Formation Constants of a
David E. Goldberg
Brooklyn The
College
cltvUniversity of New York Brooklyn
Metal-Amhe System
I
Potentiometric titration experiment
The measuremei~tofformation constants by potentiometric titration, introduced by Bjerrum (1) in 1941, has now become an extensive field of research as shown by the data included in the recent compilations of Bjerrum, Schwarzenbach, and Sillen (2) and of Yatsimirskii and Vasil'ev (3). This technique, while not the only method for the determination of formation constants, is indeed the one most often employed. However, there is no description of an experiment designed for such a determination by students, although tho experiment may easily and profitably be performed. This . paper is intended to supply such a . description. The exneriment is useful to the student in several ---. - ~r -~ways. It is an introduction to a field of research which presently is receiving great attention. It introduces him to the stepwise nature of the formation of coordination compounds, and he gets practice in the mauipulat,ion of several equilibrium expressions simultaneously. He learns the usefulness of activity coefficients, and is introduced to the method of calculation of enthalpy aud ent,ropy from the measurement of equilibrium constants a t varying temperatures. The experiment has been used successfully in both senior and graduate courses in inorganic chemistry. As a first choice, the silver-ammonia system was investigated. The copper-ethylenediamine system used later gave much better precision. Also, in the great majority of coordination reactions, the constants decrease in value; the silver-ammonia system is one of the very few in which the second constant is greater than the first (4). ~
copper(I1) perchloratr and a standard solution rontaining about 0.1 mmole of pcrchloric acid are pipetted into t,he four-necked flask. Enough boiled distilled water is added to yield a solution of precisely known volume (about 100 ml total). Thr buret is filled with the standard ethylenediamine solution, which must be protected from the atmosphrre by affixing a drying tube filled with Ascarite. The pH meter is calibrated against a buffer of known pH near 4, and checked against a second buffer near 9 or 10. After the electrodes are washed and dried with care, they are placed in the four-necked flask. I t is necessary to insure that both electrodes are placed so that their tips are beneath the surface of the solution hut are not too near the stirrer. The pH of the solution may he determined before any addition of amine to check proper operation of the system. Xitrogen is started flowing over the surface of the solution. For the most precise work, this nitrogen should be presaturated with water by first bubbling through water at the temperature of the solution, but care must be taken that no liquid water is added to the solution from the presaturator. A small portion of the amine solution is now added carefully, so that no drop adheres to the tip of the
11
?u?b+yg%kd m~. in 0.01
Arcorite
Motor-driven stirrer (motor not shown1 note large hole to permit crcope of nitrogen
The Experimenl
The apparatus is assembled as shown in Figure 1. I t is operated in a constant temperature bath at 25' or 30°C. The glass and calomel electrodes from a conventional pH meter, such as Beckmann model G, are introduced in such a way that they can he removed easily for recalibration of the meter during the course of the experiment. The following standardized solutions are necessary: 0.01 M copper(I1) ion, preferably the perchlorate, nitrate acceptable. 0.1 M perehloric acid, or less preferably nitric acid. 0.1 M ethylenediamine solution, prepared from freshly distilled ethylenediamine and freshly boiled distilled water. In addition, the necessary buffers for pH meter calibration should be provided. These are obtainable commercially, or may be prepared as given by Dole ( 6 ) .
A st.andard solution containing about 0.2 mmole of 328
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Journol of Chemicol Education
Drying containing tube
Figure 1.
Apporotur for pH titration
Nitrogen inlet
buret. The contents of the flask are mixed thoroughly, after which the stirrer is stopped and any other electrical apparatus which might affect the operation of the pH meter is shut off. The pH of the solution, t,he volume of amine added, and any color changes which might have taken place are recorded. Without addition of more amine, the stirring is started and the pH is redet,ermined after another minute has elapsed, to insure that equilibrium has been attained. If the two readings agree, further additions may he made. If the readings differ by more than 0.01 pH unit, further time must be allowed for equilibration. This check may he done about every fifth point, if equilibration is seen to be rapid, but must be done at each point if the system under investigation attains equilibrium slowly. Thr procedure is continued in this mallrrer and the results arc recorded in tabular form. It is neither necessary nor desirable that each addition be the same volume. Of greatest interest are the points at which 0.5 and 1.5 moles of amine have been added per mole of metal ion, exclusive of that which reacts with the acid. An excessive number of additions (morr than 1.5) is unnecessary; neither should there be a large gap between successive readings of pH. A normal acid-base titration curve will not be expected, since the metal ion coordinates with the excess amine and prevents rapid rise in pH. Copper ion is stable enough mith ethylenediamine to coordinate at the same or even lower concentration of amine than is required for the protonation reaction tooccur. After thcse data have been determined, the pH meter must be checked for calibration. If the meter does not check with the previous calibration, the whole procedure must be repeated. Calibration may be checked at any time throughout the titration, if desired. The valors of the protonation constants may be determined in a manner similar to that described above, with the exception that either no metal or a non-coordinating metal he added to the system. The purpose of the non-coordinating metal is to maintain the same low ionir strength that will be used in the metal ion titration. A normal acid-base tit,ration curve is obtained. Typical results for the titrations outlined above are shown in Figure 2.
ion) which does not coordinate. Hydroxide ion, formed from the reaction of m i n e with water, may cause the formation of basic species. The results of the calculations would in this case be invalid. That something was wrong mith the calculation would he shown by variable results. The determination of K,, Kz, etc., is the problem. Determination of concentrations of the various species in the solntion will yield information relative to the following expressions:
If the assumptions are made that the Debye-Hiickel Theory (6) may be applied to t,he system and that the activity coefficients of neutral species are unity, the molarity quotients may be related to the formation ronst,ants as follows:
By similar reasoning, the relationships involving the reactions of the hydrogen ion with the amine are evolved: A+HtF!AH+ AH+ + H + AH.'+, etc.
Both Bertsch (7) and McIntyre (8) found from theoretical relationships that log y+ is equal to -0.03 1 0.005 units for dipositive metal ion solutions of the concentrations called for in this experiment with ethylenediamine. The small uncertainty in log y+ is well within the experimental accuracy of the method. The calculation of formation constants is expedited by the introduction of the quantity a, which is defined as the average number of ligands bound per metal ion. total bound ligand [MA1 2[MAd ... = total metal ion - [MI + [MA] + [MA,] . . . (7)
++
+
*
Formation Constant Calculation
The reactions which take place in the addition of an amine solution to a metal ion solution are as follows: M+AF!MA MA A Ft MA?, etc.
+
(1)
All of the species are aquated, and all species which contain metal ion hear the charge of the aqueous metal ion. Equilibrium constants for the formation of the coordination products are called formation constants, and may be represented as follows:
I 2
3
4
5
6
7
8
9
1
0
ml, added
It is assumed that the metal ion is present only in the aqnated or mine-coordinated states or some combmation thereof. This assumption is reasonable since the only anion introduced is the perchlorate ion (or nitrate
Figure 2. Titrotion curves for protonation ond swrdinotion reactions. Circled poinh: Protonation reaction, 0.201 millimole of HClO. and 0.1 0 0 mmole of BalClO~hin total volume of 100 ml titroted with 0.05507 M ethylenediamine at 3O0C. Squmred points: Coordination readion, 0.201 millimde of HCIOI and 0.1 0 0 millimole CulCIOIll in total initial volume of 100 ml ot 30°C.
Volume
39, Number 7,July 1962
/
329
However, from equation (4), it is apparent that
Substitution of these values into equation (7) yields
The result of dividing both numerator and denominator by [MI is an equation called the formation function (1).
Knowledge of n values of a, together with the corresponding values of [A], makes the solution of the formation function possible by the use of simultaneous equations (9). The method for determination of [A] and then of n will be given, followed by the method of simultaneous equations. Calculation of the free ligand concentration is accomplished from a knowledge of the dissociation constants of the ligand, or more precisely, the association constants.
+
bound hydrogen ion = original hydrogen ion hydrogen ion from dissociation of water - free hydrogen ion [OH-] - [H+] bound hydrogen ion = CH (CH = moles acid/liter)
+
The bound hydrogen is also equal to the sum: bound hydrogen ion = [AH+] 2[AH?+] . . . (10) [AHt] -t2[AH2Pt] + . . . = Ca - [H+] + [OH-] qa~[Al[H+l+ ~ ~ E ~ ~ E * [ A=] Ca [ H+ ][H+] ~ [OH-] [Al(qel[Htl + 2qa~qa*[H+Is) = Ca - IH+l [OH-]
+
+ ++
Calculation of a is accomplished as follows:
-
1Z
bound ligand tot:tl metal
= -
-
total ligand - free ligand - protonated ligand total metal
where C'. = moles added metal ion/liter, and CC*= moles added mine/liter. Proceeding as before CA - [Al(l qd[H+I qalqe'[H+Ia) jj = (13) CM
+
and a may be calculated. The concentrations of amine, hydrogen ion, and metal ion originally present are determined from the stoichiometry. The hydrogen ion and hydroxide ion concentrations are determined from the knowledge of the pH, pKw, and the mean activity coefficient in the solution. The values of q d may be determined by independent titrations with no coordinating metal ion present. The formation function has been solved by simultaneous equations for each of the cases in which one, two, and three ligands are attached to the central metal ion (9). One equation is presented for the first case, one each for K , and Kpfor the second case, but a total of 12 simple equations are presented for the third case. Each of the equations involves n values of a and the corresponding values of [A]. Although mathematically, any value of n should be useable, it was found that the values should be in certain ranges, near halfintegral values, in order to get reliable results. Thus for a system with two ligands bound per metal ion, a values of about 0.5 and 1.5 should be used. The equations for the simultaneous solution involving two ligands per metal ion are given below. Equations for other cases may be obtained from McIntyre (8) or less preferably from Block and McIntyre (9). Unfortunately, typographical errors in the latter reference make it less useful than it otherwise might be.
+
J.
J.'
one obtains for n the quantity
=
ti)[AIm
(17)
(n - %')[A]"
The values of qai may be given to the students by the instructor or may be determined by the student in a separate titration with no metal ion present. The procedure for the determination is the same as is given above, with the exception of omission of the metal ion, but the calculation method is simpler since there are fewer species present in the solution. The hydrogen ion is treated as the ligand and the amine molecule as the central group. The calculations follow the same pattern. [AHf] + 2lAH,'+l = [A] + [AHf] [AHlstI
"
Substituting the value of [A] determined immediately above in equation (1I),
Defining a quantity fia
= (n -
+
With a knowledge of n sets of values of n, and [H+], the above equation may be solved by simultaneous equations for q d . Again the method of Block and McIntyre (9) is used. The hydrogen ion concentration may be determined from the pH and the activity coefficient. may be determined as follows: %a
=
=
bound hydrogen/totel amine (no metal present) Ca - [H'I [OH-]
+
(18) C* Each of the quantities on the right side of the above equation is easily found from the data.
I t may be seen that fin is the average number of protons bound per unchelated arnine molecule. Thus from a knowledge of CA, Ca, CX, and qa' and the meaaured hydrogen ion concentration, both [A] 330
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Journal of Chemical Education
Thermodynamic Constants
Complete thermodynamic constants may be calculated from the data, if formation constants have been
detcrmined a t more than one temperature. Free energy may easily be calculated from the relationship -AFo = RT In K . Enthalpy is obtained using the expression K) = Ra(ln ----
- ~0
Wl/T)
Sample Calculations
The enthalpy may be determined by plot.ting log K against 1/T and multiplying the slope of the resulting line (assuming constant enthalpy) by 2.303 R. Entropy is determined from the equation AF" = AH0 - TAS". McIntyre, Block, and Fernelius (10) list the thermodynamic data for both protonation and coordination of ethylenediarnine.
P m t a a t i a Reaction:
"V" Vwd
+
= Vi.iti.l V log [H+] = log (Hf)/r* = -pH - log r+ [H'l log [OH-] = -pK, pH - 1% r*
+
+
pH = 9.79 V =7.51 ml = 107.5 ml = -9.79 - (-0.03) = -9.76 [H+] = 1.74 X log [OH]- = -13.84 9.79 1-0021
+
[OH-] = 1.18 Xi0-l CH = 0.201 mmolej102.5 ml = 1.96 x 10-3 M (2.50 m1)(0.05507 M ) = (102.5 ml) = 1.34 X lo-' M , -- (1.96 X lo-') 1.34 X lo-'
[OH-] Ca = mmole acid/ml CA = mmole amine/ml %a = (CH - [H+I
pH = 6.88 V = 2.50 ml = 102.5 ml = -6.88 - (-0.03) = -6.85 [H+]' = 1.41 X lo-' Log [OH-] = -13.84 6.88 - (-0.03)
+ [OH-I)/C*
log qa, = 9.78 = log KHI
Note that the denominator of the above quotient is identical to the numerator of the preceding. log PHI = 6.79 log KHS= log qm 2 log qf = 6.79 2(-0.03) = 6.73
+
+
Cwrdination Reaction:
"V" Vt,t.,
= Vi.iti.1
+ -V 1%
log [Ht] = -pH
pH = 4.64 V = 2.50 ml VtOt.,= 102.5 ml log [H+] = -4.64
+ 0.03 = [Ht] = 2.46 X~O-'M
Y,
- A fil
lH+l [OH-] CE = mmole acidjml C.
=
mmole metd/ml
CA = mmole amine/ml log q ~ t [ H + l= log qm
IA1
+ log lH+l
C. [OH-] CH - lH+l = qHl[HfI 2q~IqxW+]'
+
+
J, J, J
J,'
[OH-] negligible CH = 0.201 mmole/102.5 ml = 1.96 X M Cy = 0.100 mmole/102.5 ml = 9.75 X lo-' M (0.0602 M)(2.50 ml) C* = (102.5 ml) = 1.47 x 10-3 M log qal[H+] = 9.78 (-4.61)
-
(1 (2 (1 (2
+
+
-540
[H+l = 3.98 X - ~ O -M O [OH-] negligible CH = 0.201 mmale/104.25 ml = 1.93 X lo-* M C. = 0.100 mmole/104.25 ml = 9.58 X lo-' M (0.0602 M)(4.25 ml) c* = (104.25 ml) = 2.45 X M log qaL[H+]= 9.78 (-5.40) = 4 XR
+
[A'] = 1.93 X lo-' 1.94 X lo-" 12.04 X lo6 4.49 X 10' = 1.60 X lo-$ = 4.32 X lo-'' - %)([A]) = (0.492)(4.32 X 10-'I) = 2.12 X 10-It - ir)([A])* = (1.492)(1.87 X lo-") = 2.79 X lo-" - ) ( [ A ] ' ) = ( -0.536)(1.60 X lo-') = -8.56 X = 1.19 X 10Ps - @')([A]')¶= (0.464)(2.57 X - "J2 = 2.18 X 10'0 K' = JJ,' - J,'J2 log K , = 10.34 R'J1 - G1' = 77.78 X 101 K2 = g9- J*, log K2 = 8.89
IA1
= = = =
K 17
pH = 5.43 V = 4.25 ml Vtat., = 104.25 ml = -5.43 0.03 =
=
a'
Volume 39, Number 7, July 1962
/
331
Discussion
The time required for this experiment depends on the students. If the time required seems excessive, they may be allowed to work in small groups. The experimental part of the work, including the set-up of the apparatus (not including the thermostated water bath) should not take more than three hours. Equilibration is rapid with this system. The calculations, however, ordinarily will take the students more time. An experienced worker can do triplicate calculations in two hours, but of course the students work more slowly and with less confidence. Choice of which points to calculate can be very important to the time factor. The best points to choose are those which will yield values of 7i at approximately 0.5 and 1..5. No n value should be used which differs from these values by 0.3 units. Estimation of which points to choose may be easily done. For the first point, li = 0.5, 0.5 mole of amine per mole of metal ion should be added. For the second point, li = 1.5, 1.5 moles of amine per mole of metal ion plus enough m i n e to neutralize the acid should be added. The first step of the coordination process is almost complete before appreciable acid is neutralized, but the acid is over 90% neut.ralized before the second step of the coordination reaction is half completed. With other systems, inspection of the amount of pH rise will allow approximate calculation of the quantity of amine to be added for near half-integral values of n. It should be noted specifically that each titration point need not be calculated with the use of the method of Block and McIntyre. Superior students may be assigned hexacovalent metal ions, such as nickel ion, to investigate. The procedure is the same as for the copper ion, except that a third n value must he determined. The calculations are of the same form. However, as might be expected, the simultaneous solution of three equations involves much more effort than the corresponding solution of two equations. An experienced worker can do triplicate calnulations in six hours. Calculation of the protonation constants should take no longer than an hour, for an experienced worker. The lowering of t.he titration curve (Fig. 2) by the ad-
dition of metal ion gives a qualitative indication of the stability of the coordination compound. Copper ion lowers the curve a good deal; nickel ion lowers it less; cadmium ion scarcely lowers it at all. The acid is added in the first placeto retard coordination somcwhat, so that, the first step in the reaction may be measured. In alkaline medium, the first step of the copper coordination vould be too extensive to measure. Any side reactions which affect the pH of the system would seriously affect the results of the det,ermination. Format,ion of basic species, such as CuOH+, would invalidate the results of the experiment. If such a species is formed, the values for each K will differ from those calculated using different ri values. Admission of carbon dioxide to the system will have a similar effect. A det,ailed set of directions, suitable for presentation to the students, may be ohtained from the author. Acknowledgment
The author is indebted to Dr. W. Canard Fernelius for suggesting the need for such a paper and for criticizing t,he manuscript. Literature Cited (1) BJERRUM, J., "Metal Ammine Formation in Aqueous Solution," P. Haase and Son, Copenhagen, 1941. (2) BJERRUM, J., S C H W ~ Z E N R AG., CH AND , SILLEN,L. G., "Stability Constants: I. Organic Ligands," Special Publication No. 6, The Chemical Society, London, 1957. (3) YATSIMIRSKII, K. B., A N D VASIL'EV,V. P., "Instabilitv Constants of Complex Compounds," Pergamon Press, New York, 1960. (4) AJERRUM,J., S C ~ ~ ~ ~ E N B G.,A AND C H SILLEN, , L. G., "Stability Constants: 11. Inorganic Ligands," Special Publicstion No. 7, The Chemical Society, London, 1958. (5) DOLE,M., "The Glass Electrode," New York, John Wiley ck Sons, Ine., 1941, p. 315. L , Physik. Z., 24, 185, 305 (6) DEBYE, P., AND H ~ ~ C K EE., F., "Outlin~sof P h p i c ~ l (1923); as described in DANIELS, Chemistry," New York, John Wililey and Sons, Inc.,
----.
1051. n 512 r-
(7) BERTSCH, C. R.,Doctoral Dissertation, The Pennsylvania State University, 1955. (8) MCIXTYRE, G. H., JR., Doctoral Dimertation, The Pennsylvania State University, 1952. (9) BLOCK,B. P., AND MCINTYRE, G. H., JR., J . Bm. C h m . Soc.., 75.. 5667 (1953). . . (10) MCIRTYRE, G. H., JR., BLOCK,B. P., AND FERNEMUS, \T. CONARD, J . Am. Chem. Soc., 81, 529 (1959).
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332 / Journal o f Chemical Education
