In the Laboratory edited by
Secondary School Chemistry
Erica K. Jacobsen University of Wisconsin–Madison Madison, WI 53706
Discovering the Thermodynamics of Simultaneous Equilibria An Entropy Analysis Activity Involving Consecutive Equilibria
W
Thomas H. Bindel Pomona High School, 8101 West Pomona Drive, Arvada, CO 80005;
[email protected] The following is a classroom activity in which students discover the thermodynamics of simultaneous equilibria involving two consecutive reactions.1 Specifically, the students discover that it is possible to start with an entropy-diminishing or endergonic reaction and couple it with an entropyproducing or exergonic reaction, thereby causing the entropy-diminishing reaction to occur (thermodynamic coupling). The activity is appropriate for second-year high school chemistry and AP chemistry classes. As the AP chemistry curriculum includes calculations involving simultaneous equilibria (1), a conceptual understanding of simultaneous equilibria would be of benefit. Many of the experiments in the literature dealing with simultaneous equilibria are at the physical chemistry level (2–5) and are inappropriate for the aforementioned level. In the activity, the consecutive equilibria are composed of an unfavorable equilibrium that is coupled to a variety of second equilibria that are favorable to varying degrees. The two reactions that comprise the reaction sequence (reactive system) taken together are forward progressing (∆Suniv > 0 or ∆rG < 0). They each are composed of a single elementary step that involves the transfer of a proton. The first reaction, an acid–base reaction between ammonium ion and water, + NH4 (aq) + H2O(l)
H3O+(aq) + NH3(aq) (1)
is thermodynamically unfavorable (entropy diminishing or endergonic) under standard-state conditions with ammonia as the desired product. Standard states will be used to simplify the thermodynamics and provide a venue for the calculation of equilibrium constants, if so desired. The second chemical reaction,
B−(aq) + H3O+(aq)
H2O(l) + HB (aq)
(2)
involves the transfer of a proton from hydronium ion to an aqueous base B−(aq). The hydronium ion is the common intermediate that links the two equilibria. Equation 3 represents the overall chemical reaction for eqs 1 and 2: + NH4 (aq) + B−(aq)
NH3(aq) + HB (aq)
(3)
Equations 4–9 constitute the thermodynamic variety of chemical reactions for eq 2:
OH−(aq) + H3O+(aq)
H2O(l) + H2O(l)
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(4)
3− PO4 (aq) + H3O+(aq)
2− HPO4 (aq) + H2O(l) (5)
2− CO3 (aq) + H3O+(aq)
− HCO3 (aq) + H2O(l) (6)
C2H3O2−(aq) + H3O+(aq) HC2H3O2 (aq) + H2O(l) − + H2PO4 (aq) + H3O (aq)
H3PO4 (aq) + H2O(l) 2− SO4 (aq) + H3O+(aq)
− HSO4 (aq) + H2O(l)
(7)
(8)
(9)
The reactions are all entropy-producing or exergonic reactions and vary in terms of their degree of entropy production or exergonicity. The reaction schemes, for the most part, are derived from a classroom demonstration in this Journal (6). Activity
Part I: Student Preparation Students should have some basic knowledge of thermodynamics. They should be familiar with the concepts and calculations associated with determining the thermodynamic feasibility (spontaneity) of reactions that occur under standard-state conditions. Students should also be familiar with the basic concepts of general chemical equilibrium. The activity is presented in terms of both entropy analysis and Gibbs energy analysis. This provides a framework for instructors to use whichever fits best with their curriculum. Entropy analysis involves analyzing processes and reactions exclusively in terms of entropy contributions (7). Recently, Craig (8) provided arguments in support of the use of entropy analysis. Additionally, the author has presented an entire thermodynamics teaching unit built upon entropy analysis (9). Students determine the thermodynamic tendencies for the reactions under standard-state conditions by either calculating ∆S univ(sys°) or ∆ rG° for eqs 1 and 4–9, where ∆Suniv(sys°) is the change in the entropy of the universe with the reactants and the products in standard states (10).2 The enthalpies and the entropies of various substances and the
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results of the thermodynamic calculations are available in the Supplemental Material (Tables 2 and 3).W Students consider the following question. Question: Consider an entropy-diminishing (endergonic) chemical reaction. If an entropy-producing (exergonic) reaction is coupled with it, what effect, if any, does the coupled reaction have on the entropy-diminishing (endergonic) reaction?
Students realize from the thermodynamic calculations that eq 1 is the entropy-diminishing reaction and that eqs 4–9 represent a variety of entropy-producing reactions. Students also realize that eq 1 couples with any of eqs 4–9 through the common intermediate H3O+. Their attention focuses onto the ammonia product in the entropy-diminishing reaction and the question of how to monitor its concentration. The answer is to introduce a minimal concentration of Cu2+ ion into the reaction mixtures; so that if there is little to no ammonia present, then the solutions will be essentially colorless. As the ammonia becomes present in significant quantities, the solutions take on the blue color of the tetraammine copper(II) ion [Cu(NH3)4]2+:
Cu2+(aq) + 4NH3 (aq)
[Cu(NH3)4]2+(aq) (10)
Part II: Experimenting with Consecutive Equilibria Students conduct the following experiment. Equal volumes of aqueous 0.1 M ammonium ion are placed into six separate test tubes. To each test tube is added an equal volume of one of the following aqueous ions (0.1 M): hydroxide, phosphate, carbonate, acetate, dihydrogen phosphate, and sulfate ion. Aqueous solutions of ammonium ion and ammonia are placed separately into two other test tubes. Finally, a solution of aqueous 0.1 M copper(II) ion is added dropwise to each of the eight test tubes and observations are made. The results are presented in Table 1. Part III: Analysis of Data Students compare and discuss the results of the activity to the thermodynamic calculations for the respective reactions. The test tubes containing systems with a net increase in entropy (entropy producing) give blue-colored solutions. Those that are entropy diminishing give colorless or near colorless solutions. Students calculate the equilibrium constants for the constituent reactions, using ∆S univ(sys°) or ∆ rG° {K = exp[∆Suniv(sys°)兾R ] or K = exp(᎑∆rG°兾RT )}. The equilibrium constants for the net reactions are calculated using two methods, which are algebraically equivalent and numerically the same. In the first, the net (or overall) equilibrium constant is calculated by taking the mathematical product of the constituent equilibrium constants. In the second, the net equilibrium constant is calculated from the sums of the ∆Suniv(sys°)s or ∆rG°s for the constituent reactions. Students compare the equilibrium constants for each net reaction derived from the two methods. Hazards Students must wear splash goggles. The instructor should know the hazards associated with each substance and use ap450
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Table 1. Observations of the Various Test Tubes Test Initial Composition Tube
Coupled Observations Equations
1
NH4 + + Cu2+
2
NH4 + + OH− + Cu2+
1+4
Initially a light-blue ppt forms, disappears upon agitation, and is replaced by a bluecolored solutiona
3
NH4 + + PO4 3− + Cu2+
1+5
Same as test tube 2b
4
NH4 + + CO3 2− + Cu2+
1+6
Same as test tube 2
5
2+
NH4 + C2 H3 O + Cu
1+7
Faint blue or tealcolored solution
6
NH4 + + H2 PO4 − + Cu2+
1+8
Colorless solution
7
NH4 + + SO4 2− + Cu2+
1+9
8
NH3 + Cu2+
+
–
−
–
Colorless solution
Colorless solution Same as test tube 2
a The blue color is similar to sky blue or baby blue. b A hazy solution is sometimes observed. It appears that the order of addition may make a difference. If 4 drops of aqueous copper(II) ion is added to the 3 mL of ammonium ion followed by the 3 mL of phosphate ion, a clearer solution apparently forms.
propriate safety precautions (11). Many of the substances are eye, skin, and respiratory irritants. A few are corrosive, such as sodium hydroxide, sodium carbonate, and aqueous ammonia. Copper(II) nitrate is a strong oxidizer and affects both the liver and the kidney. Discussion The author believes students should be taught entropy analysis before they are introduced to the change in the Gibbs energy. Once students grasp the concepts and calculations associated with entropy analysis, then it is appropriate to introduce them to and allow them to use the change in the Gibbs energy. Various physical changes take place when copper(II) ion is added to the reaction mixture. The blue color is from the production of tetraammine copper(II), [Cu(NH3)4]2+. Three of the reactive systems (test tubes 2–4), along with the ammonia and copper ion (test tube 8), produce observable [Cu(NH3)4]2+. When copper ion is first introduced into these test tubes, a light-blue precipitate immediately forms, which is copper hydroxide. This occurs because these solutions are the most basic. If the copper ion is only added to the aqueous solutions of the bases, a light-blue precipitate immediately forms at the outset with the hydroxide, the carbonate, and the phosphate ion. Thus, as soon as the copper hydroxide forms in the reactive systems, it reacts with the ammonia upon agitation and the blue color of the [Cu(NH3)4]2+ appears. Test tube 2 is the only one that retains a small quantity of the precipitate. Three of the six reactive systems, ammonium ion and either hydroxide, carbonate, or phosphate ion, are overall entropy producing, ∆Suniv > 0 (Table 2). These systems produce relatively high concentrations of ammonia, which in turn produce the characteristic blue color associated with [Cu(NH3)4]2+. Calculations show the equilibrium concentrations of ammonia to vary from 0.039 M to 0.049 M (80% to >99% yield; see Table 4 of the Supplemental MaterialW
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Table 3. Calculated Equilibrium Constants
Table 2. Thermodynamics of the Coupled Systems Consecutive Equations (Rxn Systems)
∆Suniv(sysº)/ (J K᎑1 mol rxn᎑1)
1+4
91
᎑27.0
6 x 104 (5 x 104)c
1+5
59
᎑17.6
1+6
21
1+7
᎑86
1+8 1+9
Equation
∆rGº/ (kJ mol rxn᎑1)
Koveralla,b
Calc Ka ᎑10
1/K
K1Kib
1
6 x 10
4
1 x 10
5
1 x 103
᎑6.2
1 x 10
1
25.6
3 x 10᎑5
8
᎑136
40.6
8 x 10
᎑8
1 x 1020 (1.4 x 102)c
1 x 10᎑20 (7.1 x 10᎑3)c
6 x 10᎑8 (8 x 10᎑8)c
᎑139
41.5
5 x 10᎑8
9
1 x 1020 (9.6 x 101)c
1 x 10᎑20 (1.0 x 10᎑2)c
6 x 10᎑8 –
a Koverall = exp[∆Suniv(sysº)/R]; Koverall = exp(∆rGº/RT ). bThe number of significant figures is the same as the number of decimal places in the exponent (12). cThe first number given is calculated from entropies and the second number, in parenthesis, is calculated from the change in the Gibbs energy.
for the calculated equilibrium concentrations.) This is rationalized by assuming a forward progression of the reactive system until the reactive intermediate is nearly depleted or is in relatively low concentration, ceasing all forward movement. In other words, as the first reaction (eq 1) spontaneously moves forward producing ammonia and the intermediate hydronium ion, the second reaction (eq 2) begins to move spontaneously forward, causing the intermediate concentration to decrease. The decreasing intermediate concentration allows the first reaction to continue to move spontaneously forward until the intermediate can no longer be replenished by the first reaction, as the reactant concentrations are low and the reaction then has no ability to spontaneously move forward. When a reactive system is no longer capable of change, then it has reached equilibrium. At this point, all species in the reactive system must have nonzero concentrations, including the intermediate. The remaining three reactive systems, ammonium ion and either acetate, dihydrogen phosphate, or sulfate ion, are overall entropy diminishing, ∆Suniv < 0 (Table 2). Equation 1 moves spontaneously forward to produce a low concentration of ammonia as the reaction has a ∆Suniv(sys°) < 0. The initial conditions allow for spontaneous forward movement of the reaction, because the product concentrations are initially zero. However, the reaction will not progress to a large extent (extent of reaction < 0.5), because the change in the standard-state entropy is less than zero (nonspontaneous under standard-state conditions). Likewise, the eq 2 goes to a small extent of reaction as it is also nonspontaneous under standard-state conditions. The consequence of this is that the intermediate concentration does not undergo much change and so, the overall reaction sequence stops at a small overall extent. The relatively low equilibrium concentration of ammonia does not produce the coloration of [Cu(NH3)4]2+. The faint blue or teal color observed in the acetate system (test tube 5) is formed from a complex of copper ion and acetate ion(s).3 If 3 mL of 0.1 M acetate ion, 3 mL of water, and 4 drops of Cu2+(aq) are mixed together, the faint blue or teal color forms. www.JCE.DivCHED.org
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–
– ᎑14
6 x 104
2 x 10
᎑13
5 x 10
1 x 103
6
2 x 1010
5 x 10᎑11
1 x 101
7
6 x 10
14 12
40
1 x 10
᎑50
2 x 10
4 x 10᎑5
a K = exp[∆Suniv(sysº)/R]; K = exp(∆rGº/RT ). bK1Ki is the mathematical product of the equilibrium constants for eq 1 (K1) and the coupled r e a c t i o n i, K i , w h e r e " i " i s t h e e q n u m b e r ( 4 – 9 ) . T h e e q u i l i b r i u m constants are calculated from either the changes in entropies [∆Suniv(sysº)] or from the changes in the Gibbs energy (∆rGº). cThe first number given is calculated from entropies and the second number, in parenthesis, is calculated from the change in the Gibbs energy.
The consecutive equilibrium activity is conducted at conditions that are not standard state. The concentration effect can be neglected (see Part IV of the Supplemental MaterialW) as it does not change the expected qualitative results. Those reactive systems that have a net increase in entropy are expected to have fairly high yields of ammonia; that is, those systems that have an entropy-producing second reaction that is greater in magnitude than the entropy-diminishing first reaction will produce relatively high concentrations of ammonia, which in turn react with most of the copper(II) ions to produce tetraammine copper(II) ions. Calculations of the equilibrium concentrations of all reaction species for the three simultaneous equilibria (eqs 1, 2, and 10) are presented in Table 5 of the Supplemental Material.W The values for {[Cu(NH3 )4]2+}eq are corroborated by a spectroscopic determination of the concentrations (see Part III of the Supplemental MaterialW). Students calculate many equilibrium constants, using the entropies [∆Suniv(sys°)] or the changes in the Gibbs energy (∆rG°) for the constituent reactions and the overall reaction of each system. The results of these calculations are presented in Tables 2 and 3. The equilibrium constant for eq 1, the inferior reaction,4 is much less than 1; while the equilibrium constants for the other reactions (eqs 4–9) are about 100 or greater. When the reciprocals of the equilibrium constants (1兾K ) are calculated for eqs 4–9 (Table 3), they correspond to the respective acid ionization constants (Ka ). The equilibrium constant, the ∆Suniv(sys°), and the ∆rG° are different ways of expressing the same extent of reaction. Students calculate the mathematical product of the equilibrium constants for the constituent reactions of each system. The results are presented in Table 3. Only the first three reactions (eqs 4–6) have K1Ki > 1, which correspond to those reactions producing a relatively high concentration of ammonia. Students compare the equilibrium constants derived from the total change in entropies (or the change in the Gibbs energy) for the overall reactions (Koverall ) to the product of the equilibrium constants for constituent reactions (K1Ki ).
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Koverall and K1Ki (Tables 2 and 3) compare favorably within the limits of rounding errors. Students realize that the mathematical product of the equilibrium constants for the constituent reactions is the same as the equilibrium constant for the overall reaction; that is, K1Ki = Koverall. This result is demonstrated algebraically through the equations that relate the entropy change to the equilibrium constant. The algebraic equivalence is demonstrated in Part VII of the Supplemental Material.W Finally, there are three remaining issues, which are presented and discussed in the Supplemental MaterialW (see the appropriate part under Additional Material). These are as follows: • Misconceptions (Parts V and VIII) that arise around overall equilibrium constants (13) and coupled reactions: Spencer (14) states, “...the concept that one reaction can drive another against its free energy has only led to the misuse and misunderstanding of coupled reactions.” • Criterion for spontaneity based upon the equilibrium constant and the reaction quotient (Part VI) • Applications of simultaneous, consecutive equilibria (Part IX).
Conclusion Students realize that an entropy inferior process can advance to a large degree if it is coupled to an entropy superior process, as long as the total change in entropy is greater than zero. They also realize that the equilibrium constant (K ), the ∆Suniv(sys°), and the ∆rG° are different ways of assessing the extent of reaction. Finally, the students understand that in a reaction sequence the mathematical product of the individual equilibrium constants is the equilibrium constant for the overall reaction. Simultaneous equilibria, such as the ones explored in this activity, are crucial to many fields of chemistry. Consequently, an understanding of the ideas and concepts associated with simultaneous equilibria is important. W
Supplemental Material
A materials list, instructions for the preparation of the activity, student handouts, instructor notes, and other supporting materials are available in this issue of JCE Online.
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Notes 1. This information was presented at the 2005 Colorado Science Convention. 2. The use of the symbol, ∆Suniv(sys°), is intended to clarify that the system is under standard-state conditions (sys°). The alternative use of ∆Suniv° is problematic in that there are no standard states for the universe. It is important to know when the system is under standard-state conditions because then the change in the entropy of the universe is directly related to the thermodynamic equilibrium constant of the system, K, through ∆S univ (sys°) = R ln K. 3. Copper(II) acetate is a green solid that is water soluble. Aqueous solutions are blue or green in color, depending upon their pH. 4. “Inferior reaction” refers to an unfavorable reaction or an entropy-diminishing reaction within a coupled system of reactions and “superior reaction” refers to a favorable reaction or entropyproducing reaction within a coupled system of reactions.
Literature Cited 1. College Board. AP: Chemistry Topics Outline. http:// www.collegeboard.com/student/testing/ap/chemistry/topic.html?chem (accessed Dec 2006); go to “Chemical Calculations” and section H, “Equilibrium Constants and Their Applications, Including Their Use for Simultaneous Equilibria”. 2. Trimm, H.; Patel, R. C.; Ushio, H. J. Chem. Educ. 1978, 56, 762. 3. Adamson, R.; Parks, P. C. J. Chem. Educ. 1971, 48, 120. 4. Ellison, H. R. J. Chem. Educ. 1971, 48, 124. 5. Bauman, J. B. J. Chem. Educ. 1977, 54, 618. 6. Anderson, M.; Buckley, A. J. Chem. Educ. 1996, 73, 639. 7. Craig, N. C. Entropy Analysis; VCH: New York, 1992. 8. Craig, N. C. J. Chem. Educ. 2005, 82, 827. 9. Bindel, T. H. J. Chem. Educ. 2004, 81, 1585. 10. Bindel, T. H. J. Chem. Educ. 2005, 82, 839. 11. Vermont SIRI. MSDS Online. http://www.hazard.com (accessed Dec 2006). 12. Schwartz, L. M. J. Chem. Educ. 1985, 62, 693. 13. McPartland, A. A.; Segel, I. H. Biochem. Educ. 1986, 14, 137. 14. Spencer, J. N. J. Chem. Educ. 1992, 69, 281
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