THEORY O F I O S I C EXVH.\SGE
-173
ISTRODCCTIOS
A theoretical discussion of ionic exchange equilibria may be approached either from a consideration of the thermodynamics or statidics of a system at equilibrium or by developing tlic kinetics of the exchange proces. Kinetic theories have been presented hy T-agcler and Kolterstlorf (8), Gapon 12), and .Jenny ( 5 ) . T-ageler and Woltersdorf attempted to apply Langn,uii 'q adsorption theory to the case of ionic exchange. Gapon assiimetl that ionic exchange is a reversible process, leading to the formation of tn o coniplexes, form-ed by adsorption of each kind of exchangeable ion. - i t equilibrium the mtei of adsorption of each kind of ion must be equal. It is assumed that the rate of each adsorption iq proportional to the concentration in the liquid phahc of the ion ~t hich is being adsorbed, to the numLer of the other kind oi ion on the colloidal curface, and t o n constant. These two theories apparently arc Lased upon the assumption that the adsorption is a simple process involi.ing a monomolecular layer on the colloidal surface. Ions are present in the system in one or the other. of tn-o discrete states.-an adsorbed state and a completely fret1 state. This idea is not entire1)- consistent 11ith the concept of a diffuse double layer. THE OYCILLATIOS VOLUME THEORY
Jenny's theory represents the ions as present in a diffuse double layer. Each ion required to neutralize the charge on the particle is oscillating in a n oscillation cell adjacent to a charged area on the particle surface. Other ions from the added electrolyte may enter the oscillation cells, or may remain in the esternal phase. A condensed statement of this theory, x i t h minor changes in terminology, is presented.
Ions o*f equal c1int.g~ The t n o ionic species in an exchange process are labeled w and b. The rate of adsorption of w ions and release of b ions from a surface containing 0 ions may be considered. Each adsorbed b ion is oscillating in the neighborhood of one or more active spots, of unit electronic charge, on the surface of the colloid particle. The numher of active spots per oscillation cell will be equal to the charge number of the b ion. The volume of the oscillation cell is c b . The total 1-olunie of the system is I-. The numbers of wandering w and b ionb present in the system are hr, and Nb, respectively. Wandering members of both ionic species w and b niay be present in the 0 5 -
cillation cell of an adv.n-'~etlb iun. The ixii111wr~oi i w h , I ( and 111. respectively, may T ary from time t o tinic The prolubilitiea. TT-, and R,. oi each number k i n g piesent, and the proldiility, of both lwing present -iinultaneouily may vaiy. The fraction, CY of t h e nmiI)er, j t , ui 11' ion+ present in tlic o+ cillation cell of a b ion \\-hick1\vi11 he a:imbxl TI ill v ~ r y ith n and 111, so that thr probability of adsorption TI ill he CY, vLTf7,LB,,A, $01 an oscillation cell containing I I and ions, respecti\ ely, vf the qpwie. and li T h total prohshility of :idsorption vi11 then 1)e
P,,
a,
=
TT, B,
=
(cy0
+
11, Z3J
n,m
+
(01
lI-i&
+
81
1KLbBi)
+
11'1 E,, 32 TT-,B, + 7 2 ll-oB2) + . . . (1) where the various coefficienth iefei to the qeneral term CY, ,,,. The individual probnliility terms, TV,& :ind R,, are iunctions oi the number3 11 and m of ions actually present u t any inonieiit and of the average nuniberb, p, and p b , present in the oscillation cells of the 0 ions. The terms plo and p b are proportional to the number 01 \T-undering ions per unit volume of the system and to the volunic of the oscillation cell. That is, pl =
(cy?
zb
-
A$-7c
TThe probability teiiiis are then 1Tr,(2b)
=
(PdT --
n!
and
- F p u alld
pb =
lb
7v b
1 -
Bm(ub)=
(Pb)"'
-__
n1 !
e-cb
Equation 1 then becnqiea
The rate of adsorption of IO ions hy the colloid is proportional t o P , the probability of adsorption for each cell; t o z b . the nuniber of cells; and to ~ ( z L ) ,a function of the velocity of thc imp. Thus, \\-e have
l y e may now consider I: surface containing t o ions. Since the number of active spots per oscillation cell is the same for the t\vo kinds of ions if they have the same charges, an expression similar t o equation 3 will hold for the adsorption of b ions in the oscillation cell. of ul ions. -It equililxiuni
or
THEORl- O F IOSIC ESC".i.XGE
475
and y 2 equal zero. FurtherConipariiig equations 1 and 2 , we note that QC,, more, since v b and r , are very small compared n-ith I-, the higher terms in equation 2 may be quite small. To a first approximation tliev may be neglected, pariiczilarly since they occur to the same powcr in both ?:timerator and denominator of the expansion of the right mcmber of equntion 4. The ratio of the esponentials is approximately unity. K e may., therefore., m i t e
Jenny has assumed that the ratios L~~~ ' z b , al/b) a1 and f ( u ) b f ( z b ) w are constants, thc latter t n o equal t o unity, so that K is a constant. Although, by elimination of the higher terms in equation 2, a result identical u i t h Gapon's equation is finally obtained, Jenny's theory is probably t o be preferred. The approsimate character of the final result is explicitly indicated. The velue K may be expected to yary somewhat c-ith variation in S, and N b . The direction, and possibly the approximate magnitude, of this variation might lie predicted in favorable cases. Gapon's theory leaI.es us no such opportunities. Furthermore, as has been stated. Jenny's theory explicitly recognizes the diffuse character of the double layers.
Ions of unequal charge Hofmann and Giese (4) state that in accordance ;\ith an investigation by Jenny they have attempted to derive an equation for ions of unequal charge on the basis of the laly of mass action. The method of derivation is not presented and it is by no means certain that they utilized Jenny's method. However, they employ the terms vcS and Z ' X H ~to symbolize the volumes which are ayailable to the calcium or ammonium ions bound to the clay, i.e., the "oscillation volumes," according to Jenny. It seenis probable that they merely set up an equation analogous to those which may be derived from the lzx of mass action considered as a chemical reaction, but applied the law of niass action to a partition between the external system and the oscillation volumes. Translated into the terms used by Jenny, their equation becomes
By utilizing the statistical method developed by Jenny, with no reference to the concept of mass action. an entirely different expression can be derived. Let 11s consider an exchange involving ionc. whose charge number equals 1, labelled w ions, and ions TI hoqe charge number equals 2, labelled b ions. The criteria for the equilibrium state cannot he that the rate of adsorption of t u ions is equal to the rate of xlsorption of b ions in the caSe of unequal charge. We may set up the simple criterion that the rate of adqorption of zu ions is equal to the rate of release of w ions nt equilibrium. Let us start with a colloid saturated with b ions and consider the rate of adsorption of w ions. The oscillation cell of a doubly charged b ion occupies a space adjacent to two actire spots. The ivandering w ions enter the oscillation
176
L h X S E S E. D.kVIS
cells of the b ions. One w ion is adsorbed \\-hen it gets Iletiveen one b ion and one of the active spots on the colloid, that is, Ivhen it breaks one of the bondi of the b ion. The w ion nil1 be completely adsorbed but the b ion will not be conipletely released. The probability of adsorption is identical with equation 2 for the case of ions of equal charge. We shall now consider a colloid saturated n-ith u: ions which occupy spaces adjacent t o single active spots. Wandering b ions may enter the oscillation cells of the w ions and release the latter. The rate of release of the w ions will he equal t o the rate a t n hich the b ions break the bonds of the w ions and become bound to a single active spot \vith one of the two available bonds. In developing the probability of these events, vie have to take into consideration additional events 11 hich could not occur n i t h ions with equal charge numbers. One or more b ions may enter the double oscillation cell of t n o adjacent w ions, and one b ion may siniultaneously break both bonds of the two w ion3 or, in other words, get in between both w ions and the particle surface. The terms Wo, Wl, W z ,etc., may represent the probabilities that a double oscillation cell shall contain none, one, tTTo, respectively, wandering w ions, and likeuise the terms Bo, B1, B?, etc., refer t o Ivandering b ions. Then for the probability equations Tve hare
P;
=
(aO’rYOBo)
+
(al’TrIB0
+ /3*’TYoB1)
+ (a*’TTT*Ro+ /3*’TTTlRl+ -{?’1T‘,Bz)
* * *
(2
The coefficients are piiined to indicate that the fractions of ions adsorbed in the double cells with 11-andering b’s may be different from those for the case of the w’s entering single oscillation cells of b ions. (The chances of one b ion getting in between the two w ions and the t n o active spots may be different from the chances of one w ion getting between one b ion and one active spot.) Application of the Poisson series leads to the expression (eliminating the esponential term) :
By definition ao’, a:, and 01; are zero. This expression gives us the probability that one b ion will release tn-o adjacent w ions. When one b ion is present in the double oscillation cell of tu-o w ions, it may replace either one or the other alone (and with equal likelihood for each), as well as both simultaneously. When one w ion has been replaced, this event will not influence the probability of the subsequent replacement of the other w ion. The two events are independent. Therefore, the probability of both occurring simultaneously is equal to the product (Le., the square) of the probabilities of each occurring alone. The probability of one event occurring alone will equal the square root of the probability of both occurring simultaneously. Thus,
477
THEORY O F IONIC ESCHASGE
the probability of one w ion being released n-hen one b ion is present in t,he double oscillation cell of two w ions will be
Similar considerations will hold for those cases in which varied numbers of w and b ions are present in the double oscillation cell of tn-o w ions. The total probability of release of w ions, one a t a time, becomes
It may be noted that, after transformation of the coefficients 2
(g)“’,
the
third term in equation 9 represents the first term of the series which would have been obtained if we had considered simply the probability of a single b ion entering the single oscillation volume of a w ion and neglected the other possible ewnts. Mathematically, the terms of higher order will be negligible in comparison with the first term. Physically, the first term represents an event of greater probability than the remaining terms, because the probability of one b ion being present somewhere in a double oscillation cell is greater than that for a &ingleoscillation cell. The chance of the b ion getting in between the particle surface and one or the other of the tn-o w ions in a double cell cannot be significantly different from the chance of so releasing the w ion from a single oscillation cell. An alternative criterion for equilibrium n-ould be that the rate of adsorption of b ions must equal the rate of release of b ions. Equation 8 represents the probability of complete adsorption of a b ion. In this case we n-ould neglect events of higher probability. Likewise, to obtain the rate of release of b ions, equivalent to the rate of adsorption of tn-o w ions simultaneously, we should have to square the terms in equation 2, also leading to the neglect of the terms with higher probability. The proper kinetics involve, for the first approximation, a process analogous to a reaction of the first order in ordinary cheniical kinetics. The higher terms in equations 2 and 8 may be neglected to a first approximation. However, it should be noted that the corresponding higher terms do not have the same power in this caw. This lack of symmetry may be expected to render the simplified equation less nearly exact than in the case of ions of equal charge. The simplified equation will he
It should he noted that there is a volume factor in this expression. The expression may be modified b y transferring the volume factor, I-, to the left side. In this case it ~vouldappear that K is a constant when the concentrations,
478
LAXXES E. DAVIS
1’ and NZL’/lT, rather than the total numbers, Xb and Nu,of the external ions are employed in the calculation?. Houever, JJ-iegner and Muller (9) found in
h-b,
an experimental study that the equilibria w r e conditioned by the total number of ions rather than by the concentrations. Dilution of a mixture of colloidal sukstrate and ordinary electrolyte did not alter the numbers of ions of each ionic species on the colloid and in the external liquid. On the other hand, Eaton and Sokoloff (1) and Melley (6) have presented evidence to indicate that dilution of a sodium-calcium soil system affects the base-exchange equilibria markedly. The total amount of sodium in the external liquid increased with dilution. Jenny has suggested that TI hen the exchanging ions are not very dissimilar, the ratio ?&/ub may not be affected by the number of migrating ions. -4s he points out, the individuel oscillation volumes I\ ill increase ~f it11 dilution because the thickness of the double layer on a colloidal particle varies with the electrolyte concentration. But the ratio of the oscillation T-ohimes may remain nearly constant. Since no volume factor is explicitly plesent in the caSe of ions of equal charge, as indicated by equation -5, the value of K should not be niarkedly affected by dilution except for differences due to hydrolysis. Since a volunie factor is explicitly present in the case of io115 of unequal charge, Tie Phould expect K to vary with dillition in general, \Then total aniounts oi ions in the external liquid me employed in the calculations. The rate of variation with dilution wou!cl depend upon the relative rate< of .;ariation of z’b and n-ith changes in the total x olunle 17. Eellye an.! Hiickel found that for dil\itc solutions of sirrple electrolytcs, the ‘.thicline
