In the Laboratory
The Nernst Equation: Determination of Equilibrium Constants for Complex Ions of Silver Martin L. Thompson and Laura J. Kateley Department of Chemistry, Lake Forest College, 555 N. Sheridan Road, Lake Forest, IL 60045
The formation of metal complexes is discussed in most introductory chemistry courses and several reports of methods for the determination of their equilibrium constants have appeared. These often involve a titration, followed by graphical analysis (1–4). A more direct approach for determining such values utilizes a concentration cell and the Nernst equation. Described previously as a demonstration (5–7), it can readily be adapted as a laboratory exercise in an introductory course. The experiment requires only simple equipment and inexpensive chemicals, and excellent results can be obtained in a relatively short time. This method can be extended to include equilibria involving a host of metal–ligand complexes, as well as the determination of Ksp values for relatively insoluble salts. General Background The formation of a metal complex in an aqueous solution can be written Mx+(aq) + nL y ᎑(aq) → ML n(x+)+n(y ᎑)(aq)
(1)
Kform = [ML n
x+
E = E ° – [RT /nF ] ln Q
y᎑
(aq)]/ [M (aq)] [L (aq)]
n
(2)
In the specific example described below, where x = 1 and y = 0, these equations can be abbreviated as Ag+ + 2 NH3 → Ag(NH3)2+
(3)
Kform = [Ag(NH3) 2+]/[Ag +] [NH 3] 2
(4)
and A knowledge of the three equilibrium concentrations clearly yields Kform. The determination in this experiment makes several assumptions: 1. The formula of the newly formed complex is known (i.e., the value of n, the coordination number, is known); 2. The reaction reaches equilibrium quickly (i.e., no kinetic complications occur); 3. All activity coefficients are assumed to be one and junction potentials are negligible; 4. A large excess of the ligand is added to the aquated metal ion; thus, its concentration is diminished negligibly in creating the complex; 5. The overall formation constant, Kform, has a relatively large value, so the equilibrium value of [Ag(NH3)2]+ is practically the same as the concentration of the stoichiometrically deficient Ag + created in preparing the sample solution; 6. Concentrations of intermediate species (those containing both water and the new ligand in the coordination sphere) are negligibly small.
Because the ligand is in large excess, the equilibrium concentration of the newly formed complex is defined by the millimoles of Ag+ used and the final volume. The concentra-
(5)
where E is the emf difference between the electrodes in a cell; E ° is the standard reduction potential for the cell in volts; R is the gas constant (8.314 J mol᎑1 K ᎑1); T is the temperature; F is Faraday’s constant, which is 9.6437 × 104 J V᎑1mol ᎑1; n is the number of moles of electrons transferred through the external circuit by molar amounts indicated in the balanced equation; and Q is the reaction quotient. At 25 °C, this reduces to E = E ° – [(0.0591)/n] log Q
where the overall formation constant is (x+)+n(y᎑)
tion of the remaining free ligand can then be determined from the balanced chemical equation. The low equilibrium concentration of the aquated metal ion is the only missing datum to determine Kform, and it can be obtained using a concentration cell and the Nernst equation. The emf (voltage) of a cell depends on the concentrations (or activities) of each species involved in the cell reaction. The Nernst equation (named for Walter Hermann Nernst, who derived it in 1889) relates the concentrations in a cell to the cell’s emf:
(6)
In this experiment, the cell used is a concentration cell; that is, the two half-cells are constructed of the same materials but differ in concentration. Using silver electrodes and Ag+ ion, the half-reactions are oxidation (anode): reduction (cathode): overall:
Ag° → Ag+(dil) + e᎑ Ag+(conc) + e᎑ → Ag° Ag+(conc) → Ag+(dil)
(7)
Since E° = 0 for such a cell, the Nernst equation reduces to Ecell = ᎑(0.0591 V) log {[Ag+(dil)]/[Ag+(conc)]}
(8)
Students check the validity of this equation by using known concentrations in both half-cells and comparing the observed and calculated voltages. Then the equation is used to calculate [Ag+(dil)] when this dilute solution at the anode contains an equilibrium mixture of silver ion, ligand, and complex ion. This concentration, [Ag+(dil)], is the missing piece of information needed to calculate Kform in the experiment. Experimental Procedure A voltmeter that reads in millivolts with a precision of ± 1 mV (or a good pH meter set to record voltage) is adjusted to read zero using the zero, calibration, or standardization knob. A 1.35-V mercury battery is used to check the voltage readings. Several types of salt bridges can be used to conduct the internal circuit of the voltaic cell. A simple one that works adequately is a strip of porous paper soaked in 1 M KNO 3(aq). This moistened paper is handled with forceps. Its two ends are dipped into small beakers containing the reference solution and the one to be tested. A short piece of
JChemEd.chem.wisc.edu • Vol. 76 No. 1 January 1999 • Journal of Chemical Education
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In the Laboratory
silver wire is placed in each beaker and connected via the plastic-covered electrode clips to the voltmeter. To check the validity of the Nernst equation, several readings are observed and recorded by each student: 1. About 20 mL of 0.010 M AgNO3 is placed in one of two small (50-mL) beakers and one of the silver wires is immersed in the solution. CAUTION : Avoid contact of skin and clothing with silver-ion solutions. Fresh stains can be removed by prompt treatment with sodium thiosulfate solution, followed by washing. A similar volume of the same 0.010 M AgNO3 solution is added to the other beaker containing its electrode and the voltage recorded. (To avoid erratic readings, the electrodes should not be in contact with the paper salt bridge.) In accord with eq 8 above, students find that the voltmeter reads nearly zero. 2. The solution in one of the beakers is poured into another container and a 5.0-mL portion of this is added to a 150-mL beaker, along with 45.0 mL of deionized H2O. The Ag wire and salt bridge are reassembled as above and an additional reading is made, which should be about 59 mV. Students are asked to define the cathode and anode and to rationalize the electrode polarities with those of the voltmeter. 3. One additional dilution (a more extreme one, decided by each student) is made, and the is voltage measured as described. The result is compared with that calculated.
The Kform for the diammine silver(I) complex is determined by adding a large but known excess of ammonia to a known amount of aqueous AgNO3. This is done as follows. The reference half-cell is left undisturbed. Into a clean, dry 50-mL beaker is added 15.0 mL of 0.100 M aqueous ammonia and 15.0 mL of 0.010 M AgNO 3. The salt bridge is reassembled and the voltage generated is measured. The concentration of the Ag+(aq) ion remaining in the beaker is determined from the Nernst equation. The millimoles of the complex formed and the free ammonia remaining, along with the final volume of the solution, yield the molarities of these two species at equilibrium. Students are asked to repeat the above procedure with different volumes of the Ag+ and NH3 solutions and obtain a second value of this formation constant. They are then asked to repeat the above procedure, using 0.10 M sodium thiosulfate as a ligand in place of ammonia. After the calculations are made for these two sets of formation constants, all solutions are discarded in the waste bottles provided. Results and Discussion Surprisingly good results are obtained with this simple procedure. Values for Kform for Ag(NH3) 2+ cluster around
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1.2 × 107 (reported value = 1.6 × 107) (8). The precision of student results are reported as logarithms of Kform values; the standard deviation is then 0.23, with an accepted value of log Kform = 7.20. The Kform values for the Ag(S2O3) 23᎑ complex averaged 1.5 × 1013, ranging from 0.20 × 1013 to 2.2 × 1013. Reported values range from 0.43 × 1013 (9) to 2.0 × 1013 (8). When logs of Kform values are used, the average result obtained was 13.18 (corresponding to a Kform of 1.5 × 1013), with a standard deviation of 0.30. These results are quite acceptable, considering that activity coefficients are ignored, dilutions are done with imprecise graduated cylinders, and initial concentrations are approximated. (The last is most serious with aqueous NH3, owing to vaporization; better results can be obtained by titrating the ammonia with standard HCl to affirm its initial concentration.) A knowledge of the values of stepwise equilibrium constants confirms the relatively low concentration of intermediate species, justifying assumption (6 ) above. In the case of Ag(NH3)2+, the two values are similar: K1 = 2.1 × 103 and K2 = 8.2 × 103. In the first determination described, the calculated equilibrium concentrations are [Ag(NH3)+] = 1.5 × 10 ᎑5 M [Ag(NH3)2+] = 5.0 × 10᎑3 M and [Ag +] = 1.8 × 10᎑7 M Thus at equilibrium, the concentrations of the monoammine complex and the Ag+ are respectively 0.3% and 4 × 10᎑3% that of the diammine. The experiment illustrates nicely the power of the Nernst equation. Calculations are straightforward and the results are satisfying to students. Finally, the data can be collected in an abbreviated laboratory session, an especially desirable aspect of an experiment in an honors or advanced placement course at the high school level, where time is always a concern. Literature Cited 1. 2. 3. 4. 5. 6.
Wolfenden, J. H. J. Chem. Educ. 1959, 36, 490. King, L. C.; Cooper, M. J. Chem. Educ. 1965, 42, 464. Guenther, W. B. J. Chem. Educ. 1967, 44, 46; 1967, 44, 427. Schultz, F. A. J. Chem. Educ. 1979, 56, 62. Schwenck, J. R. J. Chem. Educ. 1959, 36, 45. Shakhashiri, B. Z.; Direen, G. E.; Juergens, F. J. Chem. Educ. 1980, 57, 813. 7. Anderson, R. H. J. Chem. Educ. 1993, 70, 940. 8. Stability Constants, 2nd ed.; Sillen, L. G., compiler of Part I; London Chemical Society Special Publication No. 17; London Chemical Society: London, 1964; p 226. 9. Stability Constants, Supplement No. 1; Sillen, L. G., compiler; London Chemical Society Special Publication No. 25; London Chemical Society: London, 1971; p 132.
Journal of Chemical Education • Vol. 76 No. 1 January 1999 • JChemEd.chem.wisc.edu
