νg is the increase in the number of gas molecules. Pressure Perturbation of a Closed System We have

∂ ξ ∂ p

∂ A ∂ p = −

∂ A ∂ ξ

A = 0, T

ξ, T

(from eqn 15)

T, p

vrn ∂ A ∂ ξ T, p

=

(from eqn 9)

(19)

Addition or Removal of Species The addition or removal of a species is rather more difficult to treat since now the system is an open system. We have to consider the affinity as a function of the chemical amounts of the species, A = A(n1, n2, …, T, p). Suppose we add an amount dni, ext of species i at constant T and p, the amounts of all other species being held constant. In general the equilibrium will be perturbed and a new position of equilibrium will reached where once again A = A(n1´, n2´ …, T, p) = 0. We will denote by i the species that is added or removed, and j will denote any species, including i. The changes in the amounts of the species will be given by dnk = νk ξ for all species k except for i. For i we have dni = dni, ext + νi dξ. Since A = 0 for the initial and final state dA = 0 =

∑ j

The differential coefficients are evaluated for the equilibrium state. vrn is the molar volume of reaction in the equilibrium state, not the standard state. Since A decreases with ξ, it follows from eq 19 that the change of ξ with p at equilibrium and the molar volume of reaction at equilibrium have opposite signs. When the molar volume of reaction is positive an increase of pressure will decrease ξ—the equilibrium will shift in the backward direction. The case of ideal gas reactions is easiest to deal with. Here, the volume of reaction will have the same sign as the sum of the coefficients (of the gas molecules) on the right hand side of the balanced equation minus the sum of the coefficients on the left hand side. We could treat the same problem with V instead of p as the independent variable.

∂ ξ ∂ V

= − A = 0, T

= −

∂ A ∂ V ∂ A ∂ ξ

ξ, T

(from eqn 15)

T, V

∂ A ∂ ξ

T, p

= − T, V

j

=

∂ A ∂ ξ

∂ A ∂ n j

dξ + T, p , nk ≠ j

dξ + T, p

∂ A ∂ ni

dn j

(22)

T, p , nk ≠ j

∂ A ∂ ni

T, p, nk ≠ i

dni , ext (23)

dni , ext

(24)

T, p , nk ≠ i

It follows that

= −

∂ A ∂ ni

T, p , nk ≠ i

∂ A ∂ ξ

A = 0, T, p

(25) T, p

(20)

(from eqn 10)

If the addition of species were done at constant volume, the appropriate equation is

Since (∂A兾∂ξ)T,p is negative, it follows that (∂ξ兾∂V )A=0,T has the same sign as (∂p兾∂ξ)T,V. For ideal gas reactions the latter is positive if the right hand side of the balanced equation has more gas molecules than the left side. Since

∂ p ∂ ξ

∑ ν j

∂ ξ ∂ ni ,ext

T, V

∂ p ∂ ξ

=

∂ A ∂ n j

∂ V ∂ ξ ∂ V ∂ p

∂ ξ ∂ ni ,ext

= −

∂ A ∂ ni

A = 0, T, V

T, V, nk ≠ i

∂ A ∂ ξ

(26) T, V

This is as far as we can go for any general system. We consider ideal systems. For ideal gases

T, p

(21) T, ξ

and (∂V兾∂p)T, ξ is negative, it follows that (∂ξ兾∂V )A=0,T has the same sign as vrn.

µ = µp° (T ) + RT ln

p = µx° (T, p ) + RT ln x (27) p°

= µc° (T ) + RT ln

c c°


(28)

1217


and for other ideal systems µ (T , p ) = µ° (T , p ) + RT ln x

(29)

where x is the mole fraction. Using A = ᎑∑iνi µi and the equation for µi , for gases we get from eq 28

∂ A ∂ ni

= − T, V , nk ≠ i

νi RT ni

(30)

Eq 26 now becomes

∂ ξ ∂ ni , ext

− = −

νi RT ni

∂ A ∂ ξ

T, V, A = 0

= − C 2 νi

(31)

T, V

where C 2 is a positive quantity. Thus for constant volume perturbations due to external additions in an ideal gas system, addition of reactants make the reaction go forward, while addition of products make the reaction go backward. The addition of substances present in smaller concentration has a greater effect on the shift. The addition of neutral substances does not shift the equilibrium. For additions at constant pressure, for ideal gases and other ideal systems we have from eqs 13, 27, and 29 dA = − RT ∑ νj d ln x j j

(

)

= − RT ∑ νj d ln n j + RT j

= − RT ∑ νj j

dnj nj

∑ νj j

d ln ntot (32)

dn + RT ν tot ntot

where ν = ∑j νj. When only ni changes, dntot = dni, dnk = 0, k ≠ i so that ∂ A ∂ ni

= T, p, n j ≠ i

RT ν ν − i ntot xi

(33)

Substituting in eq 25 we get

∂ ξ ∂ ni , ext

T, p , A = 0

ν RT ν − i ntot xi = − ∂ A ∂ ξ T, p = C2 ν −

νi xi

(34) = − νi

ν 1 − C2 νi xi

The sign of the change of ξeq on the addition of species i at constant pressure may depend on the mole fraction. For the

1218

reaction N2(g) + 3H2(g) 2NH3(g) ν = ᎑2. The sign of the change of ξeq on the addition of N2 depends on ᎑2 − (᎑1)兾xN2, which will change sign at x N2 = 0.5. When x N2 > 0.5 the addition of N2 to an equilibrium mixture will make the reaction go in the backward direction to produce more nitrogen. For H2, the sign depends on ᎑2 − (᎑3)兾x H2, which is always positive. Conclusion There is a need in all approaches to the problem of equilibrium perturbations to carefully specify under what conditions the system is relaxing to a new equilibrium. The limitations of any rules, whether they apply to all systems or only to ideal systems, should also be stated. The first qualitative introduction to the discussion of problems of perturbation from equilibrium seems best done through the kinetic approach. It is easy to understand, and, compared to the Le Châtelier principle, does not use any confusing terminology or interpretation. The method using the comparison of K with Q can be precisely formulated. It can be presented visually and related to quantities like enthalpy of reaction and volume of reaction; the relevant equations are found even in the introductory texts. If the Le Châtelier method is used, care is needed in the formulation of the principle to avoid ambiguity. It is best to restrict the application to changes in intensive variables, since it is known that changes in extensive variables are not moderated. It does not appear to have any advantage over the comparison of K with Q. If the interest is not in illustrating the principle of moderation, there is little reason to use it. It is useful at the junior level to treat the problem quantitatively. This has been done in this article by deriving equations for the variation of ξeq with the parameter of interest. The use of ξ makes the concept of “position of equilibrium” precise. An advantage of this approach is that one is aware of the limitations of the conclusions: are they applicable to all systems or to ideal systems only? The properties that govern the direction of shift relate to the equilibrium system, not to the standard state. Two methods of a visual approach are by the use of graphs of affinity versus ξ for various values of the perturbing parameter as in Figure 3 and by drawing graphs of ξeq versus the perturbing parameter. The rules for drawing the graphs are comparable to the subsidiary rules given by textbooks in interpreting Le Châtelier’s principle. The use of this method to complement the kinetic or K–Q approach at an introductory level is certainly possible. The essential concept is that the equilibrium system can be specified by a point on the graph of ξeq versus some variable of interest. Instead of formulating the rules in terms of the extent of reaction, they may be formulated in terms of the equilibrium amount of any of the product species. A feature that makes this approach particularly attractive is that the application to specific cases is very easy and can be carried out visually as shown by Figure 2: once the graph of ξeq versus the perturbing variable has been sketched predictions of the direction of shift can be made for both positive and negative perturbations.


Information • Textbooks • Media • Resources

Acknowledgment I wish to thank Jeffrey A. Gray for valuable comments on the manuscript. Note 1. The symbol A is used for the affinity of reaction to differentiate from the symbol A used for the Helmholz free energy.

Literature Cited 1. de Heer, J. J. Chem. Educ. 1957, 34, 375–380; de Heer, J. J. Chem. Educ. 1958, 35, 133-135. 2. Treptow, R. S. J. Chem. Educ. 1980, 57, 417–420.

3. Posthumus, K. Rec. Trav. Chim. 1933, 52, 25–35. 4. Fishtik, I.; Nagypal, I.; Gutman, I. J. Chem. Soc., Faraday Trans. 1995, 91, 259–267. 5. Katz, L. J. Chem. Educ. 1961, 38, 375–377. 6. Prigogine, I.; Defay, R. Chemical Thermodynamics; Longmans: London, 1954; p 263. 7. Atkins, P.; Jones, L. Molecules, Matter and Change, 3rd ed.; W. H. Freeman and Company: New York, 1997, pp 493, 495, 498. 8. Silbey, R. J.; Alberty, R. A. Physical Chemistry, 3rd ed.; John Wiley and Sons, Inc.: New York, 2001; p 136. 9. Canagaratna, S. G. J. Chem. Educ. 2000, 77, 52. 10. Bodner, G. M. J. Chem. Educ. 1980, 57, 117–119. 11. Fernandez-Prini, R. J. Chem. Educ. 1982, 59, 550–553. 12. Brice, L. K. J. Chem. Educ. 1983, 60, 387–389. 13. Treptow, R. S. J. Chem. Educ. 1984, 61, 499–502.


1219

Approaches to the Treatment of Equilibrium Perturbations - Journal of

Recommend Documents