A DILUTION LAW FOR UNI-UNIVALENT SALTS BY BOHDAN SZYSZKOWSKI‘
The fact that the solutions of Ptrong electrolytes do not obey Ostwald’s dilution law, and consequently the law of mass action in respect of the equilibrium between the undissociated molecules and the ions, was apparent since the beginning of Arrhenius’ theory of electrolytic dissociation. As, on the other hand, the law of mass action follows directly from Planck’s characteristic thermodynamical function @ (p,T,no, nl, n2. .) by equating its variation a t constant pressure (p) and temperature (T) to zero, under the aseumption that the intrinsic energy (V) and the volume (V) are linear functions of the number of molecules no, nl, n2. . . ., all hypotheses underlying the calculation of the intrinsic energy must be considered and analysed before a definite step can be taken towards improving the theory. The numbers of molecules of different) kind, no, nl, n2. . ., where suffix o applies to thc solvent, suffix I to the undissociated molecule, and suffixes 2 and 3 to positive and negative ions, depend in the first place upon the degrees of dissociation a which are not measured directly but calculated from conductivity data or from the data of other methods, e.g., from the lowering of the freezing point. The calculation of a from the conductivity data, which are the most reliable and precise wit)hin a very wide range of concentrations, involves an arbitrary hypothesis regarding the influence of viscosity (11) upon the mobility of the ions, namely, it can be assumed that a=X/Xo, or a = X v / X o v ~ , or more
.
generally a
= Xf -where X Xof(v0)
denotes the equivalent conductivity and XO the
1
equivalent conductivity at infinite dilution. As there is no method allowing of the choice between the great variety of possible forms of the function f(v)/f (90) the values of the degree of dissociation cr remain completely undetermined, and, therefore, cannot be used as fundamental experimental data for the verification of different theories of electrolytic dissociation. This is the reason why, in spite of the greatest ingenuity displayed in that direction, no satisfactory theory explaining the anomaly of strong electrolytes has yet been forwarded. The aim of the present paper is t o show that by rejecting the classical conception of partial dissociation of strong electrolytes into ions, and by admitting Ghosh’s hypothesis of complete dissociation of salts into ions independently of the concentration of their solutions, a theory of electrolytic dissociation compatible with experimental data can be established. As the first step towards the general solution of the problem, a thermodynamieal theory of dissociation of strong electrolytes of uni-univalent type in aqueous solutions Professor of Physical Chemistry at the University of Krakow.
I94
BOIIDAN SZYSZIOWVSKI
has been worked out and verified by the author. The thermodynamical method has been chosen here because it does not imply any aswumption regarding the mechanism of electrolytic dissociation. Let UP suppose with Ghoehl that strong electrolytes are completely dissociated, and with Planck2 that their behaviour is due to the fact that their intrinsic energy is not a linear function of the number of molecules. If we denote by U, S, C and 9 the intrinsic energy, the entropy, the Gibbs’ constant of entrofy and Planck’s characteristic function of the solution, further by uo and So the intrinsic energy and the entropy of I gram-molecule of the solvent, by u and s the intrinsic energy and the entropy of one gram-molecule of the electrolyte, i.e. of the cation and anion jointly (u=u1+u2,s=S,+S~), and by no, nl=nz the number of gram-molecules of the solvent and of both ions; if we nl and take further for the definition of the concentration of ions c1=cz = c =-, no consequently calculate on this assumption the concentration of the molecules of the solvent co= I - zc; then, in case of complete dissociation, the intrinsic energy of the solution would be U = nouo+nlu, according to the classical conception that U is a linear function of the number of molecules, and Ghosh’s hypothesis would consist in the addition of a non-linear term U=nouo+nl [ ~ - G c l / ~ in ] , which expression G denotes the constant of Ghosh. Ghosh’s term, -Gc1l3, takes account, of the electrostatic field due to the ionic charges. The negative sign points to the fact that the resulting force acting between ions is attractive. As Ghosh’s constant G calculated from conductivity data does not keep constant, but first rises, passes through a maximum and then falls with dilution, it is obvious that his amimption regarding the second term of the intrinsic energy, is not correct. On the other hand, his formula does not allow either of a physical interpretation, because although the value of u - G d 3 , as c tends to the limit zero, approaches u in accordance with the classical theory, the fact however that dissociated ions do not recombine cannot be explained by a formula in which only attractive forces acting between ions are taken into consideration. But if, instead of Ghosh’s function, the following function for the intrinsic energy be adopted U=nouo+nl [u- P ( c 9 + y ( ~ 1 / 3 ) ~ ] (1) A complete agreement between the theory of electrolytic dissociation and experimental data is reached. In the last two terms of this formula expressing the energy of the electrostatic field account is taken of attractive [ - P (@)“I and repulsive [+y ( ~ l / ~ forces, and as m > n the repulsive forces fall more rapidly with the dilution than the attractive, and at the limit, as the concentration approached zero, first vanish the repulsive and then the attractive forces. With increasing concentration a point is reached when attractive and repukive forces balance each other, and consequently /3 (c1/3)”+ y ( c ~ / ~ ) ~ for = o ,still higher concen-
-
Ghosh: Z. physik. Chem. 98,211 (1921). Planck: ‘‘Vorlesungen ubcr Thermodynamik,”$273 (1921).
) ~ ]
A DILUTION LAW FOR UNI-UNIVALENT SALTS
I95
trations the sum of both terms assumes a positive value, i.e. the repulsive forces predominate and do not allow to dissociated ions to recombine into neutral molecules. By assuming with Planckl zero for the constant value of the pressure, p = o , which is equivalent to the assumption that the measurements are effected in vacuo instead of under atmospheric pressure, the differential equation of the entropy, ds= d‘+pdV’ T
can be brought t o a simpler form
du dS=,I; and consequently the entropy of the ions a t constant temperature will be
s=-u
T
The entropy of the solution will assume then the form
S =noso+na+C where C stands for Gibbs’ constant
(4)
C = - R(nologco+anJogc) (5) If we introduce further the abbrevint,ions cp for the terms which depend upon the temperature and the pressure but are independent of the numbers of molecules no,nl.
.
U
,
( cp = S- - ), Planck’s characteristic function
T
U
CP (p,T,no, n1= 8- - in virtue of equations (I), (3) and (4) assumes the form T’
for the solution, and W = nolqol (7) for the vapour or solid pha.se in equilibrium with the solution. The vapour or solid phase consists here obviously of the pure solvent. The condition of equilibrium between the liquid and vapour, or liquid and solid phase will follow from the variation of @+a1at constant temperatzire and pressure
(dW+d@,,T)
(8)
=O
As the variation? of the numbers of molecular species present are to each other in the relation
mo dno :dnl :dnlo = - I :o :mo’
* Planck: “Vorlesiingen uher Thermodynamik,”
$ 273 (1921).
196
ROHDAN SZYSZKOWSKI
where mo and m 1 0 denote the molecular weights of the solvent in liquid and vapour or solid phase, equation (8) assumes the form
mo cpo1-pO+nl Td(c1/8)n-nl-d(c1/3)m P Y +dC=o mlo T
(9)
or considering that co= I - 2c, and d C = -Rlogcod11o=2Rc and
it follows
The value of log K, a t constant temperature and pressure, is constant, and for the pure solvent, where c=o, is obviously equal to zero. On the other hand, under the same conditions tJhefreezing or boiling point of the solution (T) is evidently equal to thc freezing or boiling point of the pure solvent (To) (log K ) T = T ~ = o (1 1) Expanding the difference logK- (1ogK)T= T~ with regard to temperature into Taylor’s series and breaking with the first term,
and considering further that
( ’??)P -
1’
-- where r is the latent heat of - RTo2’
the transformation, we obtain from equation8 (IO), (I I) and r -2(T-To)=2~(~-
RTo
n B -c 3 6 RT
+ mX TyC 32)
( I 2)
the equation (13)
which allows of the calculation of the lowering of the freezing point or of the raising of the boiling point (T-To) as a function of the number of molecules present in the solution and of the constants P and T.
AF, on the other hand, the lowering or the raising of the transformation point on the basis of the clasFical theory is given by the expression RTo2 T-TO=(I+~)C-- r where a denotes the degree of dissociation, the comparison of equations (14) and (IS) yields the relation between the degree of dissociations of the old theory and t,he cogstants P and y of the new theory
A DILUTION LAW FOR
UNI-UNIVALENT SALTS
197
01'
It is assumed further, in correspondence with experimental data, that m=2n, which gives to equation (17) a very convenient form for experimental verification 3
- P- -- ;(I-a) RT
(c'/~)"
+
2
Y
-(cI'3)n
RT
(cl/s)" is taken here instead of en in order to compare more conveniently the author's conetant (18) with Ghosh's constant
Formula (18)was verified on Kohlrausch's conductivity data from which the degree of dissociation was calculated as a=X/ho. On this assumption eg. (16) takes the form h=Xo
(
I-
--
+--3
cnI3
211
p / 3
RT
It is obvious from eq. (20) that the equivalent conductivity A, and consequently the mobility of ions increases with dilution, and for c = o reaches its limiting value XO. On the other hand, in order to satisfy the condition that, at concentration c = I, h must be smaller than Lo, (h),,l /2y/ obtains in all investigated cases. The average clase value of the exponent n changes symbatically with the class value of P/RT, but within a class the oscillations of n are irregular. The experimental error A% oscillates between 2y0and 5%. Considering the wide range of concentrations from 0.0001norm. to I norm. and the fact that the theory has been decided for dilute solutions, where different definl nl nr nitions of the concentrations, such as c = -, or c = orc=no no+nl+nz.. , Liter, are practically equivalent, the agreement between the theory and experimental data can be considered just as satisfactory as for the dilution law of Ostwald. Salts undergoing hydrolytic dissociation with liberation of hydrogen (NH4C1) or hydroxyl ions (KCH3C02)behave normally down to the concen: tration 0.005, as can be infered from Tables XVII and XVIII, and possess a constant characteristic of their class; below this concentration and down to o .0002norm. the constant falls rapidly. As the degree of hydrolysis increases with dilution and is inversely proportional to the square root of the concentration-the rapid and continuous fall of the constant is due to the increasing influence of the exchange of slow salt ions for very mobile hydrogen or hydroxyl ions.
.
A DILUTION L I W FOH UNI-UNIVALENT SALTS
209
Hydrochloric acid (Table XIX) behaves differently from salts. In the first place its constant P/RT can be calculated only to 0 . 5 norm’, and even in this shorter range the oscillations OQ its value reach+ 10%; in thesecond place, the exponent n, which analogously with results obtained for salts should be less than for KI is, on the contrary, greater than for LiI03, namely I . 55. Here, obviously, formula (IS) which is characteristic of the behaviour of salts cannot be applied with the same degree of accuracy. Thus, the behaviour of strong acids is in some way different from that of salts. Physical Interpretation The exponent n/3 i p very characteristic. In Ghosh’s theory its value is I/,?, and c’’~measures the reciprocal value of the distance between ions. Bat this statical conception, as was demonstrat,cd by Debye and Hucke12, is not correct. The treatment of the problem by. the methods of statistical mechanics leads to the square root of the concentration instead of the cubic root of Ghosh’s theory i.e. to the value of the exponent I/Z instead of 113. The value of the exponent, 0.44--0.51, calculated in the present paper yields a strong experimental support to Debye’s theorys. Debye has given a scientific foundation to Ghosh’s hypothesis but did not develop it, therefore, the fact that Debye’s formula is applicable only to very dilute solutions can be considered as a definite proof of the incompleteness of Ghosh’s conception. Debye and Ghosh consider the ions as points or volumes carrying uniform positive or negative electric charges, and do not take account of the constitution of the ions which follows from Bohr’s theory and Kossel’s concept’ions according to which the ions of the elements adjacent from both sides to the group zero of the periodic system possess the same constitution as the corresponding inert gases of this group. In this way they are coneti tuted of a nucleus carrying a posit’ive charge, and, on the outside, of a very stable kinematic system,of eight negative electrons. The difference between positive and negative ions would consist on this assumption in the f a d that for positive ions the positjive charge of the nucleus is greater than the total negative charge of all the electrons of the ion, while for negative ions it is smaller. A t great distances i.e. at great dilutions the integral positive or negative charge of the ions is characteristic of their behaviour, and cations and anions can be considered as positively or negatively charged spheres attracting each other, as is assumed in Debye’s theory. But with decreasing dist,ance,i.e., with increasing concentration the repulsion of negatively charged outer rings becomes more and more prominently superposed to the attraction. I n t,he last term of the intrinsic energy of formula (I), +y(c1,9”, account is taken of this repulsion and, as in this way a very satisfactory agreement with experimental data is reached, it must be considered as an important experimental support of treating the ions of electrolytic dissociation as electrostatic spheric doublets with ‘ A t 1.0 norm. the deviation of the constant from the arerrtge value attains 30%! Debye und Hiickel: Physik. Z.24, 185 (1932). The result,s of this paper were communicated t o the Polish Chemical Congress in Warsaw, April 4th, 1923, before Debye’s paper appeared (May 1923).
210
BOIlD AN SZY SZKOWSKI
positive charge in the centre and negative on the surface of the sphere. From this point of view the stability of the exterior electronic ring of the ions is answerable for the fact that oppositely ch?rged ions cannot recombine just in the same way and on the same ground, as for the fact that atoms of inert gases do not combine into molecules with atoms of the same species or into chemical compounds. with atoms or molecules of other species. Therefore, this paper must be considered as a strong experimental support of Kossel’s hypothesis. The value of the constant /3,/RT is characteristic of the electrostatic field created by the ions. In the first class of salts the constant is the lowest and consequently the electric field the weakest. This is in perfect agreement with the fact that K, RC, Cs and C1, Br, I ions possess the highest mobility, the mobility of ions being on Born’s theory1 the greater the smaller is the electric field created by the ion. On the other hand the ions of class IV, e.g., Li possess the highest value of PIRT and the lowest mobility 33, at the same time their electrostatic field reaches the highest value. The univalent ions of groups I B and 111 B of the periodic systema s Ag and T1 behave somewhat differently; inspite of their high mobility 55 and 66 their constant P/RT is greater than for the K and Na ions (P/RT= 1.45). This points to the difference in the stability of the exterior electronic system in groups zero and eight of the periodic eystem. The exceptional constitution of the hydrogen ion which consists of a single positive elementary electric charge is answerable for the fact that acids even the strongest as the hydrochloric acid, are partially dissociated while the electrolytic dissociation of salts is complete. There is no exterior ring of electrons in the hydrogen ion preventing its recombination with anions into a neutral molecule. This is the reason why hydrochloric acid does not obey the same dilution law RS the salts (formula IS) and why the majority of organic anions recombines so greatly with the hydrogen ion as to form very weakly diesociated acids. The same tendency of the hydrogen ion to recombine with correspondingly constituted anions is answerable for the hydrolysis of salts of weak acids. Analogous constitutionnl reasons, although they are not so obvious, must apply also to the hydrolytic activity of hydroxyl ions. The physical meaning of formula (20) is obvious. It can be written in the form
where U denotes the mobility of the ion. According to Born2 the energy of a uniform electrostatic field in which ions move will be used up not only towards the contributions of velocity to ions, but also towards the reconstruction of the electric field whose uniformity is constantly disturbed by the irregular superposition of electrostatic fields due to ionic charges, and towards the setting of the axes of the electrostatic Born: Z. Elektrochem. 26,401(1920) Born: Z.Elektrochem. 26,401 (1920).
A DILUTION LAW FOR UNI-UNIVALENT SALTS
211
doublets of the molecules of the solvent which are continually deviated from the direction of the exterior electrostatic field by collisions with other molecules The work used up for this purposc will be the greater the greater are the electrostatic moments of the molecules of the solvent, but these according to the theory of Sir J. J. Thornson' will be increased under the influence of ionic charges which induce upon the molecules of the solvent an additive electrostatic moment. The stronger will be the ionic field, i.e., the greater the concentration, the greater will be also the induced moment, and consequently the greater the energy of the exterior electrostatic field required for the setting of the axes of the doublete. Therefore, the motion of ions in the electric field will be continuously hampered with increasing concentration and the ionic mobility will decrease. According to this theory the decrease of the equivalent conductivity with concentration is exclusively due to the decrease of the ionic mobility, and thus, the theory of electrolytic dissociation of salts is brought to great simplicity and clearness, and these remains no ambiguity created in the old theory by the necessity of defining and interpreting the degree of dissociation. Formula (IS) i? yet capable of further improvement. If we admit with Debye the square root of concentration as a fundamental relation, we can, instead of postulating m = zn, keep n constant and equal to 1/2, n = I / Z , and consider m as variable, 2(I-)U y O P T = -$- +2m-c (23)
y
RT
This would follow from the assumption of the function U = N ~ U , , + N ~ [ U - ~+cym] C~/~ (24) instead of function (I) for the intrinsic energy m would be here slightly different from I , and its value would be characteristic of the repulsive forces acting between the exterior rings of ions, and consequently of the constitution of the ions. As the calculations of constants presented in this paper waR effected before the appearance of Debye's article I could not make use of his square root relation and therefore could not establish if formula (23) would show as good or a better agreement with experimental data as formula (IS). At any rate, before a theory of the phenomenon based on stat,istical mechanics will be worked out the difference between both formulae must be considered as purely formal. Nevertheless the considcration of formulae (23) and (24) makes it intelligible why the exponents n calculated on the ground of formula (IS)are not constant and equal to I/Z but vary from 0.44-0.g1. Still another reason for the variation of the exponent m can be looked for in the fact that, according to Debye's theory, the constant 0 is a function of the dielectric constant of the solvent, which increases its value in a yet unknown way with progressing ionic concentration. Such a stipulation is a consequence of electrostatic moments induced upon the electrostatic doublets of the molecules of the solvent by the ionic-charges of a strong electrolyte. J. J. Thomson: Phil. Mag. 27, 757 (1914).
