Predominance-Zone Diagrams in Solution Chemistry Dismutation Processes in Two-Component Systems (M-L) A. Rojas-Hernandez, M. T. Ramirez, and I. Gonzalez Universidad Autonoma Metropolitana-lztapalapa, Departmento de Quimica, Apartado Postal 55-534,09340 Mexico, D.F. Mexico J. G. lbanez
Universidad Iberoamericana. De~artmentolnaenieria v Ciencias Quimicas, Prolongacion Reforma 880, Lomas de Santa Fe, 01210 MBX~CO, D. F. ~ e i c o The concept of disproportionation or dismutation has been frequently used to analyze the stability of intermediate oxidation states of different elements; nevertheless, i t has only been used sporadically (usually in a relatively nonsystematic fashion) in processes without electron exchange. A recent paper in this Journal explains the construction of predominance-zone diagrams (PZD's) a s a n important aid for the comprehension of aqueous solution chemistry by using a general donor/acceptor/particle treatment (1).However, the dismutation concept is not dealt with i n this reference. In this paper we show the application of the concept of dismutation i n nonredox processes and its relationship to the corresponding chemical-species distribution diagrams and PZDS. During the past few years, we have used this approach in analytical chemistry courses for students majoring i n chemical engineering, chemistry, pharmacy, and biochemistry with gratifying results a t the Universidad Nacional Autdnoma de Mexico as well as a t the Universidad Autdnoma Metropolitana-Iztapalapa. Distribution Diagrams and the Predominance of Chemical Species Systems of the Type ML, L.../ M M L
In aqueous solutions containing two components, M and L, the chemical species, ML,Mh,ML3, ...,ML,
with an associated equilibrium constant,
ie
lO,l,2,..., (i-111. j e l l , 2,..., nl
This notation includes successive and overall equilibria (2).Other formation equilibria t h a t a r e useful i n some cases are also included. Taking logs on both sides of this equation and solving for pL (defined as pL = -log [LI), one obtains the following Henderson-Hasselbalch type of equation.
In Charlot's nomenclature (31, ML, is the polydonor species i n the system; M is the polyacceptor; ML, MLz, ..., ML(,-1) are the ampholytes; and L is the particle. A common, well-known graphical representation for this type of systems is a plot of the molar fractions of the species containing M vs. pL. This i s called a chemical-species distribution diagram (4, 5). Molar Fractions of the Species The equations of the molar fractions of the species of M depend on the concentration of L a s follows.
can be produced by the following formation equilibria,
Table 1. Overall Formation Equilibria and log Values of their Equilibrium Constants for the Systems Used
System Cu(II)-NH3
Equilibrium CU'++ NH3 = CU(NH~)'+
cu2++2NH3 = CU(NH,)?
Proton-phosphates
log p 4.10 7.60
cu2++ 3NH3= C U ( N H ~ ) ~
10.50
cu2++4NH3 = CU(NH,)?
12.50
PO& 4 + H+ =
HPOK
12.34
PO:
+ 2Hi = H2POi
19.54
PO:
+ 3Hi = H3P0,
21.68
Ref 6
6
System Zn(ll)-NH,
Equilibrium
109 B
2n2++ NH, = z ~ ( N H , ) ~ + 2n2++ 2NH3 = Z~(NH,)?
2.40 2.30
zn2++ 3NH3 = z~(NH,)? zn2++ 4NH3 = z ~ ( N H , ) ~
2.60
Mn(ll)+xalates ~ n ' + + OX"= MnOx
Ref -
6
2.10 2.40
7
5.84 Fe(ll)-(ophen) ~ e ' ++ c-phen = Fe(ephen)'+ ~ e "+ 2c-phen = Fe(ephen)$+ 11.20
7
Fez++ 3c-phen = ~ e ( o p h e n ) y 21.30 Volume 72 Number 12 December 1995
1099
(4 Figure 1. Chemical-speciesdistribution diagrams for some metal cation systems as a function of pNH,. (a) Cu(II)-NH,. (b) Zn(ll)-NH,. (c) Ag(I)-NH,. (H fraction of M, A fraction of ML, x fraction of MI+, 0 fraction of ML,, and A fraction of Mb. The PZD's above each distribution diagram are obtained by projecting the intersection points of the greatest fractions over the pNH, scale.)
#c
and [MQl
f ~ 3 [MI E + [MLI + ... + LMLJ
=f&q~Y'
(5)
These functions are monotonic with respect to [LI (or to pL) for the polyacceptor a s well a s for the polydonor because the functions in their denominators are polynomials containing only integer exponentials (positive for M, negative for ML,,). However, the functions for the ampholytes present a maximum because here each polynomial presents integer exponents that are both positive and negative when the expression is simplified with the numerator.
1100
Figure 2. Distribution diagrams and PZD's for systems of the type MLdMLIMIL. (a) Limiting case, where log/#: log or log Gc,M >> 0. .( fraction of M, A fraction of ML, and x fraction of ML2.)
Journal of Chemical Education
When the fraction of MLj is larger than that of all the other species that contain M, then MLj is said to predominate in the system. The chemical-species distribution diagrams shown throughout this paper can be easily constructed by introducing eqs 4 and 5 in a spreadsheet (e.g., Excel 4.0, Microsoft) and using equilibrium constants reported in the literature (6, 7). Table 1shows the values of the equilibrium constants of the systems discussed in this paper.
Table 2. Dismutation Equilibria of the Ampholytes in the Systems
Proton-Phosphates, Manganese(ll1-Oxalates (Ox) an0 Iron(ll)-ofihoPhenantnro ine (opnen) w th Correspond ng Val~esof !he log of the E ~ Jbrun I Constants Arnpholyte
Dismutation Equilibrium
log K
HPO~~-
2 ~ ~ 0 :=-H,PO;
+ PO: ~ H P O ~ H,P04 -= + 2~0:
-5.1 -1 5.3
H2P04-
2H2POT= H3P04+ ~ ~ 0 2 -
MnOx
3H2POT= 2H3P0, + PO: 2MnOx = Mn0x: + Mn2+
-5.1 -1 5.3
Mn0xP
3MnOx = Mn0x: + 2Mn2+ 2MnOx:-= ~ n 0 $ + MnOx
-2.9
3Mn0x:-= 2Mn0$ + Mn2+
-5.0
~e(o-phen?+ 2Fe(oPhen)'+= ~ e ( o - ~ h e n+)Fez+ F ~e(o.phen)~+ = Fe(ophen): + 2Fe2+
Wo-phen)?
0.8 -1.2
-0.5 3.78
2Fe(ophen)P = Fe(o-phen)? 4.7 + ~e(o.phen)~ 9.0 3 ~ e ( o - ~ h e n=$~e(o-~hen)? + + Fez+
I n Figure 1 some distribution diagrams of metallic cations (Cu(II), Zn(II), and &(I)) with ammonia are shown. In the case of comer.. all of the soecies can predominate in a given pNH3 interval. On the contrary, in the cases of silver and zinc the soecies monoamminesilver and diamminezinc cannot predoknate anywhere even though they are present in the system; thus it cannot be said that they do not exist. Although this type of system (where not all the existing species have a predominance zone in the pL scale) is not considered in the paper mentioned above ( I ) , PZD's can be constructed for them. Suff~ceit to observe the pL interval for which the fraction of one species is larger than any other in order to construct the PZD of M-containing species in the system.
..
The Points of Intersection
The points of intersection of the fractions of the species MLi and ML, are given by the values of
accordlog to the Henderson-Hasselbalch type of equation. The uooer Dart of Fi~mreI also shows how the I'ZU:., of a n d h ( 1 1 ) systems can be obtained as the ~ u ( i i )LgU), , projections of the intersection points of the largest fractions on the pNH3 scale. At these points, the molar fractions of the predominant species become equal. Vale et al. ( I ) call these points acceptance potentials; they mark the predominance boundaries of the chemical species. Although PZD's can be constructed this way, this method does not show why some ampholytes cannot predominate in the system. Dismutation Equilibria and Ampholyte Stability Ampholytes can r e a d both a s particle donors and acceptors; this enables them to react with themselves under some circumstances. Charlot (3) has pointed out that it is useful to include the dismutation of ampholytes to simplify
(4 Figure 3. Sample distribution and PZD's for systems of the type MLJML,/MUM/L. (a) proton-phosphates. (b) Mn(ll)-Ox, (c) Fe(ll)(o-phen)..( fraction of M, A fraction of ML, x fraction of ML,, and 0fraction of ML3.) the analysis of polydonor/ampholyte(s)/polyacceptor/particle systems. This is shown below. Two-Pa~ticleDonors (MLz/MUM/L) Here, there is only one ampholyte (ML) with one dismutation equilibrium. 2ML=ML2+M (6) and with a n associated equilibrium constant given by
Two possibilities arise from this: Either the ampholyte can predominate in the system (Fig. 2a), or it cannot predominate (Figs. 2b and 2c). This information can be obtained from the value of the dismutation constant of the Volume 72 Number 12 December 1995
1101
I-
-
-
PI MLj
,
M:fi+l)
- -1 I
(4 Figure 4. Placement of the successive donorlacceptor pairs of different systems according tothe points a, b, and cof Charlot'sconvention (3). (a) proton-phosphates system on the pH scale, (b) Zn(ll)-NH, system on the pNH3 scale. (c) Fe(ll)-(0-phen) system on the p ( e phen) scale. The dotted diagonal arrows signal ampholyie dismutalion processes with equilibrium constants greater than 1. ampholyte, a s shown in Figure 2. Then, in this type of system ML can predominate in some interval of pL values if and only if 0 and thus cannot predominate in the system. (b) The ampholyie ML, dismutates with log = O and thus cannot predominate in the system. (Fhe lines joining the donorlacceptor pairs must bend because both pairs are located at the same O), then that ampholyte cannot predominate in the system. Multiparficle Donor Systems (MLd.../MUM/L) In general, for multiparticle donor systems there are (n- 1) ampholytes with dismutation equilihria of the type, (k - ilMLj = (i- i)MLk+ (k -j)MLi
and their equilibrium constants are written a s
(9)
I t is relatively easy to demonstrate that in this general case, the ampholyte MLj can have up to j(n -j) different dismutation processes; this can be shown by counting the number of species before and after this ampholyte, according to the following sequence. , -.-,Mb, M, ML, ..., MLti-l,, M L ~MLti+~13 j species (n -J) species I n this way, the total number of different dismutation processes (NDP's) is equal to the sum of the individual possible dismutation processes of each ampholyte. "-1
"-1
"-1
NDP = Cj(n -j) = n x j j=1
j4
n(n2 - 1) j2= ----6
essary to establish the type of pairs that must be placed on it. c Use of the Scale for a Polydonor System of the Type MLd.../MUM/L
c.1 Placement of the Successiue DonorIAcceptor Conjugated Pairs. Each pair of the set of independent successive equilibria (MLti.lr + L = ML,) is placed on the scale. The Henderson-Hasselbalch type equation used for each donorlconjugated acceptor pair is
(11)
j4
Only (n - 1)dismutation equilibria can be used to gather a set of n independent equilibria (8)to thermodynamically describe these systems. In addition, a s a general rule, a n ampholyte cannot predominate in a system where a t least one of its dismutation constants is greater than or equal to 1. For this reason, dismutation equilibria can be considered a s stability criteria of ampholytes. Charlot's Scale for the Prediction of Reactions The calculation of the constants of all the dismutation equilibria present in a given system to determine which ampholytes cannot predominate could become cnmbersome; the total number of equilibria could be very large. For example, for a hexadonor system of the type ML$ ...IMLIM/L n=6
Then, when [MLy - 111 = [MLjl,
For Bronsted acids and bases, pK, values (that correspond to successive dissociation equilibria) are usually reported; whereas, log p values (that correspond to global formation equilibria constants) are reported for coordination compounds. For this reason, care must be taken when using these data. Hess' law can be used when necessary to obtain successive constants from global constants and vice versa. Figure 4 shows some examples of the placement of successive formation equilibria i n different systems. As shown, the successive formation equilibria of the protonphosphate system follow the statistically expected order (121, that is,
there are 35 different dismutation equilibria, according to the calculation described above (eq 11).For this reason, Charlot (3)and Remillon (9)devised a graphic algorithm that facilitates the successive and exhaustive selection of only the dismutation equilibria of ampholytes that cannot predominate in the system. We now describe the convention for Cbarlot's scale for the prediction of reactions and its application to the construction of PZD's. Charlot's Convention for the Scale for Predicting Reactions as Applied to Construction of PZD's a. Scale
The scale for the prediction of reactions is a linear pL scale.
ML2
ML
M
In the system Zn(I1)-NH3,
b. Positioning a Donor/Acceptor Pair on the Scale
The conjugated donor/acceptor pairs like those of eq 1 are placed on the pL scale a t the point where [MLil= [MLjl because this implies that
according to the Henderson-Hasselbalch type equation (eq 3). Donors are placed on the upper part, and the acceptors are on the lower part. Then, the placement of this pair on the scale looks like
In the system Fe(IIb(o-phen), l o g S ? < l0gPdL< 10gKj$y
Here, MLj is the conjugated donor of MLi (1, 10, 11).In order for this scale to have a practical application, it is necVolume 72 Number 12 December 1995
1103
is greater that 1and so is that of the dismutation, 2Febphen),2i = ~ d o - p h e n2+) ~ + Fek-phen12+ according to Figure 4c. d. PZD's from the Scale of Prediction of Reactions
From here, two possibilities arise. d.1 All the dismutation constants corresponding to the equilibria of the type
(named here as stoichiometry 2:l:l) are less than 1.Thus, all the ampholytes can predominate according to the statistically expected order shown below. Then, the PZD is a s follows.
Figure 6.Simplification of the scale for the prediction of reactions of the system Zn(ll)-NH3 according to the points dand e of Charlot's convention (3).(a) Initial scale. (b) First and last scale simplification. (Dotted arrows show the assignment of species to their respective predominance zones.) These graphical representations, developed from the convention points described above, show some chemical reactions and the log values of their equilibrium constants a s well a s some dismutation equilibria that are described below. c.2 DirectedDistance. I t is well-known t h a t the log value of the equilihrium constant of the following dismutation equilibria,
d.2 One or more dismutation constants of 2:l:l stoichiometry, represented on the scale, are greater than 1. In these cases, a simplification procedure must be used to obtain the corresponding PZD, a s explained below. e. Simplification of the Scale for the Prediction of Reactions
e.1 Identification. b o n g the ampholytes with equilihrium constants greater than 1, the most unstable-that is, the one with the largest dismutation constant (e.g., M L , b is selected. e.2 Substitution. Those pairs containing the most unstable ampholyte, ML, are removed from the scale and substituted with the pair that involves the products (MLj and MLk) from the dismutation equilibrium that prevents ML, can be obtained by using Hess'law and combining the corfrom predominatingin the system (3,9). responding successive equilibria a s follows. e.3 Exhaustiue Repetition. Points e.1 and e.2 are then repeated until there are no more ampholytes in t h e scale with dismutation equilihrium constants greater than or equal to 1or until there log K = + ~ o ~ K+U~ $ . ML, + L = ML~,,, are no more ampholytes there. 2ML. e.4 PZD's. The PZD is then constructed from 2MLj = MLy+,, + MLti.l) lag K M L ~ ~ =, ,+logZ' ~ ~$:;+:, ~ ~ , -logK%y-", , the simnlified scale. followinp the statisticallv of species. expected order for the An example of a PZD thus obtained is The sign of the term, 2ML
log KML&ML~.,, depends upon the values of hg
Z'$yl,
and 1og*kyL Because successive equilibria have been graphically represented a t pL = log K, with the donors on the upper side and with the acceptors on the lower side, the values of - I ) their signs have also been reprelog K # ~ ~ , , M L ~ and sented, as shown in Figure 5. Here, it is also shown that this convention defines directed distances among the pairs; it defines vectors. So this scale is just a graphical representation of Hess' law like any other graphical method of prediction of reactions (13,141. Then, Figure 4b shows that the equilibrium constant for the dismutation, 1104
Journal of Chemical Education
Figure 4a shows that it is not necessary to simplify the scale of the system proton-phosphates. Figures 4b and 4c show that it is necessary to use the procedure described above for the systems Zn(IIkNH3 and Fe(1IHo-phen). Exa m ~ l e sof the sim~lification~ r o c e d u r edescribed above t i . e , point r of the convention, and the corresponding PZDk are mvcn in Fieures 6 and 7 . The simolificat~onorncess in Figure 6 is stopped when the amphoiyte ~ n ( ~ i 3 ) 2 ~ + is removed from the scale for the prediction of reactions because all the constants of the dismutation equilibria of are the other two ampholytes (zn(NHdZt and z ~ ( N H ~ ) & less t h a n 1.0n the contrary, the scale for the system
-
b of the convention) is applicable to other equilibria in addition to the dismutation cases discussed here. Other points of the convention (not treated here) allow for the placement of the particle L on the pL scale as well a s for the overlap of all the necessary reduced scales in order to describe the composition of the solution. This enables the prediction of reactions in polydonor systems that exchange the same particle L, for example, MLnI...IMLIMIL, AL,I ...IAWAL, BLb/.../BWB/L This leads to adequate models for the calculation of equilibrium compositions in systems involved in processes such a s titrations, hydrometallurgy, biochemistry, and processes in aqueous solutions in general (3,9). The complete Chadot's convention is not presented here due to space limitations and because our main objective is to emphasize the importance of the dismutation processes in the construction of PZDH. We have used Ringbom's nomenclature (15) for the labeling of the different equilibrium constants in an attempt to use one single convention because the IUPAC has not established a convention for writing equilibrium constants of dismutation processes.
Figure 7. Simplificationof the scale for the prediction of reactions of the svstem ~,~ Fe1ll)-(0-ohenl . . . , . accordina to the Doints dand eof Charlot'sconventlon (3) (a)Inma1 sca e (5F m &ale jmpltf cat on snowmg tne o smdlaton of tne ampholyte Fe(ephen1 '. (cl Second an0 last scales m p ilcallon. (Done0 arrows signal me asslgnmenl of species to their corresponding predominance zones.) ~
F e i l l ~ o - p h e nmust ) be simplified twice, as shown in Figure 7. This can be understood in the light of the above d ~ s russion bv lookinv at the corres~ondinedata in Table 2. In Fi&e 7b thz dismutation kquilib&n constant for ~~
~~
~
(not plotted in Figures 7a nor 7c) is greaterthan 1, 2t
Fe(o-phen)z* ~ ~ ( ~ ,pelf - = ~ 1.89 h > ~ 0~
)
Acknowledgment This paper is dedicated to Gaston Charlot (in memoriam) and Bernard Tr6millon for their contributions to the knowledge of solution chemistry. We acknowledge financial support from CONACyT. Literature Cited
= %~e(o-phen)?+ I?e2+ 3/2~e(o-phen)~+
log a
Conclusions The main contribution of the concept of dismutation equilibria as applied to solution chemistry is that it explains the nonpredominance of some ampholytes in a given system. The use of these equilibria combined with another graphical method of prediction of reactions (3, 9, 13, 141, can lead to the construction of PZD's without having to use the corresponding chemical-species distribution diagrams. The PZD's discussed here (and earlier (I)),describe twocomponent systems. Actually, real-life chemical processes usually involve more than two componmts; nonethelrss, the algorithm discussrd here can 11rpmcralizcd for multicomponent, multireacting systems inaqueous solutions by considering the formation of mixed complexes (16, 17), polynuclear species (18),condensed-phase equilibria (19, 20) and electron exchange (redox systems) (21).
~
Then, the constant for the equilibrium 3 ~ e ( o - ~ h e n=) ~ + Fe(o-phen)$ + 2Fe2+(which represents the same dismutation process) is also greater than 1, 2+
~ O ~ K " ~ [ ~= 3.78 $ ~ >~0~ , ~ ~ , Z ~
1. 2. 3. 4. 5.
6. 7. 8. 9.
In the simplified scales for the prediction of reactions that appear along the procedure outlined in point e of the convention, some of the dismutation processes thereby represented no longer have a 2:l:l stoichiometry. Care must be used with the interpretation of the dismutation processes represented in the simplified scales. Observations Charlot's convention for the scale for the prediction of reactions is more comprehensive and powerful than we have shown in this limited space because the pL scale contains more than the dismutation equilibria, that is, other reaction equilibria among donors and nonconjugated acceptors not discussed here. The directed distance (defined in points a and
11. 15. 16. 17. 18. 19. 20. 21.
Vale, J.: FernBndezPereira, C.; Alcalde, M. J. Chem Educ. 1993, 70,790-795. Lagowski,J. J. Modern Inorganic Chemistry;Oekker: New York, 1973; Chapter 17. Charlot. G. Chimis Anolyliyue G4ndroIe. Part I: Marson: Patis. 1967. Kdthoff I . M.; Ssndell. E . B.; Meehan. E. J.: Brnckenaiein. S. Quonliloliu~Chemien1 Analysis: MacMillan: Iandon, 1969. Hdgfddt. E. In l h s t i s e onAnalytica1 Chemistry; Kolthoff, I . M.: Eluing. P J.. Eds: Interscience:New York, 1979: Part 1. Val. 2, Section D (Continuedl, Chapter 15. Part A. Inorganic Ligands Hogieldt, E. Slobdity Constants ofMeLal-Ion Compler~s: (IUPACChemical Data Seties, No. 211; Pergamon:New York. 1979. Perrin. D. D. Stability Constonis of Metal-Ion Complaes: Palf B:Organic Ligands llUPAC Chemical Data Seties, No. 22): Pergamon:New York, 1979. Smith. W. R.: Missen, R. W. Chemicol Reaction Eouilibrium Analysis: Thwly and Algorithms: Wiley: New York. 1985. ~4millon,B.El~cfmhimi~An~lylique~LR4aelionsenSolulon. Volume I:R4anionr
1952. Fmsf. A. A. J. Am. C k m . Snr 1951, 73,2680-2683. Ringb0rn.A Compleration in Anolylicnl Chemistry: Wiley: New York, 1963. Rajas, A,: GonaBler. I.And. Chim. A d o 1986,187.279-265. Roirs-Hemindez, A ; Rarnirez, M. T.; Ihanez, J. G.: Gondlez. I. A n d Chim. Aeto i m , z a 6 . 435442. Rojas-Hemindez, A ; Ramirez, M. T;GonzBlez. I.: banes, J. G . A n d . C h m Aelcl 1992,259.95-104. Rojar-Hemindez. A,; Ramirez, M. T;Gonz8e8, lAno1. Chim Acfo 1993,278,321333. RQjas-Hernindez,A: Ramirez,M. T;GonzBen, 1.A n d Chim. Act. 1993,278,335347. Rolas-Hemindez, A ; Rarnirez. M. T;Ibmez. J. G.:GonrBler, I. J. Electmeham. Soe 1991,138,365371.
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