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Equilibrium and Transport Properties of Concentrated Electrolyte Solutions.

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Equilibrium and Transport Properties of Concentrated Electrolyte Solutions
BY PROF. DR. R. HAASE
INSTITUT FUR PHYSIKALISCHE CHEMIE DER TECHNISCHEN HOCHSCHULE AACHEN
A historical survey is followed by a discussion of’a number ojproblems associated with concentrated solutions of electrolytes containing two ion constituents: general thermodynamic
properties, dissociation equilibria, phase separation, analytical representation of the thermodynamic function5 over the entire range of concentrations, and general relationships in
electrical conductivity, diffusion, and sedimentation. A number of relationships valid at all
concentrations is formulated, and known relationships pertaining to dilute solutions are
derived from these as special cases. The discussion of concentrated solutions is illustrated
by numerical examples, most of which are based on recent results.
I. Introduction
The beginnings of physical chemistry are closely linked
with the experimental and theoretical study of the equilibrium and transport properties of electrolyte solutions,
a field which also covers investigation of galvanic cells.
Consider, for example the contributions made during
the second half of the last century by Arrhenius, Gibbs,
Helmholtz, Hittorf, Kohlrausch, Nernst, Ostwald, Planck,
and van’t Hoff:
The properties in question were described in these early
investigations on the basis of the theory of ideal dilute
solutions. In this simple approximation, the electrolyte
solutions are assumed to be so dilute that the interactions
between the ions are negligible. Consequently, the only
thermodynamic difference between such electrolyte solutions and ideal dilute non-electrolyte solutions is the
increase in the number of solute particles as a result of
dissociation in the former case. The dissociation equilibrium is described by the classical law of mass action.
An effect of the approximation is that the dependence of
the equivalent conductivity on the concentration appears
to be due only to changes in the degree of dissociation of
the electrolyte, whilst the mobilities of the ions are independent of concentration. This statement, together
with the classical law of mass action, leads to Ostwald’s
dilution law, which was confirmed for dilute aqueous
solutions of weak electrolytes, e.g. dilute acetic acid, to
within the experimental accuracy possible at that time
(Ostwald 1888). The diffusion coefficient of the electrolyte in an ideal dilute solution also can easily be related
to the ionic mobilities (Nernst 1888).
It soon became clear, however, that even at very high dilutions, solutions of strong electrolytes (e.g. hydrochloric or sulfuric acid or salts in water) exhibit such wide
deviations from the laws known at that time that the
equations for ideal dilute solutions could no longer be
regarded as being applicable. These deviations were even
observed for concentrations at which non-electrolyte
solutions still show approximately ideal behaviour. Thus
these “anomalies of strong electrolytes” are undoubtedly connected with the relatively high concentrations,
and consequent interactions, of the ions. Quantitative
Angew. Chem. internut. Edit.
1
Vol. 4(1965)
1 No. 6
description of these interactions was soon attempted. It
was not until 1923, however, that a useful molecular
statistical theory was evolved by Debye and Hiickel. This
theory was later extended and re-stated by Onsager
(1926- 1927) for application to electrical conductivity.
Some consequences of these laws had already been discovered empirically, by Kohlrausch (1869-1880) in the
case of conductivity, and by Bjerrum (1916), Bronsted
(1922), Lewis (1921-1923), and others in the case of the
thermodynamic functions.
I t is now known that the equations proposed by Debye,
Hiickel, and Onsager are themselves limiting laws: they
correspond to the first approximation beyond the laws
of ideal dilute solutions. The molecular concepts on
which they are based are as follows:
The electrolyte solution consists of a system of charged
particles embedded in a non-conducting medium with
the same density, dielectric constant, and viscosity as the
solvent. The ion concentration is so small that elimination of the charges would yield an ideal dilute solution.
The ions move in a disordered manner (Brownian movement). When transport processes take place, an ordered
motion is superposed on the thermal movement; in particular, during conduction of electricity, cations and
anions migrate in opposite directions, giving rise to relaxation and electrophoretic effects, the complete mathematical treatment of which is extremely complicated.
The present molecular theories of the equilibrium and
transport properties of electrolyte solutions still reduce
to the limiting laws of Debye, Hiickel, and Onsager at
sufficiently low concentrations. The range of validity has
been extended to include higher electrolyte concentrations by taking into account such factors as incomplete
dissociation and ion association or solvation. However,
general applicability of the equations to electrolyte solutions of any concentration is ruled out right from the
start by the fundamental simplification of the theory,
according to which ions and aggregates of ions are regarded as discrete entities and the solvent as a structureless continuum.
If it is desired to cover the entire range of concentrations
from pure solvent to pure liquid electrolyte, as is possible experimentally e.g. with the systems water/acetic
485
acid, water/hydrochloric acid, or water/sulfuric acid, we
cannot base our considerations on the theoretical or
semi-empirical equations which have been used in the
past. Instead, we must build up an entirely new picture
in which (at least for the equilibrium properties) the nonelectrolyte and the electrolyte are no longer regarded as
the “solvent” and the “solute”, respectively, but as equivalent components. In this new scheme, which is essentially empirical at the moment, and which will be developed below for thermodynamic functions, the DebyeHuckeZ limiting law retains its validity. This law, together with Onsager’s limiting law, is one of the few results of the molecular theory of electrolyte solutions
which has been accepted as reliable by all research
workers.
Our discussion of the equilibrium and transport properties of concentrated electrolyte solutions is confined to
solutions of a single electrolyte containing two ion constituents, e.g. CaC12, H2S04, or H3P04, in a n inert solvent, e . g . H2O.
For the sake of simplicity we exclude electrolytes with three
or more ion constituents (e.g. KHSOd), mixed electrolyte
solutions, and fused salts. Moreover, we cannot concern
ourselves with details of liquid-vapor and liquid-solid
equilibria or osmotic phenomena, since although these are
important for experimental detcrrninations of thermodynamic
functions, they are not limited to electrolytes. It will also be
necessary, in our consideration of transport processes, to
omit discussion of viscosity and non-isothermal processes
(thermal conduction, thermal diffusion, etc.), as well as of
galvanic cells.
saturation limit, e.g. H20/NaCI, it represents a hypothetical
state ( e . g . “fused sodium chloride a1 25°C and 1 atm”).
Nevertheless, the two components are treated in principle as
equivalent.
It is possible to calculate all equilibrium properties of the
electrolyte solution, with the exception of the dissociation constant, provided that we know the chemical potentials p~and p~of the two components or the molar
Gibbs free energy, = ( I - x)pl + xp2, of the liquid mixture as a function of T, P, and x. We need not know the
absolute values of pl,p2, and G, but simply the differences between these quantities and their values for the
pure liquid components. Let polbe the chemical potential of the pure liquid component i at the given values of
T and P, and R the gas constant. We then make use of
the dimensionless quantities defined by equations (2) and
(3), which we consider as functions of T, P, and x.
The function
is related to the “activity” a1 of component 1
(which we d o not require in this discussion) by the equation
$1 = In a ] .
Equations (4) and (5) follow from the Gibbs-Duheni
eauation.
11. Equilibrium Properties
1. General
Consider a binary liquid mixture, in which component
1 is a non-electrolyte (“solvent”), and component 2 an
electrolyte containing two ion constituents. At high
electrolyte concentrations we therefore have at least four
types of particles present : non-electrolyte molecules, undissociated electrolyte molecules, and at least one kind
of cation and one kind of anion.
We assume for the moment that the liquid is free from
electric potential or concentration gradients, erc., since we
are at present concerned only with equilibrium properties.
As independent variables with which to describe the
macroscopic state of the liquid mixture, we choose the
absolute temperature T, the pressure P, and a single concentration variable, which we take to be either the stoichiometric mole fraction x of the electrolyte or the molality m (moles electrclyte per kg solvent). The quantities
x and m are related as shown in Equation (l), where MI
is the kilogram-molecular weight of the non-electrolyte.
The other properties of the quantities $1 and $2 or I’,
e .g. those relating to the conditions for stability and coexistence, are also similar to those of the chemical potentials p~ and p 2 or the molar free energy -6.
Two other quantities are generally used in the literature
instead of 41, $2, and
these are the osmotic coefficient ‘p of thesolvent and the conventional activity ccefficient y of the electrolyte, which are considered as
functions of T, P, and m.
r:
Let vL and v _ respectively be the number of cations and
anions which result from complete dissociation of one
1;
molecule of electrolyte (e .g. H2S04: v+ = 2, vH3P04: \I+ = 3, \I_ =- I). Moreover, let p.2”be a standard
value of the chemical potential of the electrolyte, depending, like pol and ~ 0 2 only
,
on T and P.
:
The functions ‘p and y can then be defined as follows.
v.ln ( v + - m y )
E
o2+ C2
(7)
with
v
The pure solvent is described according to Eq. (1) by x = 0
or m = 0, and the pure liquid electrolyte by x = 1 or m + co.
The pure liquid electrolyte corresponds to a real state only in
systems such as H;?O/HCIor H2O/H;?S04; for systems with a
486
v,
+ v-;
v:
= V;+-J~-
(8)
and
Angew. Chem. internal. Edit. / Vol. 4 (1965) / No. 6
In the case of electrolytes such as HC1 or NaZS04, which
give only two types of ions on dissociation, irrespective
of the concentration, it is possible to use the degree of
dissociation cc of the electrolyte and the mean ionic activity coefficient y + [on the molality (m) scale] in accordance with Equation (10). The quantity which can be
measured directly is y, not y + .
y="'y+.
(10)
Whereas q, C 2 ,a n d u. a r e dimensionless,y a n d y * a s defined
in equations (7) a n d (10) have t h e s a m e dimensions a s l / m
and M I , namely kg/mola. In this connection, we use t h e
following abbreviations :
rnt
E
I mole/kg; yt
E
Mi
=
In t h e above equation, yu is t h e activity coefficient of t h e
undissociated electrolyte a n d K, t h e dissociation constant
of the electrolyte (both quantities o n t h e m scale). Krn is
dimensionless and depends only o n T a n d P.
For our example (13), we have: v -= 3 ; v f = 4113.However, the dissociation equilibria discussed below are of
the simplest type (1 5).
HN03
Here v
2; v f
=
%=
HF+NO3.
1 , so that Equation (14) reduces to
Since
1 kg/rnole.
An ideal dilute electrolyte solution is characterized by
the conditiony& = yt. Furthermore, if theelectrolyte in
such a solution is completely dissociated, then cp = 1 ;
a= l;y* - y - - y t .
Using Equations (l), (4),(6), and (7), and integrating betweenm=O(cp= I ; y - y t ) a n d m = m(cp = y ; y = = y ) a t
constant temperature and pressure, we obtain Equations
(1 I ) and (12), which are important for the evaluation of
experimental data.
rn
it follows from Equation (16) [2,3] that
Using Equation ( 1 S), therefore, we can find the dissociation constant K,, at given pressure for any temperature,
provided that y(m) and a(m) are known. The standard
values of the affinity AQ, the heat of dissociation h:p,
and the entropy of dissociation s4p (on the m scale)
can be obtained from K, by means of Equations(19),
(20), and (21).
A 0 = R T In K m
rn
(
(12)
hp,,= RT2( b InbTKm )p= --R b In K,
b(l/T) P
If both components a r e volatile, t h e quantities $1 a n d $2
can b e found f r o m t h e partial vapor pressures, a n d these
a r e used together with Equations (6) a n d (12) t o find t h e
functions q and y. If only the solvent is volatile, then vapor
pressure measurements (generally by t h e indirect isopiestic
method) lead t o $1, a n d hence t o 'p a n d y, b u t not t o $ 2 . If
only the electrolyte is volatile, t h e n only $2 can b e obtained
from t h e vapor pressure. O n e can also find q, a n d hence $1
a n d y. from freezing point determinations. E.m.f. measurements lead t o t h e quantity y, from which 'p can b e determined
by Equation (1 I), a n d hence + I by Equation (6). T h e constant
Cp for volatile electrolytes can b e obtained from y a n d +z
with the aid of Equation (7).
2. Dissociation Equilibria
A dissolved electrolyte such as HC1 or Na2SO4 yields
only two types of ions, irrespective of the concentration,
e.g.
Na2S04
;-f
2 N a + + S0,2-.
(13)
Electrolytes such as H2SO4 or H3P04, on the other hand,
dissociate in two or three stages. We shall disregard these
more complicated cases in the present discussion.
A single-step dissociation of the type (13) can be described by the general equilibrium condition (14) [l].
[ I ] R . Haase, Z. physik. Chern. N.F. 39, 360 (1963).
Angew. C h e m . internat. Edit.
/
(15)
Vol. 4 (1965)
/ No. 6
(19)
(20)
We also have
T
( '2 (
)p =
(22)
Whereas some method can always be found for the
measurement of the activity coefficient y, the experimental determination of the degree of dissociation a
presents a problem. The usual methods, particularly for
strong electrolytes, depend on assurnpions, e.g. that of a
certain concentration dependence of the electrical conductivity. It was a great advance, therefore, when several
authors, and Redlich in particular, pointed out the possibility of determining the concentrations of ions by the
relatively direct spectroscopic methods. Thus values of
the function a(m), obtained from Ranian [3] and nuclear
resonance spectra [4],are now available for aqueous nitric and perchloric acids at several temperatures. We base
our calculations on the results reported by Hood and
Reilly [4], since these appear to be the most precise so Far.
We have determined the activity coefficients from partial vapor pressure and isopiestic measurements, for the
[ 2 ] R . Haase, K . - H . Diicker, and H. A . Kippers, Ber. Bunsenges.
physik. Chern., in press.
[3] 0. Redlich and J. Bigeleisen, J. Arner. chern. SOC.65, 1883
(1943); 0. Redlich, E. K . Holt, and J . Bigeleisen, J. Amer. chern.
SOC.66, 13 (1944); 0. Redlich, Chem. Reviews 39, 333 (1946);
Mh. Chem. 86, 329 (1955).
141 G. C. Hood, 0. Redlich, and C. A. Reilly, J. chern. Physics 22,
2067 (1954); G. C. Hood and C. A . Reilly, ibid. 32, 127 (1960).
487
systems H20/HN03 and H20iHC104, as far as possible
in the concentration and temperature range for which
the degrees of dissociation are known. These activity coefficients were used to calculate K,, A e,hFp, and sp
: [2]
with the aid of equations (18) to (21) at atmospheric
pressure. The results are listed in Table 1. HNO3 and
HClO4 are strong acids of comparable strengths. The
quantity hp: and hence also [by Eq. (22)] s : ~ , is found to
be practically constant over the temperature range
studied.
Table I . Dissociation constant K, and standard values of the affinity
A@, heat of dissociation hFp, and entropy of dissociation spp,
for aqueous nitric and perchloric acids 121.
A9
Temp.
i “Cl
[kcal/moIe]
S
[kcal/molel
HNOj
HNO,
__
10
25
40
50
75
33.5
41.1
34.5
29.3
20.3
14.5
- 2.08
- 2.09
-’2.10
!- 2.10
-1.99
-1.99
- I .99
-3.51
r 1.94
-3.51
- 1.85
-3.51
g
[cal.deg-I mole-’]
-4.81
1
HC104
0.36
0.36
0.36
-4.8 I
-4.81
We then used the values of a(m), y(m), and K, for both
acids, together with Equations (10)and (16), to determine
the mean ionic activity coefficient yi (m) and the activity
coefficient y,(m) of the undissociated acid [2]. By way
of example, Fig. 1 shows the three functions y(m), y* (m),
and y,(m) for aqueous perchloric acid at 25 “C.
9-
J
i
8-
f
76
I
-~
Y”j
i
t
/
3. Phase Separation
In a two-component liquid mixture, phase separation
results in two coexistent binary liquid mixtures with different compositions in equilibrium with each other. This
phenomenon is relatively rare in electrolyte solutions,
and is usually not mentioned even in modern monographs on electrolytes.
As early as 1888, Bakhuis Roozeboom [6] observed phase
separation at high acid concentrations in the systems
H20/HCl and HZO/HBr. The former system has been
quantitatively investigated [7] over the entire miscibility
gap, together with the solid-liquid equilibria which occur
at low temperatures. We have verified these early results
with the aid of vapor pressure measurements [ 5 ] , supplemented them by quantitative investigations on the
H20/HBr system, and showed that phase separation also
occurs in the system H20/HT.
All three of these systems exhibit miscibility gaps without accessible critical temperatures for phase separation.
Thus the upper limit of the range in which the two liquids coexist is formed by the critical point for evaporation (which almost coincides with the critical point
of the pure hydrogen halide), and the lower limit is fixed
by equilibrium with a solid hydrate. All the phase separation curves are extremely unsymmetrical, one phase
always containing almost preu acid.
The miscibility gaps found for the three systems are
given in Table 2. The results relate to the pressures at
which the three phases coexist (i.e. the vapor pressure
over the two liquid phases), but the values at atmospheric pressure should be practically the same. We can
therefore construct two isobaric phase separation diagrams from the data given in Table 2 (x” and x”’ as functions of T at constant P).
Table 2. Phase separation in aqueous hydrogen halides. Stoichionietric
mole fractions x” and x”‘ of the acid in the water-rich and acid-rich
phases respectively. as functions of temperature.
X”’
System
f r o m [71
HzOjHCI
?
I0
15
20
25
30
35
HzOiHBr
Fig. 1. The system HzO/HCI04 a t 25 “C: Logarithms of the conventional
activity coefficient y. the mean ionic activity coefficienr y+, and the
activity coefficient yu of the undissociated electrolyte, vs. the square
root of the molality m, according lo [21.
(yi
I kgjrnole, m t E 1 molejkg).
=
15
20
25
30
35
HZOIHI
0.485
0.480
0.475
0.470
0.465
0.460
0.455
0.478
0.473
0.468
0.464
0.459
0.454
0.450
0.423
0.418
0.41 1
0.403
0.400
0.346
I from I51
0.9990
0 9985
0.9982
0.9980
0.9977
0.9975
0.9972
0.999
0.999
0.998
0.996
0.994
0.995
Remarkably high values ar e found for yu in concentrated
acids, in contrast t o the values of ab o u t 1 kg/mole found for
dilute acids. For example, 16 molal aqueous perchloric acid
a t 25°C has yu = 5 x 105 kg/mole. On th e other hand, the
highest conventional activity coefficient previously known, i. e .
that of 35 molal hydrobromic acid at 25”C, is y = 4800
kg/mole, obtained from vapor pressure measurements IS].
Th e occurrence of a miscibility gap in an y binary system can
be seen from t h e form of t h e function r ( x ) for given values
of T an d P. If the entire curve of r ( x ) is convex towards the
x-axis, miscibility is complete. If phase separation occurs,
the curve contains two points of inflection and a bend which
allows the construction of a double tangent (cf:Fig. 4 below).
Th e range o f concentrations over which is concave towards
the x-axis corresponds t o labile (absolutely unstable) states.
151 R . Haase, H . Naas, and H. Thumm, Z . physik. Chem. N.F. 37,
210 (1963).
[7] F. F. Rupert, J. Amer. chem. SOC.3 / , 851 (1909).
488
r
[61 H . W . Bakhuis Rooreboom, Z . physik. Chem. 2, 449 (1888).
Angew. Chem. intermit. Edit. Vol. 4 (1965) / No. 6
T h e limits of stability a r e given by t h e points of inflection.
T h e region between the points of inflection a n d t h e points
of contact of t h e double tangent correspond t o metastable
states. All other concentrations correspond to stable onephase systems. T h e points of contact o f t h e double tangent
show t h e compositions (x" a n d x"' in Table 2) of t h e tw o
coexistent liquid phases. If these points o f contact a r e found
from t h e r - x curves for various temperatures a t a given
pressure a n d transferred to a T-x diagram, t h e isobaric
phase separation diagram results.
We shall give some examples of thz shapes of I'-x curves
which have been obtained from experimental results for
aqueous electrolyte solutions at 25 OC a n d 1 atm.
electrolytes in t h e last stage of dissociation ( e . g . CaC12:
= -2). Consequently Bo
z- = 2; z- = - I ; H2S04: z.+ = I ; z_
is always known a t t h e outset.
If component 1 is water, then at 25 "C and 1 atm:
M 1 = 15.016 x 10-3 kgimole; a
=
1.176 kgl/z/molel/z
(27)
Equation (27), together with Equations (25) and (26),
gives for aqueous electrolyte solutions at 25 "C and 1 atrn
the Equations (28) and (29). Numerical values of C and
Bo are given in Table 3.
C = Cz-v.lnv+
+ v.ln
(18.016.10-3)
(28)
4. Analytical Representation of the
Equilibrium Properties
To obtain an analytical representation of the equilibrium
properties of an electrolyte solution at any concentration,
it is best to start with the function (T, P, x). All other
thermodynamic quantities can then be obtained by differentiation.
r
We express function in the form of Equation (23) [I,
81, which is valid for solutions of an electrolyte with two
ion constituents over the entire range of concentrations
fromx=Otox=l.
In equation (23), C, Bo, B I , B2, . . . are dimensionless
parameters depending only on T and P and related by
Equation (24).
The terms in Equation (23) which contain Bl, B2, . . .
etc. are empirical terms corresponding to a continuation
of the expansion of (x) in fractional powers of x. If the
electrolyte is not volatile, there are n independent parameters (B1, B2, . . . BJ, and the quantity C is determined from Eq. (24). For volatile electrolytes ( e . g . HCI,
HBr, HI, HNO3), C can be determined from vapor pressure data, so that Equation (24) gives a relationship between B1, B2, . . . B,. We then have (n-1) indepecdent
parameters. If v = 1, Bo -- 0, B2 = 0, B4 . = 0, . . . etc.,
Equation (23) reduces to the ordinary power series for
binary solutions of non-electrolytes.
r
Using Equations (l), (9,(6), and (7), it i s possible to
derive expressions for $1, $2, y, and y from Equation
(23) [1,8]. We shall confine ourselves here to giving the
equation (30) for the activity coefficient y,which is derived from ( I ) , (3,(7), (23), and (25).
v.ln
I
2
1
2
1
B1- 5 B z + 3 B 3 + 7 B 4 + 4 B g +
...=
V - l i C f
2
3
=
Cr -v.ln v -
+ v - I n MM I?
(25)
The fourth term of Equation (23), which contains Bo,
corresponds to the Debye-Huckel limiting law. Thus Bo
is related to the Debye-Huckel coefficient a by
In this equation, z+ a n d z_ a r e the electrochemical valences
of the cations a n d the anions respectively, formed from the
[8] R . Huase, H . Naos, and K . - H . Ducker, Z. physik. Chem. N.F.
39, 383 (1963).
A n g e w . Chem. internat. Edit.
Vol. 4 (1965) 1 No. 6
=
+
v.ln ( I - X ) - B ~ . X ~ I ~( ~ , - v
+ 1j.x
Bo (24)
Differentiation of (23) with respect to T or P gives the
corresponding expressions for the dependence of the entropy of mixing and heat of mixing, or of the volume effects, on concentration.
The first three terms on the right-hand side of Equation
(23) correspond to the equations for ideal solutions of
fully dissociated electrolytes. The constant C appearing
in the third term is given by the general expression (25),
where Cz has the same meaning as in Equations (7) and
(9). For volatile electrolytes, Cz can be derived directly
from theexperimental results (cf. Section 11, I), so that
we can immediately find C.
C
:'-
y7
+ (Bs
-:
B3) ex3 -
B4.x7/2 -
3
9 5 . ~ 4 . .. ,
(30)
As can be seen from Equation (30), which does not involve C, the slope of the curve of 1 n(y/yt) against xliZat
x 0 is -Bo/v. This corresponds to the Debye-Huckel
limiting law.
Table 3 shows the values of the adjustable parameters
B I , Bz, . . . etc. for aqueous solutions of 17 electrolytes
at 25 "C and 1 atm, as obtained by us [2,8] from experimental data. These are practically the only electrolyte
solutions for which reliable data covering a wide range
of concentrations exist. The constants v , C, and Bo are
also included in Table 3. The C values for the hydrogen
halides and nitric acid were determined from experimental values of C2, using Equation (28), that for perchloric acid (in brackets) was subsequently determined
from Bo, B1, B2, etc., with the aid of Equation (24). I t
can be seen that three, four, or five independent parameters are required.
Measurements extending to pure liquid electrolyte (x= 1)
are possible in only a few cases ( e . g . sulfuric acid). The
continuation of the investigations up to x - 1 usually is
prevented by fundamental or practical difficulties, e.g. a
miscibility gap in the case of the hydrogen halides, a
489
Table 3. Values of the constants in Equations (23) and (30). for
concentrated aqueous electrolyte solutions at 25 “C and I atm [2,8]
Electrolyte
I
I BO
C
I BI
I B2
~
10.33
16.17
13.99
4.15
(128.3)
17.53
17.53
17.53
17.53
17.53
91.08
17.53
17.53
17.53
17.53
91.08
17.53
17.53
17.53
17.53
91.08
91.08
191.5
206.8
238.6
151.8
161.7
476.9
117.8
163.8
142.2
134.7
1473
149.0
189.2
108.5
96.91
721.6
1469
- 9238
- 1044
-
1335
487.3
449.4
798.2
236.9
- 738.6
- 316.9
- 957.5
-13020
- 345.2
- 1115
- 563.3
- 761.2
- 2121
-12640
-
I B3
I B4
__
2908
3515
4712
768.5
719.7
490.4
250.4
2406
380.2
4483
63 720
484.1
4 449
1662
3 I71
2165
60 500
1Efx
__
- 3898
- 4786
-
6784
417.5
1765
2156
3 325
- 3660
2170
- 10320
-14430
8 996
I2090
- 8906
- 2505
- 6439
6 842
I502
5063
-- I 3 630
I 1480
With the constants listed in Table 3, a miscibility gap for
the hydrogen halides and complete miscibility for the
other systems can be predicted. In the example of Fig. 3
(perchloric acid), the course of the
curve is characteristic of complete miscibility. Fig. 4, on the other hand,
shows a typical example (hydrochloric acid) of the r - x
curve in a system which exhibits phase separation. The
r--Y,
saturation limit in the case of bases and salts, or decomposition of the electrolyte in the cases of nitric and perchloric acids.
Fig. 2 shows a plot of In(y/yf) as a function of m’iz for
hydrochloric acid. The calculated values agree closely
with the experimental results. The measurements on
which the experimental curve of Fig. 2 is based [ 5 ] extend
as fat. as the miscibility limit (cf. Table 2).
u
0
02
& 17iJj
-
06
x”O4
2
08x”’lO
-x
Fig.4. The systemHlO/HCI at 2 5 T : 1‘ (x) as a function of the stoichiometric mole fraction x of the electrolyte. according to [XI, calculated
by Equation (23) using the constants in Table 3 (x”. x”‘: values of x for
the two coexistent liquid phases).
compositions of the two coexisting liquid phases
(x” = 0.35 and x”’ = 0.84) as obtained from the points
of contact of the double tangent are reasonably close to
the values obtained by direct measurement (Table 2:
x” = 0.46 and x”’ = 0.998). Similar results were found
for hydrobromic and hydriodic acids f81.
111. Electrical Conductivity
1. General
0
rn
I
I
I
2
i,
6
fi-
Fig. 2. The system HzO/HCI at 25 ‘C: Logarithm of the conventional
activity coefficient y vs. the square root of the molality m according to
[XI. Experimental: ( 0 0 0 ) ; calculated from Equations ( I ) and (30), using
1 kg/mole,
the constants in Equation (27) and Table 3: (- -). ( y t
mt
I mole/kg).
=
In an electrolyte solution which is situated in an external
electric field, bdt which is otherwise homogeneous, i. e.
exhibits no gradients other than the potential gradient,
conduction of electricity occurs, associated with migration of the ions. If we discregard extremely high field
strengths, Ohm’s law expressed in the form (31) applies
to each volume element.
- + - +
1=x
-
+
= -x
grad
(3 I )
+
In Equation (31), the vectors 1 and ale the electric current
density and field strength, respectively, ‘p is the electric
potential, and K is the specific conductivity.
-EOOL
[.KLZiZ
02
I
I
,
I
04
06
08
10
r-
Fig. 3. The system H20/HC104 at 25 “C: I? (x) as a function of the
stoichiometric mole fraction x of the electrolyte, as calculated by
Equation (23) using the constants in Table 3.
490
The result of the mutual opposition of the electrostatic
force on the ions and the resistance of the medium (resistance due to friction and electrostatic interaction whith
other ions) is that, after a brief acceleration, an ion
moves at a constant average velocity with respect to the
solvent. If the magnitude of this velocity of migration is
divided by the magnitude of the electric field, we obtain
a positive quantity u,, the mobility of ions of type i.
Within the range of validity of Ohm’s law, u, is indeAngew. Chem. infernat. Edit.
Val. 4 (1965)
No. 6
pendent of the field strength, but is a function of the intensive variables of the system (temperature. pressure,
concentration). It is convenient to replace u i by the
ionic conductivity Ai of the ions of type i, as defined in
Equation (32); F is the Faraday constant.
hi
(32)
F.uj
E
It can be shown [9] that, provided that Ohm's law is
valid, the relationship (33) must always be satisfied. In
this equation, /zd is the magnitude of the electrochemical valence and ci the molarity of ions of type i. The
summation must be carried out over all types of ions
present.
(33)
When more t h a n t w o types of ions a r e present, the discussion of equation (33) becomes complicated. I n t h e next
section, therefore, we shall confine ourselves to solutions of
electrolytes containing t w o types of ions (e.g. HCI or NazS04).
We shall also assume, for the s a k e of simplicity, that t h e
solvent ( e . g . H z 0 ) does not f o r m ions. After t h e discussion
of transport numbers, we shall also examine whethe; t h e
ionic mobilities a n d conductivities c a n be derived from
experimcntal d a t a in this case.
2. Equivalent Conductivities
We shall now deal with electrolyte solutions containing
two types of ions, using either the molarity c or the equivalent concentration c* of the electrolyte as the macroscopic concentration variable. We then have [cf. Equation ( I )] :
C=
pim
;c*=
I t M2.m
2,
v c = z v-c
law of independent migration of ions, first formulated
by Kohlrausch in 1893 and repeatedly verified since that
time, uo (or A;) is characteristic of ions of the type i in a
given solvent at a given temperature and pressure, independent of the oppositely charged ion. Thus i\o is a
sum of contributions due to the individual ions.
For very dilute solutions of weak e!ectrolytes, we can put
hi- hp in Equation (38). We then obtain from (38) and
(39) that Ai-40 = a. This relationship leads to Ostwald's
dilution law when it is combined with the classical law of
mass action for dissociation equilibria of type (15 ) , i. e.
with Equations (10) and (16) and y* yu = yt.
For very dilute solutions of strong electrolytes, on the
other hand, a = 1, and A;, A_,and h contain terms proportional to c'" which can be calculated from the valences of the ions and properties of the solvent, using
Onsager's limiting law. An extension of the theoretical
treatment of electrical conduction in dilute solutions of
strong electrolytes beyond the square root law has recently been developed by Onsuger and FUOJS
[lo], again
on the basis of the relaxation and the electrophoretic
effects.
Concentrated electrolyte solutions show a complicated
dependence of a, A,, A_,and11 on concentration which
is at present not amenable to theoretical treatment. We
have determinedA over a wide range of concentrations
for the systems H20/HN03 and H20/HC104, for which
a-values are available from spectroscopic data [I 11.
(34)
In this equation p is the density of the solution a n d MZ is t h e
formula weight of t h e slectrolyte.
We also have
c,
= v. .LY.c;
c..
(35)
= V-'OI'C
which, together with Equation (34), gives
z+ c + =
2- 'C- = E.C*
(36)
Equation (37) defines the equivalent conductivity A
3.
E
xjc'
(37)
which can be measured. This relationship, together with
Equations (33) and (36),leads to afundamental equation
-1= FxY..(u+T
U-) =
%.(A-
-+ A_)
(38)
for the electrical conductivity in electrolyte solutions containing two types of ions.
For infinite dilution, Equation (38) becomes
in which the superscript signifies the limiting value
of the quantity in question when c + 0. According to the
[9] R . Haase: Thermodynamik der irreversiblen Prozesse. Steinkopff, Darmstadt 1963, p. 281 ff.
Angew. Cltem. internat. Edit.
Vol. 4(1965)
No. 6
vcict
Fig. 5. T h e system H 2 0 / H N 0 3at 2 5 ' C : Degree of dissociation n of t h e
electrolyte as a function of t h e s q u a r e root of t h e niolarity c according
to 141. ( c t
1 mole/i).
Fig. 5 shows the degree of dissociation a of the electrolyte, Fig. 6 the measured values of
together with the
values of (A+ i- h-) calculated with Equation (38), as
functions of cIlzfor aqueous nitric acid (at 25 "C and
1 atm). Aqueous perchloric acid presents a similar picture [I 11. The effect of incomplete dissociation at high
concentrations can be clearly recognized.
n,
In order to obtain an analytical representation for the
entire concentration range from x = 0 to x = 1 (for which
experimental values are available for aqueous nitric acid
[ 101 R . M . Fuoss and F. Accascina: Electrolytic Conductance.
Interscience, New York-London 1959.
[ I l l R . Haase, P.-F. Sauernmnn, and K . - H . Ducker, Z. physik.
Chem. N.F. 4 3 , 218 (1964).
49 1
400
h
respectively: N a occurs in the species N a ’ a n d Na2S04,
a u d sulfate occurs as SO:- a n d Na2S04.
If dissociation of an electrolyte is incomplete, it IS necessary to take into account transport of the undissociated
electrolyte moIecuIes, relative to the solvent, in the electric field. This relative migration of the undissociated
electrolyte may be either direct (association of tbe neutral electrolyte molecules with ions) or indirect (solvation of ions and consequent movement of the solvent reIative to the electrolyte molecules). Corresponding to the
mobility ui and the ionic conductance hi, therefore, we
also introduce the mobility u, of the undissociated electrolyte and the quantity h, = F.u,.
I
0
-_
m?7&
.
1
2
1
-
3
9.=
u,
u ++ u _
-
I-a
01
.-
-
uu
-
-
u+ + u _
1
l a
- (A+ i -.Au)
(40)
A,+A-
I/ c / c i
Fig. 6 . The system H*O/HNO, at 25 “ C : Equivalent conductivity A and
sum (i.+ i h . ) of the ionic conductances, according to [ I l l (units
ohm-1 cmz eq. 1 ; c t
I inole/l).
and sdfuric acid 1111, it would be necessary to find
empirical expressions for A(x) or h-(x) and A-(x) which
would reduce, at high dilution, to the equations of
Onsager and Fuoas [lo] (cf. the method used in the case
of equilibrium properties, Section 11.4). No attempt has
so far been made in this direction, partly owing to the
complicated form of the Onsager-Fuoss equations, and
partly because of the difficulties encountered in the experimental determination of the fundamental quantities
h- and 1,- in concentrated solutions (cf. Section 111.4).
The general expressions (40) and (41) can be derived for
the dimensionless quantities 8, and 9-[13].
The upper (lower) sign applies when the undissociated
electrolyte molecules migrate in the same direction as the
cations (anions). It can be seen that, when dissociation
is complete (a = I ) or if transport of the undissociated
electrolyte is negligible (u, = 0; A, = 0), 8, and 3- reduce to the Hittorf transport numbers t, and t- = (1 -t+)
of the cations and the anion:, respectively. At infinite dilution (c + 0; 01. + l), we have
3. Transport Numbers
Contrary to widely held beliefs, neither the Hittorf transport number nor the true transport number of an ion
can be measured at all electrolyte concentrations [9].
The quantity measured is the transport number of an ion
constituent. which corresponds to the Hittorf transport
number only at sufficiently high dilutions [9]. This fact
can be demonstrated for each of the methods used for
the determinatim of transport numbers, i. e. Hittorf’s
method [12], the moving boundary method [13], and
e. m.f. measurements in concentration cells [9,12] and in
gravity or centrifugal cells [9].
For the sake of simplicity, let us again consider solutions
of a single electrolyte containing two types of ions. The
transport numbers of the cation and anion constituents
respectively are 9,and 3- = (1 -8,). 8, and 3- are equal
to the number of equivalents of the cation and anion
constituents, respectively, passing through a reference
plane which is stationary with respect to the solvent,
during the passage of 1 faraday of electricity through the
solution. The term “ion constituent” refers to the parts
(atoms or radicals) of the electrolyte molecule which can
form ions, irrespective of the extent to which these components are actually present as ions.
in which$: and to are the limiting values of the transport
numbers at infinite dilution.
We have determined transport numbers for concentrated aqueous solutions of HNO3, HC104, and AgN03
at 25 “C,using the moving boundary method [14]. Fig. 7
1
078
b;
076
074
0 72
0701
0
’
’
’
1
’
2
’
’
\
3
c cict
T h u s in a n a q u e o u s solution of sodi um sulfate, t h e a t o m N a
an d t h e radical SO4 a r e t he cation a n d anion constituents
Fig. 7. The system HxO/HNO3 at 25 ‘ C : Consrituent transport number
8, of the cation IK c’I2, according to [ I I]. ( c t
I molell).
[I21 M . Spiro, J . chem. Educat. 33, 464 (1956); Trans. Fardday
SOC. 55, 1207 (1959); A . Weissberger: Physical Methods of Organic Chemistry, Part I V . Interscience, New York 1960, p. 3049.
[I31 R . Haase, 2. physik. Chem. N.F. 39, 27 (1963).
[I41 R. Haase, G . Lehnerr, and H.-J. Jansen, Z . physik. Chem.
N.F. 42, 32 (1964).
49 2
=
Angew. Chcni. intermit. Edit. 1 Vol. 4 ( 1 9 6 5 ) 1 N c . 6
shows the curve of a+ against c1I2for aqueous nitric
acid. A significant feature is the maximum, which is also
found with aqueous perchloric acid.
ing feature is the strong dependence of AT and 1,; (for
H+) on concentration, compared with that of A_*and :
A
(NO5 and C10,). This feature is related to the proton
jump mechanism of the migration of H+ ions, which becomes increasingly noticeable with increasing dilution.
4. Ionic Conductivities
We have seen that, in solutions of a single electrolyte
containing two ionic species, it is possible to measure the
following quantities: the equivalent conductivity A, the
stoichiometric transport numbers
and in some cases
the degree of dissociation u.Experimental data can also
be used to find the limiting values A0 and to.
310
a,,
f
200
A
It is therefore possible to determine the limiting values
A: and!A of the ionic conductivities, usingEquation(44),
which is derived from Equations (39), (42), and (43).
Aq
==
tt; ..lo;
=
. .4o
100
(44)
These quantities, which were mentioned earlier, are tabulated in modern monographs and reference books for
many ions at various temperatures.
I
0
3
VCICt
The equations
A:
2
1
:$4:7q
= 8+-;\;
At = 8 :IZ
(45)
can be used for all concentrations with the experimental
values of 9, and11 or of 9+,A, and a.
From (38), (40), (41), (45), and (46), we find [I 11:
Fig. 8. T h e HzO/HNO, system a t 25 “ C : T h e quantities i : ,.;.7
i.’.
and ‘7. as f u n c t i o n s of t h e s q u a r e r o o t of t h e molarity c [molelll,
a c c o r d i n s t o [ I I ] (units o h m - ’ cmz eq-1; c t T- 1 mole/l).
Sufficient data for the determination of :A over a wide
range of concentrations are available for only a few
other electrolyte systems ( e . g . aqueous hydrochloric acid
[I I], silver nitrate [14,15], and silver perchlorate [15]).
IV. Diffusion
from which we obtain the identities:
A’
f
A‘
=
A:
+ A‘
=
(51)
A,.3- A_.
(52)
In accordance with Equation (51), AT and :
A may be referred to as the equivalent conductivities of the cation
and anion constituents of the electrolyte respectively.
According to Equations (49) and (50), A; and A‘ may be
identified with the ionic conductivities A,. and A- only if
dissociation is complete (oc = I), or if the migration of
undissociated electrolyte in the electric field is negligible
(A, 0). Consequently, at high concentrations A, and
A_ cannot be separately determined. Provided that experimental data for a are available, however, it is possible to find the sum (A+ + A-), using Equation (38) or
(52) (cf. Fig. 6).
Fig. 8 shows values of AT, A;, AT, and A- for aqueous
nitric acid at 25 OC, over a wide range of concentrations.
These values were determined from the data of Figs. 5,
6, and 7, using Equations (45) and (46). A similar picture is found with aqueous perchloric acid [l I]. A strikAngew. Chem. intermit. Edit. / Val. 4(1965)
No. 6
Diffusion is the transport of matter in a mixture, caused
by concentration gradients. Let us again consider a
solution of a single electrolyte containing two ion constituents. As a result of the conditions of electroneutrality and local dissociation equilibrium (which normally
are satisfied by every volume element, even in a medium
in which transport processes are taking place), all electrolyte particle types move at the same average velocity.
Apart from the (average) velocity of the solvent molecules (vector
), therefore, there is only one other independent velocity, i. e. the common (average) velocity
+
v2 of the ions and undissociated electrolyte molecules
( ~ . gthe
. species H+,HSO,, SO2,, and H2S04 in aqueous
sulfuric acid). We also have mly one independent concentration variable, for which (as in the case of electrical
conductivity) we shall choose for the moment the molarit4 c of the electrolyte.
<
The transport of matter is characterized by a vector
) or c(
). The latter quantity is the diffusion current density of the electrolyte in the “solventfixed” frame of reference, (“Hittorf‘s reference system”)
(G
>; <
+
and is denoted by 1Jz. Thus
-f
1Jz
+ +
c.(vz-v,).
(53)
[IS] A. N . Campbell and K. P . Singh, Canad. J . Chem. 37, 1959
(1959).
493
<
I n this reference system, the velocity of the solvent
is
taken as the reference velocity. There are also other reference systems, such as that in which the reference veloc+
ity is the mean volume velocity w (Fick’s reference system). The corresponding vector for the diffusion current
density is
Hittorf’s system is related to Fick’s system by equation
(55) [9], where is the molar volume of the mixture, Vl
and V2 are the partial molar volJmes of the solvent and
the electrolyte respectively, and x is the stoichioinetric
mole fraction of the electrolyte.
v
T h e above discussion is true not only for diffusion but also
for other transport processes which occur in t h e absence of
a n electric current, e.g. sedimentation (cf. Section V) a n d
thermal diffusion. F o r t h e present, however, we shall confine
our discussion to pure diffusion, i.r. we shall ignore external
force fields, a s well a s pressure a n d temperature gradients.
In Section V we shall consider pressure gradients in addition
t o concentration gradients.
For pure diffusion in a solution of an electrolyte containing two ion constituents, each volume element obeys
Fick’s law in the form
in which D is the diffusion coefficient. Equation (56) is
analogous to Ohm’s law (31), and is always valid unless
the concentration gradients are extremely high. As the
electric conductivity x in Equation (31), so the diffusion
coefficient D in Equation (56) is a measarable positive
quantity (in stable mixtures not at their critical points),
which depends on the intensive (local) variables of state
(temperature, pressure, concentration).
It is often desirable to convert from Fick’s reference system to Hittorf’s system, and from the rnolarity c to the
molality m. From Equation (34), it is possible to derive
grad c =
C.(I-C.V2)
-. grad m
m
(57)
In the above equations, a1 1, a22, and a12 are phenomenological coefficients related to the ionic conductivities A,
and A- by Equations (62) arid (63).
A.
+
,J2 = -
D
.c.
grad m
rn
z .all t z:a12
(62)
For very dilute, but not necessarily ideal dilute solutions
(in the thermodynamic sense), the kinetic theory shows
[18] that it is possible to disregard the electrophoretic
effect, and use the approximation:
.=
A:
; A- = At
(64)
From Equation (64), together with (59)-(63), we obtain
Equation (65) [18,19],
(58)
We shall now restrict the discussion to completely dissociated electrolytes containing two ionic species. Thus
we exclude compounds such as H2S04 or H3P04, and
set tc = 1. The solution then contains only three types of
particles: non-electrolyte molecules, cations, and anions.
The condition tc = 1 stipulates dilute solutions of strong
electrolytes. Application of the thermodynamics of irreversible processes to such systems leads to the general
equation (59) for the diffusion coefficient D [9,16,17].
[I61 R . Haase, Trans. Faraday SOC.49, 724 (1953), footnote on
p. 728.
[17] R . Haase: Thermodynamlk der Mischphasen. Springer,
Berlin-Gottingen-Heidelberg 1956, p. 579.
494
VL’C
As previously mentioned, in diffusion the ions move at
the same average speed and i n the same direction, namely opposite to the direction of the chemical potential
gradient. Consequently, in contrast to the conduction of
electricity, only the electrophoretic effect is active, the
relaxation effect being absent. The coefficients a1 1, a22,
and a12 in (61) must therefore lead to elimination of the
relaxation effect which is still accounted for in Equations
(62) and (63). Thus if we have a kinetic molecular theory
of electrolytic conduction with explicit exrressions for
h, and A_,the condition of non-occurrence of the relaxation effect in Equation (61), together with Equations
(62) and (63), provide the three equations required for
the calculation of all, a22, and a12 from molecularphysical data. Consequently, any kinetic treatment of
electrical conduction in dilute solutions of strong electrolytes, combined with Equation (59), leads to an explicit expression for D. Onsager and Fuuss [18] first reported such an expression. At higher concentrations,
the molecular-kinetic theory of diffusion, like that of
electrical conduction, meets with difficulties which cannot at present be overcome.
a J 2= 0; h,
Combination of Equations (55), (56), and (57) gives
Equation (58), which is Fick’s law expressed in Hittorf’s
system.
F2
=
which relates the diffusion coefficient in very dilute solutions of strong electrolytes to the limiting values :
A and
:1 of the ionic conductances and to thermodynamic properties of the electrolyte solution.
At infinite dilution (m -+ 0), combination of (66) with
(65) gives the limiting law (67):
[18] L. Onsager and R . M . Fuoss, J. physic. Chem. 36, 2689
(1932).
[I91 G . S. Hartley, Philos. Mag. J. Sci. Z2, 473 (1931).
Angew. Chern. internat. Edit. 1 V d . 4 (1965)
1 No. 6
After sufficient time, the opposing effects of sedimentation (“seraration”) and diffusion (“mixing”) lead to the
+
establishment of a steady state characterized by 1 J l = 0,
which under the conditions assumed (uniform temperature) corresponds to an equilibrium state. According to
classical thermodynamics [9], this sedimentation equilibrium is described by
proposed by Noyes in 1908. For 1 : I electrolytes
(z+ = -z- = 1) Equation (67) reduces to
given by N e m s f in 1888.
-->
V. Sedimentation
On the other hand, by applying the condition
to Equation (73), we obtain
Sedimentation is the transport of matter in a mixture
caused by gravitational or centrifugal fields. If we confine the discussion to solutions of a single electrolyte
containing two ion constituents, then the general considerations regarding concentration variables, velocities, and diffusion current densities as discussed in
Section IV are still valid.
Consider first an electrolyte solution subject to no fields
other than the external force field (gravitational or centrifugal acceleration
and no gradients other than the
pressure gradient (grad P). We may assume local
mechanical equilibrium, so that if p is the density of the
solution, then
z),
S.
M1.m.T=D.
VI
m
1J2
. gra d m
~
0
(75)
The equilibium conditions (74) and (75) immediately
lead to
D .mz -(>yz/>-m)~.p
__ M I .
s
c.V1.(Mz-Vzp)
(76)
Equations (l), (2), (7), and (9) give the following relationships:
M I . m2
-
c
=m.(I
+ MI.m).V
(77)
4
p .g
=
grad P
Moreover, if we disregard extremely high force fields,
each volume element satisfies the equation
vz-w
and from these, together with Equation (76), we obtain
+
+ +
=
s-g
or, combining this with Equations (54) and (69):
---f
=
c.s.g
=
c
s.
F
.grad P,
(71)
or with Equations (l), (34), and (55):
Equations (71) and (72) correspond respectively to (56)
and (58). The measureable quantity s is the sedimentation coefficient. It may be either positive or negative, and
(like the diffusion coefficient D) depends on the intensive variables of the volume element.
Sedimentation in a mixture initially uniform in concentration gives rise to a concentration gradient, which in
turn leads to diffusion. In general, therefore, we must
consider the simultaneous occurrence of a pressure gradient and a concentration gradient, and hence a superposition of the two transport processes (diffusion and
sedimentation). Experience has shown that this superposition is linear. Starting with Equations (58) and (72),
we can therefore write
(73)
Equation (73) describes the simultaneous occurrence of
diffusion and sedimentation in the solvent-fixed reference system.
Angew. Chem. internor. Edit. / Vol. 4(1965)
1 No. 6
Equation (79) is valid at all concentrations, and relates
the transport coefficients D and s. The quantities on the
right-hand side can be found from equilibrium data
alone, i.e. from experimental values of p(m) and yfm).
Equivalent expressions were derived earlier by various
methods [9,20]. Our derivation [21] has the advantage
of generality and simlicity. The above form of Equation
(79) is particularly suitable for electrolyte solutions. For
other purposes, however, particularly for high molecular
weight non-eletrolytes, other forms of this relationship,
which can also be derived from Equation (76), are more
suitable [9,20].
Creeth [22] has recently confirmed experimentally a n
equation corresponding to (79) (but using the “c” scale
and not involving
and V,) for aqueous solutions of
T12S0.4 at 25°C at electrolyte concentrations of 8 to
50 g/l. The functions V,(m) and p(m) were determined
by density measurements, y(m) by e. m.f. measurements,
v
[20] For a review containing historical data, see R. Haase in J. W .
Williams: Ultracentrifugal Analysis in Theory and Experiment.
Academic Press, New York-London 1963, p. 13.
[21] R. Haase,Z.physik. Chem.N.F.25,26(1960); R. Haaseand
H. Schonert, 2. Elektrochem., Ber. Bunsenges. physik. Chem. 64,
1155 (1960).
[22] J . M . Creeth, J. physic. Chem. 66, 1228 (1962); J. M. Creeth
and B. E. Peter, J . physic. Chem. 64, 1502 (1960).
495
D(m) from diffusion experiments, and s(m) from experiments with the ultracentrifuge. The order of magnitude of the diffusion coefficient D is 10-5 cmZ/sec in this
system, and that of s is 10-13 sec.
In the limit m i
0, we find from Equations (81) and (79),
together with Equations (66), (SO), and (83):
Equation (65) applies to very dilute solutions of strong
electrolytes containing two ionic species. In this case we
can also apply the approximations
v!
Thus from Equations (65) and (79) we obtain
in which M; is the equivalent weight of tk.e electrolyte,
defined by
Since the molecular theory of sedimentation is fundamentally simpler than that of diffusion, Equation
(79) is useful in calculating D from s. On the other
hand, D is the more easily measured quantity, so that
(79) will also be used for the reverse calculation.
and in which we have made use of (60) and
V? = M z .
V2
I n Equations (84) and (85), is the limiting value of the
partial specific volume of the electrolyte at infinite dilution, and pol is the density of the pure solvent. The limiting law (85) is known as Svedberg’s equation (Svedberg
1925), particularly under the condition -I == I , (nonelectrolyte solutions). This relationship can be used to
determine the quantity M ~ / vEquation
.
(85) can also be
derived from Equations (67) and (84) using (60) and (82).
(83)
(v2 is the partial specific volume of the electrolyte).
Received: October 21st, 1964
[A 427/224 tE]
German version: Angew. Cheni. 77, 517 (1965)
Translated b y Express Translation Service, London
Chemistry and Stereochemistry of Fluorophosphoranes [*]
BY DR. R. SCHMUTZLER
UNIVERSITY CHEMICAL LABORATORY, CAMBRIDGE, ENGLAND [ * *]
Fluorophosphoranes are a recently discovered new type of phosphorus compound derived
,from phosphorus pentafluoride by substitution of fluorine atoms with various groups.
Synthesis, chemistry, and stereochemistry of this class of compounds containing phosphorus
in the comparatively rare coordination number 5 will be discussed.
A. Introduction
Until recently, relatively few examples of pentacoordination were known; s x n e halides of group V elements
were recognized as typical representatives of pentacoordinated species. For phosphorus in particular, pentacoordinated halides with trigonal-bipyramidal configuration (in the gas phase) have long been known, r.g. PBrs,
PCls (covalent form), PFs, and the covalent forms of the
halofluorides PX,F,-,
(X = C1, Br; n = 2,3,4). The
chemistry of P(V)-halofluorides, which are distinguished
in several instances by the interconversion of covalent
and ionic forms, has recently been reviewed [I -31.
[ * ] Presented in part as lectures a? Newark, Delaware (U.S.A.)
(Feb. 23rd, 1963); Argonne, Illinois (U.S.A.) (Sept. 4th, 1963);
Birmingham (England) (Dec. I3th, 1963); Cambridge (England)
(Feb. I Ith, 1964); Saarbrucken (Germany) (Feb. 18th, 1964);
Glasgow (Scotland) (April 28th, 1964); Heidelberg (Germany)
(May 22nd, 1964); Wien (Austria) (Sept. 9th, 1964).
[**I Present address: E. I. duPont de Nemours and Co., Inc.:
Experimental Station, Wilmington, Delaware, U.S.A.
[ I ] D.S . Puyne, Quart. Rev. I S , 173 (1961).
[2] R. R. Holmes, J. chem. Educ. 40, 125 (1963).
[3] L. Kolditz, Z . Chem. 2, 291 (1963).
496
In the present article, a review is presented on-chemical
and stereochemical aspects of a further class of phosphorus-halogen compounds containing pentacoordinate
phosphorus. The compounds of this type are derived
from phosphorus pentafluoride by substitution of fluorine atoms with various groups. The following types of
these “fluorophosphoranes” will be considered :
I . Fluorophosphoranes containing hydrocarbon
groups as substituents, RnPF5-, (n -- 1,2,3).
2. Aryltrifluorophosphorus hydrides, ArPF3H. Compounds of this type are not true fluorophosphoranes
within the scope of this article. Because of the stereochemical relationship to other fluorophosp horanes,
however, aryltrifluorophosphorus hydrides will also
be considered here.
3. Arylchlorotrifluorophosphoranes, ArPF3CI.
4. Alkyl(ary1)-dialkylaminotrifluorophosphoranes,
R(Ar)PF3NR’2.
5. Dialkylaminofluorophosphoranes,
(R2N),PF5-,
(n = l,2).
6. Perhaloalkyldihalodifluorophosphoranes,
RH,,PF2X2 (Hal =: F,CI; X := CI, Br).
Angew. Chem. internnt. Edit.
Vol. 4 (1965) / Nu. 6
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