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Apparent and partial molar volumes of concentrated solutions of some uni-univalent electrolytes in formamide
APPARENT AND PARTIAL MOLAR VOLUMES OF CONCENTRATED SOLUTIONS OF SOME UNI-UNIVALENT ELECTROLYTES IN FORMAMIDE Istituto
P. BRUNO and M. DELLA MONICA* di Chimica Analitica dell’Universit8 dl Bari, Italy (Recewed
23 F&ruar~
1975)
Abstract-The
densely uf solutions of NH,SCN, KSCN, NH,N03, LiN03. NH,CIO,. NH,Br in formamIde at 25‘ has been measured in the concentration range going from - 0.5 m up to the saturated solution. By the least-square method the coefficients of the straight lines relating density and concentration have been calculated. It follows that the apparent molar volumes of salts (ama) m the investigated concentration range result independent of the concentration. and close to the molar volume of the pure salt. Calculated by the V” (the partial molar volume at infinite dilution) values of the tetraalkylammonium salts and following a Robinson and Stokes suggestion, ionic partial molar volumes (pmv) at infinite dilution agree for both methods. The ionic pmu at infinite dilution follow the Hepler’s equation. the A and B coefficients for alkali metal catwns and halide anions being independent of the sign of the ionic charge. The value of the A term has hem found close to the intrinsic ionic volume plus the volume of the void space which is generated around an ion in a “random close packing distribution”. The low values of Lhe coefficient B. which takes Into accounl the eleclrostriction of the solvent, has heen explained in terms of solvent structure energetically equivalent to the structure of the solution and with the small compressibility of this solvent
INTRODUCTION
In the past few years an increasing interest in ion +,olvent mtcractions in organic solvents has been shown by electrochemists through the investigation of thermodynamic and transport properties of electrolyte solutions. Among the various quantities which characterize salts in solution, apparent and partial molar volumes of electrolytes are of great help in understanding the interactions occurring in solution between a solute particle and a solvent molecule. So, while useful informattons on the ion-ion interactions can be drawn from the study of the concentration dependence of the apparent and partial molar volumes of salts in solution. the entity of ion-solvent and solvent-solvent interactions can be deduced from the values of the pn~r of clcctrolytes at infinite dilution. The results of density measurements of a series of sodium salts in formamidc solutions were discussed in a previous paper [l]. Solutions were investigated in a concentration range going from 0.1 to 0.5 m up to the saturated solutions, and the analysis of the dcnsity data, assuming a linear dependence on the concentration of the density of all the investigated salts. gave arm almost independent of the concentration; that is the slope of the Masson equation [Z] was found to be close to zero. The aim of the present work is to invcstigatc the behaviour of salts, other than sodium salts, in formamide solution and in a concentration range going from dilute to saturated solution, where strong modifications of the structure of the solutions are expected to take place. The choice of the electrolytes. lithium, *To whom
inquiries
should
be addressed.
potassium, and ammonium salts, has been made with a view to throw light on the influence of the cation charge density on the structure of formamide solutions.
EXPERIMENTAL
SECTION
Baker Analyzed Reagent formamide was deionized by means of a mixed bed of Amherlite ion-exchange resins loaded with H ’ ions and subsequently with HCONHions. Bcforc deionizing formamide was dried with molecular sieves 3 A (Union Carbide) [3]; the product obtained and used in the density measurements had the specific conductance of lO~‘W cm-‘. Lithium and ammonium nitrate (Carlo Erbd Reagent grade) were recrystallized twice from doubledistilled water; ammonium bromide (Carlo Erba Reagent grade) was also recrystallized three times from double-distilled water, and then dried in vacua over P,O, at 110’; ammonium perchlorate (Fisher Reagent grade) was recrystallized three times from conductivity water and then kept 48 h in vacua at 130”; ammonium and potassium sulphocyanate (Fisher Reagent grade) were recrystallized twice from conductivity water and then kept 48 h in vacua at 60”. Densities were measured to an accuracy of 0.01% using a 40ml pycnometer calibrated with conductivity water and the volumes were read on the graduated capillary side arm of the pycnometer to 0.001 ml by a magnifying glass. The solutions were prepared by adding formamide to a stock solution of the different salts; a thermostat controlled at 25 + O.CO5”C was used. The preparation of solutions of the diffcrcnt concentrations, and the
Table
I. Densities
of NH,SCN,
NH,NO,,
NH,ClO,.
NH,Br,
salts in formamide
at 25°C
d (g/ml)
0.5089 1.0336 2.1054 3.3.532 4.3043 5.6718 6.5450 7.8860
1.1324 1.1370 1.1407 1.1480 11552 1 1622 1.1683 1 1747 1.1x11
I. 1489 1.1784 1.2123 1.2333 1.2536 1.2890
filling of the pycnometer were saturated with dry nitrogen. RESULTS
AND
0.1507 0.7784 1S663 3.0578 4.8934 5.8599 6.9928
performed
DlSCUSSlON
The density of solutions of NH,SCN, KSCN, NH,NO,, LiNO,, NH,ClO,, NH,Br in formamide at 25” has been measured in the concentration range from _ 0.5 mole/l to saturated solution, and the results reported in Table I. The measured density d shows a linear dependence on the salt concentration c’. The coefficients of the straight lines relating density and concentration have been calculated by the Ieastsquares method and the results reported in Table 2 with the standard deviations. The experimental densities were also used to calculate the amv 0” of electrolytes through the equation : =do-d+cM, cd,
(1)
’
where Mr is the molar mass of the solute, d, is the density of the pure solvent and d is the density of the solution of concentration c. 2. Apparent
1.2433 1.2720 1.2965
I.3560 NH,ClO,
1.1349 I.1567 1.1849 I .2377 1.3023 1.3363 1.3761
in a dry-box
molar
volumes
and
density
equation
Best equation d = 1.12968 + d = 1.129S2 + d = 1.1295, + d = 1.1295, + d = 1.1295, + d = 1.1295, +
7.3415 2.2229 4.3427 4.6424 3.5282 2.9914
x x x x x x
IO-‘c 10~‘~ 10 2r lo-“c lo-*c IO-*c
3 1.1363 1.1458 1.1586 1.1700 1.1867 1.2154
KSCN
0.4168 1.0510 I .7726 2.2507 2.6833 3.4238
”
0.2246 0.5448 0.9748 1.3589 L.9061 1.8633 3.8028 4.7741 5.5665 7.5763
1.1410 1.1524 1.1770 1.2045 1.2259 1.2560 1.2749 1.3051
NH,Br
~
LiNO
NHz,NO,
0.3783 0.9959 1.5222 2.5382 3.43 IO 4.3591 5.2318 6.1452 7.0538
NH,SCN NH,No3 NH,C’lO, NH,Br KSCN LiNO,
LiNO,
(mole/l) NH.,SCN
salt
and
c
c
(mole/l)
Table
KSCN
0.3826 0.9650 1.491 I 1.9418 2.7308 2.9493
1.1462 1.1716 1.1939 I .2142 1.2478 1.2579
The @,, values obtained in this way were plotted in Fig. I against the square-root of the concentration. For comparison, in the same Fig. 1. the umu values of some sodium salts calculated by data in a previous article [I] have also been reported. As obvious in the concentration range used and for the reported salts. the slope of the @,, us ,, ‘2 curves is close to zero. Thus, if @,: = @3 = const. the density must be given by the equation:
where the coefficient of the concentration c should be an additive property of the ions[4J_ The differences between the coefficients of the concentration c in the equation of Table 2 of this work and Table 4 of the previous work [I] relative to couple of electrolytes with a common ion give in fact constant values as shown by data reported in Table 3. The dependence of the amv of electrolytes on the salt concentration was first investigated by Masson [2_l who found that amv follow an empirlcal equation
@ =@“+s&. 0
of sc~me uni-univalent 25°C
Y
electrolytes
(3)
L
in formamide
Standard deviation
Concentration range (mole/l)
Apparent molar volumes (ml/mole)
o.OQo3 0.0003 0.0003 0.0005 o.ooo2 O.WO2
0.4-7.1 0.5-7.9 0.4-2.9 0.4-3.4 0.2-7.0 0.2z7.6
60.8 51.0 65.6 45.6 54.8 34.7
solutions
Molecular volumes (ml/mole)
48.2 60.3 42. I 29.0
at
Apparent and part&
molar volumes of concentrated
801
solutions
Table 4. Ionic partial molar volumes in formamide
1OIl
at 25 C
V’ (Ion)
V (lon)~r
(ml/mole)
(ml A/mole)
NH&N .-*_~cc.-.-.-
.-
Li’ Na’
60
KSCN
.-.-.-
.-.-
K+
-4.0 -2.3 6.1
0.60 0.95 1.33
NH; Cl_ BY I_
13.3 25.9 32.3 43.9
1.48 1.81 I 95 2.16
SUN-
3x.7 4x.7
ClO, NO;
. . . I=-.
&
.A.-.-
40
No I
.
l
a-.-NaN4
LINE
l-.-.l.C*-.-.-*-*
.
30
.
-
.-•-•-NaBr
l
I
I 0.5
I
2.5
1.5 c.
ml-I-’
Fig. 1. Apparent molar volume CD, of salts in formamide against the square root of the concentration c at 25°C. In equation (3), @‘:’ is the amo of the salt when the concentration is zero and S, is the slope of the experimental curves whose value changes with the nature of the electrolyte. Later. Redlich and Rosenfeld [S, 61. by applying the interionic-attraction theory of Debye and Hiickel[7] to the concentration dependence of the UIIW, were able to calculate the theoretical limiting slope for 1-l electrolytes at temperature T in any solvent where the dielectric constant and compressibility are LI and @ respectively, through the relationship (R: gas constant. N: Avogadro constant. r: elementary charge) S,. = NJ tp3(Sn/o”RT)“2((dlnn/dp).,
- /?/3).
(4)
From equation (4) it is evident that the limiting law dcpcnds only on the temperature and on the physical properties of the pure solvent. The confusing situation existing in the literature about this equation has been recently pointed out by Millcro [X]; he has observed that, while the Redlich-Rosenfeld equation, derived from the interionic-attraction theory of Debye and Hiickel. should bc considcrcd as a limiting equation,
Couple of electrolytes
A g/mole
NnNO,~NH,NO, NaBr-NH,Br NaCIO,-NH,CIO,
2.1 X 10 z 2.2 X 10-2 2.1 X 10-2
NnSCN~NH,SCN
2.1 x 10
2
53.8
-2.4
2.2 8.1
19.7 48.2 63.0 94.8
the Masson equation can represent the uww data over a wide concentration range. As a consequence, the extrapolation to infinite dilution by using the Masson equation can give results somewhat incorrect. The theoretical hmitmg slope of equation (3) in formamide at 25 for I-I electrolytes has been calculated by the experimentally determined Dunn [93 from (dhD/dp), and /I data as 1.104ml mole-“” I’,“. and recently confirmed for dilute solutions of KCI, KRr. KI, and KN03 [lo]. T& graphs of Fig. I showing the slope of the a,. L’SX c curves equal to zero reveal that the data of the salts investigated in this work arc too high in concentration for the limiting law to hold. In dilute solutions nor’ are calculated from smatl differences between two large quantities [SW equation(l)]: thereforenrw in the dilute rcglon become extremely sensitive to the experimental error. Unfortunately. at present, we are unable to give the density of formamide solutions with enough accuracy so as to permit the proper extrapolations to be done. Nevertheless. in consequence of the small theorcticdl S, value in this solvent, it is easy to assume that umc vdhes at infinite dilution are only slightly different from the values obtained by extrapolation of data repot-ted in Fig. 1. This statement apparently is in disagreement wItI the results of Gopal et crl[ 10. 111 who obtained a difference of 1.3 ml/mole in the @;, of KI and KNO, by extrapolating data of different concentration range. The analysis of the @V VJ ,,‘c curves. relative to KI and KNO, in formamide. derived from the work of these authors, shows on the contrary, that the discrepancy bctwcen the two extrapolated @i: values can be easily rationalized by a difference in the accuracy of the two sets of data. Thus the arnr of the salts at infinite dilution @I. have been obtained in this solvent by extrapolating to zero concentration the data of Fig. I. These values are reported in Table 2 with the molecular volume of the pure salts. The results reported in Table 2 and in Fig. 1 suggest some interesting considerations. The prno V, are related to the c~rna of salts by the relationship:
all terms in equation (5) having their usual meaning. As a consequence of this equation the apparent and the partial molar volume values of salts coincide
802
P.
BKUNO
AND
M
when d@“/d+c is zero. In formamidc this fact happens in all the wide concentration range experienced; therefore the values reported in Table 2 represent both partial and apparent molar volumes of salts. A survey of the results of Table 2 also shows that V ’ values of I- I valent electrolytes in formamide are remarkably close to the molecular volumes of the pure salts, This important result means that, except for very dilute solutions, the volume of any solution can be obtained as sum of the volume of the salt and the volume of the solvent used to dissolve it; this fact is not easy to explain if clectrostriction phenomena, as a consequence of interactions between ions and the strong dipole of formamide 1121 are taken into account.
DELLA MONICA
./ /
I
z
E
- eoo2 :” “>
./
44 In an elctrolytc solution, when the ion-pairing is not extensive, the partial molar volume of the salts follows an additive principle [13]. The various attempts made in order to obtain the individual contribution of the cations and of the anions to the partial molar volume of the salt have met the dificulty of assigning an absolute volume to the ions. Thus numerous empirical methods have been proposed by difFcrcnt workers. In aqueous solutions pnz~l of ions at infinite dilution have been calculated assuming that the plot of the V’ of the tetraalkylammonium halides us the molar mass of the cations is a straight line [14]. The pmc V? of the anions is thus obtained by extrapolating the data to zero molar mass. Alternatively the indcpendcnt variable molar mass has been replaced by the carbon number of the tetraalkylammonium ions [S]. Another set of data has been estimated in water by selecting a value of the LIWIVof H+ that would make the V> of monovalent cations and anions fall on a smooth curve as a [unction of the crystal radii cubed [IS]. and suggestion of Robinson Following a Stokes [lh], pnlc of ions at infinite dilution can he obtained by assuming that large monovalent anions, namely Brand I-, are not solvated in water. In this calculation it should be considered that the contribution to the volume of the whole system is given by the intrinsic volume of the ions plus the void space around it. which in a “random close packing distribution” has been demonstrated to be 1.83/2X times the volume occupied by the spheres [17J. Since acid substances have not been suitably investigated in this solvent ionic partial molar volumes at infinite dilution in formamide based on the V ’ of H+ Ion cannot be calculated. As far as the tetraalkylammonium salt method is concerned, an attempt to calculate ionic gmu in forrndmide has been made using the data of Gopal and coworkers [lS] relative to the iodide serie. As shown by the graph of Fig. 2 a pmr at infinite dilution of iodide ion equal to 44 ml/mole can be calculated in the report of the I/’ values of tetraalkylammonium iodides cs the carbon number of the cations. while the plot of the V” values I’S the molar mass of the salts gives an absurdly ncgativc VT (I-) value. The ionic I“ values in formamide reported in Table 4 were calculated according to the Robmson and Stokes suggestion [16] by making use of V’(Br-) and VO(Ip) equal to 32.3 and 43.9 ml/mole respectively. of Table 4 can also Alternatively the ionic V’. values
I
0
I
I
4
8 Carbon number
I
1
I2
16
of R4 N’
Fig. 2. Plot of the partial molar volume V- of tetraalkylammonium
iodides
DS the carbon
number
of cations.
bc obtained by the V’ values of the tetraalkylammonium iodides since the V”(I-) value deduced by the graph of Fig. 2 and the value calculated according to the Robinson and Stokes method coincide. Absolute pmz~ of ions in solutions have not been obtained so far. nevertheless to study the structure of the solution around an ion density data have been extensively used by a number of workers 1191. A recent interpretation of the ionic amv in terms of size and charge of the ions makes use of the equation [ZO] V’(ion)
= V”(intr)
+ V”(elect),
(6)
where the first term on the right side is the geometric contribution to V(ion) which comprehends the intrinsic volume of the ion and the void space around it; this term is proportional to the cube of the crystal radius of the ion. In equation (6) the term V”(elect) represents the volume change in the proximity of an ion due to electrostriction of the solvent. In a solvent of compressibility p. the electrostriction effect has been calculated as V”(elect)
=
r s rl
!J i0
/3dp4rtrLdr,
(7)
where p is the pressure at a distance r from the center of an ion, and a is the minimum approach distance of the ions. The assumption that the pressure. due to the charge of the ion is proportional to ?/r4 [21], gives the result V”(elect)
= - R;‘/r
(8)
where -_ is the iomc charge and B ts a constant. The substitution of equation (8) into equation (6) gives the Hepler’s equation V’ (ion)r
= Ar4 - 8;’
(9
which rcprescnts the equation of a straight line when the term L’“(ion)r is plotted against r4. Figure 3 shows these plots rclativc to the alkali metal cations and to the halogen anions. The V”(ion)r terms of ClO:, SCN-, NC3 ions have not been reported in Fig. 3 since the crystallographic radius
Apparent and partial
molar
volumes of concentrated
803
solutions
Treating ions in solution as hard spheres of radius r (in A) the volume (in ml) occupied by a mole of ions can be computed as V(cryst)
Fig. 3. Plot of the quantity cations and halide
C/“(ion)r us r4 for alkali anions in formamide.
metal
of these ions are not accurately known. A survey of Fig. 3 shows that the L’“(ion)r values of the investigated ions fall on the same straight line. The A and R parameters of equation (9) in formamide have been calculated by the least-squares method and the results reported in Table 5. In the same Table the A and B values of alkali metal and halide ions in water [22] and in dimethylformamide (DMF) [23] are also reported for comparison. A survey of Table 5 shows the following salient features. (I) In formamide like in water A and B values are independent of the sign of the ionic charge for alkali metal and halide ions; in DMF the A term relative to halide ions is greater and the B term is smaller than the corresponding terms for the alkali metal cations. (2) The value of the constant A in formamide, 4.7, coincide with the value found in water and is remarkably greater than the value found in DMF. Discussion
of the term
A
The analysis of the A term of the Hcplcr’s cquation [ZO] involves models related to the intrinsic volume of the ions and to the void space around it. Table
5. Values
of the A and B parameters
Alkali metal cations Halide anions ‘,I) ‘h’
From ref. [22]. From ref. [23].
AS)
(10)
I/ ‘(intr) = 4.35 ?.
of the Hepler‘s equation at 2S”C
B (ml A/mole) 4 4
= 2.52 r3.
(I 1)
Therefore the fact that in formamide the experimental A value for alkali metal and halide ions is close to the value of 4.35. can be attributed to the volume of the void spaces generated by spherical ions in a “random close packing distribution”. The non dependence of the A term upon the sign of the ionic charge. for monomalent cations and anions in this solvent is another important result of the data of Table S. In water and in some non-aqueous solvents this situation has been interpreted [23] by assuming that no important differences of organization of solvent molecules around cations and anions take place due to the fact that the solvent dipoles are equally accessible to the two type of ions. For the same reason A values, independent of the sign of the ionic charge have been predicted in formamidc whcrc the N atom is as accessible to the anions as the 0 atom to the cations. However, if the results in the formamide seem to agree with this prediction. (as shown in Table 5) the above assumption cannot be generalized. In water, in fact. in spite of identical A values for the two series
Formamide A (ml/mole 4.7 4.7
= 4/3 7~N. 10-24r3
In literature A values close to the throretica1 value of 2.52 are reported for alkali metal cations in DMF [23], and for tetraalkylammonium ions in solvents like water, [22] methanol. [24] DMF [23]. In the other cases the experimental A values result greater than the value predicted by equation (10) [25]. In principle an increase of the intrinsic volume of an ion in solution can be due to two factors: (a) the expansion of the ion when going from the crystal to solution; (b) the prcscncc of void space around the ions which should be added to the effective volume of the ions in the calculation of V’(intr). As P&r as the first point is concerned. the results of recent works 1261 showed that ionic radii in solution and in the crystal are nearly the same. Thus the fact that the experimental V (intr) values are greater than the theoretical values is attributed to a sort of void space packing efYect. An attempt to calculate the volume of the void space around a random distribution of hard spheres has been made by Robinson and Stokes r16]. By filling a graduated cylinder with lead spheres and adding water until all spheres were covered. these authors were able to show that in a “random close packing distribution” the void space around the spheres is 42O;, of the whole volume, or
m formamide,
in waler
and in dimelhylformamide
Will&’
(ml/rn%z 4.75 4.75
A3)
(ml Aymole) 10 10
2.5 3.8
18 2
804
P. BKUN~ AN” M.
of monovalent ions important differences of organization of solvent molecules about anions and cations are to be expected [27]. It should bc consldercd that factors other than the accessibility to the solvent dipole are involved in the A term of the Hcpler’s equation, which interprets the disorder in a region where the electrostriction of the solvent is pre-eminent. On the other hand the electrostriction phenomenon depends on structural factors which are frequently not known. It follows that due to back of information about the solvent structure and the geometry of the salvation shell around the ions, it is ditficult to distinguish the exact contribution of the void space effect from the electrostriction of the solvent.
The term R of the Hepler’s equation takes into account the clcctrostriction of the solvent molecules around an ion. This phenomenon, due to electrostatic interactions, gives rise to a tighter packing of the solvent molecules in solution than in the pure solvent. a protic solvent. the interactions In formamide, between the solvent molecules and the cations and anions follow two different patterns: alkali metal cations coordinate the solvent with interactions which :LIC mainl! 01‘ 111~ iota- dipole type; while in the interactlons between halide ions and solvent molecules hydrogen bonding is involved. As a consequence of this fact one could expect a substantial dillerence in the organization of the solvent around a cation and around an anion. assumption also supported by the results of conductivity measurements [2X]. The results of density measurements reported in this work however, seem to contrast this point as the B term of the Hepler’s equation have the same value for alkali metal cations and halide anions, as shown in Table 5. Data of Table 5 also show that the B term relative to the halide ions in formamide is smaller than in water, but thal it is almost equal to the value found in DMF, while the B term relative to alkali metal cations is smaller in fromamide when compared to the corresponding value in water and in DMF. The above results can be exnlained hv taking into account the small compressibility exhibited by this solvent and the polar forces caused by large unsymmetrical charge distribution in the formamide molecules. When an electrolyte is dissolved in a given solvent a breakdown in the lattice of the crystal occurs at first; afterwards the ions strongly coordinate around it some solvent molecules so that a solvodynamic entity is formed. Therefore the energy consumed in breaking up the lattice and the structure of the solvent. and the heat of reaction between ions and solvent dipoles. namely the solvation energy, are involved at the same time m the dissolution process. Calculation of the mean dipole-dipole interaction energy in formamide, and the comparison with the mean interaction energy of an ion with the dipole of this solvent show that the rcplacemcnt of a formamide molecule with an ion near a solvent molecule does not involve large energy change [2Y]. In other words the breaking of the structure of the solvent in the immediate neighhourhood of an ion (the region where the electroslriction is very high) and its conse-
DELLA
MONIVA
qucnt organ&&ion in the first salvation shell occur without relevant energy change. These considerations and the small compressibility of the formamide molccule also account for the closeness of the prnr values at infinite dilution to the molecular volume values of uni-univalent elcctrolytcs in this solvent. At the same time the constancy of the UIHL’ of uni-univalent salts in formamide in the extensive experienced concentration range can be also explained. CONCLUSIONS
Density measurements of uni-univalent electrolytes in formamide at 25’ show the following interesting features: (1) the slope of the curves relating the ~ntu and the square root of the conccntratlon is almost zero for all of the investigated salts; (2) the extrapolntion of the anrr‘ L’Sv,‘c graphs at infinite dilution gives @: values close to the molecular volume of the pure salts; (3) the ionic ~WW at infinite dilution have been found to follow the Hepler’s equation with the A and B coefficients independent of the sign of the ionic charge for alkali metal cations and halide anions; (4) the A term which comprehends the geometric contribution to V’(ion) has been found close to the sum of the intrinsic Ionic volume, and the volume of the void space which is generated around the ions in a “random close packing distribution”: (5) the low value of the B term in this solvent has been explained in terms of solvent structure energetically equivalent to the structure of the solution. _ .
REFERENCES
6. 7. 8. 9. IO II I2 13
14 I5 16 17
18 19 20 21.
P. Bruno and M. Della Monica. J. ohms. C’hrm. 76. 3034 (19721. D. 0. Masson. Pllil. M
Apparent 22. R. Zana and E. B. Yeager. 424 I (I 967). 23. F. Kawaizumi and R. Zana. (1974). 24. F. J. Millero. J. phys. Ch@m. 25. F. J. Millero, J. phys. Chem. 26. P. Mukeqee. J. phys. Chem.
and
partial
molar
volumes
.I. phys.
Chrm.
71.
521.
J. phys.
Chrm.
78.
1099
73. 2417 (1969). 72. 3209 (1968). 70. 2708 (1966); S. W. Renson and c’. J. Copeland. J. phys. Chem. 67. 1194 (1963); J. F. Desnoyers. R. E. Verrall and R. E. Corway
of concentrated
solutions
805
J. phys. Chew. 43. 423 (1965); R. H. Stokes, J. Am. chrm. Sot. 86. 979. 982 (1964). 27 H. S. Frank and W. Y. Wen, Discuss. Faruday Sot. 24. 133 (1957) 28. P. Bruno and M. Della Monica, J. phys. Chem. 76. 1049 (I 972). 29 P. Bruno. C. Gatti and M. Della Monica. I;/c< t~~~lrrrrr 4L.fU 20. 533 (1975).
P. BRUNO and M. DELLA MONICA* di Chimica Analitica dell’Universit8 dl Bari, Italy (Recewed
23 F&ruar~
1975)
Abstract-The
densely uf solutions of NH,SCN, KSCN, NH,N03, LiN03. NH,CIO,. NH,Br in formamIde at 25‘ has been measured in the concentration range going from - 0.5 m up to the saturated solution. By the least-square method the coefficients of the straight lines relating density and concentration have been calculated. It follows that the apparent molar volumes of salts (ama) m the investigated concentration range result independent of the concentration. and close to the molar volume of the pure salt. Calculated by the V” (the partial molar volume at infinite dilution) values of the tetraalkylammonium salts and following a Robinson and Stokes suggestion, ionic partial molar volumes (pmv) at infinite dilution agree for both methods. The ionic pmu at infinite dilution follow the Hepler’s equation. the A and B coefficients for alkali metal catwns and halide anions being independent of the sign of the ionic charge. The value of the A term has hem found close to the intrinsic ionic volume plus the volume of the void space which is generated around an ion in a “random close packing distribution”. The low values of Lhe coefficient B. which takes Into accounl the eleclrostriction of the solvent, has heen explained in terms of solvent structure energetically equivalent to the structure of the solution and with the small compressibility of this solvent
INTRODUCTION
In the past few years an increasing interest in ion +,olvent mtcractions in organic solvents has been shown by electrochemists through the investigation of thermodynamic and transport properties of electrolyte solutions. Among the various quantities which characterize salts in solution, apparent and partial molar volumes of electrolytes are of great help in understanding the interactions occurring in solution between a solute particle and a solvent molecule. So, while useful informattons on the ion-ion interactions can be drawn from the study of the concentration dependence of the apparent and partial molar volumes of salts in solution. the entity of ion-solvent and solvent-solvent interactions can be deduced from the values of the pn~r of clcctrolytes at infinite dilution. The results of density measurements of a series of sodium salts in formamidc solutions were discussed in a previous paper [l]. Solutions were investigated in a concentration range going from 0.1 to 0.5 m up to the saturated solutions, and the analysis of the dcnsity data, assuming a linear dependence on the concentration of the density of all the investigated salts. gave arm almost independent of the concentration; that is the slope of the Masson equation [Z] was found to be close to zero. The aim of the present work is to invcstigatc the behaviour of salts, other than sodium salts, in formamide solution and in a concentration range going from dilute to saturated solution, where strong modifications of the structure of the solutions are expected to take place. The choice of the electrolytes. lithium, *To whom
inquiries
should
be addressed.
potassium, and ammonium salts, has been made with a view to throw light on the influence of the cation charge density on the structure of formamide solutions.
EXPERIMENTAL
SECTION
Baker Analyzed Reagent formamide was deionized by means of a mixed bed of Amherlite ion-exchange resins loaded with H ’ ions and subsequently with HCONHions. Bcforc deionizing formamide was dried with molecular sieves 3 A (Union Carbide) [3]; the product obtained and used in the density measurements had the specific conductance of lO~‘W cm-‘. Lithium and ammonium nitrate (Carlo Erbd Reagent grade) were recrystallized twice from doubledistilled water; ammonium bromide (Carlo Erba Reagent grade) was also recrystallized three times from double-distilled water, and then dried in vacua over P,O, at 110’; ammonium perchlorate (Fisher Reagent grade) was recrystallized three times from conductivity water and then kept 48 h in vacua at 130”; ammonium and potassium sulphocyanate (Fisher Reagent grade) were recrystallized twice from conductivity water and then kept 48 h in vacua at 60”. Densities were measured to an accuracy of 0.01% using a 40ml pycnometer calibrated with conductivity water and the volumes were read on the graduated capillary side arm of the pycnometer to 0.001 ml by a magnifying glass. The solutions were prepared by adding formamide to a stock solution of the different salts; a thermostat controlled at 25 + O.CO5”C was used. The preparation of solutions of the diffcrcnt concentrations, and the
Table
I. Densities
of NH,SCN,
NH,NO,,
NH,ClO,.
NH,Br,
salts in formamide
at 25°C
d (g/ml)
0.5089 1.0336 2.1054 3.3.532 4.3043 5.6718 6.5450 7.8860
1.1324 1.1370 1.1407 1.1480 11552 1 1622 1.1683 1 1747 1.1x11
I. 1489 1.1784 1.2123 1.2333 1.2536 1.2890
filling of the pycnometer were saturated with dry nitrogen. RESULTS
AND
0.1507 0.7784 1S663 3.0578 4.8934 5.8599 6.9928
performed
DlSCUSSlON
The density of solutions of NH,SCN, KSCN, NH,NO,, LiNO,, NH,ClO,, NH,Br in formamide at 25” has been measured in the concentration range from _ 0.5 mole/l to saturated solution, and the results reported in Table I. The measured density d shows a linear dependence on the salt concentration c’. The coefficients of the straight lines relating density and concentration have been calculated by the Ieastsquares method and the results reported in Table 2 with the standard deviations. The experimental densities were also used to calculate the amv 0” of electrolytes through the equation : =do-d+cM, cd,
(1)
’
where Mr is the molar mass of the solute, d, is the density of the pure solvent and d is the density of the solution of concentration c. 2. Apparent
1.2433 1.2720 1.2965
I.3560 NH,ClO,
1.1349 I.1567 1.1849 I .2377 1.3023 1.3363 1.3761
in a dry-box
molar
volumes
and
density
equation
Best equation d = 1.12968 + d = 1.129S2 + d = 1.1295, + d = 1.1295, + d = 1.1295, + d = 1.1295, +
7.3415 2.2229 4.3427 4.6424 3.5282 2.9914
x x x x x x
IO-‘c 10~‘~ 10 2r lo-“c lo-*c IO-*c
3 1.1363 1.1458 1.1586 1.1700 1.1867 1.2154
KSCN
0.4168 1.0510 I .7726 2.2507 2.6833 3.4238
”
0.2246 0.5448 0.9748 1.3589 L.9061 1.8633 3.8028 4.7741 5.5665 7.5763
1.1410 1.1524 1.1770 1.2045 1.2259 1.2560 1.2749 1.3051
NH,Br
~
LiNO
NHz,NO,
0.3783 0.9959 1.5222 2.5382 3.43 IO 4.3591 5.2318 6.1452 7.0538
NH,SCN NH,No3 NH,C’lO, NH,Br KSCN LiNO,
LiNO,
(mole/l) NH.,SCN
salt
and
c
c
(mole/l)
Table
KSCN
0.3826 0.9650 1.491 I 1.9418 2.7308 2.9493
1.1462 1.1716 1.1939 I .2142 1.2478 1.2579
The @,, values obtained in this way were plotted in Fig. I against the square-root of the concentration. For comparison, in the same Fig. 1. the umu values of some sodium salts calculated by data in a previous article [I] have also been reported. As obvious in the concentration range used and for the reported salts. the slope of the @,, us ,, ‘2 curves is close to zero. Thus, if @,: = @3 = const. the density must be given by the equation:
where the coefficient of the concentration c should be an additive property of the ions[4J_ The differences between the coefficients of the concentration c in the equation of Table 2 of this work and Table 4 of the previous work [I] relative to couple of electrolytes with a common ion give in fact constant values as shown by data reported in Table 3. The dependence of the amv of electrolytes on the salt concentration was first investigated by Masson [2_l who found that amv follow an empirlcal equation
@ =@“+s&. 0
of sc~me uni-univalent 25°C
Y
electrolytes
(3)
L
in formamide
Standard deviation
Concentration range (mole/l)
Apparent molar volumes (ml/mole)
o.OQo3 0.0003 0.0003 0.0005 o.ooo2 O.WO2
0.4-7.1 0.5-7.9 0.4-2.9 0.4-3.4 0.2-7.0 0.2z7.6
60.8 51.0 65.6 45.6 54.8 34.7
solutions
Molecular volumes (ml/mole)
48.2 60.3 42. I 29.0
at
Apparent and part&
molar volumes of concentrated
801
solutions
Table 4. Ionic partial molar volumes in formamide
1OIl
at 25 C
V’ (Ion)
V (lon)~r
(ml/mole)
(ml A/mole)
NH&N .-*_~cc.-.-.-
.-
Li’ Na’
60
KSCN
.-.-.-
.-.-
K+
-4.0 -2.3 6.1
0.60 0.95 1.33
NH; Cl_ BY I_
13.3 25.9 32.3 43.9
1.48 1.81 I 95 2.16
SUN-
3x.7 4x.7
ClO, NO;
. . . I=-.
&
.A.-.-
40
No I
.
l
a-.-NaN4
LINE
l-.-.l.C*-.-.-*-*
.
30
.
-
.-•-•-NaBr
l
I
I 0.5
I
2.5
1.5 c.
ml-I-’
Fig. 1. Apparent molar volume CD, of salts in formamide against the square root of the concentration c at 25°C. In equation (3), @‘:’ is the amo of the salt when the concentration is zero and S, is the slope of the experimental curves whose value changes with the nature of the electrolyte. Later. Redlich and Rosenfeld [S, 61. by applying the interionic-attraction theory of Debye and Hiickel[7] to the concentration dependence of the UIIW, were able to calculate the theoretical limiting slope for 1-l electrolytes at temperature T in any solvent where the dielectric constant and compressibility are LI and @ respectively, through the relationship (R: gas constant. N: Avogadro constant. r: elementary charge) S,. = NJ tp3(Sn/o”RT)“2((dlnn/dp).,
- /?/3).
(4)
From equation (4) it is evident that the limiting law dcpcnds only on the temperature and on the physical properties of the pure solvent. The confusing situation existing in the literature about this equation has been recently pointed out by Millcro [X]; he has observed that, while the Redlich-Rosenfeld equation, derived from the interionic-attraction theory of Debye and Hiickel. should bc considcrcd as a limiting equation,
Couple of electrolytes
A g/mole
NnNO,~NH,NO, NaBr-NH,Br NaCIO,-NH,CIO,
2.1 X 10 z 2.2 X 10-2 2.1 X 10-2
NnSCN~NH,SCN
2.1 x 10
2
53.8
-2.4
2.2 8.1
19.7 48.2 63.0 94.8
the Masson equation can represent the uww data over a wide concentration range. As a consequence, the extrapolation to infinite dilution by using the Masson equation can give results somewhat incorrect. The theoretical hmitmg slope of equation (3) in formamide at 25 for I-I electrolytes has been calculated by the experimentally determined Dunn [93 from (dhD/dp), and /I data as 1.104ml mole-“” I’,“. and recently confirmed for dilute solutions of KCI, KRr. KI, and KN03 [lo]. T& graphs of Fig. I showing the slope of the a,. L’SX c curves equal to zero reveal that the data of the salts investigated in this work arc too high in concentration for the limiting law to hold. In dilute solutions nor’ are calculated from smatl differences between two large quantities [SW equation(l)]: thereforenrw in the dilute rcglon become extremely sensitive to the experimental error. Unfortunately. at present, we are unable to give the density of formamide solutions with enough accuracy so as to permit the proper extrapolations to be done. Nevertheless. in consequence of the small theorcticdl S, value in this solvent, it is easy to assume that umc vdhes at infinite dilution are only slightly different from the values obtained by extrapolation of data repot-ted in Fig. 1. This statement apparently is in disagreement wItI the results of Gopal et crl[ 10. 111 who obtained a difference of 1.3 ml/mole in the @;, of KI and KNO, by extrapolating data of different concentration range. The analysis of the @V VJ ,,‘c curves. relative to KI and KNO, in formamide. derived from the work of these authors, shows on the contrary, that the discrepancy bctwcen the two extrapolated @i: values can be easily rationalized by a difference in the accuracy of the two sets of data. Thus the arnr of the salts at infinite dilution @I. have been obtained in this solvent by extrapolating to zero concentration the data of Fig. I. These values are reported in Table 2 with the molecular volume of the pure salts. The results reported in Table 2 and in Fig. 1 suggest some interesting considerations. The prno V, are related to the c~rna of salts by the relationship:
all terms in equation (5) having their usual meaning. As a consequence of this equation the apparent and the partial molar volume values of salts coincide
802
P.
BKUNO
AND
M
when d@“/d+c is zero. In formamidc this fact happens in all the wide concentration range experienced; therefore the values reported in Table 2 represent both partial and apparent molar volumes of salts. A survey of the results of Table 2 also shows that V ’ values of I- I valent electrolytes in formamide are remarkably close to the molecular volumes of the pure salts, This important result means that, except for very dilute solutions, the volume of any solution can be obtained as sum of the volume of the salt and the volume of the solvent used to dissolve it; this fact is not easy to explain if clectrostriction phenomena, as a consequence of interactions between ions and the strong dipole of formamide 1121 are taken into account.
DELLA MONICA
./ /
I
z
E
- eoo2 :” “>
./
44 In an elctrolytc solution, when the ion-pairing is not extensive, the partial molar volume of the salts follows an additive principle [13]. The various attempts made in order to obtain the individual contribution of the cations and of the anions to the partial molar volume of the salt have met the dificulty of assigning an absolute volume to the ions. Thus numerous empirical methods have been proposed by difFcrcnt workers. In aqueous solutions pnz~l of ions at infinite dilution have been calculated assuming that the plot of the V’ of the tetraalkylammonium halides us the molar mass of the cations is a straight line [14]. The pmc V? of the anions is thus obtained by extrapolating the data to zero molar mass. Alternatively the indcpendcnt variable molar mass has been replaced by the carbon number of the tetraalkylammonium ions [S]. Another set of data has been estimated in water by selecting a value of the LIWIVof H+ that would make the V> of monovalent cations and anions fall on a smooth curve as a [unction of the crystal radii cubed [IS]. and suggestion of Robinson Following a Stokes [lh], pnlc of ions at infinite dilution can he obtained by assuming that large monovalent anions, namely Brand I-, are not solvated in water. In this calculation it should be considered that the contribution to the volume of the whole system is given by the intrinsic volume of the ions plus the void space around it. which in a “random close packing distribution” has been demonstrated to be 1.83/2X times the volume occupied by the spheres [17J. Since acid substances have not been suitably investigated in this solvent ionic partial molar volumes at infinite dilution in formamide based on the V ’ of H+ Ion cannot be calculated. As far as the tetraalkylammonium salt method is concerned, an attempt to calculate ionic gmu in forrndmide has been made using the data of Gopal and coworkers [lS] relative to the iodide serie. As shown by the graph of Fig. 2 a pmr at infinite dilution of iodide ion equal to 44 ml/mole can be calculated in the report of the I/’ values of tetraalkylammonium iodides cs the carbon number of the cations. while the plot of the V” values I’S the molar mass of the salts gives an absurdly ncgativc VT (I-) value. The ionic I“ values in formamide reported in Table 4 were calculated according to the Robmson and Stokes suggestion [16] by making use of V’(Br-) and VO(Ip) equal to 32.3 and 43.9 ml/mole respectively. of Table 4 can also Alternatively the ionic V’. values
I
0
I
I
4
8 Carbon number
I
1
I2
16
of R4 N’
Fig. 2. Plot of the partial molar volume V- of tetraalkylammonium
iodides
DS the carbon
number
of cations.
bc obtained by the V’ values of the tetraalkylammonium iodides since the V”(I-) value deduced by the graph of Fig. 2 and the value calculated according to the Robinson and Stokes method coincide. Absolute pmz~ of ions in solutions have not been obtained so far. nevertheless to study the structure of the solution around an ion density data have been extensively used by a number of workers 1191. A recent interpretation of the ionic amv in terms of size and charge of the ions makes use of the equation [ZO] V’(ion)
= V”(intr)
+ V”(elect),
(6)
where the first term on the right side is the geometric contribution to V(ion) which comprehends the intrinsic volume of the ion and the void space around it; this term is proportional to the cube of the crystal radius of the ion. In equation (6) the term V”(elect) represents the volume change in the proximity of an ion due to electrostriction of the solvent. In a solvent of compressibility p. the electrostriction effect has been calculated as V”(elect)
=
r s rl
!J i0
/3dp4rtrLdr,
(7)
where p is the pressure at a distance r from the center of an ion, and a is the minimum approach distance of the ions. The assumption that the pressure. due to the charge of the ion is proportional to ?/r4 [21], gives the result V”(elect)
= - R;‘/r
(8)
where -_ is the iomc charge and B ts a constant. The substitution of equation (8) into equation (6) gives the Hepler’s equation V’ (ion)r
= Ar4 - 8;’
(9
which rcprescnts the equation of a straight line when the term L’“(ion)r is plotted against r4. Figure 3 shows these plots rclativc to the alkali metal cations and to the halogen anions. The V”(ion)r terms of ClO:, SCN-, NC3 ions have not been reported in Fig. 3 since the crystallographic radius
Apparent and partial
molar
volumes of concentrated
803
solutions
Treating ions in solution as hard spheres of radius r (in A) the volume (in ml) occupied by a mole of ions can be computed as V(cryst)
Fig. 3. Plot of the quantity cations and halide
C/“(ion)r us r4 for alkali anions in formamide.
metal
of these ions are not accurately known. A survey of Fig. 3 shows that the L’“(ion)r values of the investigated ions fall on the same straight line. The A and R parameters of equation (9) in formamide have been calculated by the least-squares method and the results reported in Table 5. In the same Table the A and B values of alkali metal and halide ions in water [22] and in dimethylformamide (DMF) [23] are also reported for comparison. A survey of Table 5 shows the following salient features. (I) In formamide like in water A and B values are independent of the sign of the ionic charge for alkali metal and halide ions; in DMF the A term relative to halide ions is greater and the B term is smaller than the corresponding terms for the alkali metal cations. (2) The value of the constant A in formamide, 4.7, coincide with the value found in water and is remarkably greater than the value found in DMF. Discussion
of the term
A
The analysis of the A term of the Hcplcr’s cquation [ZO] involves models related to the intrinsic volume of the ions and to the void space around it. Table
5. Values
of the A and B parameters
Alkali metal cations Halide anions ‘,I) ‘h’
From ref. [22]. From ref. [23].
AS)
(10)
I/ ‘(intr) = 4.35 ?.
of the Hepler‘s equation at 2S”C
B (ml A/mole) 4 4
= 2.52 r3.
(I 1)
Therefore the fact that in formamide the experimental A value for alkali metal and halide ions is close to the value of 4.35. can be attributed to the volume of the void spaces generated by spherical ions in a “random close packing distribution”. The non dependence of the A term upon the sign of the ionic charge. for monomalent cations and anions in this solvent is another important result of the data of Table S. In water and in some non-aqueous solvents this situation has been interpreted [23] by assuming that no important differences of organization of solvent molecules around cations and anions take place due to the fact that the solvent dipoles are equally accessible to the two type of ions. For the same reason A values, independent of the sign of the ionic charge have been predicted in formamidc whcrc the N atom is as accessible to the anions as the 0 atom to the cations. However, if the results in the formamide seem to agree with this prediction. (as shown in Table 5) the above assumption cannot be generalized. In water, in fact. in spite of identical A values for the two series
Formamide A (ml/mole 4.7 4.7
= 4/3 7~N. 10-24r3
In literature A values close to the throretica1 value of 2.52 are reported for alkali metal cations in DMF [23], and for tetraalkylammonium ions in solvents like water, [22] methanol. [24] DMF [23]. In the other cases the experimental A values result greater than the value predicted by equation (10) [25]. In principle an increase of the intrinsic volume of an ion in solution can be due to two factors: (a) the expansion of the ion when going from the crystal to solution; (b) the prcscncc of void space around the ions which should be added to the effective volume of the ions in the calculation of V’(intr). As P&r as the first point is concerned. the results of recent works 1261 showed that ionic radii in solution and in the crystal are nearly the same. Thus the fact that the experimental V (intr) values are greater than the theoretical values is attributed to a sort of void space packing efYect. An attempt to calculate the volume of the void space around a random distribution of hard spheres has been made by Robinson and Stokes r16]. By filling a graduated cylinder with lead spheres and adding water until all spheres were covered. these authors were able to show that in a “random close packing distribution” the void space around the spheres is 42O;, of the whole volume, or
m formamide,
in waler
and in dimelhylformamide
Will&’
(ml/rn%z 4.75 4.75
A3)
(ml Aymole) 10 10
2.5 3.8
18 2
804
P. BKUN~ AN” M.
of monovalent ions important differences of organization of solvent molecules about anions and cations are to be expected [27]. It should bc consldercd that factors other than the accessibility to the solvent dipole are involved in the A term of the Hcpler’s equation, which interprets the disorder in a region where the electrostriction of the solvent is pre-eminent. On the other hand the electrostriction phenomenon depends on structural factors which are frequently not known. It follows that due to back of information about the solvent structure and the geometry of the salvation shell around the ions, it is ditficult to distinguish the exact contribution of the void space effect from the electrostriction of the solvent.
The term R of the Hepler’s equation takes into account the clcctrostriction of the solvent molecules around an ion. This phenomenon, due to electrostatic interactions, gives rise to a tighter packing of the solvent molecules in solution than in the pure solvent. a protic solvent. the interactions In formamide, between the solvent molecules and the cations and anions follow two different patterns: alkali metal cations coordinate the solvent with interactions which :LIC mainl! 01‘ 111~ iota- dipole type; while in the interactlons between halide ions and solvent molecules hydrogen bonding is involved. As a consequence of this fact one could expect a substantial dillerence in the organization of the solvent around a cation and around an anion. assumption also supported by the results of conductivity measurements [2X]. The results of density measurements reported in this work however, seem to contrast this point as the B term of the Hepler’s equation have the same value for alkali metal cations and halide anions, as shown in Table 5. Data of Table 5 also show that the B term relative to the halide ions in formamide is smaller than in water, but thal it is almost equal to the value found in DMF, while the B term relative to alkali metal cations is smaller in fromamide when compared to the corresponding value in water and in DMF. The above results can be exnlained hv taking into account the small compressibility exhibited by this solvent and the polar forces caused by large unsymmetrical charge distribution in the formamide molecules. When an electrolyte is dissolved in a given solvent a breakdown in the lattice of the crystal occurs at first; afterwards the ions strongly coordinate around it some solvent molecules so that a solvodynamic entity is formed. Therefore the energy consumed in breaking up the lattice and the structure of the solvent. and the heat of reaction between ions and solvent dipoles. namely the solvation energy, are involved at the same time m the dissolution process. Calculation of the mean dipole-dipole interaction energy in formamide, and the comparison with the mean interaction energy of an ion with the dipole of this solvent show that the rcplacemcnt of a formamide molecule with an ion near a solvent molecule does not involve large energy change [2Y]. In other words the breaking of the structure of the solvent in the immediate neighhourhood of an ion (the region where the electroslriction is very high) and its conse-
DELLA
MONIVA
qucnt organ&&ion in the first salvation shell occur without relevant energy change. These considerations and the small compressibility of the formamide molccule also account for the closeness of the prnr values at infinite dilution to the molecular volume values of uni-univalent elcctrolytcs in this solvent. At the same time the constancy of the UIHL’ of uni-univalent salts in formamide in the extensive experienced concentration range can be also explained. CONCLUSIONS
Density measurements of uni-univalent electrolytes in formamide at 25’ show the following interesting features: (1) the slope of the curves relating the ~ntu and the square root of the conccntratlon is almost zero for all of the investigated salts; (2) the extrapolntion of the anrr‘ L’Sv,‘c graphs at infinite dilution gives @: values close to the molecular volume of the pure salts; (3) the ionic ~WW at infinite dilution have been found to follow the Hepler’s equation with the A and B coefficients independent of the sign of the ionic charge for alkali metal cations and halide anions; (4) the A term which comprehends the geometric contribution to V’(ion) has been found close to the sum of the intrinsic Ionic volume, and the volume of the void space which is generated around the ions in a “random close packing distribution”: (5) the low value of the B term in this solvent has been explained in terms of solvent structure energetically equivalent to the structure of the solution. _ .
REFERENCES
6. 7. 8. 9. IO II I2 13
14 I5 16 17
18 19 20 21.
P. Bruno and M. Della Monica. J. ohms. C’hrm. 76. 3034 (19721. D. 0. Masson. Pllil. M
Apparent 22. R. Zana and E. B. Yeager. 424 I (I 967). 23. F. Kawaizumi and R. Zana. (1974). 24. F. J. Millero. J. phys. Ch@m. 25. F. J. Millero, J. phys. Chem. 26. P. Mukeqee. J. phys. Chem.
and
partial
molar
volumes
.I. phys.
Chrm.
71.
521.
J. phys.
Chrm.
78.
1099
73. 2417 (1969). 72. 3209 (1968). 70. 2708 (1966); S. W. Renson and c’. J. Copeland. J. phys. Chem. 67. 1194 (1963); J. F. Desnoyers. R. E. Verrall and R. E. Corway
of concentrated
solutions
805
J. phys. Chew. 43. 423 (1965); R. H. Stokes, J. Am. chrm. Sot. 86. 979. 982 (1964). 27 H. S. Frank and W. Y. Wen, Discuss. Faruday Sot. 24. 133 (1957) 28. P. Bruno and M. Della Monica, J. phys. Chem. 76. 1049 (I 972). 29 P. Bruno. C. Gatti and M. Della Monica. I;/c< t~~~lrrrrr 4L.fU 20. 533 (1975).