The Ammonia Synthesis Reaction: An Exception to the Le ChÃ¢telier

Research: Science and Education edited by

Advanced Chemistry Classroom and Laboratory

Joseph J. BelBruno Dartmouth College Hanover, NH 03755

The Ammonia Synthesis Reaction: An Exception to the Le Châtelier Principle and Effects of Nonideality

W

Mark J. Uline and David S. Corti* School of Chemical Engineering, Purdue University, West Lafayette, IN 47907-2100; *[email protected]

At equilibrium, the chemical potentials of the components participating in a given chemical reaction must satisfy the following relation (1)

∑ νi µ i

(1)

= 0

i

where νi is the stoichiometrically balanced coefficient of species i and µˆ i is the chemical potential of species i in the mixture. For a given temperature and pressure, equilibrium compositions may then be calculated using eq 1 for all relevant reactions. If the temperature, pressure, or composition of one of the components subsequently varies, the equilibrium compositions usually change. The direction that the reaction takes to reach these new equilibrium compositions, whether to the right or left, that is, producing more products or reactants, respectively, can be determined by direct computation of the new equilibrium state. Observations of the directions that reactions took to return to equilibrium after various perturbations were introduced led to the formulation of a general statement referred to as the principle of Le Châtelier (2), or sometimes as the principle of Le Châtelier and Braun (3). Le Châtelier’s principle can be stated as follows: In a system at equilibrium, a change in one of the variables that determines the equilibrium will shift the equilibrium in the direction counteracting the change in that variable (2).

The above statement is useful in inferring, without direct calculation, the effects of changes in a system initially at equilibrium. For example, according to Le Châtelier’s principle, the further addition of a particular component will cause the reaction to shift in the direction that reduces the total number of moles of the system. Nevertheless, the principle of Le Châtelier, despite its seemingly intuitive nature, has not been rigorously proven (4). In fact, several exceptions are known to occur and have been discussed in the literature for some time (1–7). Consider, for example, the ammonia synthesis reaction

N2 + 3H2

2NH3

(2)

in which equilibrium has been established at a given temperature T and pressure P. With T and P held fixed, Le Châtelier’s principle predicts that the addition of more nitrogen into the reaction vessel will cause the reaction to shift to the right, that is, more ammonia will be produced. Yet, as has been shown before (1, 2, 4–7), if the initial equilibrium mole fraction of nitrogen exceeds 0.5, and the given T and P 138

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are held fixed, the addition of more nitrogen causes the reaction to proceed to the left, producing more nitrogen. The value of 0.5, as shown later, is calculated assuming ideal gas behavior. This shift of the reaction to the left is a clear exception to the principle of Le Châtelier.1 Le Châtelier’s principle can be reformulated in a more general way that becomes universally valid (4, 5), which is more appropriately called the principle of moderation. An excellent overview, and proof, of this new general statement is given by de Heer (4). The principle of moderation, however, is valid only for infinitesimal changes from the initial state (6, 7). Both Liu et al. (6), and Corti and Franses (7), by analyzing the shifts in the direction of the ammonia synthesis reaction, demonstrated that the principle of moderation is violated in some cases for finite additions of nitrogen. For finite changes, no universally valid statement on the direction that a reaction follows can be formulated. In such cases, instructors should advise that each reaction be considered individually with the shift in the equilibrium state being determined directly from the relations of chemical equilibrium.2 Although the exception to Le Châtelier’s exhibited by the ammonia synthesis reaction has been discussed before, there has been no discussion, to the authors’ knowledge, of the effects of nonideality on the direction that the reaction shifts upon the addition of more nitrogen. Previous analyses assumed ideal gas behavior so that the value of 0.5 for the mole fraction of nitrogen was obtained using the relations for an ideal gas mixture. Whether this critical value of the mole fraction of nitrogen deviates from 0.5, and to what extent, when nonideal effects are included has not been determined previously and is the focus of this article. We also consider finite additions of nitrogen, but unlike refs 6 and 7, nonidealities are accounted for in the analysis. Overall, this article can form the basis of several stimulating lectures and problem-solving sessions on chemical equilibria. The analysis combines a known exception to the principle of Le Châtelier with the procedures required to incorporate more complex (at least beyond those of the ideal gas) thermodynamic expressions for the behavior of real gases. The resulting experience gained in determining the direction of shifts in the reaction should provide students with a deeper understanding of chemical reaction equilibria. The article is organized as follows. First, the set of relations (applicable to both ideal and nonideal gas mixtures) needed to determine the direction in which the reaction shifts upon the addition of more nitrogen is provided. Next, the relations obtained in the last section are applied to the ideal gas mixture. Also included is a discussion of why 0.5 emerges

Vol. 83 No. 1 January 2006

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Research: Science and Education

as that value of the mole fraction of nitrogen above which an exception to the principle of Le Châtelier arises. An analysis of the ammonia synthesis reaction with nonideal effects is given in the next section. The mole fraction of nitrogen above which the reaction shifts to the left is calculated for various temperatures and pressures. We also discuss why this value of the mole fraction differs from the ideal gas mixture result of 0.5. Next, the direction in which the reaction shifts is again considered but now for the case of finite additions of nitrogen. Conclusions are given at the end. General Formulation of the Ammonia Synthesis Reaction The set of relations required to analyze the ammonia synthesis reaction is provided below. For the interested reader, detailed derivations of the equations presented in this section are contained in the Supplemental Material.W Consider the ammonia synthesis reaction outlined in eq 2. Let species 1 represent nitrogen, species 2 hydrogen, and species 3 ammonia. The chemical potential, µˆ i, of each species i in the mixture is given by (1) µi = Gi ° + RT ln

fi f i °

− ∑ νi Gi °

2

f1 f2

3

= K a = exp

i

RT

= exp

− ∆G rx° RT

(4)

where ∆Grx is the standard state Gibbs free energy change upon reaction. For gases, ∆Grx and Ka are only functions of T. In general, fî is a function of T, P, and two out of the three mole fractions (e.g., y1 and y2). Now, let the system be at equilibrium at a given T and P. At this equilibrium state, there are n1, n2, and n3 moles of each species with mole fractions y1, y2, and y3 satisfying eq 4. Next, we consider the addition of ∆ moles of nitrogen (1), while keeping T and P constant. As the system reestablishes equilibrium, the final mole fractions are given by (7) y1 =

y ° + ∆ ′ − ξ ′ n1° + ∆ − ξ = 1 1 + ∆ ′ − 2ξ ′ n ° + ∆ − 2ξ

y2

y2° − 3ξ ′ = 1 + ∆ ′ − 2ξ ′

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y3° + 2ξ ′ 1 + ∆ ′ − 2ξ ′

(5c)

in which n = n1 + n2 + n3, ∆´ ≡ ∆兾n and ξ´ ≡ ξ兾n; ∆´ and ξ´ are dimensionless quantities. ξ is the extent of reaction starting from the above initial equilibrium state, with ξ defined to be positive if the reaction proceeds to the right. Equations 4 and 5 are valid for all values of ∆´ and can be used to determine the value of ξ´ for a given choice of ∆´. The sign of ξ´, however, determines the direction in which the reaction shifts, and its value determines the extent of the reaction triggered by the addition of more nitrogen. Initially, we consider how ξ´ varies when an infinitesimal amount of nitrogen, that is, ∆´ → 0, is added. One can solve eq 4, along with eq 5 for various values of ∆´ and then determine the sign of ξ´ as ∆´ → 0. Since ξ´ → 0 as ∆´ → 0, however, we instead determine dξ´兾d∆´ for ∆´ → 0. To proceed, we differentiate the logarithm of both sides of eq 4 with respect to ∆´ while holding T and P constant. Equation 5 is then used to determine the derivatives of the mole fractions with respect to ∆´. In the limit as ∆´ → 0, where ξ´ → 0 and y1 → y1 and y2 → y2, we find upon rearrangement that

(3)

where Gi is the chemical potential of pure component i in its standard state at the temperature T, fî is the fugacity of species i in the mixture, fi is the fugacity of pure i in its standard state at the temperature T, and R is the ideal gas constant. For gases, both real and ideal, the standard states of all components are chosen as the pure materials in an ideal gas unity fugacity state (3). At equilibrium, the chemical potentials of the components participating in the chemical reaction must satisfy eq 1, where ν1 = 1, ν2 = 3, and ν3 = 2 (see eq 2). Thus, for the ammonia synthesis reaction, substitution of eq 3 into eq 1 yields upon rearrangement (1)

f3

y3 =

η =

α 2 (2 y 2° − 3) + α1 (2 y1° − 1)

(6)

where η = dξ´兾d∆´ and αj =

∑ νi i

∂ ln f i ∂ yj

(7) T , P , yi ≠ j

The direction that the reaction takes upon an infinitesimal addition of nitrogen is determined by the sign of η in eq 6. If η > 0, the reaction shifts to the right; if η < 0, the reaction shifts to the left. Since α1 and α2 are functions of y1, y2, T, and P, the sign of η is also dependent upon y1, y2, T, and P.3 If η changes sign, such that the reaction shifts direction from right to left, or vice versa, then at some state point η = 0. From eq 6, the condition of η = 0 implies that

α1 (1 − y1° ) = α 2 y2°

(8)

Note that eqs 6 and 8 are general results, in that they are valid for all chosen expressions for the fugacities. Ideal Gas Mixture: Exception to the Principle of Le Châtelier To illustrate the use of eq 6, let us first consider the case of an ideal gas mixture, where fî = yi P, so that fˆ1 = y1P, fˆ2 = y2P, and fˆ3 = y3P = (1 − y1 − y2)P. Therefore,

(5a)

(5b)

•

α 2 y2 ° − α1 (1 − y1° )


α1 =

y 2° − y1° − 1 y1° (1 − y1° − y2° )

α2 =

3 y1° + y2° − 3 y2° (1 − y1° − y2° )

•


(9)

139


For the ideal gas, α1 < 0 and α2 < 0 and are not functions of T and P. Substituting eq 9 into eq 6 yields, after some algebra, y 2° y 3° (1 − 2 y1° )

η =

4 y1° y 2° (1 − y 3° ) + 9 y1° y 3° + y 2° y 3°

(10)

Since the denominator is always positive, the sign of η is given by the sign of the numerator in eq 10 or, more specifically, by the sign of (1 − 2y1). Thus, for infinitesimal additions of nitrogen, we conclude that (7)

when y1°
0

reaction proceeds to the right, consistent with the (11a) Le Châtelier principle

when y1° >

1

2, η < 0

reaction proceeds to the left, not consistent with (11b) the Le Châtelier principle

when y1° =

1

2, η = 0

no reaction takes place, not consistent with the (11c) Le Châtelier principle

The exception to Le Châtelier’s principle occurs for the ideal gas mixture when the initial equilibrium mole fraction of nitrogen is equal to or exceeds 0.5 and is independent of the mole fractions of the other species or of the values of the temperature and pressure. Note that eq 11c also could have been derived from eq 8. The conclusions drawn from eq 10 are consistent with previous analyses of the ammonia synthesis reaction (assuming ideal gas behavior). Though correct, the steps leading to eq 10 do not, however, provide any readily apparent insights into why y1 = 0.5 emerges as a transition point (nor why, if the analysis were redone, Le Châtelier’s principle is not violated for additions of hydrogen and ammonia). Before we begin our discussion of the effects of nonideality, we therefore provide a more transparent analysis of the ideal gas mixture. Begin by introducing a new variable, the affinity, A, in which (1) A = −∑ νi µ i = µ1 + 3 µ 2 − 2 µ 3

where eq 1 requires that A = 0 at equilibrium. Let us now start from a system at equilibrium, where A = 0, and add a given quantity of one of the components. If A becomes positive (i.e., µˆ1 + 3µˆ 2 > 2µˆ 3 ), A must therefore decrease back to zero in order to reestablish equilibrium. A decrease in A requires that (µˆ 1 + 3µˆ 2) decreases while 2µˆ 3 increases. In general, this is accomplished by having the reaction proceed to the right (ξ > 0) where the consumption of nitrogen and hydrogen decreases (µˆ 1 + 3µˆ 2) and the generation of ammonia increases 2µˆ 3.4 If, on the other hand, A became negative (µˆ 1 + 3µˆ 2 < 2µˆ 3 ), the reaction would proceed to the left (ξ < 0), generating more nitrogen and hydrogen (increasing µˆ 1 + 3µˆ 2) and consuming ammonia (decreasing 2µˆ 3) so that A returned to zero. In other words, the reaction must occur (1) so that Aξ > 0. Journal of Chemical Education

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dA (RT ) d∆ ′

d µ1 (RT ) d∆ ′

= T, P

d µ2 (RT ) d∆ ′

+ 3

T, P

− 2 T, P

d µ3 (RT ) d∆ ′

(13) T, P

The chemical potential of a component in an ideal gas mixture is given by (8)

µi = Γ i (T ) + RT ln ( yi P )

(14)

where Γi(T ) is the chemical potential of pure i as an ideal gas at the given temperature (and a reference pressure always chosen to be unity). Thus,

dA (RT ) d∆ ′

= T, P

1 d y1 y1° d∆ ′ +

3 y2°

dy 2 d∆ ′

2 dy3 y3° d∆ ′

−

(15)

Now, let ξ´ = 0 (i.e., no reaction) in eq 5. Differentiating the resulting expressions and then setting ∆´ = 0 (infinitesimal addition of nitrogen) yields d y1 d∆ ′

= 1 − y1° ;

dy 2 d∆ ′

= − y 2° ;

dy 3 d∆ ′

= − y 3° (16)

As expected, y1 increases upon the addition of nitrogen while y2 and y3 both decrease. Consequently, eq 15 reduces to dA (RT ) d∆ ′

(12)

i

140

Now consider the (dimensionless) addition, ∆´, of an infinitesimal amount of nitrogen at fixed T and P without any reaction occurring. According to eq 12, A兾RT will vary in the following manner

= T, P

1 − y 1° 1 − 3 + 2 = − 2 y1° y1°

(17)

There are two competing influences on A. First, A varies by an amount equal to (1 − y1)兾y1. Second, each of the remaining terms change A by an amount given by that component’s (not including nitrogen) stoichiometric coefficient (and is independent of composition). A will become positive upon the addition of nitrogen for y1 < 0.5 (so that the reaction will proceed to the right) and will become negative for y1 > 0.5 (so that the reaction will proceed to the left). No reaction will occur for y1 = 0.5 since the value of A, having started from a zero value, does not change upon the addition of nitrogen. Note that the 2 appearing on the far right side of eq 17 is the sum of all the stoichiometric coefficients participating in the reaction (see eq 2). A more general derivation of eq 17 that is valid for any reaction and for the infinitesimal addition of any component is provided in the Supplemental Material.W In general, an


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exception to Le Châtelier’s principle occurs for any species j in an ideal gas mixture that satisfies

yj ° ≥

νj

(18)

∆ν

where ∆ν is the sum of all stoichiometric coefficients. Meaningful values of yj are therefore generated only when νj has the same sign as ∆ν and for |νj| < |∆ν|. For the ammonia synthesis reaction, ∆ν = 2. Hence, exceptions to Le Châtelier’s principle will only occur for nitrogen, where ν1 = 1. Though hydrogen has a negative stoichiometric coefficient (ν2 = 3), |ν2| exceeds |∆ν|. No exception occurs for ammonia, where ν3 = 2, so the reaction will always proceed to the left upon addition of more ammonia. Nonideal Gas Mixtures The ideal gas value of y1 at which the reaction shifts from right to left is independent of T and P because α1 and α2 are not functions of T and P. Other ideal gas reactions that generate exceptions to Le Châtelier’s principle (5) will yield mole fractions of one or more components for which η = 0 that are independent of T and P as well, since for ideal gases the corresponding αi are independent of T and P. For nonideal gases, however, α1 and α2 are, in general, functions of T and P. Thus, the value of y1 that yields η = 0, beyond which the reaction shifts in the direction that violates Le Châtelier’s principle, will be a function of T and P. How strongly dependent is the value of y1 at which η = 0 upon T and P is the focus of this section of the article. For the ammonia synthesis reaction, the value of y1 at which η = 0 will only be a function of T and P. The condition of chemical equilibrium, eq 4, imposes a constraint on the intensive variables T, P, y1, and y2 in that one of these variables becomes dependent upon the others. Once T, P, and y1 are chosen, the value of y2 is fixed by eq 4. Furthermore, eq 8, the relation that results from setting η = 0, imposes another constraint on the system, reducing further the degrees of freedom of the system. From the initial set of four intensive variables (T, P, y1, and y2), eqs 4 and 8 reduce the number of independent intensive variables from four to two. Thus, the value of y1 at which η = 0 will only be a function of T and P. This conclusion also applies to the ideal gas mixture. But for the ideal gas, all values of T and P yield y1 = 0.5 at η = 0. The investigation of the significance of nonideal effects requires that the fugacities of the components in the mixture be known. Various equations of state (1, 8, 9) can be used to generate expressions for the fugacities of each component. To keep our analysis somewhat simple, while still accounting for some degree of nonideal behavior, we make use of the fugacities generated from the truncated virial equation (8) that includes nonideal corrections only up to the second virial coefficient. More accurate (and therefore usually more complex) expressions for the fugacities appear in the literature (1, 9). Nevertheless, our chosen expressions, while accounting for some level of deviation from ideality, should at least qualitatively capture, if not quantitatively for some conditions, the relevant trends. In addition, the truncated www.JCE.DivCHED.org

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virial equation should provide a reasonable description of the properties of nitrogen and hydrogen (and most likely ammonia as well) at the temperatures and pressures we consider in our analysis. Using the truncated virial equation, the fugacity of component i in the mixture is given by (9)

ln f i = ln ( yi P ) +

P 2∑ yj Bij − Bmix RT j

(19)

where Bij is the second virial coefficient for the interaction between species i and j and is a function of temperature and Bmix =

∑ ∑ yi yj Bi j i

j

(20)

in which Bij = Bji. Both α1 and α2 can be evaluated using eq 19. The required partial derivatives are listed in the Supplemental Material.W The values of Bij were estimated from the Pitzer correlation (8). Furthermore, the determination of y2 via eq 4, once T, P, and y1 are chosen, requires that the equilibrium constant Ka(T ) be known at the given temperature. How Ka was determined at different temperatures, as well as the Pitzer correlation, are discussed in the Supplemental Material.W The ammonia synthesis reaction is run in commercial reactors whose operating conditions (3) are at pressures of about 350 bar and at temperatures from 350 C to 600 C. In our analysis, we considered a range of pressures between 0 bar and 1000 bar for temperatures between 500 K and 800 K and a range of pressures between 0 bar and 100 bar for temperatures of 298 K and 400 K. Since the critical temperature of ammonia is 405.7 K, we did not choose pressures that exceeded the critical pressure of ammonia, which is 112.80 bar, for temperatures less than 405.7 K. All the temperatures considered were above the critical temperatures of nitrogen and hydrogen. Hence, we neglected the possibility of condensation of any of the components in our analysis (a reasonable assumption for the chosen conditions). We did not, however, perform phase equilibrium calculations to confirm that none of the components condensed, particularly ammonia, at our chosen state points. The value of y1 at which η = 0 now designated as y–1, was determined directly from eq 8. For a given temperature and pressure, an initial guess of y1 was chosen. The equation of chemical equilibrium, eq 4, was then used to solve for the corresponding value of y2. Both y1 and y2 were then substituted into eq 15 to see if that equation was satisfied. If not, a new guess for y1 was chosen, whereby the new value of y2 was again generated via eq 4. This procedure was continued until eq 8 was satisfied. At this particular value of y1, or y–1, no reaction occurs upon the addition of an infinitesimal amount of nitrogen with the given temperature and pressure held fixed. When y 1 < y– 1 , the addition of an infinitesimal amount of nitrogen keeping the temperature and pressure fixed causes the reaction to shift to the right (as predicted by Le Châtelier’s principle), producing more ammonia. When y1 > y–1, the addition of an infinitesimal amount of nitrogen causes the reaction to shift to the left (opposite to the prediction of Le Châtelier’s principle).


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Figure 1 shows a plot of y–1 versus pressure for isotherms of T = 298.15 K, 400 K, 500 K, 600 K, 700 K, and 800 K. For all temperatures and pressures considered y–1 ≤ 0.5. As P → 0 (where ideal gas behavior is recovered), y–1 → 0.5 for all the chosen isotherms. Likewise, for all pressures, y–1 → 0.5 as T → ∞ (again as expected). Overall, the value of y–1 is sensitive to the system temperature and pressure. For example, for T = 298.15 K, y–1 reaches a value of 0.175 at P = 100 bar. Also, for T = 600 K, y–1 decreases from 0.5 at P = 0 bar to 0.163 at P = 1000 bar. Even at P = 500 bar and T = 600 K, a state point that is comparable to the conditions found in a commercial ammonia synthesis reactor (3), y–1 = 0.288 (a 42% deviation from the ideal gas result of 0.5). This value is much lower than 0.5, demonstrating the strong effects of nonideality on the behavior of the system. Each isotherm in Figure 1 also serves to divide the plot into two separate regions. Below each curve are the values of y–1 and P for which the reaction shifts to the right upon the addition of an infinitesimal amount of nitrogen. Above each curve, the reaction shifts to the left, in violation of Le Châtelier’s principle. The isotherms denote the point at which η = 0, or where no reaction occurs upon the addition of nitrogen. Since all the isotherms shown indicate that y–1 ≤ 0.5, the reaction shifts to the left over a much broader range of conditions than would be expected assuming only ideal gas behavior. Overall, Le Châtelier’s principle is violated more often than it is satisfied. One could even say that the reaction shifting to the right, consistent with Le Châtelier’s principle, is in fact the exception (and not the normal behavior of the reaction). Why y–1 is always less than or equal 0.5 can be more clearly seen from the following rearrangement of eq 8. Both α1 and α2 in eq 8 contain derivatives of the fugacities of each species in the mixture,5 which when evaluated and substi-

tuted into eq 8 yield the following expression 1 − 2 − Φ (T , P , y1° , y 2° ) = 0 y1°

(21)

The term in square brackets is simply the ideal gas term that also appears in eq 17. The remaining nonideal effects are included in Φ(T, P, y1, y2) where Φ (T , P , y1° , y 2 ° ) = −

2P RT

−5B13 − 3B23 + 4B33 + B11 + 3B12 + (9B13 + 3B23 − 6B33 − B11 − 3B12 ) y1° + (3B13 − 3B22 + 9B23 − 6B33 − 3B12 ) y2° (22) 2

+ (2B33 − 4B13 + 2 B11 ) ( y1° )

+ (2 B22 − 4B23 + 2B33 )( y 2 ° )2 + (4B 33 − 4B13 − 4B23 + 4 B12 ) y1° y 2°

Equation 21 provides an implicit equation for y–1, which can be rewritten in the following manner

y1° =

1 (23)

2 + Φ (T , P, y1° , y 2° )

For the given system of interest and state points considered, we found that Φ ≥ 0 (Figure 2), with Φ going to zero as the system approached ideal gas conditions (P → 0 or T → ∞). In addition, eq 23 indicates that for a given temperature and composition, Φ increases with an increase in pres-

ideal gas 0.5

(T

ⴥ)

800 K

500 K

4

0.4

y1°

700 K

400 K

3

Φ

y1°

0.3

600 K

0.2

600 K

2

y1°

298 K 500 K

0.1

700 K

1

y1°

0

0.0 0

100

200

300

400

500

600

700

800

900

0.1

1000

Figure 1. The initial mole fraction of nitrogen for which the reaction does not shift in any direction upon an infinitesimal addition of nitrogen, y–1, versus the system pressure along several isotherms. The ideal gas result of y–1 = 0.5 is independent of the temperature and pressure.

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0.2

0.3

0.4

0.5

0.6

0.7

0.8

y1°

Pressure / bar

•

Figure 2. Dependence of the function Φ (eq 22) on the initial mole fraction of nitrogen y1 and temperature T for a pressure P = 500 bar. The trends shown in the figure are similar to what is observed at other pressures. Also note that Φ ≥ 0. Values for y–1 are also marked on the figure for the systems shown.


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sure. Likewise, for a given pressure and composition, Φ increases with a decrease in the temperature (see Figure 2). Hence, y–1 ≤ 0.5, with y–1 decreasing (away from a value of 0.5) as the pressure increases or the temperature decreases. Although the second virial coefficients in eq 22 obtained with the Pizter correlation5 were, in general, both positive and negative, the dominate term at all temperatures was found to be B33, which models the ammonia–ammonia interaction. For the temperatures and pressures considered, B33 was always negative with a magnitude that was significantly greater than the other second virial coefficients. In fact, setting all Bij = 0 except for B33 in eq 22 yields values of y–1 (via eq 23) that are relatively close to those shown in Figure 1. In fact the largest error associated with only using B33 in the analysis was calculated to be 25%. Finite Additions of Nitrogen Figure 1 along with eq 6 and eq 8 are only valid for infinitesimal additions of nitrogen at constant T and P. Equation 4, however, is also valid for finite additions of nitrogen at fixed T and P. To determine how the reaction shifts upon addition of a finite amount of nitrogen, one must solve eq 4 and eq 5 directly to determine the value (and sign) of ξ´ for a given value of ∆´. The physically relevant value of ξ´ must be such that the final mole fractions of eq 5 lie between 0 and 1. For small to moderate values of ∆´, the physically relevant solution of eq 5 is therefore small. Equations 4 and 5 were solved by iteration with an initial guess of ξ´ = 0. Convergence to the one physical root was readily achieved. Values of ξ´ were generated for a range of ∆´ for various values of the initial mole fraction y1 at a given T and P.6 For the case of the ideal gas (7), the specific value of T and P is not required for the determination of ξ´. The ideal gas relations used for finite additions do not require that one specify the pressure and temperature explicitly. Figure 3 displays a plot of ξ´ versus ∆´ for different values of y1 at T = 500 K and P = 100 bar. The initial slope of each curve (based upon the infinitesimal addition of nitrogen, that is, ∆´ → 0) must be consistent with Figure 1. For

example, at T = 500 K and P = 100 bar, Figure 1 indicates that η = 0 as ∆´ → 0 for y1 = y–1 = 0.422. Thus, the initial slope of ξ´ is positive for y1 < 0.422, negative for y1 > 0.422, and equal to zero for y1 = 0.422. Figure 1, however, no longer applies to finite additions of nitrogen. As seen in Figure 3 for y1 ≥ 0.422, the reaction proceeds to the left, that is, ξ´ < 0, for any finite addition of nitrogen. The value of ξ´ is quite small, remaining so as ∆´ → 0. The reaction shifts to the left, but only by a relatively small extent, even if nitrogen is added in an amount equal to the total number of moles of all the species initially present (e.g., ξ´ ≈ 0.00271 for ∆´ = 1.0 and y1 = 0.422). For y1 < 0.422, each curve has an initial slope that is positive, as required by Figure 1. Each curve also displays a maximum, indicating that for sufficiently large ∆´, the curves eventually yield negative values of ξ´. For example, for y1 = 0.3, the reaction proceeds to the right for small additions of nitrogen, but shifts to the left when a sufficient quantity of nitrogen is added (∆´ > 0.671). The same trend is observed for y1 = 0.35, but now the reaction shifts to the left for ∆´ > 0.322. This critical value of ∆´, where ξ´ first becomes negative, approaches zero as y1 → 0.422 (where, as required by Figure 1, the initial slope is equal to zero). Although ξ´ is always positive for ∆´ ≤ 1 when y1 = 0.2, the appearance of a maximum suggests that the reaction will eventually proceed to the left (ξ´ < 0) if a large enough amount of nitrogen is added to the reaction vessel. Figure 4 also displays a plot of ξ´ versus ∆´ for different values of y1 but now at T = 600 K and P = 900 bar. At these conditions, the initial slope of ξ´ is zero for y1 = 0.181 (the more precise value is y1 = 0.1806). Similar trends as in Figure 3 are observed. Both figures imply that for all values of y1, and at all temperatures and pressures, the reaction will shift to the left when an appropriate amount of nitrogen is added to the system (in some cases, the amount of additional nitrogen needed to cause the reaction to shift to the left is quite large). This conclusion was also reached in ref 7 for the ideal gas mixture. The only difference between the current calculations and the ideal gas results is that the value of ∆´ required to yield a negative value of ξ´ is much less in the nonideal case than com-

3

3

0.2 2

2

0.05

0.3

ξ′ / 10ⴚ3

ξ′ / 10ⴚ3

1 1

0.35

0 −1

0 −1

0.1

−2 −3

0.422 0.6

−2

−4

−3 0.0

0.1

0.2

0.3

0.4

0.5

0.181 0.2

−5 0.6

0.7

0.8

0.9

1.0

−6

0.0

0.1

∆′

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

∆′

Figure 3. Extent of reaction, ξ´, versus the amount of nitrogen added, ∆´, at T = 500 K and P = 100 bar for different values of the initial mole fraction of nitrogen, y1.

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0.2

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Figure 4. Extent of reaction, ξ´, versus the amount of nitrogen added, ∆´, at T = 600 K and P = 900 bar for different values of the initial mole fraction of nitrogen, y1.


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pared to the ideal gas calculations (the precise amount being a function of the temperature and pressure). Conclusions If a mixture initially at chemical reaction equilibrium is subject to a perturbation away from the equilibrium state, the system, when reestablishing equilibrium, does not always respond in a way qualitatively consistent with the Le Châtelier principle. The addition of one reactant (N2), as in the ammonia synthesis reaction, may cause the reaction to proceed in a direction that produces more of the added ingredient (N2). The direction of the reaction depends on whether the added amount is infinitesimal or finite. These results are perfectly consistent with the laws of thermodynamics. The present analysis also illustrates the effects of nonideality on the determination of chemical reaction equilibrium. Although the qualitative trends obtained for the nonideal system are similar to the ideal gas case, the quantitative results are quite different. The current analysis provides an interesting example of the effects of nonideality along with an exception to Le Châtelier’s principle that instructors can apply towards the development of lectures or problems appropriate for students at various skill levels. These calculations also suggest ways for experimentally testing the predictions and demonstrating the thermodynamic laws. Acknowledgments Partial support of this work was obtained from an Academic Reinvestment Proposal, from the Purdue Research Foundation. DSC also acknowledges helpful discussions with E. Franses. WSupplemental

Material

Detailed derivations of the equations presented in this article are available in this issue of JCE Online.

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Notes 1. Le Châtelier’s principle is not violated when hydrogen or ammonia are added to the system at fixed temperature and pressure (2–5). 2. We note here that the principle of moderation does appear to be universally valid for finite changes in temperature and pressure (7). 3. Note that only three out of these four variables are independent, owing to the condition of chemical equilibrium. (See eq 4.) 4. One can show (1) that the change in chemical potential of each species is related to the sign of νi ξ. 5. See the derivations in the Supplemental Material.W 6. The final mole fraction of nitrogen will be different from y1.

Literature Cited 1. Tester, J. W.; Modell, M. Thermodynamics and Its Applications, 3rd ed.; Prentice Hall PTR: Upper Saddle River, NJ, 1997. 2. Levine, I. N. Physical Chemistry, 3rd ed.; McGraw-Hill Book Co.: New York, 1988. 3. Sandler, S. I. Chemical Engineering Thermodynamics, 3rd ed.; John Wiley & Sons, Inc.: New York, 1999. 4. de Heer, J. J. Chem. Educ. 1957, 34, 375–380. 5. Katz, L. J. Chem. Educ. 1961, 38, 375–377. 6. Liu, Z.-K.; Agren, J.; Hillert, M. Fl. Phase Equil. 1996, 121, 167–177. 7. Corti, D. S.; Franses, E. I. Chem. Eng. Educ. 2003, 37, 290– 295. 8. Smith, J. M.; Van Ness, H. C.; Abbott, M. M. Introduction to Chemical Engineering Thermodynamics, 6th ed.; McGraw Hill: New York, 2001. 9. Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria, 3rd ed.; Prentice Hall PTR: Upper Saddle River, NJ, 1999.


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The Ammonia Synthesis Reaction: An Exception to the Le ChÃ¢telier

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