Understanding Chemical Equilibrium Using Entropy Analysis: The

In the Classroom

Understanding Chemical Equilibrium Using Entropy Analysis: The Relationship Between ΔStot(syso) and the Equilibrium Constant Thomas H. Bindel Pomona High School, 8101 West Pomona Drive, Arvada, Colorado 80005 [email protected]

This article is intended for undergraduate general chemistry instructors who want to use entropy as a basis for understanding chemical equilibrium. Instructors are provided with (i) the theoretical and mathematical formulations needed for the application of entropy analysis to extents of reaction and (ii) visuals in the form of entropy graphs to help students make the direct connection between the second law of thermodynamics and chemical equilibrium, along with the necessary equations needed to generate the graphs. In addition, the mathematical relationship in eq 1, ΔStot ðsyso Þ ¼ R ln K

ð1Þ

is derived exclusively using entropy analysis as opposed to using the Gibbs function. Equation 1 relates the total entropy change during the transformation of one mol rxn of reactants to (the equivalent amounts of) products, under standard-state conditions, to the equilibrium constant. ΔStot is the total entropy change, syso refers to the system under standard-state conditions1 (1, 2), R is the universal gas constant, and K is the thermodynamic equilibrium constant. When introducing the thermodynamics of physicochemical processes to students, it is important to make a strong connection to the second law of thermodynamics. The method of entropy analysis is a good approach. Entropy analysis involves assessing all of the entropy contributions as a result of a process (3, 4). Entropy analyses of physicochemical processes have been presented in this Journal, including the dissolution of a solid in an ideal solvent (4), equilibrium vaporization (5), osmosis (4, 5), the Haber process (6), oxidation of sulfur dioxide (4, 5), electrochemical reactions (4, 7, 8), solution-phase reactions (9), and consecutive equilibria (2). In addition, the entropy analysis of thermal energy transfer is presented (8, 10). Lastly, the author has presented the lesson plans for an entropy-analysis unit at the first-year level and beyond (8). It is important for students to understand that a spontaneous process undergoes change (transformation) until a maximum in the entropy change, ΔSmax, is achieved or until entropy production ceases. For many chemical reactions that start out with stoichiometric quantities of reactants, the reaction will advance and achieve a maximum in entropy at some point before its complete conversion to products; that is, the extent of reaction, ξ, that corresponds to ΔSmax is bounded by 0 < ξ < 1 mol rxn. Interestingly, ΔSmax is usually the result of changes in concentration.2 At the point of reaction corresponding to ΔSmax, the reaction is no longer capable of advancing or reversing as seen macroscopically. This corresponds to a state of chemical equilibrium in which there is a mixture of reactants and products. 694

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At the introductory college level, it is important for students to see this visually, as opposed to mathematically, as the mathematics are too involved. A good visual is a graph of the total entropy change as a function of the extent of chemical reaction. Students should be able to see readily that a maximum in entropy occurs during a physicochemical process, corresponding to a state of chemical equilibrium.3 It is also important to compare the equilibrium constant derived from ξ at maximum entropy to the equilibrium constant derived from experiment, keeping in mind that deviations may result from non-ideality of gases and solutions. A few graphs of the total entropy change versus extent of reaction appear in this Journal. These include nitric oxide reacting with nitrogen dioxide to produce dinitrogen trioxide (11), the Haber process (6), and the dissolution of solute in a solvent (4). General Approach: The Total Entropy Change as a Function of the Extent of Reaction Assume a system is capable of chemical transformation, is composed of reactants in amounts equal to their stoichiometric coefficients, and is in contact with an infinite thermal reservoir (the surroundings) at a temperature of 298.15 K. Also, assume the reactants are in their standard states. The total entropy of the combination of the surroundings and the system, before any chemical transformation (ξ = 0 mol rxn), is given by Stot, ξ ¼0 ¼ Ssurr, ξ ¼0 þ Ssys, ξ ¼0

ð2Þ

where Stot,ξ=0, Ssurr,ξ=0, and Ssys,ξ=0 represent the total entropy, the entropy of the surroundings, and the entropy of the system all at ξ = 0 mol rxn, respectively. As the system undergoes chemical transformation, the total entropy change is given by eq 3, where ΔStot,ξ, ΔSsurr,ξ, and ΔSsys,ξ represent the total entropy change, the entropy change of the surroundings, and the entropy change in the system all at ξ mol rxn, respectively: ΔStot, ξ ¼ ΔSsurr, ξ þ ΔSsys, ξ

ð3Þ

The total entropy change is the result of changes in the entropy of the surroundings and the entropy of the system, as a result of chemical transformation.4 The entropy change in the surroundings at ξ mol rxn is related to the entropy change in the surroundings for complete reaction, ΔSsurr, by eq 4 (ΔSsurr = ΔSsurr,ξ=1): ð4Þ ΔSsurr, ξ ¼ ξΔSsurr In other words, a reaction that advances to ξ = 0.1 mol rxn will cause the entropy of surroundings to change by an amount equal

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Vol. 87 No. 7 July 2010 pubs.acs.org/jchemeduc r 2010 American Chemical Society and Division of Chemical Education, Inc. 10.1021/ed100192q Published on Web 05/04/2010

In the Classroom

to 10% of the entropy change in the surroundings realized over the entire course of reaction, ξ = 1 mol rxn. The entropy change of the system is given by eq 5, ΔSsys, ξ ¼

o ξΔr Ssys

þ ΔSdil, ξ

ð5Þ

where ΔrSsys is the standard entropy change for reaction and ΔSdil,ξ is the entropy of dilution at a given extent of reaction. Substitution of eqs 4 and 5 into eq 3 gives eq 6: o

o ΔStot, ξ ¼ ξðΔSsurr þ Δr Ssys Þ þ ΔSdil, ξ

ð6Þ

If the reaction is under constant temperature and pressure, the sum of ΔSsurr and ΔrSsyso is replaced by ΔStot(syso), as in eq 7:5 ΔStot, ξ ¼ ξΔStot ðsyso Þ þ ΔSdil, ξ

ð7Þ

This equation is most useful. If the reactants and products also remain in their standard states throughout the entire process, then eq 7 reduces to eq 8, ΔStot, ξ ¼ ξΔStot ðsyso Þ

ð8Þ

that is, ΔSdil,ξ = 0. The total entropy change for a given extent of reaction is directly proportional to the total entropy change for the chemical system under standard-state conditions, ΔStot(syso). The Entropy of the System as a Function of Extent of Reaction (Ssys,ξ) The system entropy at a given extent of reaction is the sum of the mathematical products of the amount of each substance i and its molar entropy: X ni, ξ Sm, i ð9Þ Ssys, ξ ¼ i

The molar entropy for substance i is the sum of the molar entropy of substance i in its standard state, Sm,io, and a molar entropy term that accounts for the substance not being in its standard state as a result of dilution (ΔSdil,m,i = -R ln Xi), X Ssys, ξ ¼ ðni, 0 þ ξνi ÞðSmo , i - R ln Xi Þ ð10Þ i

Expanding the equation results in X X o Ssys, ξ ¼ ni, 0 Smo , i - R ni, 0 ln Xi þ ξΔr Ssys i i X -R νi ξ ln Xi ð11Þ i

The first two terms represent the initial entropy of the system (ξ = 0 mol rxn), Ssys,ξ=0. Note, the second term results from the initial dilution of the reactants. The last term represents the entropy of dilution as a result of reaction. The entropy of dilution is a function of both ξ and Xi. Equation 11 simplifies to eq 12, Ssys, ξ ¼

o Ssys, ξ ¼0 þ ξΔr Ssys

þ ΔSdil, ξ

ð12Þ

where ΔSdil,ξ is the entropy of dilution at a given extent of reaction. ΔSdil,ξ is directly responsible for chemical equilibrium, causing an entropy maximum to occur between 0 < ξ < 1 mol rxn.6 The last two terms in eq 12 correspond to the entropy change for the system (ΔSsys,ξ) progressing from ξ = 0 to ξ mol rxn, as represented in eq 5. Equation 5 is most valuable in the derivation of the total entropy function, ΔStot,ξ.

r 2010 American Chemical Society and Division of Chemical Education, Inc.

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Systems Three types of physicochemical systems are explored. Each system is important within the chemistry curriculum, starting with the simplest and ending with the most complex. Type I consists of systems capable of undergoing only a change in physical state, specifically, from the liquid state to the gaseous state. The vapor pressure above the liquid is calculated as a function of the extent of reaction. The vapor pressure over a liquid is an important concept to the general chemistry curriculum. Type II consists of chemical reactions that occur exclusively in the gaseous state. The dimerization of nitrogen dioxide is presented as it is a reaction that is commonly used to illustrate Le Ch^atelier's principle. In addition, gaseous reactions are explored, in general, in terms of the entropy function. Type III consists of chemical reactions occurring in solution phase. The autoionization of water, a reaction fundamental to acid-base chemistry, is presented, along with a general aqueous-phase reaction.7 All of the thermodynamic calculations are at 298.15 K and use the following standard states: All substances are at a pressure of 1 bar; liquids and solids are pure; gases are in the ideal gas state; and solutions are ideal with unit activity (molality scale). A good knowledge of the chemical system and its standard states is needed when interpreting reaction quotients and equilibrium constants. Type I Systems: Vaporization of a Liquid at Constant Pressure and Constant Volume A cylinder equipped with a frictionless piston contains 1 mol of liquid A and 1 mol of an inert nondissolving8 ideal gas B. The cylinder walls are thermally conductive. The piston exerts a constant pressure of 1 bar to the contents of the cylinder. The surroundings are an infinite thermal reservoir at a temperature of 298.15 K. Liquid A is allowed to evaporate to give ideal gas A: AðlÞ f AðgÞ

ð13Þ

As an infinitesimal amount of gas A forms, an equivalent amount of gas B is removed through a semipermeable membrane that is connected to a large reservoir of gas B.9 The membrane is permeable to gas B, but not gas A; thus, a constant volume occurs in the cylinder before and after each increment of evaporation. The total entropy change as a function of extent of reaction, the entropy function, is written as PAðgÞ, ξ o þ ΔSres - ξR ln ΔStot, ξ ¼ ξΔSsurr þ ξΔr Ssys Po PB, ξ - ð1 - ξÞR ln o ð14Þ P where ΔSres is the change in the entropy of gas B in the reservoirs, and PA(g),ξ and PB,ξ are the partial pressures of gases A and B at ξ mol rxn, respectively. The last two terms represent the ΔSdil,ξ for gas A and gas B in the cylinder, respectively. Equation 14 is rewritten as eq 15. ΔStot, ξ ¼ ξΔStot ðsyso Þ - ξR ln ξ þ ξR

ð15Þ

Note, eq 15 is for ξ > 0 mol rxn, as the natural logarithm is undefined when the argument is zero. This equation is the same as the equation that results from the evaporation of a liquid at

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In the Classroom

Figure 1. ΔStot,ξ versus ξ for the vaporization of various liquid A's at a constant P of 1 bar, at a constant V, and a constant T of 298.15 K. Table 1. Calculated Extents of Reaction and Equilibrium Vapor Pressures for the Vaporization of Various Substances at 298.15 K ξeq/mol rxn

Calculated Peqa/bar

Experimental Peq/bar

Water

0.0317

0.0317

0.031690b; 0.03167c

Ethanol

0.081

0.081

0.0787b,c

Cyclohexane

0.125

0.125

0.130b,c

Acetone

0.322

0.322

0.308b; 0.306 c

Diethyl ether

0.67

0.67

0.717b; 0.669c

Liquid A

Peq represents the pressure of gaseous A at chemical equilibrium. This corresponds to the pressure of A at ξeq. b Values are taken from ref 12. c Calculated using the values of A, B, and C (Antoine equation parameters) from NIST and a temperature of 298.15 K.

constant volume (isochoric process) without any pressuring gas (see the supporting information). Equation 15 is plotted, ΔStot,ξ versus ξ, for various liquid A's (Figure 1). The liquid A's include water, ethanol, cyclohexane, acetone,10 and diethyl ether.11 Each liquid A gives a curve with a maximum occurring between 0 < ξ < 1 mol rxn. The maxima in terms of extent of reaction (ξeq) are presented in Table 1.12 The extent of reaction is labeled ξeq, as the maximum corresponds to a state of chemical equilibrium. The calculated equilibrium vapor pressure of each gas A (PA(g),eq) matches fairly well with the experimentally derived equilibrium vapor pressure. Larger ΔStot(syso) correlates with substances having greater equilibrium vapor pressures. ΔStot(syso) is the result of two opposing entropy effects: (i) the entropy in the surroundings is diminishing as thermal energy flows into the system (endothermic process) and (ii) the entropy of the system is increasing as the liquid becomes gas. In all of the systems studied, the net result of the two effects, over the course of complete reaction, is a decrease in entropy. This happens because the compounds studied have a normal boiling point (NBP) greater than 25 °C. For compounds with a NBP less than 25 °C, the net result is an increase in entropy. Finally, a compound with a NBP of 25 °C has no net change in entropy. (See Section IIb and Figure 9 of the supporting information.) As already stated, dilution is responsible for chemical equilibrium for 0 < ξ < 1 mol rxn. This is most evident in Figure 2. The straight-line relationship given by “c” represents


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Vol. 87 No. 7 July 2010

the evaporation of acetone at constant T and at a constant P of 1 bar without any dilution. As stated before, if the reactants and products remain in their standard states throughout the reaction, then there is no entropy of dilution and chemical equilibrium is not possible.13 The curvilinear relationship given by “b” represents only the dilution effect (ΔSdil). The sum of these two gives the curve represented by “a”, which is ΔStot,ξ versus ξ. This clearly shows the contribution of ΔSdil to the entropy maximum, which is responsible for chemical equilibrium for 0 < ξ < 1 mol rxn. Type II Systems: Reactive Gaseous Systems

a

696

Figure 2. Entropy changes for the vaporization of acetone as a function of extent of reaction at constant T and P.

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Two systems are presented. The first is the dimerization of nitrogen dioxide and the second is a generalized system. The first chemical system is composed of 2 mol of gaseous NO2 at a temperature of 298.15 K, at a constant pressure of 1 bar, and in contact with an infinite thermal reservoir at 298.15 K. The system is capable of reacting (dimerizing) according to 2NO2 ðgÞ f N2 O4 ðgÞ

ð16Þ

The gaseous reactants and products are assumed to be ideal gases. The total entropy change as a function of the extent of reaction is ΔStot, ξ ¼ ξΔStot ðsyso Þ - ð2 - 2ξÞR ln

ð2 - 2ξÞ ξ - ξR ln ð2 - ξÞ 2-ξ

ð17Þ Note, this equation is for 0 < ξ < 1 mol rxn, as the natural logarithm is undefined when the argument is zero. A maximum in entropy is observed at ξeq = 0.81 mol rxn (Table 2), corresponding to K of 6.7 (Po is 1 bar). The literature experimental value is Katm = 6.79 (13), which converts to Kbar = 6.70.14 The value for the equilibrium constant derived from entropy analysis is in good agreement with the experimentally derived value (see Table 2). The second system represents a general entropy function for the reactions involving only gases:15 2

ΔStot, ξ ¼ ξΔStot ðsyso Þ 0

13

6X B n i, 0 þ ν i ξ C 7 - R4 ðni, 0 þ νi ξÞ ln@P A5 ðni, 0 þ νi ξÞ i


ð18Þ

i

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In the Classroom Table 2. Equilibrium Extents of Reaction and Corresponding Equilibrium Constants at 298.15 K ξeq/mol rxn

System

Ka

Kexpb

2 NO2(g) f N2O4(g)

0.81

6.7 (Po/1 bar)

6.70 c (Po/1 bar)

2H2O(l) f H3Oþ(aq) þ OH-(aq)

1.00 10-7

1.00 10-14

1.008 10-14d

a

The equilibrium constant is calculated from ξeq and is referred to as the thermodynamic equilibrium constant, which is sometimes written as Ko. b The equilibrium constant is derived from experiment. c See ref 13. d See ref 15.

Table 3. The First and Second Partial Derivatives of ΔStot,ξ with Respect to ξ for Each Reactive System System A(l) f A(g) at constant P and V

The First Partial Derivative ofΔStot,ξ with respect to DΔS ξ Dξtot, ξ

The Second Partial Derivative 2 ofΔStot,ξ with respect to D ΔS , ξ ξa,b Dξtot 2

ΔStot(syso) - R ln Q

-R ξ -R ξðξ þ 1Þ -R ξ - 2R ξð1 - ξÞð2 - ξÞ - 2Rð55:5Þ ξð55:5h- 2ξÞ i - R 1 -a ξ þ 1 -b ξ þ ξc þ dξ

o

A(l) f A(g) at constant P

ΔStot(sys ) - R ln Q

A(l) f A(g) at constant V

ΔStot(syso) - R ln Q ΔStot(syso) - R ln Q

2NO2(g) f N2O4(g) þ

-

2H2O(l) f H3O (aq) þ OH (aq)


aA(aq) þ bB(aq) f cC(aq) þ dD(aq)


a When ξcrit is substituted into each of the second derivatives, a negative value results, confirming that the extremum is concave downward. b For the last entry, corresponding to the general aqueous system, assume 1 > ξeq > 0 mol rxn.

The second term represents the dilution effect as a function of the extent of reaction.

given by ΔStot, ξ ¼ ξΔStot, ξ ðsys Þ þ R ln o

Type III Systems: Reactive Homogeneous Solutions Two chemical systems are presented. The first is the autoionization of water and the second is a general aqueous system. The first chemical system consists of 55.5 mol16 of liquid water (1.00 kg) at a constant pressure of 1 bar. The system is in constant contact with an infinite thermal reservoir at 298.15 K and is allowed to react according to 2H2 OðlÞ f H3 Oþ ðaqÞ þ OH - ðaqÞ

ð19Þ

The aqueous-ion solutes and the liquid water are assumed to be ideal. The standard state for each ion is unit activity (based on the molal scale, mo = 1 m). (See the supporting information for a discussion of the standard-state entropies of ions.) ΔStot,ξ for the autoionization of water is given by ΔStot, ξ ¼ ξΔStot ðsyso Þ - 55:5R ln XH2 O ! aH3 Oþ aOH - ξR ln XH2 2 O

ð20Þ

Note, this equation is for ξ > 0 mol rxn, as the argument of the natural logarithm cannot be zero. At equilibrium, the maximum total entropy change occurs at ξeq = 1 10-7 mol rxn, which corresponds to hydronium and hydroxide ion concentrations of 1 10-7 m (mol/kg) for each. This also corresponds to a concentration of 1 10-7 mol dm-3, as these solutions are dilute (Table 2). The second system is a general aqueous system. The chemical system is composed of “a” moles of substance A and “b” moles of substance B dissolved in aqueous solution. The solution is assumed to be an ideal solution. The chemical system is at a constant temperature of 298.15 K, a constant pressure of 1 bar, and is capable of transforming according to aAðaqÞ þ bBðaqÞ f cCðaqÞ þ dDðaqÞ

ð21Þ

The total entropy change as a function of the extent of reaction is


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- ξR ln

XAa , ξ ¼ 0 XBb, ξ ¼ 0

XCc XDd XAa XBb

!

!

XAa XBb ð22Þ

for 0 < ξ < 1 mol rxn, where Xi is the mole fraction of solute i. Conclusion A derivation of eq 1 is most desirable from an entropy point of view. The partial derivatives of ΔStot,ξ with respect to ξ, for all of the physicochemical processes in the three systems described, are all the same DΔStot, ξ ð23Þ ¼ ΔStot ðsyso Þ - R ln Q Dξ where Q is the reaction quotient (Table 3). If this derivative is set equal to zero, Q is substituted for K (Q = K at ξeq), and the equation is solved for ΔStot(syso), then eq 1 is obtained. The entropy maxima are easily observed from the graphs of the entropy functions; alternatively, the maxima correspond to mathematical critical points (ξcrit), which are found by setting the partial derivatives of the entropy functions equal to 0 and solving for ξ. This approach gives only extrema, and consequently, the second partial derivatives are needed to verify that the extrema correspond to maxima. The second partial derivatives of the entropy functions give negative values at ξcrit mol rxn, which are consistent for curves that are concave downward (maxima). For a given value of Q, the partial derivative of the entropy function can be calculated. If the result is less than zero, then the process will proceed spontaneously in the reverse direction toward chemical equilibrium. On the other hand, if the result is greater than zero, then the process will proceed spontaneously in the forward direction toward chemical equilibrium. In a similar way, if the values of Q and K are known, then the


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697

In the Classroom 4. Note, the “ΔS” symbol with a subscript ξ stands for the entropy change for an extent of reaction from 0 mol rxn to ξ mol rxn; however, an “S” symbol with a subscript ξ stands for the absolute entropy at ξ mol rxn. 5. At constant temperature and pressure, ΔSsurr = -ΔrHo/T. 6. If 1 mol of A transforms according to A f B and ΔSdil,ξ is plotted versus ξ, then a curve (concave down) results with the maximum at ξ = 0.5 mol rxn. Of course, the end points (ξ = 0 mol rxn and ξ = 1 mol rxn) have a value of ΔSdil,ξ = 0, as there is no dilution. 7. In a draft of this manuscript, the esterification of acetic acid and ethanol was included. A reviewer pointed out that the reaction is complicated by the immiscibility of the products, the ester and the water. 8. Gas B does not dissolve into liquid A. 9. As the piston moves upward from the formation of “dn” moles of gas A, an electrical connection is completed, which opens a port in the side wall of the cylinder to expose the semipermeable membrane. The large reservoir of gas B is at a pressure such that “dn” moles of gas B will spontaneously diffuse across the membrane, causing the piston to return to its original position. There are an infinite number of reservoirs of gas B. 10. Propanone. 11. Ethoxyethane. 12. If the amount of gas B is increased, the ξeq will increase, and likewise, if the amount of gas B is decreased, the ξeq will decrease. 13. It should be pointed out that the condition of “reactants and products remaining in their standard states” is not the only way of eliminating the entropy of dilution. For example, consider a reaction involving only gases. If the partial pressures of the gases remain constant throughout the entire reaction, then there is no entropy of dilution. 14. The value is calculated using the equation presented in ref 14 and a T = 298.15 K. 15. 0 1 XB ni, 0 C Stot, ξ ¼ Ssurr, ξ ¼0 þ ξΔStot ðsyso Þ þ R @ni, 0 ln P A ni, 0 i

derivative can be calculated. Equation 19 represents the relationship and is easily derived ! DΔStot, ξ K ¼ R ln ð24Þ Q Dξ from eqs 1 and 23. If Q is less than K, then the partial derivative of the entropy function will be greater than zero, and if Q is greater than K, then the derivative will be less than zero. Finally, it has been a common practice to express the partial derivative as a “delta quantity”, ð25Þ ΔStot, ξ ¼ ΔStot ðsyso Þ - R ln Q leading to a great deal of confusion. This is also seen in the analogous eq 26. ð26Þ Δr G ¼ Δr G o þ RT ln Q Bent described this in a heading as “A Weed in the Field of Thermodynamics” (16). Many others have proposed other symbols to alleviate this dilemma (11, 17-20). Summary Entropy analyses of eight chemical systems, including a general aqueous system, as a function of the extent of reaction give graphs showing a single maximum between 0 < ξ < 1 mol rxn. These maxima occur at the points corresponding to chemical equilibrium. In every system, the natural logarithm of the equilibrium constant is directly related to the total entropy change with the system under standard-state conditions. This relationship is derived exclusively from entropy analysis. Acknowledgment I would like to thank the reviewers for their time and efforts in making this a much improved manuscript. I would also like to express my gratitude to Mary Katherine Bindel for checking the mathematical derivatives. Notes

2

1. The reactants are separate, pure, and in their standard states. Likewise, products are separate, pure, and in their standard states. 2. If the reactants and the products remain in their standard states throughout the reaction, then the maximum in entropy will occur at one of the end points, either ξ = 0 mol rxn or ξ = 1 mol rxn. On the other hand, if during a reaction a substance does not remain in its standard state, such as a gas changing in partial pressure or a solute changing in concentration, then the maximum in entropy will occur between 0 < ξ < 1 mol rxn. This is the direct result of a change in partial pressure (concentration). As the reaction proceeds, the reactants become more dilute and the products become more concentrated, resulting in entropy changes. I have referred to this collectively as the “entropy of dilution”, even though the products actually become more concentrated as the reaction progresses. 3. An interesting question is “what would happen if there is both a local maximum and an absolute maximum?” The absolute maximum would still represent the position of chemical equilibrium (point of stability), as it is the highest point in entropy.

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-R

i

0

13

X6 B ni, 0 þ νi ξ C7 4ðni, 0 þ νi ξÞ ln@P A5 ðni, 0 þ νi ξÞ i i

16. Molar mass of water is 18.02 g mol-1.

Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Bindel, T. H. J. Chem. Educ. 2005, 82, 839. Bindel, T. H. J. Chem. Educ. 2007, 84, 449. Craig, N. C. Entropy Analysis; VCH: New York, 1992. Craig, N. C. J. Chem. Educ. 1988, 65, 760. Craig, N. C. J. Chem. Educ. 1996, 73, 710. Brosnan, T. J. Chem. Educ. 1990, 67, 48. Bindel, T. H. J. Chem. Educ. 2000, 77, 1031. Bindel, T. H. J. Chem. Educ. 2004, 81, 1585. Bindel, T. H. J. Chem. Educ. 1995, 72, 34. Gislason, E. A.; Craig, N. C. J. Chem. Educ. 2006, 83, 885. Gerhartl, F. J. J. Chem. Educ. 1994, 71, 539.


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In the Classroom

12. Handbook of Chemistry and Physics, 87th ed.; Lide, D. R., Ed.; CRC Press, Inc.: Boca Raton, FL, 2006; pp 6-9; 15-13. 13. Yin, Y.-j. College Chemical Handbook; Shandong Science and Technology Press: Shandong, China, 1985; p 830. 14. Treptow, R. S. J. Chem. Educ. 1999, 76, 212. 15. Harned, H. S.; Robinson, R. A. Trans. Faraday Soc. 1940, 36, 973. 16. Bent, H. A. J. Chem. Educ. 1973, 50, 323. 17. MacDonald, J. J. J. Chem. Educ. 1990, 67, 380. 18. David, C. W. J. Chem. Educ. 1988, 65, 407.


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19. Craig, N. C. J. Chem. Educ. 1987, 64, 668. 20. Spencer, J. N. J. Chem. Educ. 1974, 51, 577.

Supporting Information Available Thermodynamic quantities; spreadsheets and graphs; additional supplemental material for each of the systems (graphs of Stot,ξ versus ξ; extent of reaction; entropy of dilution or expansion; standard-state entropies of ions). This material is available via the Internet at http:// pubs.acs.org.


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Understanding Chemical Equilibrium Using Entropy Analysis: The

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