Geometrical Description of Chemical Equilibrium and Le Châtelier's

Article Cite This: J. Chem. Educ. XXXX, XXX, XXX-XXX

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Geometrical Description of Chemical Equilibrium and Le Châtelier’s Principle: Two-Component Systems Igor Novak* Charles Sturt University, Orange, New South Wales 2800, Australia ABSTRACT: Chemical equilibrium is one of the most important concepts in chemistry. The changes in properties of the chemical system at equilibrium induced by variations in pressure, volume, temperature, and concentration are always included in classroom teaching and discussions. This work introduces a novel, geometrical approach to understanding the chemical system at equilibrium with the example of a simple twocomponent system and shows how the equilibrium changes under the influence of external perturbations. The paper discusses how thermodynamic factors (contained in the equilibrium constant K) and the conservation of mass principle govern the equilibrium state. The equilibrium and its changes are described using the geometrical representation in concentration space. The goal of this work is to help students better understand the basis behind the well-known Le Châtelier’s principle and equilibrium in general. KEYWORDS: First-Year Undergraduate/General, Upper-Division Undergraduate, Physical Chemistry, Continuing Education, Equilibrium

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beset with possible misconceptions.10 The LCP statements are often given in qualitative terms which raise ambiguities, thus leading to student misconceptions, especially when studying the behavior of open thermodynamic systems which contain equilibria. Advanced and rigorous thermodynamic analysis of open equilibria is possible in terms of the extent of reaction (ξ).4 However, such approaches are not suitable for general chemistry subjects. Nevertheless, the use of ξ has been recommended for the teaching of chemical equilibrium in order to avoid misconceptions and to obtain a degree of visualization of the behavior of equilibrium system. Even the reaction rate can be described in terms of ξ.8 This work presents a novel, geometrical approach to the teaching of chemical equilibrium that avoids the pitfalls typical for LCP, but which also relies on visualization and avoids a calculus based approach. In this way, we hope to present an approach that is complementary to existing ones.

INTRODUCTION The discussion of chemical equilibrium is a must in most general and physical chemistry courses starting from the secondary school level.1,2 This being the case, the chemical education literature describing teaching and learning of chemical equilibrium is very extensive. Nevertheless, although the fundamental, rigorous descriptions of chemical equilibrium have been reported,3,4 the learning and teaching of the concept of chemical equilibrium involve some pitfalls and difficulties. The graphical approaches to teaching and learning chemical equilibrium described to date include cartoon style diagrams of reactants and products, the use of ICE Tables, and the majority and minority species strategy.5 The very concept of chemical equilibrium and factors which govern the equilibrium state is often tied up and explained via the concept of the extent of reaction (ξ).6−8 While ξ is a well-defined quantitative descriptor, another principle which is usually invoked when dealing with chemical equilibrium is Le Châtelier’s principle (LCP). The principle itself is very general and can be stated in various forms.9 Here we use the following form: When a system at equilibrium is subject to perturbation the equilibrium always shifts in the direction that will partially relieve the effects of perturbation. LCP applies to all chemical systems which enhances its significance.9 The principle entails an open thermodynamic system and is inseparable from the understanding of factors which govern the existence of the equilibrium state in the first place. The thermodynamic criterion for the existence of chemical equilibrium state is (∂G/∂ξ)T,p = 0 from which the expression ΔG° r = −RT ln K, often mentioned in thermodynamics, can be derived. G is the Gibbs free energy and K the equilibrium constant. However, the use of LCP is © XXXX American Chemical Society and Division of Chemical Education, Inc.

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DISCUSSION We start by defining concentration space, which is a type of phase space. The concentration space is a set of points where the Cartesian coordinates of each point represent molar concentrations of reacting chemical species. Some of the points will represent equilibrium states of the system, but most will not. The changes in the reaction system (changes of concentration, volume, temperature) can then be represented by shifting the system from point to point in concentration space. In order for the point to be a valid representation of the Received: August 28, 2017 Revised: November 6, 2017

A

DOI: 10.1021/acs.jchemed.7b00665 J. Chem. Educ. XXXX, XXX, XXX−XXX

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Equation 4 is the equation of a straight line with constant slope −(FA/FB) and varying intercept (M/FBV), which depends linearly on M/V. The form of eq 4 reveals that, for a given M/V ratio, CB and CA must lie on a specific straight line. Varying the M/V ratio will produce straight lines that have the same slope (they will be parallel to each other), but different intercepts. Equations 1−4, 6, and 7 show that concentrations are determined by the M/V ratio, rather than by varying M and V independently. Geometrically speaking, changing M/V corresponds to moving the system (in concentration space) from the point on one line to the point on another, parallel line. For our two-component system we have

system, we assume that the reaction is slow compared to the time it takes to vary the components of the system (short mixing time) and also slow compared to the time it takes to uniformly change the temperature of the system (fast thermal equilibration). We also make an assumption that aqueous solutions are so dilute that the activity coefficients of individual components are close to unity or if components are gases that they behave ideally. We also assume that, because of low concentrations, the total volume V is determined by solvent alone so that V is constant and independent of particular chemical reactions. In general in chemical reactions in solution we may expect small changes of V to occur. Three constraints pertain to the equilibrium state and govern the accessibility of points in concentration space. The first constraint derives from the nature of chemical concentrations which must be positive, real numbers. The second constraint is derived from thermodynamics and is embedded in the equilibrium constant K. The third constraint (condition) is conservation of the total mass of reactants and products. We can express the conservation of mass in terms of molar concentrations of reactants and products as M ∑ CiFi = V i

KC =

2 (CA )eq

(5)

where the eq subscript denotes concentration in the equilibrium state. Combining eqs 4 and 5 and rearranging, we get quadratic eq 6, which allows us to obtain coordinates of the equilibrium point (CA and CB) in concentration space: 2 K C(CA )eq +

(1)

FA M (CA )eq − =0 FB FBV

(6)

We select the physically meaningful (positive) solution of eq 6, which is

where ci is the molar concentration of the ith component, M the total mass of reaction components, Fi the formula weight of the ith component, and V the total volume of the equilibrium system. If our equilibrium system is in the gas phase and all of its components are gases, we can write a similar expression (assuming ideal behavior for gases): MRT = ∑ pF i i V i

(C B)eq

(CA )eq =

⎡ F − FA + ⎢ B ⎣

2

( ) FA FB

2K C

+

1/2 4K CM ⎤ FBV ⎥ ⎦

(7)

Subsequently, a combination of eqs 4 and 7 allows us to calculate (CB)eq. The plot of (CB)eq versus (CA)eq using eq 5 gives a parabolic curve (Figure 1).

(2)

where pi is the partial pressure of the ith component, T the temperature, and R the universal gas constant. Equations 1 and 2 are polynomials that can, under the assumption of constant M, T, and V, be represented geometrically as straight lines in concentration space. Because of difficulties in graphical visualization of systems with three or more components (see Conclusion), we describe in this work only systems consisting of two components: one reactant and one product, for example, two ideal gases present in the system of constant temperature and volume (see the Application Example which follows).

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APPLICATION EXAMPLE An example of such a two-component reaction system would be 2A ⇌ B, but analogous reasoning would also be valid for any equilibrium system of the type aA ⇌ bB. Using molar concentrations of A and B and eq 1 we get M = CAFA + C BFB V

(3)

Figure 1. Concentration space for the reaction system 2NO2(g) ⇌ N2O4(g). Kc = 3; M/V in g/L.

Upon rearranging it becomes CB =

F M − A CA FBV FB

(4)

In order to illustrate our approach, we shall describe the numerical analysis of the simple system 2NO2(g) ⇌ N2O4(g), where CA = [NO2] and CB = [N2O4]. FA and FB then have values of 46 and 92 g/mol, respectively. We assume that V = 1 L and that KC = 3. Plugging these values into eq 7, we get

We could have used partial pressures for A and B and eq 2 instead of molarities (but assuming ideal gas behavior); the results would be analogous up to the constant factor RT/V since pi = niRT/V. B

DOI: 10.1021/acs.jchemed.7b00665 J. Chem. Educ. XXXX, XXX, XXX−XXX

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(CA )eq =

1 1 − 2 + ⎡⎣ 4 +

Article

12M ⎤1/2 92 ⎦

6

(8)

and when plugging FA and FB into eq 4 we get CB =

M 1 − CA 92 2

(9)

The diagram in Figure 1 shows the results for four different M/V ratios (in g/L). The equilibrium states (points) corresponding to these four ratios are C, E, G, and H. The dependence of CB on CA is evident in the diagram. Le Châtelier’s principle can now be nicely illustrated by inspection of the points C, D, E, F, and G. If we perturb the system in its initial equilibrium state C by increasing concentration of A (NO2), we reach nonequilibrium point D. The system subsequently reverts to the new equilibrium state at point E following the trajectory in concentration space along the straight line (arrow) from D to E. The system travels along this trajectory by converting some (but not all) of the excess A (NO2) into B (N2O4). In the language of Le Châtelier’s principle, the equilibrium state shifts to the right, that is, toward the product B (N2O4). This is the direction required to relieve the perturbation introduced by increasing the concentration of NO2 (A). If, on the other hand, we decrease the concentration of NO2 in the initial equilibrium state C, we reach the nonequilibrium point F. From this point the system reverts to the new equilibrium state G along the straight-line trajectory (arrow) connecting F and G. In this case, some (not all) of B (N2O4) is converted to A (NO2). We say that the equilibrium shifts toward the reactants, to the left as predicted by Le Châtelier’s principle. Analogous reasoning can be employed for the case of varying concentrations of product B (N2O4) or any combined variation in concentrations of A and B. A point to note is that, upon perturbation, the system evolves toward the new equilibrium state, not the old, initial one. The temperature change also influences the equilibrium system. The value of Kc changes with temperature depending on ΔH of the reaction as is well-known.11 In this example, we have taken experimental KP values of the 2NO2(g) ⇌ N2O4(g) system 12 and converted them to Kc at two different temperatures (using KP = KC(RT)Δn expression). Kc values at 298 and 350 K are 162.12 and 6.35 L/mol, respectively. We then solve the system of three equations for M/V = 1.288 g/L. 162.12 =

6.35 =

Figure 2. Effect of temperature change on the reaction system 2NO2(g) ⇌ N2O4(g) shown in concentration space as a plot of [NO2] vs [N2O4]. The curves represent eqs 10 and 11 while the M/V line represents eq 12.

[NO2 ] = 0.0219 and [N2O4 ] = 0.00304 at 350 K

The 2NO2(g) ⇌ N2O4 reaction is exothermic with ΔH°r = −57.2 kJ/mol. Le Châtelier’s principle predicts that an increase in temperature leads to a decrease in [N2O4] and an increase in [NO2] that was, indeed, observed in the geometric analysis (Figure 2). The concentration space diagram in Figure 2 shows what happens to the equilibrium system when there is a temperature change imposed on the system. The system is initially in the equilibrium state at the black point on the T = 298.16 K curve. After the temperature is increased, the system follows trajectory in concentration space indicated by the arrow until it reaches the new equilibrium state represented by the black point on the T = 350 K curve. The total mass of reaction components remains the same during the temperature perturbation so there is now only a single straight line present in the diagram. The reverse process, N2O4(g) ⇌ 2NO2(g), is endothermic so the arrow in Figure 2 would lie on the same line but point in the opposite direction. The relevance of our approach to teaching and learning chemistry is outlined below. First, the diagrams in Figures 1 and 2 show clearly what happens to the system after perturbation by identifying the exact trajectory of the system in concentration space. The detailed postperturbation events are not discussed in standard chemistry textbooks nor is the fact emphasized that system must revert to the equilibrium state if the perturbation is not continuous. However, the system does not evolve toward the same equilibrium state (point) as before like the mechanical “coiled spring” analogy implies. The system does not “bounce back”; rather, it “bounces forward” and goes to the new equilibrium state (point)! The system evolves following the fundamental and general laws of thermodynamics and the conservation of mass. Second, the diagram clarifies qualitative and ambiguous statements such as “If we increase the concentration of reactants, the equilibrium shifts to the right, i.e., toward

[N2O4 ] [NO2 ]2

(10)

[N2O4 ] [NO2 ]2

46[NO2 ] + 92[N2O4 ] = 1.288

(11) (12)

The equilibrium concentrations are given in Figure 2 as intersections between the line representing conservation of mass and the two curves representing dependence of [NO2] on [N2O4] and Kc at two temperatures. The coordinates of the intersection points in concentration space (equilibrium concentrations) are [NO2 ] = 0.00788 and [N2O4 ] = 0.0100 at 298.16 K

and C

DOI: 10.1021/acs.jchemed.7b00665 J. Chem. Educ. XXXX, XXX, XXX−XXX

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ACKNOWLEDGMENTS The author thanks Charles Sturt University for financial support of this work and the referees for excellent suggestions.

products”. Increasing the concentration of reactants leads us from the initial equilibrium point C to the nonequilibrium point D. What happens after this perturbation is that we end up at the new equilibrium point E. Point E, indeed, has a larger concentration of product B and smaller concentration of reactant A than before the perturbation, as predicted by Le Châtelier’s principle. However, the equilibrium does not “shift”; it is re-established under different conditions; the system has evolved and adapted.

REFERENCES

(1) Ghirardi, M.; Marchetti, F.; Pettinari, C.; Regis, A.; Roletto, E. Implementing an Equilibrium Law Teaching Sequence for Secondary School Students To Learn Chemical Equilibrium. J. Chem. Educ. 2015, 92, 1008−1015. (2) Ghirardi, M.; Marchetti, F.; Pettinari, C.; Regis, A.; Roletto, E. A Teaching Sequence for Learning the Concept of Chemical Equilibrium in Secondary School Education. J. Chem. Educ. 2014, 91, 59−65. (3) Silverberg, L. J.; Raff, L. M. Are the Concepts of Dynamic Equilibrium and the Thermodynamic Criteria for Spontaneity, Nonspontaneity, and Equilibrium Compatible? J. Chem. Educ. 2015, 92, 655−659 and references therein.. (4) Solaz, J. J.; Quilez, J. Changes of Extent of Reaction in Open Chemical Equilibria. Chem. Educ. Res. Pract. 2001, 2, 303−312. (5) Davenport, J. L.; Leinhardt, G.; Greeno, J.; Koedinger, K.; Klahr, D.; Karabinos, M.; Yaron, D. J. Evidence-Based Approaches to Improving Chemical Equilibrium Instruction. J. Chem. Educ. 2014, 91, 1517−1525. (6) Moretti, G. The ‘‘Extent of Reaction’’: A Powerful Concept To Study Chemical Transformations at the First-Year General Chemistry Courses. Found. Chem. 2015, 17, 107−115. (7) Vandezande, J. E.; Vander Griend, D. A.; DeKock, R. L. Reaction Extrema: Extent of Reaction in General Chemistry. J. Chem. Educ. 2013, 90, 1177−1179. (8) Schmitz, G. What Is a Reaction Rate? J. Chem. Educ. 2005, 82, 1091−1093. (9) Gündüz, Ö .; Gündüz, G. The Phase Space Interpretation of the Le Châtelier−Braun Principle and Its Generalization as a Principle of Natural Philosophy. Phys. Essays 2014, 27, 404−410. (10) Cheung, D. The Adverse Effects of Le Châtelier’s Principle on Teacher Understanding of Chemical Equilibrium. J. Chem. Educ. 2009, 86, 514−518. (11) Vargas, F. M. A Simple Method To Calculate the Temperature Dependence of the Gibbs Energy and Chemical Equilibrium Constants. J. Chem. Educ. 2014, 91, 396−401. (12) Hisatsune, I. C. Thermodynamic Properties of Some Oxides of Nitrogen. J. Phys. Chem. 1961, 65, 2249−2253. (13) Borge, J. Reviewing Some Crucial Concepts of Gibbs Energy in Chemical Equilibrium Using a Computer-Assisted, Guided-ProblemSolving Approach. J. Chem. Educ. 2015, 92, 296−304. (14) Hanson, R. M. A Unified Graphical Representation of Chemical Thermodynamics and Equilibrium. J. Chem. Educ. 2012, 89, 1526− 1529. (15) Weltin, E. Are the Equilibrium Concentrations for a Chemical Reaction Always Uniquely Determined by the Initial Concentrations? J. Chem. Educ. 1990, 67, 548. (16) Powers, J. M.; Paolucci, S. Uniqueness of Chemical Equilibria in Ideal Mixtures of Ideal Gases. Am. J. Phys. 2008, 76, 848−855 and references therein..

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CONCLUSION Representations of chemical equilibrium often do not involve graphical presentations. Some representations showing the dependence of the amounts of individual reaction components on the extent of reaction or the relationship between thermodynamic functions have been described in chemical education literature.13,14 Graphical representations are useful to students because they visualize functional dependences among the parameters that describe the system, especially if these dependencies are nonlinear. The graphical description of chemical equilibrium, quantitative analysis of Le Châtelier’s principle, and explicit introduction of conservation of mass is, to the best of the author’s knowledge, novel. By following this approach, students become aware that chemical equilibrium is governed by two fundamental principles: laws of thermodynamics (in the form of K), and the conservation of mass. Another novelty introduced here is that students can follow trajectory in the concentration space, that is, follow in detail how the perturbed equilibrium system relaxes back into the equilibrium state, albeit not the same as the initial one. The aim was not to replace the existing educational approaches to chemical equilibrium (extent of reaction, M&M strategy, or rigorous thermodynamic analysis); rather, it was to provide a novel perspective that adds clarity and general validity. This work discussed two-component systems only. For higher component systems, geometrical representation becomes unfeasible because curves are replaced by hypersurfaces and straight lines by hyperplanes. For example, for threecomponent reactions such as A + B → C, the equilibrium concentrations are points embedded on a 3D surface, and the straight line expressing the conservation of mass becomes a plane in the three-dimensional concentration space. For fourcomponent reactions such as A + B → C + D, everything happens in hyperspace where geometrical visualization is impossible; a hypersurface now embeds the equilibrium points, and a hyperplane describes the conservation of mass. As the number of components increases the question arises whether the hyperplanes and hypersurfaces will still share only a single common point (locus). They will, because of the proven assertion that a unique chemical equilibrium state exists under a given set of conditions (amounts of components, temperature, pressure, volume).15,16

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Igor Novak: 0000-0002-3413-2605 Notes

The author declares no competing financial interest. D

DOI: 10.1021/acs.jchemed.7b00665 J. Chem. Educ. XXXX, XXX, XXX−XXX