RESTRICTIONS ON THE USE OF JOB'S METHOD

When a measured property is not a strictly linear function of the concentrations of the species involved, the deviation of this property from additivi...
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RESTRICTIONS ON THE USE OF JOB'S METHOD

August, 1958

1005

RESTRICTIONS ON THE USE OF JOB'S METHOD BY MARKM. JONES AND K. KEITHINNES Department of Chemistry, Vanderbilt University, Nashville, Tennessee Received May 10, 1968

There are two restrictions, inherent in Job's method, which must be considered before it can be used on a system. The first arisqs from the effect which activity coefficients can have on the molar ratio a t which the concentration of the complex IS a maximum. The second arises as a limitation on the choice of physical measurements which can be used. When a measured property is not a strictly linear function of the concentrations of the species involved, the deviation of this property from additivity may show extrema which do not coincide with the maximum concentration of complex. For such systems, Job's method in its original form is not satisfactory. Several physical properties which have been used with this method are considered in light of these restrictions and criteria for such methods are suggested.

Job's method (the method of continuous variations, or isomolar solutions) is one of the methods most frequently used to study complexes in solution. Although the method was discovered prior Job's work, Job's presentation5 was of greater generality and especially convenient for use with spectrophotometric measurements. The great increase in the ease with which such measurements can be made has led to the extension of Job's method to more complex equilibria than were treated in the original presentation6-7 as well as to its presentation in a form suitable for graphical use.8 Its striking success when used with measurements of light absorption has led to its use with a number of other physical properties of solutions. The results in many cases have been ambiguous and frequently indicate the existence of complex species whose presence can be detected only by the measurement of a particular physical property of the solution. Such a situation is certainly unsatisfying, and it is the purpose of this paper to show suitable criteria for a system and measuring procedure if Job's method is to be used in the customary manner. In the following treatment the symbolism and method of setting up the equations follow that of Vosburgh.GJ The first restriction on the use of this method can be seen by examining the manner in which the stoichiometry of the complex is related to its point of maximum concentration. For a system in which only one complex species, AB,, is important, the equilibrium for its formation will be A+nBzABn

(1)

Cz = Mx

- nCs

(111)

While these solutions are commonly made up by mixing (1 z)V ml. of an M molar solution of A with zV ml. of an M molar solution of B to give a final volume of solution V , the occurrence of any volume changes in this process does not invalidate the use of the method of continuous variations. It merely makes this particular method of preparing the isomolar solutions unsuitable for such a system. The thermodynamic equilibrium constant for the equilibrium is

-

(IV)

K = %a1aP

where the a's are the activities of the various species. The usual method of establishing the stoichiometric point (where the ratio of A t o B in the solution is the same as that found in AB,) involves differentiation of this expression with respect to x and taking the point where C3 is a maximum (Le., dCa/dx = 0). Here, we will carry out a similar process, but emphasizing the effect of the activities on the value of n when Cs is a maximum. From I V Now, d In K/dx is zero, and it can be shown for the condition of interest By the use of this and equation 11,we can obtain an explicit solution of n as

This system is then studied using solutions in which the sum of the number of moles of A and B added is a constant, M . These are prepared by making a solution which contains (1 - x)M moles of A per liter, and XM moles of B per liter. The concentrations of A, B and ABn will be desig- or, with activity coefficients yi and rearrangement nated CI,CZand Ca, respectively, and are related by n = -CZ

--

C1

M(l

- Z) - Ca

.

(11)

(1) 1. Ostromisalensky, J . Rzass. Phys. Chem. S O ~ .42, , 1332, 1500 (1910); Be?., 44, 288 (1911). Chem., 23, 1830 (2) 0. Ruff, Cham. Ztg., 13, 1003 (1910): 2. (1910); 2.physilc. Chem.. 76, 21 (1911): Ber., 44, 548 (1911) (3) E. Corneo and G. Urbain, BuEI. BOC. chim.. 26, 215 (1919). (4) Y. Shibata, T. Inouye and Y. Nakatsuka, Jopan. J . Chem., I, 1 (1922); C. A.. 16, 2075 (1922). (5) P . Job, Compt. rend., 160, 928 (1925); Ann. chim., [lo], B, 113 (1928): L111, 6, 97 (1936). (6) W.C. Voeburgh and G. R. Cooper, J . Am. Chem. Soc., 68, 437 (1941). (7) R. K. Gould and W. C. Vosburgh, ibid.. 64, 1630 (1942). (8) P. Hagernuller Compt. rend., 230, 2190 (1950).

Cr

ongem.

.

.

. .

(VIII)

If we can assume that the solutions are sufficiently dilute that the Debye-Huckel limiting law can be used in the form In yi = kZiaI'/r

(1x1

where IC is a constant, Zi is the charge on the

MARKM. JONES AND K. KEITH INNES

1006

ith ion, and I is the ionic strength of the solution, then (XI

From this it can be seen that even in the favorable situation which is found when A and AB,, are ions of the same charge type and B a neutral species, the solutions must be dilute enough to follow the limiting law if the stoichiometric point is t o coincide with the maximum concentration of complex (cf. ref. 6). Thus, for a situation where 21= 2, ZZ= -1 and Z3 = -4 (e.g., MXs-4, with Mf2) and CI = 0.001 and C2 = 0.006, n = 0.97C~/C1or 5.82. For the same complex with C1= 0.01 and CZ= 0.06, n = 0.89Cz/C1 or 5.34. I n the usual application, Z1 = Z3 and under these circumstances equation X becomes n = -4

c1

if the Debye-Huckel limiting law is obeyed. If the activities are all unity and independent of concentration, equation VI11 will also reduce to equation XI. Other forms in which equation X I is commonly met are n =

- nC3 - X ) - C3

Mx

M(l

(XII)

or n=-

X

1 - 2

Vol. 62

possible to select a wave length at which only the complex absorbs. I n this case the determination of the value of x which gives the maximum concentration of complex is very easy, The use of physical properties other than light absorption introduces new complications. Job5 has shown that when a physical property of the solution is a linear function of the concentration for each solute species present, the deviation of this property from strict additivity (calculated on the assumption of no reaction) is an extremum when C3is a maximum. Most physical properties of a solution are no2 strictly linear functions of the concentrations of the solutes. Since Job's method has often been used with such properties, it is of some importance t o determine what the relation is, in such cases, between the point where C3is a maximum and that where the deviation from additivity is a maximum. For this purpose we can write a fairly general expression for the deviation, Y , of a physical property from additivity, using anotation similar to that of Vosburgh, as y

= fdCd

+

fZ(C2)

+

fS(C8)

+

fl(M(1

-M M X ) - 2)) - f d C S ' ) (XIV)

fS(Cd

Here the terms have the same meanings as before and C, and C,' represent the concentration of the solvent before and after the preparation of the mixtures. We will assume the f's are of the form CakCik,where the possible values of k are the same k:

(XIII)

Thus when the restrictions on activity coefficients and charge type are met, C3 will reach a maximums at the stoichiometric point. The assumption that the non-electrolytes in such systems have activity coefficients of unity must be used with care.l0 This effect of the activity coefficients would seem to limit the use of the method to systems where the activity coefficients can either be shown to be unity or to be taken into consideration in an explicit manner. That this does not cause difficulties in the use of the method is due to the following two facts: (1) there are a very small number of chemically possible values for n in the systems commonly studied; and (2) measurements of C3 are generally made a t wide intervals, of perhaps 0.1 in 2 apart-a system might actually have a maximum value of C3 at say x = 0.52 but measurements carried out only at x = 0.3,0.4,0.5, 0.6, etc., would show a maximum of 2 = 0.50. Fortunately the displacement, due to -yif 1, is very small for spectrophotometric or other measurements which can be carried out on dilute solutions. Many physical methods require more concentrated solutions, however, and thus must face this complication as well as another mentioned below. Once the requirement of using dilute solutions has been met, a suitable physical or chemical measurement must be selected to determine when the concentration of the complex is a maximum. When spectrophotometric methods are used, it is often (9) That this is a maximum and not a minimum or inflection point follow8 in a straightforward manner from equation XII. (10) F. A. Long End W. F. McDavit. Cbm. Rays.. 61, 119 (1962).

for each substance, and the coefficients a, b and c used for A, B and AB,, respectively. If we assume further that C, and C,' are identical constants, then differentiating this Y function with respect to 2 and use of equations 11,111,VI and VI1 results in

- Z k U k [ M ( l - - Calk-' + (M + 2) 9 + MZkak [ M ( 1 - x)lk-l ax 2)

SL!kCsk-l

M Z l c b b ( M ~ ) ~ - '(XV)

Thus, in general, for an arbitrary physical property of the solution which can be expressed in the above form,,an extremum in Y does not occur a t the same value of x as a maximum value of C3 (i.e., dY/dx = 0 and dC3/dx = 0 do not occur at the same value of x). When the physical property used is a strictly linear function of the concentration the index IC takes only the value 1and d- Y=

(CI - a1 - nbl) dC3

(XVI)

and the necessary and sufficient condition for an extremum in Y is that dC3/dx = 0 (unless C1 = al nbJ,so an extremum in Y will be found at the same value of x as the maximum value of Ca. Under these conditions, and when the activity coefficients are either unity or cancel out, Job's method can be used without modification to establish the stoichiometry and stability (or instability) constant of the single complex in solution. Apart from those cases where k = 1, it is possible to investigate only suffcient conditions for

+

August, 1958

RESTRICTIONS ON

THE

USE OF JOB’S METHOD

1007

extrema in Y . If, in equation XV, dCg/dx is set equal t o zero, one obtains

wave length h and temperature 1. (The usual symbol for the molar optical rotatory power, M , is not used here to avoid confusion with M , the total dY M Z { ~ ~ [ ( MZ nCa)b-’ - ( M % ) ~ - ’ I molar concentration.) dy = Other properties of this first class might be named, a b [ ( M ( l - X) - Ca)”-’ - ( M ( 1 - x))“-’]) (XVII) e.g., nuclear magnetic resonance in some cases, Only for n = 1 and all a k b k will dY/dx = 0. etc., but they are not properties commonly deOf the various physical properties of solutions, the termined for such systems. density seems t o be one property for which this The number of the second group of properties latter condition might be fulfilled. is also small; the only common examples being the If equation XVII is differentiated again and magnetic susceptibility, the refraction and posdCa/dx set equal to zero sibly the optical dispersion. It is possible t o evaluate the contribution of the solvent to these dap - M2Zlc(bk(k - l)[(MX - nc3y-2 d 7properties and t o eliminate it in a straightforward (Mz)’-’]a k ( l c - l ) [ ( M ( I - X ) - Ca)’-’ manner. The measurements can thus be used (M(1 - z))’--”] 1 with Job’s original treatment where the contribuIf the f’s are of the form ale1 U Z Cand ~ ~ blc2 ~ O C Z ~ , tion of the solute species is a linear function of the but with a1 # bl, and az # bz, then concentration. The Y functions for these properties are day - -0

+

+

+

dxe

i e , , complex formation is a suflcient condition for an inflection in the curve. For otherf-polynomials, inflection points will be found only for special conditions on the a’s and b’s. The large number of physical properties which might be or have been suggested as of possible use with this method may be divided into three groups, vix. : (1) physical properties which are characteristic properties of the ions rather than of the solution as a whole; (2) physical properties characteristic of the entire solution but which can be unequivocally (or nearly so) apportioned to contributions due t o each of the various species in the system, the contributions being proportional t o the Concentration of the species; (3) physical properties characteristic of the entire solution which cannot be divided up into contributions due to each species. These latter are especially those properties in which the solvent plays the chief role, such as surface tension. There are very few properties which fall in the first class, the most readily accessible being the absorption of light a t a given wave length and the optical rotatory power. Fortunately, both of these properties are strictly linear functions of the concentration of the species in sufficiently dilute solutions. For light absorption, solutions which follow the Beer-Lambert law and for optical rotatory power, solutions which follow the corresponding relation,11~12are suitable for study by Job’s method as it was originally presented. I n both cases the solutions must be dilute for the strictly linear concentration dependence to hold. The Y functions in these cases are for light absorption

+

+

Y = Z(E1Cl ezcz eaca - C l l l l ( 1 - 5) - e2Mx) where 1 is the length of the light path and the e’s are the molar absorptivity coefficients of the respective solute species a t the wave length used. This equation was first given in this form by Vosburgh and Cooper.6 for optical rotatory power Y = l((ri4 4-a2Cs aaCa - RIM(1 5) cydW5) where the a’s are the molar optical rotations at a

+

- -

(11) P. Walden. 2. phyaih. Chsm., 15, 196 (1894). (12) bT. bmpbell, Tars JWUAY SS, 1143 (lsSl).

magnetic susceptibility Y = XlCl XzC2 XaCa

+

+

+XLM(1 XaC. - - X&X - XXa’ 2)

Here the X’s are the molar magnetic susceptibilities. For dilute solutions the contribution of dissolved salts to the magnetic susceptibility is proportional to the concentration. 13-15 for refraction Y = RlCl+ RzCz

+ RjCI + RBC, - R I M (1 - Z) - R ~ M xRaCa’

where the R’s are the molar refractions of the various species. The molar refraction is a constant over almost the entire range of attainable concentrations for most speciesls and is changed when the species is subjected to strong interaction^.^' The use of refractive index rather than refraction is commonly found in the literature.18-20 For solutions concentrated enough for accurate measurements, the deviation of activity coefficients from unity will usually be large. Measurements which depend upon the fourth decimal place in the refractive index will frequently contain experimental errors of sufficient magnitude to invalidate any conclusions as t o the nature of the complexes present. Tahvonen21 has shown that the results of Spacu, et al., on the KC1-BaClz system are based upon errors of this kind. The investigations of Asmus and Reichl9 on the chloro-complexes of copper may be consulted for a system where the difficulty of accurately predicting the refractive indices of solutions makes any interpretation of (13) G. W.Brindley and F. E. Hoare, Trans. Faraday Sac., 38, 268 (1937). (14) W. R. Angus and D. V. Tilston, ibid., 43, 221 (1947). (15) L. 8. Brant, Phys. Reu., 17, 678 (1921). (16) H.Falkenhagen “Electrolytes,” Oxford Univ. Press., 1934, p. 316 ff. (17) N. Bauer and K. Fajans, “Refractometry” in A. Weiasberner, “Physical Methods of Organic Chemiatry,” Vol. 11, Interscienoe Publishers. Inc., New York. N. Y.,1949. pp. 1141-1240. (18) G. Spacu and E. Popper, Bull. sac. Stiinfe Cluj, 7 , 400 (193i3); Z . physib Cham., BI5, 460 (1934); 8 3 0 , 113 (1935); B35,223 (1937); B41, 112 (1938): G. Spacu. J. G . Murgulescu and E. Popper, 2. physile. Chem., 862, 117 (1942). h‘.a (19) E. Aamua and J. G. Reich, Aneew. Cham., 61, 208 (1949). (20) B. V. Ioffe, Zhur. Obshchei Rhim. 26, 3259 (1956): C. A . , 61, 13632 (1957). (21) P. E. Tahvonen, Ann. Acad. Sei. Fennicae, A49, Nos. 6 and 7 (Urn; C. d,,58, 7176 (1989).

1008

MARKM. JONES AND E(. KEITH INNES

Vol. 62

experimental results difficult. Bauer and Fajansl? these it might be possible to show that they can be point out that the refractive index is rarely, if used in special cases. ever, a strictly linear function of the concentration A further source of confusion may be encountered and may show maxima due to strong, non-specific, in the frequent examples where the method of coninteractions, as in the acetone-water system. tinuous variations is applied together with ideas As a last member of this class one might have the reminiscent of the hydrate theory of solutions. dispersion of the refractive index. Because of the The most commonly found of these is the identidifficulties of measuring small variations in this fication of every maximum or minimum in a propproperty for dilute solutions, it does not appear t o erty-concentration curve with the presence of a be of practical interest. compound whose composition is the same as that of The third class of properties is a very large one as the solution a t that point. So far, the method of many of the common properties of a solution are continuous variations has been used successfully on characteristic of the system as a whole. They are, systems where more than one complex is formed in general, not linear functions of the concentration. only in conjunction with spectrophotometric measSeveral of these will be considered but only one, ~rements.68~For such systems the maximum the heat of mixing, appears to be suitable for use deviations from additivity do not necessarily occur with Job’s method. at stoichiometric points, but are displaced from Ebulliometry, cryoscopy and related colligative them by a predictable amount. Where many properties have been used with this meth0d~~J3extrema are found for a Y function this may be (or systems containing complexes24J6) in several indicative of complexing, but the positions of the cases. While they undoubtedly can be used t o extrema, in general, do not give the composition of establish the presence of complexes in solution, in the complexes in solution. Where the property most cases the concentration of solute species is so is measured on concentrated solutions the addigreat that the activity coefficients must be con- tional complication is introduced that the maximum sidered. These points are clearly indicated in the concentration of the complex is not found at the stoichiometric point. work of Stokes.26 The heat of mixing has been used with a few SllmmRIll systems26--28and analyzed in some detail by SidCriteria for the evaluation of deviations from dhar~ta.~?It appears to be satisfactory for some additivity of a physical property of a solution can systems where the partial molal heat contents of be listed as follows. the various species are independent of the concen1. It is probable that significant deviations tration. The effect of the activity coefficients from additivity of a physical property of a dilute would lead to an error in this case, were not the solution in water are due t o complex formation. possibIe stoichiometric points so limited in number. The expected value may be obtained by estimating Under favorable circumstances, then, this property the properties in an equivalent non-reactive mewill fall in our second classification. dium. What deviations are considered “signifiThere are a number of other properties which cant” will depend on the experimental method and have been used with this method in one form or the nature of the combinations considered to be another. Most of these are either definitely not complexes (e.g., are ion-pairs considered to be linear functions of the concentration or are deter- complexes?). For some physical properties this mined to the greatest extent by the state of the sol- additivity in the general sense will not be found vent. These are the viscosity,29Auidityj6the den- (e.g., the surface tension**). sity,30 the electrical c o n d u ~ t a i i c e , ~the ’ ~ ~surface ~ 2. For properties which are strictly linear functension,33 compressibilitys4 and dielectric con- tions of the concentration, Job’s method may be ~ t a n t . ~ With s none of these properties has it used where only one complex is important and the been established that they are both linear functions activities may be set equal to the concentrations of the concentration of the solute and accurately without serious error. Where two complexes are measurable in solutions of sufficient dilution to have found, the method of Vosburgh and Cooper6 may activities equal to concentrations. With some of be used, other restrictions being met. This last is capable of use with suitable properties other than (22) E.Cornec and U. Urbain, Bull. 800, chim., 85, 215 (1919). (23) F. Bourion and E. Rouyer, Ann. Chim., [IO] 10, 182, 263 light absorption. (1928). 3. For properties which are not strictly linear (24) R.H.Robinson and R. H. Stokes, Trans. Faraday SOC.,41,752 functions of the concentrations, or systems where (1945). the activities are significantly different from the con(25) R. H.Stokes, ibid., 44, 137 (1948). (25) Chavenet, P. Job and G. Urbain, Compt. rend., 171, 855 centrations, the use of Job’s method in its original (1920). form is not warranted. I n such cases the extreme (27) 8.K. 6iddhanta, J . Indian Chem. Soc.. 96, 579,684 (1948). deviations from additivity will generally not occur (28) 8.K.Siddhanta and M. P. Gluha, ibid., 88, 355 (1955). a t stoichiometric points. An exact evaluation of (29) E. Irany, J . Am. Chsm. SOC.,611, 1392 (1943). (30) A. Tian, Bull. soc. chim. [5],18, 407 (19451. the conditions for a maximum concentration of com(31) Y.Wormaer, dbid., [5].15, 395 (1948). plex and for an extremum in the deviation from (32) P.Hagenmuller, Ann. Chim.. [I21 6, 5 (1951). additivity must be carried out for each such situa(33) H. J. Kaai and C. M. Desai, Cuwenf Science (India), 89, 16 tion. (1053); J . Indian Chsm. Soc., 80, 287, 290.291, 424,426 (1953). See 61, 482 however E. A. Heintz and D. N. Hume. THISJOURNAL, This work was supported by a grant from the (1957). Research Corporation. (34) V. 8. Venkatesubramanian, Current Science (India), 90, 13 (1951). (35) N. Trin, Compt. rend.. S16, 403 (1948).

(85) E. A. Guggenheim, “Thermodynamics,” North-Holland Publishing Co., Amsterdam, 3rd ed., 1957, pp. 367-371.

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