Zinc Chloride
0.16 Zinc Sulphate and Po-
tassium Chloride
0.83 Potassium Sulphate and
Zinc Chloride
0.17 Zinc Sulphate and Po-
tassium Chloride
COEFFICIENT OF
PARTITION
0.91 to 0.92
0.83 to 0.84
* Malaguti, Ann. Chim. Phys., (3) xxxvii, 198 (1853).
70 METHODS OF ANALYTICAL CHEMISTRY
Starting from either of the two opposing systems, the limits
reached are the same.
The study of the esterification of alcohols by acids, comparable to
the double decomposition of salts, undertaken from 1860 to 1863 by
Berthelot and Pean de Saint-Gilles* enables one to obtain in a very
clear manner the limits corresponding to the equilibrium. This
limit is almost independent of the temperature, but is attained very
slowly in the cold. It is reached in a few hours at 200. With
acetic acid and alcohol, there is obtained, as a limit, 67.3 per cent
of ethyl acetate at the end of twenty-eight hours at 200, and, in
starting with the opposite system, ester and water, 69.3 per cent
undecomposed ester is obtained.
In his researches, Malaguti occupied himself especially with
verifying the existence of a limit and in measuring it by employing
equivalent mixtures. This method gives no information concern-
ing the influence of the reacting masses upon the final state of equi-
librium and, in order to obtain this, it would have been necessary to
vary successively the masses of the reacting substances contained in
unit volume, that is, their concentration. In these experiments,
Berthelot and Pean de Saint-Gilles observed carefully what occurs
when an equivalent of acid is mixed with several different equiva-
lents of alcohol, and reciprocally. But the very simple result at
which they arrived in these special reactions, namely : the propor-
tionality of the quantity of the ester produced, first to the quantity
of the alcohol employed, and, second, to the quantity of the acid,
had hidden from them the exact expression of the mathematical
relation of the phenomenon, which, in this case, is particularly com-
plicated, because the ester produced is not soluble in water which
dissolves only the alcohol and the acid, and because, consequently,
one does not deal with a homogeneous liquid system.
Work of Guldberg and Waage. A short time after the re-
searches of Berthelot and Pean de Saint-Gilles on esterification, two
Norwegian scholars, C. M. Guldberg and P. Waage presented in a
memoir, entitled Etudes sur les affinites chimiques, published in
French in Christiana in 1867 the results of their researches upon
the double decomposition between barium sulphate and potassium
carbonate limited by the reverse action of potassium sulphate upon
the barium carbonate produced, as the previous experiments of
* Berthelot and Pean de Saint-Gilles, Ann. Chim. Phys., (3) Ixv, 385
(1862); (3) Ixvi, 5, in (1862); (3) Ixviii, 225 (1863).
BASED UPON CHEMICAL REACTIONS 71
Dulong and of H. Rose had proved. In their investigation, Guld-
berg and Waage studied with the greatest care the influence of
concentration and of temperature upon equilibrium as well as the
speed of reaction and the influence of foreign bodies added to the
system. They also investigated equally thoroughly the speed of
evolution of hydrogen in the action of acids on metals. The con-
clusions of their researches are that, in a double decomposition,
limited by a reverse reaction, the force which produces the trans-
formation of the system, of the bodies A+B into a system of
bodies A'+B' is proportional to the product of the active masses of
the two bodies A and B, and to a certain coefficient of affinity pe-
culiar to the reaction A+B = A'+B'. If p and q represent the
active masses of the bodies A and B in unit volume, the force in
question is equal to k-p-q.
If we call p' and q' the active masses of A' and B', and k f the
coefficient of affinity peculiar to the reverse reaction
A'+B' = A+B,
we will have, likewise, for the force which produces the trans-
formation, the value k'-p'-q'.
If, then, the temperature being constant, we consider the
moment when the equilibrium is established, the two opposing forces
ought to be equal and consequently we should have :
(I) k-p-q = k'-p'-q'.
By determining the active masses p, q, p' and q' by direct experi-
ment, we can find the ratio between the coefficients k and k' and
prove that this ratio is constant, by varying the initial masses of the
reacting bodies :
BaSO 4 +K 2 CO 3 <H BaCO 3 +K 2 SO 4 .
If we designate by P, Q, P' and Q' the absolute initial quantities
of the four bodies A, B, A' and B' before the reaction; by x, the
number of molecules of A and B that are transformed into A' and B' ;
and by V, the total constant volume of the solution ; we will have :
V
By introducing these values into equation (i) and multiplying
by V 2 , we obtain :
72 METHODS OF ANALYTICAL CHEMISTRY
(2) (P *) (Q *)= (?'+*) (QH-*).
If we suppose with the authors that the active masses of the
insoluble or very slightly soluble bodies A (barium sulphate) and
A' (barium carbonate) remain constant, at least approximately,
which is justifiable, since the solution being always necessarily
saturated to these bodies, their concentration in the solution is
constant, the equation (2) is reduced, p and p' being constant, to
(3) : Q- *=-%(&+*), :
from which we derive :
Q-jQ'
(4) * = ~
and
(5)
k
X being the concentration, C K2 co 3 of potassium carbon-
ate, and Q'+Jtr the concentration, C K2 so 4 , of potassium sulphate,
the mathematical law to which Guldberg and Waage arrived
amounts to the expression:
4
4 =^ = C (constant).
In general, if we call C and C' the concentrations respectively
of the bodies A and B, of the first system, C" and C" that of the
bodies A' and B', of the secondary system, equation (2) takes the
form:
OC k?
P ; , C// , = = Constant.
k 1
In the reaction studied, the value for a particular experiment
was found to have a value of 4. The experiments were made at
100 and since boiling water dissolves glass, silver or platinum
flasks were employed. Into these flasks pure barium sulphate was
introduced, then solutions of different concentration of either potas-
sium carbonate or potassium carbonate and potassium sulphate. It
BASED UPON CHEMICAL REACTIONS
73
was sufficient to determine the sulphuric acid in the liquid and the
carbonate of barium in the insoluble residue, when the equilibrium
was established, in order to deduce the number of molecules, x
transformed. The following table compiled by the authors, gives
the values of x, which were on the one hand observed and on the
fei
other calculated by formula (4) , taking = 4, for the different
K
values of Q and Q', (expressed in molecules of salt dissolved in
500 molecules of water) :
Q
Q'
x OBSERVED
x CALCULATED
2.0
0.395
0.400
2-5
0.500
0.500
3-5
o
0.719
0.700
2.5
3-o
0.25
0.25
0.300
0.408
0.300
0.400
2.O
0.50
0.005
0.000
The agreement between the calculated and observed values of x
is very satisfactory, but the authors, in order to compile this table,
have employed only the figures corresponding to concentrations
which are but very slightly different.
From the experimental results of Guldberg and Waage, which
represent wider ranges of concentrations, ' Van't Hoff* obtained the
following table which shows less satisfactory agreement:
Q
Q'
x OBSERVED
CALCULATED
Q/_J_^ ^ AI
3-5
0.719
3.87
2.5
0.500
4.00
2.0
0.395
4.07
1.0
0.176
4.68
2.O
0.25
O.20O
4.00
2-5
0.25
0.300
4.00
3-0
0.25
0.408
3-94
3-8
0.25
0-593
3.8o
2.O
0.50
4.00
The extreme values of the ratio of the coefficients of affinity,
3.80 and 4.68, represent a difference of 12 per cent of the average
ratio. This difference is certainly greater than the experimental
errors because the studies made by Guldberg and Waage, upon the
* Van't Hoff, memoir on the Lois de Vequillbre chimique dans I'etat dilu
gazeus ou dissous, Stockholm, p. 30, 1886.
74
METHODS OF ANALYTICAL CHEMISTRY
speed of reaction show that with a very close approximation, the
limit is attained in general, at the end of three or four days. As an
example, one of the diagrams published in their paper is repro-
duced (Figure II). The curve represents diagrammatically the re-
sults obtained at 100, when one molecule of barium sulphate was
acted upon by one molecule of potassium carbonate dissolved re-
spectively in 67.67 molecules of water, curve I ; 100 molecules of
water, curve 2; 500 molecules of water, curve 3; 1,000 molecules of
water, curve 4. The abscissas represent the time and the ordinates
the values of x (the percentage of the insoluble body decomposed).
10
30 40
FIG. II
50
70 HOURS
The study of the form of these curves shows that they can be
represented by an equation of the form
dx
dt
=*(*
where x represents the quantity of the insoluble body transformed
at the end of the time t, I the limit of the transformation, and k
a constant depending upon the concentration. The regularity of
these curves and definiteness of the position of the asymptote x =
are such that one must necessarily attribute the differences between
fei
the values of the ratio - to the fact that the law of equilibrium, de-
K
duced by Guldberg and Waage, from their experiments, is not
entirely exact.
Although still imperfect and teaching nothing concerning the
yf
variation of the coefficient -7 with the temperature, the law of
K
Guldberg and Waage, nevertheless, marked a very important step
in the discovery of the law of equilibrium in double decompositions,
BASED UPON CHEMICAL REACTIONS 75
in that it stimulated careful investigation, by means of which the
more exact form, deduced from the principles of Thermodynamics,
could be submitted to numerous experimental verifications.
Very soon Guldberg and Waage, themselves, perceived, after
the publication of their paper in 1867, that the general law which
GC
they had formulated, = C (constant) , should be modified by
V^ "v_/
introducing an exponent, indicating for each concentration, the
number of molecules taking part in the reaction. Then, if the reac-
tion is represented by the equation nA+n'B <r* ri'^+n'''^, the
mathematical expression of the equilibrium is
O-C' M '
"n"."ii'" =C (constant)*
an equation in which all of the exponents reduce to unity in the
fundamental reaction of their theory :
BaS0 4 +K 2 C0 3 ^ BaC0 3 +K 2 S0 4 ,
in which only one molecule of each of the substances is involved.
Verifications of the Law of Guldberg and Waage. It is toward
the verification of the final equation of Guldberg and Waage that
the later investigations have been directed with different degrees
of success, the experiments giving at times perfect verification and
again, on the contrary, flagrant disagreements.
One of the first researches undertaken with this object in view,
was the investigations of Thomsen on the equilibrium which is
established by adding sulphuric acid to a dilute solution of sodium
nitrate: H 2 SO 4 +NaNO 3 <-> NaHSO 4 +HNO 3 , a reaction whose
limit of transformation is easily determined by the calorimeter and
the results are in perfect accord with the law of Guldberg and
Waage.f
On the other hand, the studies made shortly after, in 1872, by
SchloesingJ upon the transformation of insoluble alkaline earth
carbonates into bicarbonates, gave results in complete disagreement
with the equation of Guldberg and Waage. In the reaction,
* Jour, prakt. Chem., (2) xix, 69 (1879).
t Thomsen, Thermochemische Unterschungen I.
+ Schloesing, C. R., Ixxiv, 1552 (1872) and Ixxv, 70 (1872).
76 METHODS OF ANALYTICAL CHEMISTRY
BaCO 3 +CO 2 ( dissolved ) * Ba (HCO 3 ) 2
the above mentioned equation would give
=C (constant)
C B a (HC0 3 ) :
or, x = k-y, where x represents the pressure of the carbon dioxide
proportional to its concentration, y, the quantity of carbonate dis-
solved in a definite volume of water, and k, a constant. Schloesing
showed that the phenomenon follows an entirely different law repre-
sented by the equations :
^0.38045 fry f or the barium carbonate ;
^0.37866 fry f or calcium carbonate.
There exists then, a deficiency in the mathematical law of Guld-
berg and Waage. It could only be supplied by the application of
the principles of Thermodynamics to the reversible phenomena of
chemistry. The first attempt to do this has been made almost
simultaneously by Moutier and Peslin in France, and by Horstmann
in Germany, who were to pave the way later for Van't Hoff and
Le Chatelier to complete the equation of Guldberg and Waage by
terms which experiment alone had been powerless to reveal.
3. Analogous Law Drawn from the Principles of Thermo-
dynamics
Equation of Van't Hoff and H. Le Chatelier.]. H. Van't Hoff,
in his memoir on the Laws of Chemical Equilibrium in the Dilute
State, Gaseous and in Solution, presented to the Royal Academy of
Sciences of Sweden, October 14, 1885*, gave a mathematical expres-
sion of the law controlling equilibrium in double decompositions,
completing the equation of Guldberg and Waage by the introduction
of a temperature factor. Without entering into the details of the
calculations, we will only indicate the course pursued by him.
The starting point of Van't Hoff is the law of Osmotic Pressure,
n v*= RT
*A Memoir printed in French in 1886 at Stockholm (Royal Printing
Office) and in the Archives Neerlandaises, Vol. 20.
BASED UPON CHEMICAL REACTIONS 77
for bodies in aqueous solution, combined with the law of Mariotte-
Gay Lussac,
for molecular weights of gases. In the first formula n represents
the osmotic pressure of the dissolved salt molecules, i is a constant
coefficient, peculiar to every dissolved substance, deduced from the
experiment of Pfeffer and de Vries to which we will return subse-
quently, R, a constant equal to 845, expressed in meter-kilogram
units.
Van't Hoff makes use of the conception of the semipermeable
membrane, used by Pfeffer for the measurement of osmotic pres-
sure, and conceives a dilute solution as describing a series of cyclic
reversible transformations, first at a constant temperature and then
at a variable temperature. He thus arrives, basing his conception
upon the principle of Carnot-Clausius, at the following equations:
(i) CZ ^ = k = C (constant)
C^*A
for an isothermal transformation and
(^ *iog* = g
<tT ART 3
for a transformation at a variable temperature.
In these equations C represents the concentration of the sub-
stances in the initial system, when the equilibrium is established, Q
the concentrations of the substances of the second system ; n, n lt the
number of molecules participating in the reaction; i, i lf the coeffi-
cients of the law of osmotic pressures specific for each of the sub-
stances of the two systems; A is the reciprocal of the mechanical
equivalent of heat; finally, q represents the heat liberated by the
transformation of n lf n\, molecules of the second system into n, n',
molecules of the first.
Shortly after the publication of Van't Hoff's memoir, Le Chate-
lier gave a different demonstration of the same mathematical law
in his memoir, entitled Experimental and Theoretical Researches on
Chemical Equilibrium, published in 1888.* The starting point of
*H. Le Chatelier, Ann. dks Mines, (8) xiii, 157-382 (1888).
78 METHODS OF ANALYTICAL CHEMISTRY
this demonstration is the formula of Clapeyron-Clausius, upon the
laws of the pressure of saturated vapors applied to the phenomena
of simple dissociation of heterogeneous systems (calcium carbonate,
for example). This law is extended to the phenomena of the dis-
solution of gases in water with the help of Henry's Law : P = fcC*
at constant temperature, then to the dissociation of homogeneous
systems, finally, on the assumption that all substances are volatile,
to saturated solutions, and finally to double decomposition of salts.
The form of the law thus reached by Le Chatelier is expressed as
follows :*
J f LJ'p
AirJ -^r~-=
(constant)
in which the terms C, n, A, R, and T have the same significance as
in Van't Hoff's equation. The coefficients i, however, are those
obtained by the cryoscopic measurements of Raoult (identical with
those given by the osmotic pressure method). Finally, L is the heat
liberated in the transformation of n, n' molecules of the first system
into n", n"' , molecules of the second system (that is to say, equal
to the terms of Van't Hoff's equation with the sign reversed).
The equations of Van't Hoff and of Le Chatelier are then
identical. Both are established for very dilute solutions and assume
that Mariotte's law or the law of osmotic pressure holds without
change of coefficient in all the changes of concentration studied.
Besides, the amount of the solvent is supposed to remain constant
and does not, therefore, explicitly appear in the equation. Insoluble
substances, whose concentration may be regarded as constant, do
not enter either into the equation, which is the same thing as consid-
ering their volume in the solid state negligible in comparison to their
volume occupied in the gaseous state. However, although this law is
only approximate, we will see that it is verified experimentally up to
higher concentrations than one has to use in analytical chemistry.
L being, in general an unknown function of the temperature, we
;T JT
but, inasmuch as only reactions
occurring at a definite temperature need to be considered in analyt-
ical chemistry, it is not necessary to know the form of this function
in order to discuss the equation and deduce from it the most favor-
*H. Le Chatelier had previously given C. R., ci, 1484 (1885), the same
equation for equilibrium in gaseous systems.
BASED UPON CHEMICAL REACTIONS
79
able conditions in analysis, and we can content ourselves in general
with the equation reduced to the case of isothermal transformations :
= C (constant)
rn"i"r"'n"'i'"
the value of the term under the J sign remaining constant in such
transformations. J
Experimental Verification of the Approximate Law of Equilib-
rium in Double Isothermal Decompositions. The law of Guldberg
and Waage differs, definitely, from the preceding expression only
in the coefficients i. It can then be understood that, according as
these coefficients are but slightly different from each other or, on
the contrary, very different, excellent agreement or considerable
variation has been obtained between the data of the experiments
and the results calculated by means of the Guldberg- Waage equation.
Let us take up again the experiment previously considered. In
the reaction
BaSO 4 +K 2 CO 3 <-> BaCO 3 +K 2 SO 4
the coefficients i are almost equal, 2.26 for K 2 CO 3 and 2.11
K 2 SO 4 , so that Van't Hoff's equation becomes
P 2.26 p 1.07
K 2 C 3 ' K 2 C 3 C (constant)
for
2.II
C
K 2 S0 4
an equation differing but little from that of Guldberg and Waage.
In calculating again by means of this corrected equation the con-
stant of the table compiled by Van't Hoff from the data of the
experiments of Guldberg and Waage, we obtain a much more satis-
factory agreement between the experimental and the calculated
values as shown by the accompanying table.
Q
Q'
X
Q *
(Q -*) 1 ' 07
Q!+*
(Qi+f)
3-5
0.719
3.87
4.16
2-5
0.500
4.00
4.20
2.0
o
0.395
4.07
4.20
I.O
o
0.176
4.68
4.62
2.0
0.25
O.2OO
4.00
4.17
2-5
0.25
0.300
4.00
4-23
3-0
0.25
0.408
3-94
4.21
3-8
0.25
0-593
3.8o
4.13
2.O
0.50
o.ooo
4.00
4.20
8o METHODS OF ANALYTICAL CHEMISTRY
While the difference between 4.68 and 3.80 is 12 per cent of the
average value 4.09, with the formula of Guldberg and Waage, it is
only 8 per cent with the extreme values, 4.13 and 4.62, by compari-
son with the average 4.23, which is given by the thermodynamic
equation.
Likewise, we see why the Guldberg and Waage formula may
have conformed exactly with the experiments of Thomsen upon the
equilibrium :
Na 2 SO 4 +HNO 3 <-> NaHSO 4 +NaNO 3 .
The values of i are, in fact, almost identical for the substances
of the system in equilibrium; 1.91 for Na 2 SO 4 , 1.94 for HNO 3 , 1.88
for NaHSO 4 and 1.82 for NaNO 3 .
On the contrary, the formula of Guldberg and Waage is in
complete disagreement with the reaction studied by Schloesing,
BaCO 3 +CO 2 (dissolved)*Ba(HCO 3 ) 2 (dissolved)
where the coefficients i are very different; i.o for CO 2 , 2.66 for the
bicarbonate of barium and 2.56 for the bicarbonate of calcium.
In this case, Van't Hoff's formula leads to the equations
X o.376 = ky f or BaCO 3 , X- 390 = ky for CaCO 3 ,
which is absolutely in accord with the analogous equation previously
indicated, and deduced from the experimental data.
Other verifications of the thermodynamic law have been made
and the agreement has always been satisfactory. We will content
ourselves by giving here the following two experimental verifica-
tions which relate more especially to cases frequently considered in
analytical chemistry:
( i ) The experiment of Ostwald upon the decomposition of zinc
sulphide by sulphuric acid*
In the reaction of equilibrium,
H 2 SO 4 +ZnS <-* ZnSO 4 +H 2 S,
the thermodynamic equation, replacing i by its corresponding values
and taking into account the fact that the sulphide of zinc is insoluble,
becomes,
* Ostwald, four, prakt. Chem., (2) xix, 480 (1879).
BASED UPON CHEMICAL REACTIONS 81
C 2 '^
" H 2 S0 4
pO.gS -C i-04
U ZnS0 4 H 2 S
As in the initial system, there is neither sulphate of zinc nor
hydrogen sulphide, the concentrations of these two substances are
necessarily equal and the equation reduces to
2.06
t =C (constant).
Here is the comparison of the observed and calculated values of
one of the experiments which was taken as a basis in the calculation
of the constant :
H 2 S0 4
C (OBSERVED)
H 2
S
C OBSERVED
C CALCULATED
0.250
0.125
0.062
0.031
0-595
0.296
0.150
0.077
0.600
0.300
(poor)
0.075
The agreement is absolute.
(2) Experiments of Le Chatelier upon the hydrolysis of mer-
curic sulphate.
In the reaction studied,
3 HgSO 4 +2H 2 O <* HgSO 4 -2HgO+2H 2 SO 4
i is equal to 0.98 for HgSO 4 and to 2.06 for H 2 SO 4 , the basic
sulphate is insoluble and the concentration of water is assumed to be
constant. The equation of the isothermic equilibrium then gives,
4 , r i-40
H 2 S0 4
Here is the table of comparison between the experimental and
calculated values, compiled from this author's data obtained at 13.
6
82
METHODS OF ANALYTICAL CHEMISTRY
HgS0 4
NUMBER OF EQUI-
H,S0 4
NUMBER OF EQUIVALENTS IN 10
VALENTS IN IO
LITERS
LITERS
C OBSERVED
C CALCULATED
3-i8
7-52
6.33
3.88
8.80
7-29
470
9.70
8.36
5-92
11.50
9.86
8.90
14.20
13.20
12.90
17.20 (poor)
13.00
17.50
17.29
18.00
21. OO
21.82
20.20
24.00
23.70
The agreement is satisfactory, for the result of the experiments
may be represented by the equation :
instead of
1 ' 58 k
~ ?
pi-40 _
which is given by the thermodynamic formula.
We can, then, in fact, consider the equation of Guldberg and
Waage corrected by the introduction of the coefficient i, as a
theoretical and experimental law of the equilibrium in double
decompositions of salts and sufficiently exact for its application to
chemical analysis, where it is a question especially of determining
the direction in which the reaction should proceed in order to make
it as complete as possible. As experience has already indicated for
each method the approximate degree of precision of which it is
capable, it is unnecessary, in general, to repeat for each complete
study of the equilibrium as in the examples just given, and we can
content ourselves with verifying whether the method of operation
conforms to the deductions which can be drawn from the general
law of equilibrium or if we should, on the contrary, make it undergo
changes according to these deductions. In this respect, it is not
even necessary to know accurately the constant of the law of
equilibrium in each particular case, nor the coefficient * in order to
be able to take into account the manner in which the temperature
and concentration influence the direction of the double chemical