1737
Partial Molal Volumes of Electrolytes in Sodium Chloride
The Partial Molal Volumes of Electrolytes in 0.725 m Sodium Chloride Solutions at 25 OC Frank J. Millero,' Arthur L. Laferrlere, and Peter V. Chetlrkln The Rosenstiel School of Marine and Atmospheric Science, University of Miami, Mlami, Fiorida 33 149 (Received February 22, 1977) Publication costs assisted by the National Science Foundation and the Office of Naval Research
The apparent molal volumes of 29 electrolytes have been determined in the medium 0.725 m NaCl from precise density measurements at 25 "C. The values of 9~ have been extrapolated to infinite dilution by using a least-squares fit of the data and by using Young's rule. The two methods yield values of Poin 0.725 rn NaCl that agree to within 0.2 cm3mol-l. The experimentallydetermined values of Y*O are compared to those predicted from binary solution data by using the ionic strength principle, Young's rule, and the specific interaction model. The values of Popredicted by using the specific interaction model and Young's rule are similar and in better agreement (f0.3cm3mol-') with the experimentalresults than those predicted by using the ionic strength principle (f0.4 cm3mol-'). For electrolyteswithout a common cation (Na+)or anion (Cl-),the specific interaction model gives the best estimates. The volumes of mixing the electrolytes with a common cation (NaX + NaC1) and anion (MC1+ NaC1) at I = 0.725 determined from our results correlate very well with the enthalpies of mixing and various properties of the uncommon ion (M+ and X-).
Introduction In recent years a number of workers have de~elopedl-~ and used6-11various ionic interaction models to estimate the activity coefficients of electrolytes in various ionic media (e.g., seawater). Progress has also been made in using these models to estimate the effect of pressure on the activity coefficients or the partial molal volumes of electrolytes in ionic media.12-17 Owen and BrinkleyI2 estimated the partial molal volumes of electrolytes in seawater by assuming the P s were equivalent to the values in binary solutions a t the ionic strength of seawater (-0.725 m). They defined 0.725 m NaCl as being equivalent to "sea salt" and estimated the P of electrolytes in sea salt from the measurements of Wirth." In recent years, we have examined the P s of electrolytes in 0.725 m NaC113-15and seawater,13-15by using a simple hydration rn0de1.l~ We have interpreted the deviations from this model in terms of ion pairing.13-15 Leelg has measured the P s of electrolytes in NaCl solutions and also showed how Young's ruleBVu could be used to estimate the partial molal volumes. Leyendekker~'~ has recently used these methods to estimate the partial molal volumes of electrolytes in seawater. In a recent paper, we hav? used the specific interaction modeP to estimate the V of electrolytes in NaCl and seawater solutions. At present it is difficult to state with certainty which of these methods gives the most reliable estimates for the partial molal volumes of electrolytes in an ionic media because of the paucity of reliable experimental data. In the present paper we will present our experimental results for measurements of 29 electrolytes in 0.725 m NaCl. We will use these experimental results to examine the most reliable method that can be used to estimate the partial molal volumes of electrolytes in an ionic media.
(n
Experimental Section The density measurements were made with a vibrating densimeter that is described in detail elsewhere.25 The densimeter measures the relative densities of aqueous solutions (d - do) to a precision of *3 x 10-6 g cm-3.25726 The densimeter was calibrated by using ion-exchanged water and dry nitrogen. The reliability of the densimeter was checked by measuring the densities of standard
seawater solutions weight evaporated or diluted with water. The measured densities agree on the average to f 2 ppm26 with the values calculated from the equation of state of seawater.27 The temperature of the densimeter is controlled to fO.OO1 "C and set to h0.005 "C with a platinum resistance thermometer (calibrated by the National Bureau of Standards on the 1968 IPTS temperature scale) and a G-2 Mueller bridge. All of the salts used were Baker reagent grade. The salts that did not decompose were heated in vacuo a t 110 O C for at least 1h. Stock solutions of these salts (KBr, LiCl, RbC1, CsC1, NaI, KI, NaBr, KC1, NaN03, KNOB,Na2C03, K2C03, Na2S04,K2S04, NaF, KHCO3, NaHC03,NH4C1, and NH,Br) were made by weight in 0.725 m NaC1. Approximately 1m stock solutions of the electrolytes that could not be dried (HC1, NaOH, KOH, KF, MgC12,CaC12, SrC12, and BaC12) were made. The molalities of these solutions were determined by measuring the density (HC1, NaOH, KOH, MgSO,, and KF) or by titrating with AgN03 (MgC12,CaC12, SrC12,and BaC12). The NaCl was added to these solutions to make them 0.725 m NaCI. Dilute solutions of the salt mixtures (salt and NaC1) were prepared by adding a weighted amount of 0.725 m NaC1. Duplicate density measurements were made on most of the solutions. The two measurements agreed on the average to h3.5 ppm which represents the precision of the measurements. The density of the NaCl medium for all of the experiments was 1.025810 f 0.0000025g mL-'. This density is equivalent to a molality of 0.72525 h 0.00006 mol (kg of H20)-' as determined from the relationship 103(d - do) = 1.011 + 38.227 m (valid from 0.7 to 0.8 m). The combined error in density was -6 ppm. This error is equivalent to an error of f0.006 cm3 mol-' at 1.0 m and k0.06 cm3 mol-' at 0.1 m in the apparent molal volumes of the electrolytes. Results and Calculations The densities at 25 "C for the various electrolytes (NaF, KF, HC1, LiC1, NaCI, KC1, RbCl, CsC1, NH4C1, MgClz CaC12,SrC12,BaC12,NaBr, KBr, NH4Br NaI, KI, NaOH, KOH, NaN03, KNOB,NaHC03, KHC03, Na2C03,K2CO3, Na2S04,K2S04,and MgSO,) at various molalities (m3)in The Journal of Physical Chemistry, Vol. 81, No. 18, 1977
1738
g
-8
-./
23
-
22
-
24
21 20
F. J. Millero, A. L. Laferrlere, and P. V. Chetirkin
27 26
~
-
t
l9 181 0.00
Lee
I
.% 25
I
*Ours '
' 0.20
'
'
0.40
'
'
'
0.60
-8
j
24
'
0.80
23
6 27'0
/
COCI,
1
22 0.00 0.20 0.40 0.60 0.00
NaBr
1.00
1.20 1.40
1.00
1.20 1.40
12
IO
4 *Ours
2 25 55 0000
020
040
060
080
100
8
120
"7' 6 -8
JTii;
4
Figure 1. The apparent molal volumes of Na2S04and NaBr in 0.725 m NaCl at 25 O C plotted vs. the square root of molality.
2
0.725 m NaCl are given in Tables I-XXIX.2s As with binary solutions, it is convenient to examine the volume properties of ternary electrolyte solutions in terms of the apparent molal volume (4v). If one of the electrolytes 3 is considered to be dissolving in a solvent of electrolyte 2 in water, one can define the apparent molal volume of 3 in this ionic medium by
0.00 0.20 0.40 0.60 0.00 f
;3
*->
where Vsolnand Vmed are the volumes of the solution and medium, and n3 is the number of moles of electrolyte 3. For a molal solution the volume of the solution and the medium are given by VSoh = [lo00 + m2M2+ rn3M3]/d
+ m2M2]/d2 = l O O O / d o + mz4v(2)
8-
-
22
-
2I
-
*Ours
(2)
18
I 000
(3)
where m, is the molality and M,is the molecular weight of solute i, d is the measured density of the mixed solution, d2is the density of the medium, dois the density of and 4v (2) is the apparent molal volume of the medium. Substituting eq 2 and 3 into eq 1 gives
3
Lee
Vmed = [ l O O O
4v *(3) =
24
i
0.4 0
0.8 0
I20
I60
4-K Figure 2. The apparent molal volumes of CaCI,, MgS04, and MgCI, in 0.725 m NaCl at 25 OC plotted vs. the square root of molality.
XXX along with the standard deviations. The partial molal volume of electrolyte 3 in the medium can be determined from
1800(d2-d)+-+--M 3 m2M2 m2M2 ddzma d m3d m3d2 (4)
The values of (bv*(3) determined from eq 4 are given in Tables I-XXIX.2s The variation of the 4v*(3)vs. rn3ljz for a 1-1 electrolyte (NaBr),three 2-1 electrolytes (Na2S04, CaC12,MgC12),and the 2-2 electrolyte MgS04are shown in Figures 1 and 2 along with the results of Lee.lg These values of 4 ~ * ( 3have ) been fitted to the equation $+*(3) = GV*O(3) + Am31/2+ Bm3
(5)
The values of the coefficients &*O(3) = V3*0,the infinite dilution partial molal volume, A and B determined by a weighted least-squares fit of the data are given in Table The Journal of Physical Chemistry, Vol. 81, No. 18, 1977
The differentiation of eq 5 gives -
V* =
v3*' + 1.5Am31/2+ 2Bm3
(7)
where V3*0is the infinite dilution partial molal volume of electrolyte 3 in the medium. Although the &~*(3)data for the electrolytes studied can be fitted to eq 5 with standard deviations less than 0.1 cm3 mol-', the errors in V3*O are slightly larger (as much as f0.2 cm3mol-l for 2-1 and 2-2 electrolytes). To test the extrapolation procedure, we have made volume measurements on solutions formed by adding weighted amounts of NaCl to 0.725 m NaC1. The ex-
1739
Partial Molal Volumes of Electrolytes in Sodium Chloride
TABLE XXX: Coefficients of Eq 5 for Various Electrolytes in 0.725 m NaCl a t 25 Ca Electrolyte
@v*O
A
B
oLee
0
0.02 NaF 0.303 -0.015 1.630 KF 0.336 0.01 10.612 0.883 HCl 19.606 0.01 0.198 0.028 19.192 0.02 LiCl 0.595 -0.178 0.056 19.080 NaCl 0.701 0.01 0.339 29.343 0.01 KC1 0.557 0.04 0.608 34.654 RbCl 0.7 58 41.742 CSCl 0.03 0.208 37.905 NH4Cl 0.02 0.372 1.754 18.942 0.03 1.408 MgC1, 0.689 22.973 0.05 1.931 CaCl, 1.967 0.03 23.169 SrCl, 1.593 1.474 1.959 28.917 0.04 BaCl, 0.521 25.620 0.03 NaBr 0.299 0.03 0.208 36.165 0.526 KBr 0.02 0.011 44.715 0.295 NH,Br 36.917 0.02 0.215 0.343 NaI 0.03 47.424 KI 0.625 0.04 0.647 -2.417 1.178 NaOH 0.04 KOH 0.796 0.860 7.896 0.02 0.428 0.567 30.549 NaNO, 0.01 0.445 0.501 40.861 KNO, 0.699 1.155 NaHCO, 26.772 0.02 0.491 1.310 0.01 KHCO, 37.200 1.616 1.398 0.03 1.945 Na,CO, 0.06 1.261 1.188 22.599 K,CO3 0.186 2.340 0.05 Na,SO, 20.794 0.04 2.191 4.739 41.239 K,SO4 0.09 7.081 1.007 1.608 MgSO4 a The values of @v* have the units of cm3 mol''. Since the values of @v*a t 1 m are ten times more reliable than at 0.1 m for a given error in density; the values of $v* have been weighted by the factor 1 0 m .
*Ours
/ 0.00
0.40
NaBr
0.80
1.20
1.60
m3
Figure 3. The mean apparent molal volumes of NaBr and Na2S04In 0.725 m NaCl at 25 OC plotted vs. molality.
26 24 22
-- 2 0 c!
6-18 16
trapolated value of +v*o= 18.08 cm3mol-' for NaCl agrees very well with the value of Y = 18.06 cm3 mol-1 at 0.725 m calculated from apparent molal volume data. The volume properties of these ternary electrolyte solutions can also be examined in terms of the mean apparent molal volume defined by
14
12
1 000
040
080
1.20
160
m3
where VHzo = 1000/do is the volume of water in the solution and m2+ m3 is the total molality of electrolytes in solution. Substitution of eq 2 into eq 8 gives
1000(do- d ) m2M2+ m3M3 7 =~ d' o d ( m 2+ m 3 ) (m2+ m3)d +
@
~
(
~
(9)
This equation can be simplified by defining the total molality, mT = m2+ m3and mean molecular weight, MT = (mi% + m3%)/(mz + m3)
Flgure 4. The mean apparent molal volumes of CaCI,, MgCI,, and MgSO, in 0.725 m NaCl at 25 OC plotted vs. molality.
The Y3*can also be determined from the concentration dependency of aV(2,3). By differentiating eq 11 with respect to m3 and substituting into eq 6, we have v3* =
@v(2,3)+ (m2 + m3)[a@,(2,3)/am,l
(13)
Since &(2,3) = 4"(2) and (m2+ m3)= m3 at m3 = 0, the limiting partial molal volume Y3*0is given by -
V3*' = h ( 2 ) + m2[a@v(2,3)/am31,,=o
The values of @~(2,3) for the various electrolyte mixtures calculated from eq 9 at various molalities are also given in Tables I-XXIX.28 The variation of the @"(2,3) vs. m3 for a 1-1 electrolyte (NaBr), three 2-1 electrolytes (Na2S04, MgC12, CaC12),and the 2-2 electrolyte MgS04 are shown in Figures 3 and 4 along with the results of Lee.lg From the two definitions of 4v*(3) and @"(2,3),cane can relate these two quantities by
(14)
The derivative 8@"(2,3)/8m3can be obtainedlg from the concentration dependence of @"(2,3)
aV(2,3)= $ v ( 2 )+ A'm, + B'm:
+ ...
(15)
At m3 = 0 the derivative of a@"(2,3)/am3equals A', and the partial molal volume is given by
V3*0= qiV(2)+ m2A'
(16)
The values of $*O calculated from eq 16 are not very reliablelg because the value determined for A'is quite dependent upon the degree used to fit the data and reliable values of 9~(2,3)in dilute solutions are not available. The Journal of Physical Chemistty, Vol. 81, No. 18, 1977
F. J.
1740
Millero, A. L. Laferriere, and P. V. Chetlrkin
TABLE XXXI: Coefficients for Eq 21 for Various Electrolytes in Water at 25 ’C Electrolyte
@V“O
Ai
Bi
Ll
Ref
NaF KF HCl LiCl NaCl KCl RbCl CSCl NH,C1 MgCl, CaCl, SrCl, BaCl, NaBr KBr NH,Br NaI KI NaOH KOH NaNO, KNO, NaHCO, KHCO, Na,CO, K*CO, Na,SO,
- 2.371
1.839 1.724 1.460 1.700 1.811 1.839 1.818 1.843 1.717 3.155 3.611 3.941 4.576 1.689 1.403 1.965 1.593 1.645 2.048 1.809 2.586 3.103 3.662 3.691 7.710 5.525 6.750 7.464 17.765
0.561 0.509 - 0.307 - 0.198 0.094 0.087 0.191 0.162 -0.147 0.106 -0.117 - 0.038 - 0.360 0.076 0.480 - 0.498 - 0.126 0.044 0.984 0.987 -0.292 -0.712 -0.115 - 0.024 - 0.383 0.371 0.087 -0.285 -18.462
0.02 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.03 0.03 0.05 0.04 0.02 0.02 0.02 0.03 0.02 0.05 0.05 0.05 0.05 0.05 0.07 0.05 0.05 0.03 0.03 0.05
30, 31, 32 31, 33, 34 18b, 35-37 31,33, 38-40 18b, 19, 38,40-43 18a, 31, 38,41,44-45 31,38 31, 38 40 19, 36, 40, 46 19, 36, 40, 41, 46 40, 46, 47 36, 40, 41, 46 19, 31, 32, 39, 48, 49 31, 39, 41 40, 48 31, 39, 49 31, 39, 41 31,50 40 40, 51 36, 40 40, 52 40 40, 44, 52 40 19, 36, 44 18a, 3 6 , 4 0 19,40
K2S04
MgSO, a a
7.844 17.854 16.961 16.613 26.850 31.940 39.166 35.815 14.522 17.858 17.975 23.208 23.504 33.725 42.521 35.018 45.229 - 5.246 4.929 27.805 37.979 23.118 33.427 -6.195 14.526 11.559 32.024 -7.198
For MgSO, the terms 12.316Z3”
-
3.0541, must be added t o eq 2 1 t o fit the data.
Another method that can be used to determine the V3*0 of the electrolytes in the medium is to use Young’s rule23p24
~ a t e r ~(from ~ J0 ~to ,1m) ~ were ~ ~ fitted ~ to equations of the form
GV’(i) = GVo((i)
+ AiI1” + BJ
(21)
The values of &O(i), Ai, and B; for the various electrolytes are given in Table XXXI. The first two terms of eq 20 are given by where 4”’(2) and 4v’(3)are the apparent molal volumes of electrolytes (2) and (3) in binary solutions at the ionic strength of the mixture ( I = w2m2+ w3m3,where w, is a valence factor equal to 1.0, 3.0, and 4.0,respectively, for 1-1, 2-1, and 2-2 electrolytes) and AVmis the volume of mixing the two electrolyte solutions at the same ionic strength. By combining eq 12 and 17, we obtain
@ v * ( 3 )= ( m z / m d @ v ‘ ( 2-) @v(2)1 + @ v‘(3) + (Avm/m3)
(18)
The partial molal volume of electrolyte (3) can be obtained from eq 18 by differentiation and substitution into eq 6
The value of V3* at m3 = 0 can be evaluated from
+
L21m3=0
To evaluate V3* as well as calculate AVm from eq 17, the apparent molal volumes of the electrolytes studied in pure The Journal of Physical Chemlstty, Vol. 81, No. 18, 1977
[4v1(3)Im3=o= 4 v 0 ( 3 ) + A3m21’2+ B3mz
(22)
The value of 4”’(3) at m3 = 0 is just the value of 4”(3) in itself at the ionic strength of the NaCl medium (0.725 m). The value of m2[a4v’(2)/am3] at m3 = 0 is simply equal to V2 - 4 ~ ( 2= ) m2[a4,(2)/am2] for NaCl (0.839) at 0.725 m times the valence factor w3 for the added solute. If AV, = 0, the V3*0is simply given by -
V3*0 = qjV0(3)+ A3m2’I2+ B3m2+ w3[B2mz + (A2/2)m21’21 (24)
To calculate (aAVm/am3),,=o, one must examine the concentration dependence of AV,. The volume of mixing electrolyte solutions at a constant ionic strength have been studied by a number of worker~.5~-~’ For the mixing of two 1-1 electrolytes at a constant ionic strength, the AV, is given by A v m = Y 3 ( 1 - Y3)[VO + u l ( 1 - 2Y3)I (25) where y3 = m3/(m2 + m3), (1- y3) = y2 = m2/(m2 +.m3!, and uo and u1 are adjustable parameters related to ionic interactions. The differentiation of eq 25 with respect to m3, evaluated at m3 = 0 gives m,=O
- -uo + -u1 m~ m2
(26)
Since the measurements made in this study were made at
Partial Molal Volumes of Electrolytes in Sodium Chloride
1741
v*,for
TABLE XXXIII: Comparison of the Values of Various Electrolytes in 0.725 m NaCl Obtained b y Extrapolation and by Using Young's Rule
TABLE XXXII: Values of ( a A Vm/am,) at m3 = 0 for Mixing Various Electrolytes with NaCl at 25 " C and I = 0.725 m Electrolyte NaF KF HCl LiCl NaCl KCl RbCl CSCl NH4Cl MgCl, CaCl, SrCl, BaCI, NaBr KBr
(aA Vm/ am,),,=,
-0.18 0.17 -0.04, -0.02, 0.00 0.02, 0.16 0.11 -0.07, -0.64 -0.45 - 0.66 -0.42 -0.07, 0.10
Electrolyte NH,Br NaI KI NaOH KOH NaNO, KNO, NaHCO, KHCO, Na,CO,
-
V*O, cm3 mol-'
(aAv,/
-0.02, -0.13 -0.07, 0.39 0.11 -0.01, -0.01, -0.12 -0.13 - 0.66 0.68 0.82 0.73 - 2.26
K2C03
Na,SO, K2S04
MgSO,
Electrolvte
changing ionic strengths, it is not possible to fit the values of AV, calculated from eq 17 to eq 25. We have subsequently fitted the values of AV, calculated from
AV, = (m*+ m3)[@,(2,3) - Y34v '(311 to equations of the form AV, = A t B m 3 + Crn? +
- YZGV
'(2) (27)
.,
(28) The values of (aAVm/am3)at m3 = 0 (B in eq 28) evaluated from the experimental results are given in Table XXXII. It should be pointed out that the values of (aAVm/am3)at m3 = 0 are equal to (uo + ul)/rnz. For most of the 1-1 electrolytes studied (aAVm/am3),,,, is less than 0.2 cm3 mol-l; while for the 2-1 electrolytes the values are less than 0.7 cm3mol-'. The value of (aAVm/am3)ma=0 for MgSO, is quite large, probably due to ion pairing effects.5s The values of V*O calculated from eq 20 using the values of AaV,/em, at m3 = 0 are given in Table XXXIII along with those obtained from eq 7. The results of V*O obtained by the two methods agree to within 0.2 cm3 mol-l for all of the electrolytes studied (which we feel is the maximum error). We will use the average values of V*O obtained by these two methods in all of our further calculations. The reliability and internal consistency of our values of V*O can be determined by examining the additivity of the various salt pairs. For the difference between Na+ and K+, our results give V*O(Na+)- V*O(K+) = -10.36 f 0.07 cm3 mol-l from the salt pairs NaC1-KC1, NaF-KF, NaBr-KBr, NaI-KI, NaOH-KOH, NaN03-KN03, NaHC03-KHC03, 1/2Na2C03-L/2K2C03, and 1/2Na2S04-1/2K2S04(the maximum difference was 0.13 cm3mol-l). The difference between C1- and the various anions obtained from our results are: V*O(Cl-) - V*O(F-) = 18.74 f 0.04 cm3 mol-', V*O(Cl-) - V*O(Br-) = -6.75 f 0.11 cm3 mol-l; V*O(Cl-) V*'(I-) = -17.99 f 0.10 cm3 mol-l; V*O(Cl-) - V*O(N03-) = -11.54 f 0.01 cm3 mol-'; V*O(Cl-) - P*O(OH-) = 21.435 f 0.01 cm3mol-l; V*O(Cl-) - V*(HCO
