EQUILIBRIA JAMES J. FRITZ Pennsylvania State College, State College, Pennsylvania
.. ", of simultaneous ionir equilibria (except for the succes-
minor species present. Before doing so, it will be desirable to examine some of the coucepts to be used. The introduction of activity coefficients will not he examined in great detail, since these may he brought in as soon as the major ionic concentrations are known, and have the effect merely of converting the thermodynamic equilibrium constant to concentration units. Starting Materials. We shall, in general, choose as starting materials (i. e., reactants) those substances which would be expected to be present in major proportions immediately after solution. For strong electrolytes, these will usually be the primary ions; for weak electrolytes the undissociated material; and in all cases, water. Sometimes there will he an immediate reaction which goes substantially to completion; if this occurs, we shall consider the products of this reaction as our starting materials. Such reactions might be neutralization (as in the mixing of strong acid and strong base), replacement (as in the additioii of a strong acid to the salt of a weak one), or complete hydrolysis (as in the solution of (NH&P04). Types of Reaction. The problems of equilibrium calculation are greatly eased by the choice of appropriate reactions to express the possible equilibria. We may classify these reactions (for convenience) into several distinct types. (a) Dissociation of acid, base, or water, with equilibrium constants KG,Kb, and K,. Where successive dissociations occur, we shall designate the constants K.,, K.,, etc., with the subscript 1 referring to the primary dissociation. (b) Neutralization of weak acids and bases and its reverse, hydrolysis. Taking as examples NHIOH and HCN, we may write for the neutralization NH,OH + HCN NH,+ + CK- + H20
sive dissociations of neak electrolytes) is seldom discussed. If more complicated cases are considered, the methods described ofteu become too cumbersome to be of any practical value In order to obtain meaningful information about a complex situation (such as the equilibrium in an aqueous mixture of ammonium dihydrogen phosphate and ammonium hydrogen tartrate), it is usually necessary to adopt stringent methods for limiting the amount of ~vorkrequired. Otherwise the calculation may become a major mathematical exercise, and little more. For any chemical mixture of several components one can write a very large number of possible chemical reactions, and, given suitable thermodynamic data, tabulate equilibrium constants for them. In practice, of course, one considers only those reactions which from the standpoint of rate or equilibrium coustant have some reasonable degree of importance. Among the limited number remaining, it will generally happen that one of these, by itself, will express the major course of the over-all reactions; the others may theu be considered as side reactions of varying importanre. We shall call the reaction which by itself expresses the major part of the chemical change the Principal Reaction, and refer t o it frequently below. We may also limit the effort involved in solving the equilibrium problem by considerations of the inherent accuracy of the work. Only a very few equilibrium constants for ionic reactions are knovn to one part in 1,000, most to a few per cent, and some to no better than 10 per cent. Alack of knowledge of the activity coefficients of the ions will often limit still further the accuracy of our final answers. Finally, in many cases, one needs only an approximate ansn-er; an estimation to 10 per The equilibrium constant for this react,ion is given by cent of the pH of a buffer solution, for example, may he K" = K N E , O E K ~ N / K ~ more than accurate enough. For all these reasons we can invoke mathematical approximations when useful, For the reverse reaction, hydrolysis, we may write taking care that we do not destroy thereby the internal K* = K ~ l K ~ n , o a K ~ c x consistency of the ralcnlations. If we adopt the artifice of calling the dissociation conMETHODS OF CALGULATION stant of a strong acid or base unity, we may write The methods of calculation described below will con- general expressions sist essentially of (a) the determination of the Principal K n = K.KsIK, and Kb = K.dK.Ks Reaction; (b) a calculation of the equilibrium expressed by this reaction; (c) calrulation of the various "side( c ) Displacement of hydrogen or hydroxide ion from reactions"; and (d) correct~on,if needed, of the original one weak acid or base by another. If we add acetic calculation, and calculation of the concentrations of acid (HAc) to KCN solution the net reaction is:
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SEPTEMBER, 1953
HAG + CN-
-
443
HCK
+ Ac-
Kd
=
-
-
K E A ~ I K B O N set onlv reactions involving our startine materials.
since only such a reaction can express the gross stoichiometry of the over-all chemical change. HA + X - - H X + .%Calculation of the Equilibria: The steps in calculation A particularly interesting and useful form of this reac- will be as follows: (a) Choose the Principal Reaction. tion occurs when the species on the left are identical, as: (b) Calculate the maior concentrations in the soluHA- + HAHIA A-Ks = K,,IK., tion as if this were the only reaction, using suitable simFor example, in NaHCOa solutions, the major reaction plified relationships for balance of mass and charge. is (c) Calculate the remaining concentrations by use of any convenient set of equilibrium constants. HCO-3 + HCOa- + HzCOa COa-(d) As a second approximation, repeat the calcula, where K,, and K., above refer to the first and second tion (b), substituting into the mass and charge conditions (as constants) the minor concentrations obtained dissociation constants of carbonic acid. We shall frequently find that reactions of type (b) or in (c). (e) Repeat step (c). (c) are more convenient to use than a chemically equiva(fl may be lent set of simple dissociation reactions. .. ... I n rare cases additional approximations Coditions Necessary Jor Solution of the Equilibrium required. Problem: The equilibrium constants alone will not (g) As a check on algebra, verify the fact that all provide sufficient information for a complete solution of equilibrium constants and other conditions are satisfied our problems. The additional data, may be obtained t o sufficient accuracy by the final answers. Step (a) is in fact the critical one. If difficulty is enfrom two types of conditions. These are: (1) One or more equations expressing the fact that the mass of the countered further on, this nsually means that the origsvstem remains constant (balance of mass). (2) An inal reaction chosen was not the principal one. equation expressing the ele&ical neutrality of the solu- EXAMPLES tion (balance of charge). We may illustrate these conThe hydrolysis of0.1 M NH4CN: ditions by application to the dissociation of an acid KEoN = KNa,oa = Kw = H2A,of initial concentration C. I n this case we have five unknown concentrations-[HzA], [HA-], [A--1, [H+] The starting materials are NH4+, CN-, and HzO. and [OH-]-and only three independent equilibrium The possible reactions are: K.,, and K,. The two additional equaconstants K,,, . tions required for the solution are: Kt = 5.5 X lO-'D (1) NH4+ Hx0 NKOH + H+ or more generally
-
+
+
-- +++ + + + +
CNH20 HCN NH4+ CNH20 N&OH HCN H20 H+ OH-
and
+ 2[A--1 + [OH-] = [H+l
OH-
K* = 1.4 X
(2)
KI K,
(3) (4)
= 0.79 = 10-14
Equation (3) obviously represents the Principal Reaction. Equation (I) expresses the fact that the total amount of A case involving two competing equilibria--0.01 M the atom or Group A cannot change. Equation (2) NH4HC03: For ammonium bicarbonate, we may write states that the solution must remain neutral, i. e., that six reactions involving the starting materials, of which the total concentration of negative charge must equal two are significant. Taking the total concentration of positive charge. I n doing K., = 4.5 X lo1; KO, = 5.6 X lo-" t.his, as indicated, each ion concentration must be multiplied by the size of the ion charge; e. g., the con- and other constants as before: centration of a doubly charged ion must be doubled. (Note that equation (2) is unique, whereas equation (1) might be expressed in alternate forms, as might the equilibrium constants.) Considering (1) as the Principal Reaction, we obtain Choice of the Principal Reaction: To choose the proper approximation: Principal Reaction, we must list (mentally at least) all [P\TH*OH]= [H2COI]= 3.42 x lo-' possible equilibria whose reactants involve only our starting materials, together with their equilibrium con- alld stants. Unless the original concentrations of the [NH,+l = [HC03-] = 0.00966 various starting materials differ greatly, the Principal Reaction will be the one with the laxest constant. I n Bv use of the other constants, we obtain any event, a rapid calculation will enable us to choose rl V nVr I) ~ -,--~ ".= 2 n v >" ~n-s pr~perly. ordinarily one of the possible reactions will which is not negligible compared to [HzCOz] although hn.ve a, nonst,ant nmat,er ~-~~~~ bv several orders of magnitude [H+]and [OH-] are. than all others. To begin our second approximation, we temporarily It is most important, however, to consider a t the out[HA-]
(2)
, \
'
~
~
~
-
~~
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JOURNAL OF CHEMICAL EDUCATION
444
fix [COB--] a t the value calculated above. Then, the relation for electrical neutrality, neglecting the very small concentrations of H + and OH-, is [NHdil = [HCOa-I
+ 2[COa--1
= [HCOa-I
+ 6.8 X
It may readily he shown that these concentrations fit all relationships to within the accuracy mhich they are given. The Introduction of Activity Coeficients. In actual fact we are not justified in recording most of the conand centrations to the precision given, sinre we have not iu[NHIOH] + [NH4+l= 0.01 troduced activity coefficients into the calculation. For Then, if we set [NHIOHI = x, successive application of mixed electrolytes at high concentrations, such as in the last example, we are unlikely to have knowledge of the the three equations above yield activity coefficients or any way to estimate them aecurately. This may introduce errors of many per cent in some of the concentrations. If the concentration is Substituting these expressions into (I), and solving for lorn, as in the second example, or if we have a simple other concentrations as before, we obtain: species, as in the first, we may be able to get good activity coefficients. [ N H 4 0 H ] = 3.19 X lo-' [HCO1-I = 9.61 X [NHatI = 0.68 X lo-* [H'] = 1.7 X Measured activity coefficients may be introduced [OH-] = 6 X 1 0 ' lH43Oal = 3.59 X lo-' wherever convenient. Coefficients estimated from the ICOa--1 = 3.6 X lo-& Debye-Hiickel theory can he ralculated only after we These results are as precise as our knowledge of the have some knowledge of the ionic strength; this is usequilibrium constants justifies. ually after we have made a preliminary calculation. A Mixture 1 M i n ammonium dihydrogen phosphate In the second example given, the first approximate and ammonium hydrogen tartrate: For H3POa,take: calculation indicates that the ionic strength of the solution is about 9.7 X M , a t mhich me may estimate KP = 6.2 X K3 = lo-'* K, = 7.5 X the artivity coefficients of the univalent ions as about For tartaric acid (H,Ta), 0.90. Wemay then introduce these activity coefficients into the second calculation. I n the case in point, this K S =-I X 1 0 K, = 9.7 x 10-' reduces the calculated concentrations of XHaOH, In this case we may write 12 reactions, of which t ~ are o H2C03,and H + by about 10 per cent without 'hanging of major import,ance: materially the larger concentrations. The use of activity coefficients may require additional mathematics, since it may be necessary to adjust the values of these coefficients if the calculated ionir strength changes apFrom equation (I), neglecting all else, we obtain preciably on their introduction. In addition, from a mass balance on COa and NH4
REMARKS
By subsequent use of (3, we get
The only other concentration above 10-2is that of NHl+ (2.00 M). If we now- substitute in the equation for electrical neutrality the values for [?u'H,+]and [H2POn-1,me get: [HTa-]
Setting [Ta--1
=
+ 21Ta--1
= 1.025
x, [HTn-I = 1.025 - 22
From a mass balance on tartrate, [H2Ta] = r - 0.025
Substitution of the new expressions into K , and solution of the quadratic gives [Ta--I = 0.157
[HzTxl = 0.132
[HTa-] = 0.711
From K* and the dissociation constants of the two arids, we may evaluate the remaining concentrations.
It is not suggested that the methods given above will always be the most convenient ones, since in many equilibrium problems it is possible to find specifir shortcuts to the information desired. I n the more complex applications of these methods it is often possible to proceed after the first step in a variety of ways, some much more convenient than others. I t is believed that these methods are quite general in their applications to complex situations, that they will ordinarily supply romplete information about the solution considered with remarkably little effort, and that barring mathematical errors they can never give appreciably incorrect answers. With experience it is possible to adapt the methods as needed for the greatest efficiency. The examples above have hinted at some of the ways of doing this. I t should he pointed out that although the methods have been described entirely in terms of ionic equilibria, the same techniques are equally applicable to any situation in which two or more reactions are at equilibrium simultaneously.
