J . C . M C C O U B R E Y 747
SPECIFIC INTERACTION OF IONS
BY E. A. GUGGENHEIM AND J. C. TURGEON Chemistry Dept., The University, Reading
Received 8th September, 1954
After a thorough revision of the best values for ionic interaction coefficients in aqueous golutions at 0" C and at 25" C, the theory of specific interaction has been applied quanti tatively to several acidbase equilibria.
The formulae of Debye and Huckel 1 for the thermodynamic properties of electrolyte solutions are based on the model of rigid ions with diameters all equal. Sometimes one meets formulae of the same type with different values of the diameter relating to different ions in the same solution, but such formulae contra vene the requirements of thermodynamics. A modified set of formulae was pro posed by one of us 2 having the following properties.
(i) They are thermodynamically mutually consistent. (ii) They take account of the specific differences between electrolytes of the
same charge type present in the same solution. (iii) They enable the thermodynamic properties of mixed electrolytes to be
deduced from those of single electrolytes. (iv) They embrace Bronsted's 3 principle of specific interaction. (v) They represent the facts with a useful degree of accuracy, at least for uni
univalent, biunivalent and unibivalent electrolytes, up to an ionic strength of about 0.1.
These formulae contain parameters called '. interaction coefficients " Is,, x for cach combination of a cation R and an anion X. The value of each interaction coefficient /?R,x is most simply determined fiom measurements on a solution containing only the electrolyte whose ions are R and X. The values of /lR,x so determined can then be used for estimating the thermodynamic properties of mixed electrolytes.
Values of the coefficient /3R,x for numerous electrolytes in aqueous solution have been determined 4 from the experimental data and have been tabulated.5 Most of the values being derived from freezingpoint measurements, relate to 0" C . A few, derived from e.m.f. measurements, relate to 25" C or other temperatures.
The main object of the present paper is to apply the theory to some experi mental measurements of acidbase equilibria. It is, however, expedient at the
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748 SPECIFIC I N T E R A C T I O N O F IONS
same time to revise the previously published values of the interaction coefficients. The need for this arises from the following ca~ises.
(i) Whereas the interaction coefficients originally calculated nearly twenty years ago related to activity coefficients and osmotic coefficients defined on the mole fraction scale (" rational coefficients ") it is in fact customary to use activity coefficients and osmotic coefficients defined on the molality scale (" practical coefficients ") .
(ii) The most precise determination of the heat of fusion of ice leads to a value of the cryoscopic constant of water slightly different from that formerly used.
(iii) Improved values of the general physical constants and of the dielectric constant of water Iead to modified values of the fundamental coefficient in the formulae of Debye and Huckel.
(iv) Thanks to isopiestic measurements we can now assign values to p at 25" C for many miunivalent electrolytes for which previously only values at 0" C had been determined.
(v) The method of analyzing the experimental data has also been improved.
NOTATION
The following symbols will be used : R, R' cations, x, X' anions, ZR = z,. number of positive charges on cation,  zx = z number of negative charges on anion, v+ number of cations per molecule of electrolyte, V number of anions per molecule of electrolyte, v_ harmonic mean of v+ and v, mR molality of cation, mX molality of anion, m molality of electrolyte, I YR, X
4
PR, X A = a/In 10, B EE 2$3/ln 10, 0 freezingpoint depression, A, b E e.m.f ., E" standard e.m . f., KO acidity constant, Kw ionization constant of water.
ionic strength (on moIality scale), activity coefficient (on molality scale) of electrolyte having cation R and anion X, osmotic coefficient (on molality scale), fundamental coefficient in DebyeHuckel formulae, interaction coefficient between cation R and anion X,
rx
coefficients in cryoscopic fonnula,
BASIC FORMULAE
We begin by quoting, without derivation, the basic formulae 6 relating to the The mean activity coefficient yR, x of the electrolyte having cation molality scale.
R and anion X is given by
which for a solution of a single electrolyte reduces to
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E. A . GUGGENHEIM AND J . C . TURGEON 749
where the subscripts R, X have been dropped. It is sometimes convenient to use the corresponding formulae involving decadic logarithms, for example, for a single electrolyte,
logy = 
where A G cc/ln 10, B = 2$/ln 10.
It is then given by
where
We shall use the osmotic coefficient 4 only in solutions of a single electrolyte.
1  4 = * cc z+zI* o(I+)  ypi11,
and tables of (T are availabIe.7
been calculated and tabulated.8 In particular : Values of the DebyeHuckel constant cc in water at various temperatures have
at 0" C, A = a/ln 10 = 0.4883 kg3 mole%, 4 cc = 0.374 kga mole), at 25" C , A E a/ln 10 = 0.5085 kgs mole*.
VALUES OF INTERACTION COEFFICIENTS
The revised values of the interaction coefficients calculated from measurements on solutions of single uniunivalent electrolytes are collected in table 1 . Columns (a) contain values at 0" C obtained from measurements of freezingpoint. Columns (b) contain values at 25" C obtained from measurements of e.m.f. Column (c) contains values obtained from isopiestic measurements relative to NaCl. Table 2 relates to biunivalent and unibivalent electrolytes and, for the reason given later, contains only values obtained from measurements of freezingpoints. We shall now indicate as briefly as possible how these values of /3 have been obtained.
electrolyte
HCI HBr HI HC104 HNO3 LiCl LiBr LiI LiC103 LiC104 LiNO; Li02CH LiOAc NaF NaCl
NaBr NaI NaC103
TABLE 1 .+ VALUES FOR UNIUNIVALENT ELECTROLYTES (0) (0) (b) (b )
ref. B ref. B k g mole1
025 a1 0.27 bl 0.33 b2
0" c 25" C kg mole1
0.16 0.20 0.30
0.25 035 0.23 0.1 1 0.19
0.1 1 0.1 1 0.20
0
a2 a3 a3
a4 a4 a5 a6 a6
a3 a7 a3
a4
0.1 5 b3
(4 25" C
B kg mole1
036 0.30
0.22 0.26 0.35
0.34 0.21
0.18 0.07
(0.15)
0.17 0.2 1 0.1 0
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750 SPECIFIC INTERACTION O F IONS
electroIyte
NaC104 NaBrOj NaI03 NaN03 Na02CH NaOAc NaCNS NaH2P04 KF KCl
KBr KI KC103
KBr03 KTO3
KO;?CH KOAc KCNS
RbCl RbBr RbI RbN03 RbOAc CSCl CsBr CSI CsN03 CsOAc AgN03 TIC104
KH2P04
TI NO^ TlOAc
TABLE 1 .(contd.)
(0) (a) (6) (6)
ref. B ref. kg mole]
B kg mole]
0" c 25" C
005 a4
 0.41 a8  0.04 a5
0.13 a6 026 a6
0.05 a9 0.04 a3, a10 0.10 b4 0.00 a l l 0.06 a3
0.19 a4  0.30 a9  0.55 a4  0.55 a9
 0.43 a8  0.43 a9  0.26 a5  0.3 1 a10
0.1 5 a6 0.30 a6
 0.14 b5
25" C B
kg mole1
0.13 0.0 1
0.04
0.23 0.20
 0.06 0.13 0.10
0.1 1 0.15
 0.04
 0.07  0.07
 0.1 1
0.26 0.09
 0.16 0.06 0.05 0.04
 0.14 026 0 0
 0.01 0.15 0.28
 0.14 . 0.17  0.36  0.04
(a) values at 0" C from freezing points. (b) values at 25" C from e.m.f. (c) values at 25" C from isopiestic measurements, relative to NaCI, by Robinson
and Stokes.
TABLE 2 . 4 AND B VALUES OF BIUNIVALENT AND UNIBIVALENT ELECTROLYTES FROM FREEZING POINTS
ref. B kg mole] B
kg mole] electrolyte
BaC12 + 07 + 0.8 a8 Ba(N03)2  0.5  0.6 a12 coc12 t 1 . 1 + 13 a8 K2S04 0.1 0.1 a8
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E . A . GUGGENHEIM A N D J . C . TURGEON 75 1
FREEZING PoiNTsThe osmotic coefficient $ is calculated from the freezing point depression 0 by means of the formula
S(1 + be) = (v+ + v)hm+, where the cryoscopic constants A and b have the values (see appendix) :
h = 1.860 deg. mole1 kg, b  4.8 x 104 deg.*
We dcfine a quantity +st by
We then have p t = 1  $ cc z+z I b ( 2 3 ) .
S(I + be)  (v+ + v) Am@' == (v+ + v)hn?(+  +st) = (v+ + v)$?Xm2 = 2v+v/3hm2,
so that, if we plot S(l + be)  (v+ + v)Amg!St against m2, we should obtain a straight line of slope 2v+v/3h, from which /l can be calculated. This procedure is illustrated in fig. 1 which relates to measurements on KCl by Adams,9 by
2
0
 2
 10  12 
0 10 20 30 4 0 5 0 6 0 70 80 90 100 10 120 lo" m2/mole2 kg'
FIG. 1.
Scatchard and Prentiss 10 and by Lange,ll and to measurements on KC103 by Scatchard, Prentiss and Jones12 and by Lange and Herre.13 The accuracy attained in freezingpoint measurements varies widely. An accuracy of i 00002" C has been attained but rarely. We observe from fig. 1 that the three sets of measure ments on KC1 are in fair agreement, but that in the case of KC103 there is a glaring discrepancy since the two straight lines have slopes corresponding to ,B = 0.19 kg mole1 and  0.30 kg mole1. Adams 9 calibrated his thermocouple by comparison with a 24 junction element calibrated by the Bureau of Standards. Scatchard, Prentiss and Jones 12 calibrated their thermocouples against standard ized platinum resistance thermometers. Lange,ll by contrast, states that he calibrated his thermocouple against a Beckmann thermometer, calibrated at the Reichsanstalt, at the KNO3 + H20 eutectic ( 284' C) and at the K2Cx207 + H20 eutectic ( 0.63" C), but gives no details. Lange and Herre 13 give no information at all about their calibrations and we consider them suspect. Until this dis crepancy is cleared up we must reluctantly admit the possibility of other /3 values being wrong by as much as 0.1 kg mole1.
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752 SPECIFIC I N T E R A C T I O N OF IONS
E.M.F. OF CELLS WITHOUT TRANSFERENCE.we shall illustrate the procedure on the e.m.f. measurements of the cell
Pt, H2 I H a (m) 1 HgzC12 I Hg made by Hills and Ives 14 at 25" C. We define a quantity E"' by
RT RT a i d E"' 3 E + 2  Inn2  2   F 1 + nia F
and we plot E"' against M, obtaining a straight line of intercept E" on the nz = 0 axis and of slope 2PRT/F, from which we calculate p. This plot is shown in fig. 2.
2 72
27 I
270
2 69
> 268
lu 267
2 66
265
I I 1 I I I I I I I I I
' 0
 

7 2
71
7 0
69
60
67
66
65
It is possible to draw a straight line from which only the points for the three lowest concentrations and one other point below m = 0.1 mole kg1 deviate by more than 0.01 mV. This straight line leads to the value B = 0.228 kg mole1 or p = 0.262 kg mole1. By attaching greater weight to the three lowest con centrations we can obtain the alternative value B = 0.234 kg mole1, or /3 = 0.269 kg mole1. These measurements are almost unique in their accuracy and we consider them to be the only measurements which justify discussion of the third significant figure in the value of /I.
According to the measurements of Harned and Ehlers 15 on the cell
Pt, H2 I HCl (n~) I AgCl 1 Ag &.I,, is within their experimental error independent of temperature throughout the range 0" C to 35" C and has the value /3 = (0.270 & 0.005) kg mole1 or B = (0.235 + 0*005) kg mole1. These measurements also determine the standard e.m.f. E". The value at 0" C is E" = 236.55 (abs) mV and at 25" C is E" = 2225 (abs) mV.
Similar measurements by Harned, Keston and Donelson 16 on the cell
Pt, H2 I HBr (172) I AgBr 1 Ag lead to the value /3 = 0.33 kg mole1 or B = 0.29 kg mole1 at 25" C and these values are insensitive to temperature over the range 0" C to 25" C .
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E . A . G U G G E N H E I M AND J . C. T U R G E O N 753
E.M.F. OF CELLS WITH TRANSFERENCE.we shall illustrate the method by the e.m.f. measurements of the cell
made by Brown and MacInnes17 combined with measurements of the cation transport number tf by Longsworth.18 The e,m.f. E of this cell is given by
FE  17iyf RT ldl y"  _  2 t + In __
where denotes the value of tf averaged over the range In my from rn = rn' to m = mr'. Provided this range is not too great this average is readily determined by using rough values of y, or if necessary by successive approximation. Having thus determined t', we calculate values of In (y'"''), and so for each value of rn a vahe of k + log y, where k is an arbitrary unknown constant whose value will be determined later. We plot
against m obtaining a straight line of slope B = 2P/In 10 with intercept k on the m = 0 axis. The plot is shown in fig. 3 in which the ordinate scale has been adjusted so that k = 0.
14
12
10 A
3 6
4 0
2
0 0 I 2 3 4 5 6 7 8 9 10 I I
10 'm/mole kg'
FIG. 3.
VAPOUR PREssuREs.Isopiestic measurements cannot yet give accurate values of the osmotic coefficient at molalities less than 0.1 and our formulae are not valid at ionic strengths greater than 0.1. Consequently for uniunivalent electro lytes we can estimate values from the value of the osmotic coefficient at the single concentration 0.1 mole kg1 by means of the relation /3 = 104  919 at 25" C. Even so the experimental uncertainty in $ is at least 0.001 so that the uncertainty in the derived value of p is at least 0.01 kg mole1. Apart from this uncertainty, we consider that there is a small systematic error in the values of 4 at 25" C tabulated by Robinson and Stokes,l9 The isopiestic method gives only relative values and therefore to obtain absolute values one has to assume 4 values for at least one standard substance. Robinson and Stokes, following Robinson,aO assume that 4 = 0.932 for 0.1 molal NaCl at 25" C . This value was obtained by applying the GibbsDuhem relation to activity coefficients
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754 SPECIFIC INTERACTION O F IONS
obtained by e.m.f. measurements. By the same procedure we find 4 == 0.934. It is only a small difference, but it raises all the C$ values to the same extent and so raises all /3 values by 0.02. As our source of 4 values at nz == 0.1 mole kg1 we have used throughout the values tabulated by Robinson and Stokes,lY thus corrected.* It should be noted that the 4 values for HCI and HBr in the tables of Robinson and Stokes are derived from e.m.f., not isopiestic, measurements. Their tables also include values for the alkali hydroxides obtained by extra polation from e.m.f. measurements at higher molalities of cells containing amalgam electrodes. These values are not much more than guesses and for the purpose of our tables we disregard them.
For electrolytes other than miunivalent the isopiestic measurements do not extend down to an ionic strength 0.1 and consequently cannot yield useful estimates of 18. For biunivalent and unibivalent electrolytes the measurements of Robinson and Stokes 19 extend down to a molality 0.1 which is an ionic strength 0.3. If p values were estimated from these measurements they would be uncertain to about & 0.1 kg mole1 and we have not considered it worth while tabulating such estimates. If required they are immediately obtainable by applying the formula /3 = 7.5 (c$  0.782) at molality 0.1.
ACIDBASE EQUILIBRIA
Although the applicability of the theory of specific interaction to acidbase equilibria was described by one of us,21 no previous detailed comparison is known between our formulae and experimental measurements.
ACETIC ACID/ACETATE.MeaSUrefnenfS Of e.m.f. Of the Cell HOAc a NaOAc b AgCl Ag,
PtyH2 1 NaCI c ! where a, 6, c denote molalities and the ratios a : b : c were maintained constant, have been made by Harned and EhIers.22 The e.m.f. E of this ccll may be expressed in the form
I
where Eo is the standard e.m.f. of the cell Pt, H2 1 HCI I AgCl 1 Ag
and Ka is the acidity constant of acetic acid. ion concentration and occurs as a small corrcction. experiments given with sufficient accuracy by
Thc symbol h denotes the hydrogen Its value is thioughout these
hh/a = K, = 17 x 105 molc kg1. We define E' by
If we plct E' against b + r , we expect to obtain a straight line with intercept E'  (RTIn KJF on the in = 0 axis and of slope
* Since submitting this text for publication we have had the advantage of an exchange of views by correspondence with Prof. Robinson. We are now agreed that in onetenth molal sodium chloride the best value for 4 lies between 0.933 and 0.934 and that in one tenth molal potassium chloride the best value lies between 0.928 and 0.929. Values for all other 1 : 1 electrolytes at this molality determined by the isopiestic method are affected accordingly.
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E . A . G U G G E N H E I M A N D J . C . T U R G E O N 755
Thc measurements of Harned and Ehlers 22 cover a wide range of temperature. We choose the measurements at 0" C because these can be correlated with freezing point measurements. The measured e.m.f.s have been converted from int. mV to (absolute) mV and, according to Birge's values of the general constants, (RT In lO)/F = 54.199 mV at 0" C. The computed values of E' have been plotted against b + c in fig. 4. According to the freezingpoint measurements on sodium acetate and :odium chloride, we have
2 BNa, OAc  BNa, C1 = 2 j o (FNa, OAc  PNa, C1) = 0.13 kg rn0lel
a 1 1 d thc predicted slope of the straight line is RT hi 10
F (BNa,  BNa, c1) = 542 x 0.1 3 mV kg mole1 = 7.0 mV kg mole1. 
496.5 I I
> /
495.5 ' I I I 0 005 0.10
(b + C)/mo I e kg' FIG. 4.
Thc line drawn in fig. 4 has this slope. From the intercept at in = 0 we deduce RT F E"   In K, = 495.6 mV.
Wc also obtain from Harned and Ehler's measurements, converted to absolute millivolts, E" = 236.55 mV. We deduce
E I n K, =  259.0, F log K , =  259*0/54*199  4.779,
K ,  1665 A 105 inole kg1 for acetic acid at 0" C.
1 KCI c ! I FORMtC AClD/FoRMATE.MeaSUrementS Of e.m.f. Of the CCII
H02CH a Pt, Hz K02CH b AgCl Ag,
wherc n, b, c denote molalities and a = 0.80216 = 0.9268~ throughout,
have been made by Harned and Embree.23 Their measurements at 0" C have been analyzed in a manner precisely analogous to those on acetic acid/sodium acetate buffers. E' is plotted against b + c in fig. 5. In the present case the correction h is rather more important, but is readily evaluated by successive approximations. According to the freezingpoint measurements on potassium foi mate and potassium chloride, we have
2 B K , OzCH  BK, CI = 2.30 (FK, 02CH  F K , C1) = 0.10 I
756 SPECIFIC INTERACTION OF IONS
The line drawn in fig. 5 has this slope. From the intercept at m = 0 we deduce
E"  :In  K, = 4415 mV,
RT In K, = (236.5  441.5)mV =  205.0 mV, F
log K, =  205*0/54.199 =  3.782, K, = 1.65 x 104 mole kg1 for formic acid at 0" C .
442.5 1 T
0 0.05 0.10 (b + c)/mle kg'
FIG. 5.
WATER/HYDROXIDE IN PRESENCE OF cHLoR1DE.Measurements of e.m.f. of the cell
where R denotes an alkali metal and a, b denote molalities, have been made by Harned and his collaborators.24 The e.m.f. E of this cell may be expressed in the form
where E" denotes the same standard e.m.f. as before and Kw is the thermodynamic ionization constant of water. The term  0.036 (a i b) takes account of the slight decrease in the activity of H20 as the total molality a + b increases. We define K' by
HIE E") b  In K ' = ___ Rr + I n  .
If we plot  log K' against c( + 6, we expect to obtain a straight line of' intercept  log Kw on the in = 0 axis and of slope
2  (/?R,oH  PR,CI + 0018 kg mole1) = &,OH  B R , c ~ + 0.016 kg mole1. I n 10 We use the measurements at 25" C . Having converted the e.m.f.s from int. mV to mV we use the values (RT In lO)/H = 59.159 mV and E" = 222.5 mV. The computed values of  log K' have been plotted against a + b in fig. 6. In the case of NaCl the measurements recorded for a + b = 0.05 mole kg1 in fact relate 25 to a + b = 0.06 mole kg1. From the straight lines in fig. 6 we deduce in the first place
and we consider this estimate more reliable than Harned's estimate 1008 x 1014. From the slopes of the straight lines we obtain the values of PR, OH  /3R, c1 given in the second column of table 3. In the third column of this table are given the values of p ~ , ~1 obtained from isopiestic and e.1n.f. measurements. By addition we obtain the values of FR, OH given in the fourth column. From these we have
log Kw=  13.999, Kw = 1.002 x 1014 mole2 kg2,
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E . A . GUGGENHEIM A N D J . C . TURGEON 757
TABLE 3
'$R, OH m = 01 R and S
$R, OH
G and T R pR,OH SR,Cl PR,CI PR, OH m = 0 . 1 kg mole1 kg mole1 kg mole1
Li  0.47 0.22  0.25 0894 0.920 Na  0.05 0.15 0.10 0.929 0.925 K 0.04 0.10 0.14 0.933 0.944 cs 0.35 0 035 0.954 0.942
calculated values of # for ROH at 0.1 mole kg1 and these values are given in the fifth column. In the sixth column we have given for comparison the values tabulated by Robinson and Stokes.19 As already mentioned, these were ob tained by extrapolation from e.m.f. measurements of cells with amalgam electrodes and they can easily be wrong by 2 %.
14.04
14.03
14.02
I4 01
< 1400 0
13.99
13.98
13.97
13.96
13.95 0 2 4 6 8 1 0 1 2
10 *m/m o I e kg ' FIG. 6.
WATER/HYDROXIDE IN PRESENCE OF ~ ~ o ~ ~ ~ ~ .   S i m i l a r measurements on cells containing bromide instead of chloride have been made by Harned and Hamer.26 These cannot profitably be analyzed in exactly the same way because, whereas E" for the chloride cell is known to within 01 mV there is for the corresponding bromide cell a difference of about 0.5 mV between the value of E" found by Harned and Hamer and that found by most other workers.27 Following Harned and Hamer we can eliminate this uncertainty in E" by considering the e.m.f. of the double cell
a 1 AgBr I Ag I AgBr 1 HBr a I H2, Pt Pt' H2 1 EF ma
where a = 0.01 mole kg1 throughout and m is varied. may be expressed in the form
RBr ma The e.m.f. E of this cell
4 1 7 1 ?"'H+Y'OHY''Br  In Kw $ 0.036 I?? 1 In _I_ i In  11, FE
RT == ??I  a Y'Br
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758 S P E C I F l C INTERACTION O F IONS
where the single dash relates to the alkaline solution and the double dash to the acid solution. We define K by
a2m 2Am9  log ___ +  FE  log K = _____ RTIn 10 m  a 1 + m +
We have then
Consequently if we plot  log K against m we should obtain a straight line of slope
and of intercept  log Kw + a(&, B~  BR, Br) at m = 0. This intercept is not, as assumed by Harned and Hamer, equal to  log Kw but differs from it by the small amount a(BR,B,  B H , B ~ ) . In fig. 7 the values
 log K  log Kw f 0.016 m f Br  &, Br) f m(BH, Br + BR, OH).
(BH, Br + BR, OH f 0.016 kg
1405
14.04
14.03
1402 0 I
I
1401
I400
13.99 / 0 2 4 6 8 1 0 1 2
lo2 m/mole kq FJG. 7.
of  log K are plotted against m for both sodium bromide and potassium bromide. From e.m.f. the isopiestic measurements we know with amply sufficient accuracy, since a = 0.01 mole kg1,
a(&, Br  BN,, Br) = 0.01 x 0.12 = 0.0012, a(BH, Br  B K , Br) = 0.01 x 0.17 = 0.0017.
The straight lines drawn have intercepts at nz = 0 corresponding to these values when we assume for Kw the value obtained from the measurements in chloride solutions.
PH, Br f PNa, OH = 0.36 kg mole PH, Br k &, OH = 0.45 kg
Combining these with the e.m.f. value P H , B ~ = 0 3 3 kg mole1 we obtain PNa, OH = 0.03 kg mole1 as compared with the value 0.10 kg moIe1 obtained from the measurements in chloride solutions. Similarly we obtain PK, OH = 0.12 kg mole1 as compared with the value 0.14 kg mole1 obtained from the measurements in chloride solutions.
From the slopes in fig. 7 we deduce
Our final estimates for NaOH and KOH are PNa, OH = (006 5 0.03) kg mole1, PK, OH = (0.13 0.03) kg mole*.
From these we deduce at molality 0.1 the values YNa, OH = 0.765 and 1 ; ~ , OH = 0.776 and we believe these estimates to be incomparably more reliable than any based
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E . A . G U G G E N H E I M A N D J . C . T U R G E O N 759
on e.m.f.s of cells with amalgam electrodes. It so happens that the estimate made by Robinson and Stokes 19 is close to ours for NaOH but differs widely for KOH.
SOLUBILITIES
Bronsteds theory of specific interaction, of which the present theory is a development, had as its main experimental basis solubility measurements. Details can be found in Bronsteds 3 papers and we shall not recapitulate them.
Quite recently the theory has been applied28 with success to correlating the solubility of AgCl in 00286 molal KNO3 determined by electrometric titration by Brown and MacInnes29 with the thermodynamic solubility in pure water determined from various e.m.f. measurements. An apparent small discrepancy between these two quantities was thus shown to be spurious.
HEATS OF DILUTION
Our formulae provide a solid basis for analyzing measurements of heats of dilution and correlating these with e.m.f. measurements over a range of tem perature. They have recently been thus applied 30 to measurements on sodium chloride.
IONIC ASSOCIATION
Our interaction coefficients take care of specific differences in the sizes, shapes and polarizabilities of the ions. These are the specific properties which determine the extent of ionic association, which we have hitherto not mentioned. Provided the degree of association is small, our formulae are adequate to take care of it. For uniunivalent electrolytes Davies 31 considers that B = 0.1 kg mole1 is characteristic in the absence of association. He regards B < 0.1 kg mole1 as evidence of association, but he does not give any views on B > 0.1 kg mole1. Possibly we may reinterpret Davies classification somewhat as follows :
B = (0.1 & 0.2) kg mole1, association unimportant ; B <  0.1 kg mole1, association important.
For biunivalent and unibivalent electrolytes Davies regards B = 0.6 kg mole1 as characteristic in the absence of association, but according to this view the value B = 1.1 kg mole1 of CoC12 is not explained.
In bibivalent electrolytes association is much more pronounced and our formulae are often not applicable. Davies32 views on the extent of association in such electrolytes have been confirmed by recent freezingpoint measurements made in this laboratory by Prue and Brown on the sulphates of several bivalent metals. Detailed discussion of these results must await their publication in the near future.
We are grateful to Dr. J. E. Prue for many profitable discussions and to Dr. M. L. McGlashan for constructive criticism.
APPENDIX
CRYOSCOPIC CONSTANTS OF WATER
We use the following notation To the freezingpoint of pure water, T  0 the freezingpoint of the solution, AfH the molar heat (enthalpy increase) of fusion of ice to pure
water at the temperature To, Cl molar heat capacity of liquid water (constant pressure) at To, C, molar heat capacity of ice (constant pressure) at To, M I the molar mass of water, m molality of solution.
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760 SPECIFIC 1 N T E R A C T l O N O F I O N S In the formula
6(l + b0) == (v, + v)Am$, the constants A and b are defined by
We have the experimental values AjH/Ml = (333.5 f 0.2)J g1 at T ;= To (Osborne, J. Res. Nat. Bur. Stand., 1939,
23, 643), RT" = 2.27115 x lO3J mole1,
T" = 273.16" K, Cl/M1 = 4.2176 J g1 deg1 (Osborne, Stimson and Ginnings, J . Res. Nat. Bur.
C, = 9.055 x 41832 J mole1 deg1 (Giauque and Stout, J . Ainer. Chem. SOC., Stand., 1939, 23, 197),
1936,58, 1144, with a small extrapolation). From these we deduce
MiRT"2 2.27115 x 103 X 273.16 deg mole1 kg A =   1 03 AfH  103 x 333.5
= (1.860 i 0.001) deg mole1 kg. Further we have
so that deg1 = 4.8 x 104 deg1,
1 b = ( 2 m 6  > % 5
and we notice that the value of b is about ten times smaller than might be expected owing to fortuitous cancellation between the two terms.
We are not aware of any published computations of A or b during the past thirty years. The values estimated by Lewis and Randall in 1923 are :
A = 1.858 deg mole1 kg, b = 5.3 x 104 deg1.
TEXT
1 Debye and Huckel, Physik. Z . , 1923, 24, 185. 2 Guggenheim, Rep. Scaiidinaviaiz Sci. Congr. (Copenhagen, 1929), p. 298 ; Phil.
3 Bronsted, Kgl. Danske Vid. Selsk., Mat. fys. Medd., 1921, 4 (4) ; J. Amer. Chem.
4 Guggenheim, Phil. Mag., 1935,19, 588 ; 1936,22, 322. 5 Guggenheim, Thermodynamics (NorthHolland Publishing Co., 1949), p. 3 18. 6 Guggenheim, Thermodynamics (NorthHolland Publishing Co., 1949), $ 9.23. 7 Harned and Owen, Physical Chemistry of Electrolyte Solutions (Reinhold, 2nd ed.,
8 Manov, Bates, Hamer and Acree, J. Amer. Chem. SOC., 1943, 65, 1766. 9 Adams, J. Amer. Chem. SOC., 1915, 37, 481. 10 Scatchard and Prentiss, J. Amer. Chem. SOC., 1933, 55,4355. 11 Lange, 2. physik. Chem. A , 1934,168, 147. 12 Scatchard, Prentiss and Jones, J. Amer. Chem. SOC., 1934, 56,805. 13 Lange and Herre, 2. physik. Chem. A , 1938, 181,329. 14 Hills and Ives, J. Chem. SOC., 1951, 305, 311, 318. 15 Harned and Ehlers, J . Amer. Chem. Suc., 1932, 54, 1350. 16 Harned, Keston and Donelson, J. Amer. Chem. SOC., 1936, 58, 989. 17 Brown and MacInnes, J. Amer. Chem. SOC., 1935, 57, 1356. 18 Longsworth, J. Amer. Chern. Soc., 1932, 54, 2741.
Mag., 1935, 19, 588.
Soc., 1922,44,877; 1923,45,2898.
1950), p. 597.
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76 1 E . A . GUGGENHEIM AND J . C. TURGEON
19 Robinson and Stokes, Trans. Furaday SOC., 1949, 45, 612. 20 Robinson, Trans. Roy. SOC. N.Z., 1945, 75, 203. 21 Guggenheim, Phil. Mag., 1935, 19, 639. 22 Harned and Ehlers, J . Amer. Chem. SOC., 1932, 54, 1350. 23 Harned and Embree, J. Amer. Chem. SOC., 1934, 56, 1042. 24Harned and Copson, J . Amer. Chem. Soc., 1933, 55, 2206. Harned and Mann
Harned and Hamer, J. Amer. Chem. Harned and Schupp, J . Amer. Chem. SOC., 1930, 52, 3892.
weiler, J. Amer. Chem. SOC., 1935, 57, 1873. SOC., 1933, 55, 2194.
25 Harned, private communication. 26 Harned and Hamer, J . Amer. Chem. SOC., 1933, 55,4496. 27 Janz and Taniguchi, Chem. Rev., 1953, 53,430. 28 Guggenheim and Prue, Trans. Furaday SOC., 1954, 50, 231. 29 Brown and MacInnes, J. Amer. Chem. Soc., 1935, 57, 459. 30 Guggenheim and Prue, Tram. Faraduy Soc., 1954, 50, 710. 31 Davies, J. Chem. SOC., 1938, 2093. 32 Davies, Ann. Reports, 1952, 49, 30.
TABLES 1 AND 2 a 1. Randall and Vanselow, J . Amer. Chem. SOC., 1924, 46, 241 8. a2. Hartmann and Rosenfeld, 2. physik. Chem. A, 1933, 164, 377. a3. Scatchard and Prentiss, J. Amer. Chem. SOC., 1933, 55,4355. a4. Scatchard, Prentiss and Jones, J. Amer. Chem. SOC., 1934, 56, 805. a5. Scatchard, Prentiss and Jones, J . Amer. Chem. SOC., 1932, 54, 2690. a6. Scatchard and Prentiss, J . Amer. Chem. SOC., 1934, 56, 807. a7. Harkins and Roberts, J . Amer. Chem. SOC., 1916, 38, 2676. a8. Hall and Harkins, J. Arner. Chem. SOC., 1916, 38, 2658. a9. Lange and Herre, Z. physik. Chem. A, 1938,181, 329. a10. Adams, J. Amer. Chem. SOC., 1915, 37,481. a l l . Lange, 2. physik. Chem. A, 1934, 168, 147. a12. Randall and Scott, J. Amer. Chem. SOC., 1927, 49, 647. bl. Hills and Ives, J . Chem. SOC., 1951, 305, 311, 318. b2. Harned, Keston and Donelson, J. Amer. Chem. SOC., 1936, 58, 989. b3. Brown and MacInnes, J. Amer. Chem. Soc,, 1935, 57, 1356. b4. Shedlovsky and MacInnes, J . Amer. Chem. SOC., 1937, 59, 503. b5. MacInnes and Brown, Chem. Rev., 1936, 18, 335.
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