In the Classroom
Species Distribution Diagrams in the Copper–Ammonia System: An Updated and Expanded Demonstration Illustrating Complex Equilibria
W
Adam R. Johnson,* Tyrel M. McQueen, and Kit T. Rodolfa Department of Chemistry, Harvey Mudd College, Claremont, CA 91711; *
[email protected] The reaction chemistry of aqueous copper(II) ions with ammonia is commonly used in both general chemistry and inorganic chemistry texts to illustrate the equilibria of complex ions in solution (1–4). Although the system initially seems simple, further analysis of the chemical species involved shows that it is in fact quite complicated. First, ammonia is a weak base and its basic equilibrium reaction must be taken into account. Second, although the aquated copper(II) ion is the most prevalent ion in solution before ammonia is added, this species is itself a weak acid. Third, a series of four coordination complex equilibria are established once ammonia is added. Finally, sparingly soluble copper(II) hydroxide precipitates as the hydroxide concentration rises owing to the ammonia base equilibrium. Typically, when the reaction chemistry of copper(II) with ammonia is discussed, the acid–base and precipitation equilibria are ignored. Although the mathematical description of the full system is unwieldy, it can be solved numerically using a software program such as MathCad (5). Such a treatment provides a thorough description of the real chemical system, complete with precipitation and redissolution of the copper hydroxide precipitate. This seemingly simple system encompasses three major classes of equilibrium chemistry commonly taught in inorganic chemistry: acid–base, complex ion formation, and solubility. This updated classroom demonstration investigates the copper–ammonia system and explains the chemistry using species distribution plots. These plots show the change in concentration of a species in solution as a function of some variable such as pH. Since this demonstration shows the titration of a solution of copper(II) with ammonia, the species distribution plots are shown as a function of the volume of titrant added. Demonstrations using this system have been previously reported (6), and the copper–ammonia system is also the subject of a common general chemistry laboratory experiment (7, 8). This updated demonstration emphasizes the fact that coupled equilibria, while complicated and difficult to solve exactly, can be readily understood and explained by simple algebraic equations. It is also a useful summative demonstration that could end an entire unit on chemical equilibria. Selected portions of this demonstration have been used successfully at both the introductory and advanced undergraduate level. The Demonstration
Materials • 100 mL of 0.10 M CuSO4 (2.50 g of the pentahydrate per 100 mL)
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• 100 mL of 2.0 M ammonia (13.8 mL of 28% solution per 100 mL) • 50-mL buret • 400-mL beaker • stirbar • stirplate • pH meter, pH electrode, standard buffer solutions (only if pH is monitored)
Procedure Place 100 mL of the 0.10 M copper sulfate solution in the 400-mL beaker. Place several mL of the ammonia solution in the buret to rinse it, then fill the buret with the ammonia solution. Calibrate the pH electrode using the buffer solutions and monitor the pH of the solution. While gently stirring the copper solution, add the ammonia solution to the copper solution from the buret. Add the first 2–3 mL dropwise to emphasize the fleeting dark blue color of the copper(II) ammonia complexes due to locally high ammonia concentrations, as well as the copious amounts of the bluish-white precipitate of copper(II) hydroxide. The formation of the precipitate continues until approximately 12 mL have been added. Continue adding ammonia until the precipitate dissolves, the dark blue color returns, and there are no further changes. This will require approximately 35–40 mL. Alternatively, the pH data may be collected ahead of time (pH data are provided in the Supplemental MaterialW) and the titration can be used to show the color changes and precipitation. Variations There are a number of alternative investigations that could be pursued if time allows: • use less or more concentrated NH3 solution • use Cu(NO3)2 (6) • add the copper solution to the ammonia solution • add HCl, NaOH, or Na2SO4 solutions to see what effect it has on the color changes or precipitate, for example, show that the precipitate is not (NH4)2SO4 • collect the precipitate that forms and investigate the supernatant or solid separately, for example, the preparation of CuO by heating (6)
Disposal All solutions used in this demonstration can be rinsed down the sink with water.
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In the Classroom Table 1. The System of Mathematical Equations for the Addition of Ammonia to Aqueous Copper Sulfate De s cript ion
Equat ion
Kw
[H 3O+][OH −] = 1.00 × 10᎑14
Kb
[OH −][N H 4+]/[N H 3] = 2.09 × 10᎑5
Ks p
[C u(H 2O)62+][OH −]2 ≤ 2.20 × 10᎑20
K1
[C u(N H 3)2+]/[C u2+][N H 3] = 1.74 × 104
K2
[C u(N H 3)22+] /[C u(N H 3)2+][N H 3] = 3.89 × 103
K3
[C u(N H 3)32+]/[C u(N H 3)22+][N H 3] = 9.33 × 102
K4
[C u(N H 3)42+]/[C u(N H 3)32+][N H 3] = 1.59 × 102
Sulf at e dilut ion
[SO42−] = 0.01/(0.1 + V)
Ka
[C u(H 2O)5(OH )+][H 3O+]/[C u(H 2O)62+] = 4.57 × 10᎑8
C harge balance
[N H 4+] + [H 3O+] + 2[C u(H 2O)62+] + 2[C u(N H 3)2+] + 2[C u(N H 3)22+] + 2[C u((N H 3)32+] + 2[C u(N H 3)42+] + [C u(H 2O)5(OH )+] = 2[SO42−] + [OH −]
N it roge n mas s balance
(2V)/(0.1 + V) = [N H 3] + [N H 4+] + [C u(N H 3)2+] + 2[C u(N H 3)22+] + 3[C u(N H 3)32+] + 4[C u(N H 3)42+]
C oppe r mas s balance
0.01/(0.1 + V) = [C u(H 2O)62+] + [C u(N H 3)2+] + [C u(N H 3)22+] + [C u(N H 3)32+] + [C u(N H 3)42+] + [C u(H 2O)5(OH )+] + {(amount C u(OH )2)/(0.1 + V)}
N OTE: In t his t able [X ] re pre s e nt s t he nume ric v alue of t he conce nt rat ion of t he s pe cie s X e x pre s s e d in mol/L , V re pre s e nt s t he nume ric v alue of t he v olume of ammonia e x pre s s e d in L , 0.1 is t he nume ric v alue of t he init ial v olume of C uS04 s olut ion, and 0.01 is t he nume ric v alue of t he amount of C u2+ or SO42− in t he s olut ion f or t he e x pe rime nt as de s cribe d in t he t e x t .
Discussion
Full Mathematical Solution In aqueous solution, copper sulfate dissolves completely to form the delicate blue aquated Cu(II) ion. The exact formulation of this ion is most likely the six-coordinate octahedral complex, Cu(H2O)62+, although it is almost certainly highly distorted with the two axial Cu⫺O interactions significantly longer than the four remaining in the square plane (9). This species is a complex ion, an ionic coordination complex. The pH of a 0.1 M copper sulfate solution is approximately 4.17 (calculations for this and other values are in the Supplemental MaterialW), showing that the acid equilibrium occurs: 2+
Cu(H2O)6 (aq) + H2O(l)
+
+ H3O (aq)
+ Cu2 (aq) + NH3(aq)
+ Cu(NH3)2 (aq) (2)
+ Cu(NH3)2 (aq) + NH3(aq)
Cu(NH3)22+(aq) (3)
Cu(NH3)22+(aq) + NH3(aq)
Cu(NH3)32+(aq) (4)
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+
Cu(NH3)42 (aq) + 6H2O(l)
(1)
The acid dissociation constant (Ka) for Cu(II) is 4.57 × 10᎑8 (10). In this article we use the convention that the concentration equilibrium constants have no units, that V represents the numeric value of the volume in liters, and that [X] represents the value of the concentration of species X expressed in mol兾L. Water behaves as a ligand for copper in the aquated Cu(II) ion, but is not a very good ligand and can be readily displaced by ammonia (as well as bidentate nitrogen ligands, ref 6 ) according to the following stepwise formation reactions:
Cu(NH3)42+(aq) (5)
Since the coordination number of the copper ion changes from six to four during these equilibria, only ammonia ligands are indicated. The range of values observed for the stepwise formation constants suggests the experimental difficulties in determining these values (10, 11): K1 = 1.73–2.04 × 104, K2 = 3.89– 4.68 × 103, K3 = 0.933–1.09 × 103, K4 = 1.58–2.00 × 102. The overall reaction can be simplified by the overall formation equilibrium
Cu(H2O)62+(aq) + 4NH3(aq)
+
Cu(H2O)5(OH) (aq)
Cu(NH3)32+(aq) + NH3(aq)
(6)
where the overall formation constant, β4, is 0.992–2.08 × 1013 (β4 = K1K2K3K4). If ammonia only behaved as a ligand, then the chemical equations above would fully describe the system. However, when ammonia is dissolved in water, the base equilibrium also occurs:
NH3(aq) + H2O(l)
NH4+(aq) + OH−(aq) (7)
The Kb for the reaction is 2.09 × 10᎑5, where Kb is the base ionization constant for ammonia (11). There is not very much hydroxide ion formed, but the solubility product constant of copper hydroxide is very low (9, 10), with a value of 0.22– 5.0 × 10᎑20. The solubility product equilibrium for Cu(OH)2 is shown as + Cu2 (aq) + 2OH−(aq)
(8)
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Cu(OH)2(s)
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In the Classroom
Before any ammonia is added, the copper hydroxide precipitate does not form because the hydroxide concentration (due to the acidic solution) is not great enough for the reaction quotient to be greater than the Ksp (see Supplemental MaterialW for this calculation). However, as the addition continues, the concentration of hydroxide rises to the point that copper hydroxide precipitates. When using CuSO4, the bluish-green basic sulfate CuSO4⭈3Cu(OH)2 often forms (6). There are other possible precipitates, including Cu3(OH)4(SO4) (12, 13). A discussion of the synthesis and characterization of copper(II) hydroxide gels using ammonia as the precipitant has been reported (14). Although these researchers suggest that the addition of ammonia to copper sulfate does not lead to gels, we have observed gelatinous precipitates in this system. The use of copper nitrate allows only the formation of solid copper hydroxide (6). For the sake of simplicity, this derivation focuses only on the copper(II) hydroxide precipitate. Since this is a real system that has a single equilibrium state for any quantity of ammonia added, we expect there to be but a single real solution (e. g., all concentrations ≥ 0) for any given volume added. The 12 variables that can change in this system are: [NH3], [H3O+], [OH−], [Cu(H2O)62+], [Cu(H 2 O) 3 (NH 3 ) 2+ ], [Cu(H 2 O) 2 (NH 3 ) 2 2+ ], [NH 4 + ], [Cu(H 2O)(NH 3) 32+], [Cu(NH 3 ) 42+], [SO 42−], chemical amount of Cu(OH)2, and [Cu(H2O)3(OH)+]. For mathematical simplicity, the amount of copper(II) hydroxide precipitate is expressed as a molarity by taking the chemical amount of Cu(OH)2 and dividing by the total solution volume. All of these variables can be expressed using the adjustable parameter V, which is the volume of NH3 added, by use of the 12 equations given in Table 1. Also given in Table 1 are the values for the equilibrium constants that were used for the remainder of this derivation. For any given volume of ammonia that is added to the solution (V ), the system is completely described by these 12 equations and 12 variables. In principle, the system of equations could be solved numerically. In practice, it is very difficult to directly solve these equations as they are nonlinear and the concentrations span many orders of magnitude. Because of the limited precision of floating-point computer hardware, numerical solutions are defined not when the equalities hold exactly, but when they hold within some specified tolerance. This causes problems with solutions that span many orders of magnitude: when the tolerance is too large it is difficult to get meaningful numbers for small values; however, when set too low it is difficult to converge to a solution for large values. Manipulation of the equations is necessary to mitigate this effect and make it possible to use a single tolerance across the entire range of concentrations. Fortunately, it is possible to reduce the above system to three equations and three variables by suitable mathematical rearrangement. Derivation is given in the Supplemental Material.W The reduced system of equations can be solved numerically using the computer program MathCad (5). The following discussion and associated species distribution diagrams are based on this solution. Before progressing to the mathematical solution, it is worthwhile to make some quali-
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tative predictions about the system based on chemical intuition. Before any ammonia is added to the copper solution (V = 0), the pH of the solution should be less than seven as a result of the acidic nature of Cu(H2O)62+. As ammonia is added to the solution (V > 0), the hydroxide that forms will precipitate the copper(II) ions as Cu(OH)2. Therefore, the pH of the solution should not change very much. Since the Ksp for this solid is so small, the NH3兾NH4+ equilibrium (eq 7) will be shifted to the right, and the concentration of NH3 should not build up appreciably. At the point when exactly twice as many moles of NH3 have been added to the solution as there were moles of Cu(II) present initially (V = 0.010), no more precipitate can form, and the concentrations of OH- and NH3 will begin to rise. After this point (V > 0.010), the pH of the solution should rise above seven, and the copper–ammonia complexes should become the dominant copper containing species. The mathematical solution that follows is consistent with the intuitive predictions stated above. The concentration of the conjugate base of aquated copper(II), Cu(H2O)5(OH)+ starts at ca. 10᎑5 before any ammonia is added. The concentration remains at ca. 10᎑4 during the initial addition and then drops rapidly, reaching 10᎑6 when V = 0.010, and 10᎑8 when V = 0.011. During this region of the ammonia addition, [Cu(H2O)62+] drops steadily from ca. 0.10 to 10᎑8, as shown in Figure 1. Meanwhile, the copper hydroxide precipitate builds up steadily; when V = 0.010, it is the dominant species in the system, accounting for 99.7% of the copper. During this region there is a buildup and decay of Cu(NH3)2+ in the region from V = 0.003 to 0.008 (Figure 2), reaching a maximum concentration of 2.5 × 10᎑3, while the concentration of Cu(NH3)22+ reaches its maximum value of 0.1 mM at V = 0.010. After this point, the copper ammonia complexes form at the expense of the precipitate. The concentration of Cu(NH3)32+ rises rapidly from 10᎑6 as V exceeds 0.010, reaching its maximum value of 2.6 × 10᎑3 at V = 0.016 (Figure 2). The concentration of Cu(NH3)42+ starts low, with a value below 10᎑9 when V < 0.010, but rises rapidly after this point. Eventually, Cu(NH3)42+ becomes the dominant species (36 mM) when V = 0.025 (Figure 1). Based on the results of the calculations, the copper hydroxide precipitate should never fully disappear, although it accounts for only 12% of copper when V = 0.080 and 0.086% of copper when V = 0.300. The species distribution diagram of ammonia and ammonium ion is given in Figure 3. There is essentially no buildup of NH3 until after the copper hydroxide precipitate forms; the concentration stays below 10᎑5 until V is greater than 0.010. In this region, the ammonia reacts completely with water to form ammonium ion and hydroxide, which then precipitates. Only after two equivalents of ammonia have been added (V = 0.010) does the ammonia concentration rise appreciably. The concentration of NH4+ builds up rapidly, reaching a value of 0.18 at V = 0.010, but after this point its concentration decays, reaching a value of 14 × 10᎑3 when V = 0.080. The concentrations of other ions can be calculated using the equations given in the Supplemental Material.W
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In the Classroom 0.10
Simplified Mathematical Description Drastic simplifications can be made to the system by judicious use of chemical intuition. There are three principal regions of the system: (i) before the addition of ammonia, (ii) formation and precipitation of Cu(OH)2(s), and finally, (iii) redissolution of the precipitate and formation of the copper–amine complexes. Although this involves great simplification from the full system, it is possible to obtain an approximate species distribution diagram by considering the three regions independently.
Cu 2ⴙ Cu(OH)2 Cu(NH3)42ⴙ
0.08
[Species]
0.06
0.04
0.02
0
Region I: Before the Addition of Ammonia 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
V Figure 1. Concentrations of Cu2+, Cu(OH)2, and Cu(NH3)42+ as a function of V, the numeric value of the volume (in liters) of ammonia added.
3.0
[Species] / 10ⴚ3
2.5
Cu(NH3)2ⴙ Cu(NH3)2 2ⴙ ⴙ Cu(NH3)3 2
2.0
1.5
1.0
0.5
0 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
V Figure 2. Concentrations of Cu(NH 3 ) 2+ , Cu(NH 3 ) 2 2+ , and Cu(NH3)32+ as a function of V, the numeric value of the volume (in liters) of ammonia added.
0.7 0.6
NH3 NH4ⴙ
[Species]
0.5 0.4 0.3 0.2 0.1 0 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
V NH4+
Figure 3. Concentrations of and NH3 as a function of V, the numeric value of the volume (in liters) of ammonia added.
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Before any ammonia has been added to the system, the problem simplifies to that of a weak acid, Cu(H2O)62+ in water, although the possible precipitation of Cu(OH)2(s) must be considered. The six chemical species that are present are H3O+, OH−, Cu(H2O)62+, Cu(H2O)5(OH)+, SO42−, and potentially Cu(OH)2. The two chemical equations that must be considered are eqs 1 and 8 and the six mathematical expressions are Kw, Ksp, charge balance, copper mass balance, Ka, and sulfate dilution (Table 1). All of the nitrogen-containing terms in the charge balance and copper mass balance equations are set equal to zero in this region. When this simplified system of equations is solved (see the Supplemental MaterialW) the following values are obtained: [H 3O +] = 6.76 × 10 ᎑5 ; [OH −] = 1.48 × 10 ᎑10; [Cu(H 2O) 62+] = 9.993 × 10 ᎑2; [SO 42−] = 1.00 × 10 ᎑1; [Cu(H2O)5(OH)+] = 6.76 × 10᎑5; and amount of Cu(OH)2 = 0. The product of [OH−]2[Cu(H2O)62+] is 2.19 × 10᎑21, which is slightly less than the Ksp value so Cu(OH)2 should not precipitate from solution. This is an exact solution for the initial state of the system. The only simplifications that have been made are the use of amount concentrations (molarities) rather than activities. Region II: Precipitate Formation During the initial addition of ammonia (0 < V < 0.010), Cu(OH)2 becomes the dominant species at the expense of aquated copper(II). In this region, the ammonia base equilibrium (eq 7) is coupled to the copper hydroxide precipitation (eq 8). Only a small quantity of hydroxide needs to form to raise the product of [OH−]2[Cu(H2O)62+] above the Ksp value of 2.2 × 10᎑20, thereby precipitating Cu(OH)2. To maintain the value of Kb, more ammonia must react to form ammonium and hydroxide, which causes more precipitation. Therefore, there is almost no buildup of hydroxide or ammonia, the pH of the solution remains acidic, and the concentration of NH 4+ rises. It is possible to simplify the complete system of equations to give a more manageable subset of functions that depend only on V, the numeric value of the volume (in liters) of ammonia added. Since essentially no buildup of NH3 is expected, the coordination complex equilibria are neglected (eqs 2–5). Since Cu(H 2O) 62+ is such a weak acid, the concentration of Cu(H2O)5(OH)+ is also neglected. Thus, the seven chemical species remaining are Cu(H2O)62+, SO42−, NH3, NH4+, OH−, H3O+, and Cu(OH)2. The system is described by eqs 7 and 8, and mathematical expressions for Kw, Kb, Ksp, sulfate dilu-
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tion, charge balance, nitrogen mass balance, and copper mass balance. After making several reasonable assumptions (e. g., [H3O+]
