In the Classroom
Why Equilibrium? Understanding the Role of Entropy of Mixing Mary Jane Shultz Department of Chemistry, Pearson Laboratory, Tufts University, Medford, MA 02155;
[email protected] Equilibrium is a major topic in both introductory and physical chemistry curricula. This paper grew out of a problem in conveying a fundamental understanding of this important concept to students. Typical introductory courses begin with a discussion of reactions that go to completion, for example, Na with water or displacement of one metal by another. These reactions are complete because one metal (or hydrogen) is more active than another. Then students are introduced to equilibrium: reactions that produce a mixture of reactants and products. Forces that lead to reactions going to completion are put aside and a new quantity, the equilibrium constant, is introduced. Students become expert at utilizing the equilibrium constant via numerous equilibrium calculations, including acid–base, solubility, complex ion formation, and buffer problems. However, the factors that give rise to equilibrium are often not addressed. Physical chemistry courses focus on the connection between equilibrium and thermodynamic quantities, particularly the Gibbs free energy. Equilibrium is recognized as a minimum in the Gibbs free energy vs reaction progress curve for reactions at constant temperature and pressure (1–3). However, the origin of that minimum remains largely unaddressed, although the recent edition of the text by Atkins recognizes the importance of mixing (2). The enthalpic and entropic contributions (the source of the minimum) are not separated; thus an opportunity is missed to reinforce the concept of entropy—specifically entropy of mixing—and its importance. We have found the following example, the dimerization of NO2, to be very helpful in making these abstract contributions more concrete. Depending on local interest, other reactions can be substituted. The two essential characteristics that enable explicit calculation of the Gibbs free energy minimum as well as equilibrium concentrations are that the reaction takes place in the gas phase and that ∆G θ for the reaction is relatively small. It helps simplify the calculation if the number of reactants and prod-
Table 1. Thermodynamic Data at 25 8C Molecule
∆Gfθ/ kJ mol {1 ∆Hfθ/ kJ mol {1
S θ/J K {1 mol {1
NO2(g)
51.31
33.18
240.06
N2O4(g)
97.89
9.16
304.29
NOTE: Data from ref 6.
ucts is also small. The dimerization of NO2 fulfills all these requirements and has the added benefit of being familiar to many students either from laboratory exercises (4, 5) or lecture demonstrations. A moderate ∆G θ is important because it results in an equilibrium constant on the order of unity and an easily identifiable minimum in the plot of ∆G versus reaction progress. Thermodynamic data for dimerization of NO2 are given in Table 1. Notice that the reaction’s ∆G θ, {4.73 kJ,1 is moderate. Both reactants and products are gases, and to a good approximation, they may be treated as ideal gases as long as the total pressure is on the order of an atmosphere or less. In this case, all the components of ∆G can be calculated explicitly and the origin of the minimum is readily identified. Imagine the reaction as follows. The initial stage consists of two moles of NO2 at one atmosphere pressure in the left half of a cylinder divided by a transforming partition. The transforming partition merely changes 2NO2 into N2O4 with no other change. The completion stage has one mole of N2O4 in the right half of the cylinder at one atmosphere pressure. Any intermediate stage has 2 – 2x moles of NO2 on the left and x moles of N2O4 on the right, both at one atmosphere. As shown below, the equilibrium balance of NO2 and N2O4 occurs when x = 0.82. Equilibrium actually occurs when the partition is removed and the gases mix. This is illustrated in Figure 1, which also lists results discussed below.
Figure 1. Schematic of system discussed in text.
JChemEd.chem.wisc.edu • Vol. 76 No. 10 October 1999 • Journal of Chemical Education
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In the Classroom
S nθonmixing(x) = [2 × 240.06(1 – x) + 304.29x] J K{1 (2) Since ∆G = ∆H – T∆S, entropy as a function of reaction progress is multiplied by minus T, {T S nθonmixing as a function of reaction progress is plotted in Figure 3. The sum of two linear functions is also linear. Thus the nonmixing portion of ∆G is a linear function of reaction progress, as shown in Figure 4, and without mixing, reactants simply slide down the ∆G slope to products and there is no equilibrium. This picture changes when mixing is added. The enthalpy change contains no contribution from mixing because reactants and products are ideal gases, for which ∆Hmixing = 0. However, the entropy of mixing is nonzero. Entropy of mixing is a classic quantity calculated in most physical chemistry texts and is (3)
where n is the total number of moles of gas and XA and XB are the mole fractions of gases A and B. To apply eq 3 to the problem at hand requires expressing the number of moles of NO2 and N2O4 in terms of reaction progress n = nNO2 + nN2O4
(4)
nN2O4 = x; nNO2 = 2 – 2x; n = 2 – x
(5)
The entropy of mixing versus reaction progress is ∆S mixing(x) = 2 – x R
2 – 2x 2 – 2x ln + x ln x 2–x 2–x 2–x 2–x
0 0.00
0.50
1.00
Reaction Progress / mol N2O4 Figure 2. Enthalpy as a function of reaction progress for dimerization of NO2. -80
–T x (Snonmix) / kJ
∆Hinter(x) is plotted in Figure 2. This ∆Hinter is ∆Hf of the components of the mixture, not ∆H for conversion of some fraction of reactants to products: ∆Hreaction = ∆H(1) – ∆H(0). Notice that ∆Hinter is a linear function of reaction progress. The entropic contribution to reaction Gibbs free energy is separated into two parts: that due to the difference in the entropy of the reactants and the products, and that due to mixing. Like enthalpy, the nonmixing contribution to the entropy is a linear function of reaction progress.
∆Smixing = { nR(XA ln XA + XB ln XB)
35
(1)
-100
-120
-140
-160 0.00
0.50
1.00
Reaction Progress / mol N2O4 Figure 3. Nonmixing contribution to reaction entropy × ({T ) (298.15 K) as a function of reaction progress.
-75
∆Gnonmix / kJ
∆Hinter(x) = [2 × 33.18(1 – x) + 9.16x]kJ
70
∆Hinter / kJ
Since the molecules of an ideal gas do not interact, ∆Hmixing = 0 and the nonmixing enthalpy change is a linear function of reaction progress. Specifically, letting x be the reaction progress (i.e., the number of moles of N2O4), x = 1 corresponds to complete conversion of two moles of NO2 to one mole of N2O4. Then enthalpy as a function of reaction progress is
-80
-85 0.00
0.50
1.00
Reaction Progress / mol N2O4 Figure 4. Nonmixing ∆G as a function of reaction progress.
(6)
1392
–T x ∆Smixing / kJ
0
This function times {T is plotted in Figure 5. Two observations are in order. First, {T∆Smixing is not very large. Second, the position of the minimum in {T∆Smixing depends on the number of moles of reactants versus products—that is, the reaction stoichiometry. In this example, the number of moles of reactants is greater than the number of moles of products; hence the minimum is on the product side (>50% completion). Addition of {T∆Smixing to the nonmixing ∆G results in a minimum in the Gibbs free energy as a function of reaction progress, as shown in Figure 6. With this plot, students easily see the minimum in the Gibbs free energy. The minimum occurs at about 0.8 mol N2O4 produced. This is a very concrete illustration of the statement that chemical reactions
–2
–4 0.00
0.50
1.00
Reaction Progress / mol N2O4 Figure 5. Mixing contribution to reaction entropy × ({T ) (298.15 K) as a function of reaction progress.
Journal of Chemical Education • Vol. 76 No. 10 October 1999 • JChemEd.chem.wisc.edu
In the Classroom -75
∆G / kJ
Minimum -80
-85 0.00
0.50
1.00
Reaction Progress / mol N2O4 Figure 6. Reaction Gibbs free energy vs reaction progress for dimerization of NO2.
roll to a minimum in the chemical potential just as a ball comes to rest at a minimum potential energy. This visual conclusion about equilibrium can be confirmed by calculating the actual equilibrium concentrations. From ∆G θ = {RT ln Keq
(7)
the equilibrium constant is 6.74. Note that Keq is KP, since the ∆G θ are for gases. Solving for the number of moles at equilibrium, nNO2 = 0.36 and nN2O4 = 0.82 at equilibrium, just as determined from the graph. NOTE: For the mathematically inclined, the equilibrium amounts can also be calculated explicitly by differentiating ∆G with respect to x and setting the result equal to zero (equilibrium condition): θ
θ
θ
0 = d∆G = ∆H – T ∆S + RT ln KP = ∆G + RT ln KP (8) dx Doing this explicit calculation with an actual reaction makes the concept of equilibrium more concrete. Furthermore,
the concept of mixing entropy and its role in equilibrium becomes more tangible. To cement these concepts, students are asked to produce graphs such as those in Figures 2–6 at other temperatures. The enthalpy plot is unchanged, the nonmixing entropy slope increases with temperature, and the depth of the minimum in the mixing entropy plot increases. The net result is that the minimum slides toward the reactants, consistent with the exothermicity of this reaction. We have found that this calculation for a real system helps students understand entropy of mixing and the origin of equilibrium. These concepts are further reinforced if students both determine the temperature dependence of the equilibrium constant in the laboratory and graphically evaluate the minimum in a plot of the Gibbs free energy as a function of reaction progress at several temperatures. Note 1. From ∆G θ = ∆H θ – T∆S θ, ∆G θ = 4.78 kJ/mol. The difference between this calculated ∆G θ and the reported value reflects uncertainty in the experimental measurements. See ref 6.
Literature Cited 1. Alberty, R. A.; Silbey, R. J.; Physical Chemistry, 2nd ed.; Wiley: New York, 1997. 2. Atkins, P. Physical Chemistry, 5th ed.; Freeman: New York, 1994. 3. McQuarrie, D.; Simon, J. Physical Chemistry: A Molecular Approach; University Science Books: Sausalito, CA, 1997. 4. Shooter, D. J. Chem. Educ. 1993, 70, A133–A140. 5. Hennis, A. D.; Highberger, C. S.; Schreiner, S. J. Chem. Educ. 1997, 74, 1340–1342. 6. Wagman, D.; Evans, W. H.; Parker, V. B.; Schumm, R. H.; Halow, I.; Bailey, S. M.; Churney, K. L.; Nuttall, R. L. The NBS Tables of Chemical Thermodynamic Properties: Selected Values for Inorganic and C1 and C2 Organic Substances in SI Units; American Chemical Society and the American Institute of Physics: Washington, DC, 1982; Vol. 11.
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