In the Classroom
Campbell‘s Rule for Estimating Entropy Changes in Gas-Producing and Gas-Consuming Reactions and Related Generalizations about Entropies and Enthalpies Norman C. Craig Department of Chemistry, Oberlin College, Oberlin, OH 44074-1083;
[email protected] Of the subjects that all chemists learn, thermodynamics is the least used in qualitative reasoning. Structural concepts, mechanisms, molecular orbital applications, spectroscopic interpretations, and kinetic analyses are typically more widely used. The likely reason for the deficit in using thermodynamics is that students are impressed by the complexities of thermodynamics associated with doing accurate calculations and fail to learn how to apply semiquantitative reasoning. The mystery surrounding entropy is also an impediment and is often compounded by casual talk about spatial disorder, which obscures the centrality of thermal effects on entropy. Making good estimates of thermodynamic quantities not only broadens the application of thermodynamics, but doing so makes the subject more accessible and useful to a wide range of chemists. Estimates often provide insights into why a process occurs or does not occur. The main purpose of this article is to describe several useful estimates of entropies of reaction and to identify patterns in molar entropies. Working with various entropies also helps dispel any mystery about entropy. We will also give some attention to estimating enthalpies for gas-phase reactions. When doing entropy analyses of chemical reactions, whereby the entropy contributions to the change in total entropy, ∆Stot, are considered (1, 2),1 it is especially helpful to be able to make estimates of entropies. Because of systematic relationships among entropies, good estimates can be made (2, 3). One useful new rule for estimating an entropy change in a chemical reaction under standard state conditions, ∆rS ⬚, may be called Campbell’s rule in honor of J.
Arthur Campbell, who first called attention to the essence of this generalization in a paper in this Journal (4).2 Campbell was content to relate the sign of ∆rS ⬚ to the net change in moles of gas in a reaction and did not propose a numerical value. Davies was one of the first authors to emphasize the qualitative relationship between the sign of ∆rS ⬚ and the extent of gas formation (5).3 Campbell’s Rule Campbell’s rule says that the entropy of reaction, ∆rS ⬚, can be estimated as 140 J(K mol rxn) for each net mole of gas produced in the balanced chemical equation. If net moles of gas are consumed, then the entropy change is ᎑140 J(K mol rxn) for each net mole of gas consumed. This rule is true because gases have large entropies compared to solids and liquids. The entropies of gases are dominated by the translational contribution, a characteristic that is shared roughly equally by all gases. As far as contributions of other degrees of freedom to entropies go, these tend to cancel between the gases and the species from which they are produced under the stoichiometric constraints of a reaction. Entropies of reaction for selected examples of reactions with different numbers of net moles of gas formed or consumed are shown in Table 1.4 The column labeled, ∆ng, gives the net change in moles of gas for the reaction as written. The last column gives ∆rS ⬚∆ng, the Campbell’s rule ratio, for each example. Of course, we recognize that ∆rS ⬚ ≈ 0 for reactions for which ∆ng is zero and do not divide by zero for
Table 1. Standard Entropies of Reaction at 298 K as a Function of the Net Change in Moles of Gas ∆rS ⬚/ J (K mol rxn)᎑1
∆ng / mol
(∆rS ⬚/∆ng )/ J (K mol rxn)᎑1(mol)᎑1
CaCO3(c) → CaO(c) + CO2(g)
160.6
1
16 1
H2(g) + 1/2O2(g) → H2O(g)
᎑ 44.4
᎑ 0.5
89
0
–––
Reaction
C(c) + O2(g) → CO2(g) H2O2(l) → H2O(g) + 1/2O2(g) C(c) + 2F2(g) → CF4(g)
2.9 186.8 ᎑ 149.9
1.5 ᎑1
121 150
Fe(c) + 5CO(g) → Fe(CO)5(l)
᎑ 677.7
᎑5
135
CH4(g) + 4Cl2(g) → CCl4(l) + 4HCl(g)
᎑ 114.7
᎑1
115
2C(c) + 3H2(g) + 1/2O2(g) → C2H5OH(l)
᎑ 345.4
᎑ 3.5
99
᎑ 97.4
᎑ 0.5
195
HCCH(g) + 5/2O2(g) → 2CO2(g) + H2O(g) HgO(c) → Hg(l) + 1/2O2(g)
103.4
0.5
207
NH4NO3(c) → N2O(g) + 2H2O(l)
208
1
20 8
NOte: Data are from Appendix D in ref 2.
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Journal of Chemical Education • Vol. 80 No. 12 December 2003 • JChemEd.chem.wisc.edu
In the Classroom
these cases. From the examples given in Table 1 we see that the numerical value for Campbell’s rule is good to about ±60 J (K mol rxn)⫺1(mol)⫺1 (±40%). For reasons discussed in the section “Other Generalizations”, it is best not to apply Campbell’s rule to reactions involving net formation or consumption of aqueous ions. For the most part Table 1 gives rather different examples than those cited in Campbell’s paper. The average Campbell’s rule value for the eighteen reactions with ∆ng ≠ 0 is his table is 136 J (K mol rxn)⫺1(mol)⫺1. The average value for the ten reactions with ∆ng ≠ 0 in Table 1 is 148 J (K mol rxn)⫺1(mol)⫺1. Thus, we choose the rounded value of 140 J (K mol rxn)⫺1(mol)⫺1 for the rule.5 Relationship between Campbell’s Rule and Trouton’s Rule A well-known rule for estimating entropies of vaporization is Trouton’s rule. According to Trouton’s rule, the entropy of vaporization is about 90 J(K mol) for a normal liquid vaporizing at the normal boiling point to give one atmosphere of vapor pressure. Trouton’s rule is quite good and gives useful estimates of enthalpies of vaporization from boiling points, which are widely available. This mode of estimation is illustrated in general and physical chemistry textbooks.6 Why is the Campbell’s rule value greater in magnitude than the Trouton’s rule value for each mole of gas formed, and why does Campbell’s rule have a larger range of uncertainty than Trouton’s rule? In chemical reactions, when gases are produced or consumed, they arise from changes in chemical composition and the production of new species. As a consequence, structural changes in addition to gas production or consumption are reflected in the entropy change. The entropy change per mole is greater, and the variation of entropies of different gases enters in. Nonetheless, Campbell’s rule is sufficiently general to permit making semiquantitative estimates of ∆rS ⬚ and most certainly the sign of this term unless ∆ng = 0. Patterns in Molar Entropies Other patterns in molar entropies and entropies of reaction are discernible. It is a useful exercise to provide students with an extensive table of molar entropies and ask them to find patterns among molar entropies (1, 2). As starting points, students should consider the effect of phase, of molecular complexity, of mass, and of bond strength.7 The effects of shape and of unpaired electrons are more subtle considerations that a few students may discover.8 All molar entropies and entropies of reaction cited in this article are for 25 ⬚C. If no phase change occurs and no ions in solution are involved, entropies of reaction are a weak function of temperature above about 250 K. One outcome of the search through the table (2) is that entropies increase significantly as a substance goes from solid to liquid to gas. The change from liquid to gas is the dominant change as is reflected by the 90 J(K mol) value of Trouton’s rule. Other comparisons need to be restricted to species in the same phase and with the same type of bonding, that is, simple molecular, macromolecular, or ionic. A second generalization is that entropy increases with increase in molecular complexity. Thus, molar entropy increases in
the series of increasingly complex gases, in J(K mol): Kr, 164.1; Cl2, 223.1; SO2, 248.2; and SiF4, 282.5. We have kept the mass approximately constant in this series. The effect of mass on molar entropies is illustrated by the sequence of noble gases, in J(K mol): He, 126.2; Ne, 146.2; Ar, 154.8; Kr, 164.1; and Xe, 169.7. Thus, entropy increases with mass. The effect of changing bond strength on entropies is harder to assess because changes in bond strength are associated with changes in elements and thus entangled with changes in mass. However, the effect of a change in bond strength can be uncovered by considering molar entropies of a series solids in the same family in the periodic table, in J(K mol): C (graphite), 5.7; Si, 18.8; Ge, 31.1; Sn (white), 51.6; and Pb, 64.8. An alternative illustration is seen in molar entropies of a series of solid compounds with a common element (in JK mol): LiCl, 59.3; NaCl, 72.1; KCl, 82.6; RbCl, 95.9; and CsCl, 101.2. In general, the entropies in these series increase faster than would be expected for the entropy increase owing to mass as reflected in the noble gas series.9 Thus, an additional entropy increase due to bond weakening, as the atoms get larger, must also be occurring. There are underlying quantum mechanical and molecular reasons for the patterns in molar entropies. If the students have the background to appreciate these effects, these reasons should be explored (1, 2, 6). An important consequence of discovering patterns in molar entropies is to conclude that we can make very good estimates of molar entropies by keeping the phase, molecular complexity, mass, and type of bonding as similar as possible to those of a more widely known species. Thus, we expect the molar entropy of the reactive intermediate CF2(g) to be close to the molar entropy of OF2(g). The values are 241 and 247 J(K mol), respectively. The entropy of CuCl(c), a relatively uncommon material, is expected to be close to the entropy of KCl(c). The values are 86 and 83 J(K mol), respectively. Other Generalizations about Entropies of Reaction Some generalizations about entropies of reaction in addition to Campbell’s rule are worth noting. These are more qualitative than the patterns described so far. However, knowledge of the sign of an entropy change is useful in understanding chemical processes and in making qualitative predictions. For a liquid phase reaction in which a molecular substance dissociates into molecules, ∆rS ⬚ is positive. The dissociation of N2O4 in carbon tetrachloride as an example is given in Table 2. It is very important not to extend the generalization about the production of molecules by dissociation of molecules to solutions involving the production or consumption of ions. The solvation of ions, which causes an entropy decrease associated with “freezing” solvent molecules around the ions, is a dominant effect. Thus, when hydronium ions and acetate ions form by dissociation of acetic acid in water, ∆rS ⬚ is significantly negative, as shown in Table 2. Indeed, this negative ∆rS ⬚ term is the principal reason that aqueous acetic acid is a weak acid near room temperature. The entropy change in the thermal surroundings, ∆Sθ, is nearly zero at 298 K because ∆rH ⬚ is nearly zero. Overall, ∆Stot is negative, and the equilibrium constant is small.10 Thus, when ions are
JChemEd.chem.wisc.edu • Vol. 80 No. 12 December 2003 • Journal of Chemical Education
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In the Classroom Table 2. Signs and Magnitudes of Various Entropies of Reaction ∆rS⬚a/ J (K mol rxn)᎑1
Reaction N2O4(CCl4 soln) → 2NO2(CCl4 soln) HOAc(aq) + H2O(l) → H3O+(aq) + OAc–(aq) NH4+(aq) + H2O(l) → H3O+(aq) + NH3(aq)
133b ᎑ 92.1 ᎑ 2.1
H2C=CH2(g) → 1/n(–CH2–CH2–)n(s)
᎑ 166c
H2C=CH2(l) → 1/n(–CH2–CH2–)n(s)
᎑ 76d
a
Unless otherwise indicated, the data are from Appendix D in ref 2.
b
Ref 8.
c
Estimated from the entropies of ethylene and octadecane(c). The entropy of octadecane came from ref 9. d Estimated from Trouton's rule and the previous estimate for the reaction with gaseous ethylene.
phase to the liquid phase to the gas phase. Molar enthalpies increase with temperature. However, thermal enthalpies of the reactants and products almost cancel out in the enthalpy of reaction. Thus, the enthalpy of reaction is largely a reflection of the differences in bond strengths and intermolecular interactions in reactants and in products. In addition, ∆rH ⬚ is so weakly dependent on temperature that it may be regarded as a constant. The principal area for seeing patterns in molar enthalpies is in bond dissociation enthalpies (2). Furthermore, these patterns and approximate values learned for a few bond dissociation enthalpies may be used to estimate the enthalpy of reaction for gas phase reactions. Unfortunately, two units for enthalpies, kcalmol and kJmol, remain in wide use—a troublesome complication when learning these quantities. We suggest learning: ∆H ⬚(H2) ≈ ∆H ⬚(HCl) ≈ ∆H ⬚(OH) ≈ 450 kJmol
produced in aqueous solution, ∆rS ⬚ is significantly negative. ∆rS ⬚ is even more negative when the ions produced have charges greater than one. A different example of the effect of ions on entropies of reaction is the dissociation of aqueous ammonium ions to form hydronium ions. In this case the reaction involves no change in number of ions and little change in ionic character. Thus, ∆rS ⬚ ≈ 0 as shown in Table 2. The reason that the ammonium ion is a weak acid in water at 298 K is that the formation of hydronium ions from ammonium ions is endothermic, and consequently ∆Sθ is substantially negative and so is ∆Stot. We see that the reason for weak acidity of ammonium ions is quite different from the reason for weakness of carboxylic acids. This discussion about the effect of solvation can be extended to an understanding of the chelate effect, in which complex ions formed from multidentate ligands are favored over complex ions formed with monodentate ligands (2). Another qualitative generalization about entropies of reaction applies to addition polymerization processes. When, for example, polyethylene forms from ethylene monomers, ∆rS ⬚ must be significantly negative, as indicated in Table 2. Because a gas is consumed, this example is consistent with Campbell’s rule. Were a polymerization to occur wholly within the liquid phase, ∆rS ⬚ would also be less than zero as is estimated in Table 2 for the polymerization of ethylene. Thus, favorable addition polymerization processes involving gaseous or liquid monomer have ∆Stot greater than zero by virtue of a substantially exothermic reaction, which makes ∆Sθ positive and ∆Stot positive as well. Some other types of processes that have significant entropy changes are not represented in Table 2. One is the entropy increase when a molecular solid dissolves in a molecular liquid. Melting and dissolution contribute to the increase in entropy. Another is the entropy increase when a solute undergoes dilution.
∆H ⬚(C⫺H) ≈ ∆H ⬚(N⫺H) ≈ 400 kJmol ∆H ⬚(C⫺C) ≈ ∆H ⬚(C⫺O) ≈ ∆H ⬚(C⫺Cl) ≈ 350 kJmol ∆H ⬚(Cl2) ≈ 250 kJmol ∆H ⬚(O⫽O) ≈ 500 kJmol ∆H ⬚(C⫽C) ≈ 600 kJmol ∆H ⬚(CC) ≈ ∆H ⬚(C⫽O) ≈ 800 kJmol ∆H ⬚(NN) ≈ 950 kJmol
The strongest bond is ∆H ⬚(CO) ≈ 1100 kJmol. As a result of electron crowding, the strongest multiple bonds are weaker than the corresponding multiple of the single bonds in most cases. Exceptionally strong single bonds involve fluorine. Thus, ∆H ⬚(H⫺F) ≈ 550 kJmol, and ∆H ⬚(C⫺F) ≈ 500 kJmol. Bonds between small electronegative atoms with unshared pairs of electrons are weak. Thus, the bond in F2, where each fluorine atom may be regarded as having three unshared pairs of electrons, is even weaker than the bond in Cl2. Not only are bonds between halogen atoms weak but so are O⫺O and N⫺N single bonds. Also, the O⫽O bond is weaker and the N⫽N bond is yet weaker than the C⫽C bond. As noted above in the discussion of patterns in molar entropies, in most cases bond strengths decrease down a family in the periodic table as the atoms get bigger. Armed with the numerical values and relationships for bond dissociation enthalpies given in the previous paragraph, we estimate the enthalpy of reaction for a few examples.12 In drawing qualitative conclusions about contributions of bond strengths, we will use the bond in H2 as the reference. We first consider the reaction of hydrogen and oxygen to make water in the gas phase as shown in the following equation: H2(g) + h
Estimating Enthalpies of Reaction For the most part, useful qualitative patterns do not exist in molar enthalpies or in enthalpies of reaction.11 Of course, molar enthalpies increase appreciably in going from the solid 1434
1 1
2 O2(g)
H2O(g)
2(1.1h)
2h
∆r H ° ≈ −0.45h ≈ −200 kJ/(mol rxn)
Underneath each species we give an estimate of the bond strength in H2 bond units of “h”, which equals about 450
Journal of Chemical Education • Vol. 80 No. 12 December 2003 • JChemEd.chem.wisc.edu
In the Classroom
kJmol. The combustion of hydrogen is substantially exothermic. This estimated result can be compared with the exact value computed from tabulated molar enthalpies, which is ᎑241.8 kJ(mol rxn) (2). Of course, good estimates of the enthalpy of reaction may be made from the accepted bond dissociation enthalpies. We consider another familiar example before making an estimate for a reaction involving a less wellknown species. The analysis of the reaction of hydrogen and nitrogen to make ammonia is as follows:
3H2(g) + N2(g)
2NH3(g)
3h
6(0.9h)
2h
∆r H° ≈ −0.4h ≈ −180 kJ/(mol rxn)
This reaction is also predicted to be exothermic. The correct value for the enthalpy of this reaction ᎑92.2 kJ(mol rxn) (2). The third example is for a case in which an estimate will be needed unless an extensive table of thermodynamic data is at hand. This reaction is an important step in preparing tetrafluoroethylene for use in making Teflon polymer. This example also illustrates the simplification of not considering the bonds that do not break, the CF bonds in this case: CF2ClH(g) 0.9h + 0.8h
CF2(g) + HCl(g) 0
h
∆r H° ≈ +0.7h
≈ +315 kJ/(mol rxn) The observed value is 208.5 kJ(mol rxn) (2, 9). This reaction is substantially endothermic, and consequently ∆Sθ is significantly negative. The reaction is run at elevated temperatures where the reduction in the magnitude of ∆Sθ due to its 1T dependence and the substantial contribution of ∆rS ⬚ ≈ 140 J(K mol rxn), by Campbell’s rule, help make ∆Stot positive and thus to favor products over reactants. This last example illustrates how an estimate of the enthalpy of reaction and an estimate of an entropy of reaction may be used together to assess the overall spontaneity as a function of temperature. The reader may think that we have put too fine a point on having specific knowledge of bond dissociation enthalpies. We note, however, that the process of grouping bond dissociation enthalpies and seeing patterns in them has benefits even if the numerical values other than for ∆H ⬚(H2) are not recalled. One can still decide qualitatively whether a reaction is endothermic, exothermic, or almost thermally neutral.
This rule enables estimating entropies of reaction from the net change in moles of gas whenever the formation or combination of aqueous ions is not involved. A number of patterns in molar entropies are identified. These patterns are the basis for making good estimates of molar entropies of unknown or transient species from entropies of known species of the same phase, same bond type, same shape, similar composition, and similar mass. For several other types of processes, including reactions involving ions in aqueous solution, the sign of ∆rS ⬚ is considered. For molar enthalpies and enthalpies of reaction we have fewer patterns than for entropies on which to build. The principal exceptions are recognizing the patterns in bond dissociation enthalpies and estimating enthalpies of reaction for gas phase processes. A simple scheme for estimating enthalpies of reaction for the gas phase is described. Acknowledgments The author thanks several referees for helpful comments and suggestions and his colleague Terry Carlton who drew attention to the Davies reference. Notes 1. In an entropy analysis, we compute ∆Stot, which equals ∆rS ⬚ + ∆Sθ for a chemical reaction, where θ stands for the thermal reservoir in the surroundings. The entropy change in the thermal surroundings is always given by ∆S θ = ᎑∆ rH ⬚T. The sign and magnitude of ∆Stot tell about the direction and extent of change to reach equilibrium. 2. Campbell’s discussion focused on applications of the Le Châtelier principle using the signs of ∆rH ⬚ and ∆rS ⬚. Except for two cases, we agree with Campbell’s values for ∆rS ⬚ within 1 J(K mol rxn). The exceptions are: C2H4(g) + H2(g) → C2H6(g) for which ∆rS ⬚ = ᎑124.9 JK mol rxn; and 2NOCl(g) → 2NO(g) + Cl2(g) for which ∆rS ⬚ = 121.3 JK mol rxn. 3. While the manuscript of this paper was being reviewed, Watson and Eisenstein reported the application of density functional theory to the computation of entropies for a variety of species in the gas phase (6). For reactions occurring entirely in the gas phase, they noted an average ∆rS value of 132 JK mol for one net mole of gas produced, which is consistent with Campbell’s rule. 4. Data used to compute thermodynamic quantities quoted in this article came from the extensive table in ref 2. These data came principally from NIST, CODATA, and other sources of critical data. 5. The use of “mol rxn” for degree of advancement (ξ) is advocated in ref 7.
Summary Learning how to estimate thermodynamic quantities and how to use estimates greatly extends the usefulness of thermodynamic analyses of chemical reactions. Estimating helps chemists understand how chemical reactions occur and how the outcomes change with temperature. A new generalization, named Campbell’s rule, where ∆rS ⬚ ≈ 140 J(K mol rxn) per net mole of gas produced in a reaction, is introduced.
6. The enthalpy of vaporization is estimated as ∆rH ⬚ ≈ Tbp[90 J(K mol)]. 7. To enliven the search for patterns in molar entropies, this hunt can be compared to going to the “entropy zoo” in the spirit of Francis Bacon, who classified animals and plants based on their observable properties. 8. As an example of the effect of shape on molar entropies,
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In the Classroom CO2(g), which is linear has an appreciably smaller entropy than SO2(g), which is bent. As an important example of the contribution of the degeneracy due to unpaired electrons, the molar entropy of O2(g) is appreciably greater than the molar entropy of N2(g). This thermodynamic evidence correlates with the paramagnetism of triplet oxygen, which is often demonstrated. 9. The total mass, including the contribution of chlorine, must be considered in making the comparisons. 10. ∆Stot° = R ln K. 11. See, however, ref 4, where the sign of ∆rH ⬚ is linked to the sign of ∆rS ⬚. 12. Students have trouble remembering that the signs differ for computing ∆ rH ⬚from enthalpies of formation and estimating ∆rH ⬚from bond dissociation enthalpies. A comparison of the thermodynamic cycles involved in the two calculations, as shown in ref 2, p 29, helps students to understand and remember this difference.
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Literature Cited 1. Bent, H. A. The Second Law; Oxford: New York, 1965. 2. Craig, N. C. Entropy Analysis; VCH: New York, 1992; Table is Appendix D. 3. Moore, J. W.; Stanitski, C. L.; Wood, J. L.; Kotz, J. C.; Joesten, M. D. The Chemical World, Concepts and Applications, 2nd ed.; Saunders: New York, 1998; Vol. 1, pp 274–277. This text is a recent example of presenting some patterns in entropies. 4. Campbell, J. A. J. Chem. Educ. 1985, 62, 231. 5. Davies, W. G. Introduction to Chemical Thermodynamics; Saunders: New York, 1972; p 131. 6. Watson, L. A.; Eisenstein, O. J. Chem. Educ. 2002, 79, 1269. 7. Craig, N. C. J. Chem. Educ. 1987, 64, 668. 8. Redmond, T. F.; Wayland, B. B. J. Phys. Chem. 1968, 72, 1626. 9. Handbook of Chemistry and Physics, 81st ed., Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 2000.
Journal of Chemical Education • Vol. 80 No. 12 December 2003 • JChemEd.chem.wisc.edu
