4796
J . Phys. Chem. 1987, 91, 4796-4799
Thermodynamfcs of Concentrated Electrolyte Mixtures. 9. Activity Coefficients in Aqueous NaBr-CaBr, Mixtures at 25 O C A. V. Usha,+ Krishnam Raju,+ and Gordon Atkinson* Department of Chemistry, University of Oklahoma, Norman, Oklahoma 73019 (Receiued: December 19, 1986)
The activity coefficients of NaBr and CaBr, in NaBr-CaBr, aqueous mixtures have been measured at 25 "C over the range of ionic strengths 0.1 to 7.5 m. The measurements were made with a dual electrometer amplifier using Na and Ca ion selective electrodes against a Br ion selective electrode. The data were fitted to Harned's equation of the form log yi = log yp - PAgBZ to compute the trace activity coefficients and were also analyzed by using the Pitzer formalism. Deviations from the Pitzer formalism were observed for the experimental activity coefficients in this mixture.
Introduction There is a growing interest in the properties of multicomponent electrolyte mixtures in water. Such solutions are important in various biological, geological, and industrial processes. In order to facilitate the use of mixed electrolytes in these processes, it is desirable to know accurately the activity coefficients of both the components in the mixture. Pitzer and c o - ~ o r k e r s ' -have ~ developed a series of equations for calculating the thermodynamic properties of electrolyte solutions. These equations have been applied extensively to a variety of mixed electrolyte solutions and it has been found that Pitzer's equations yield accurate representation of the experimental data for several binary mixtures of electrolytes containing a common i0n.~3~Atkinson and co-workers6*' from our laboratory and a number of investigators8-'8 from other laboratories have successfully used these equations to calculate the activity coefficients of one or both of the components in mixed electrolyte systems. The work reported in this paper was undertaken to provide thermodynamic data for the NaBr-CaBr,-H,O system over a range of ionic strength from 0.1 to 7.5 m at 25 OC.
Experimental Section All reagents were obtained from Fisher Scientific Co. in ACS certified grade. Calcium bromide was prepared from the carbonate by dissolving C a C 0 3 in hydrobromic acid followed by crystallization. Sodium bromide was recrystallized from water and dried appropriately just before use. Stock solutions of calcium bromide were analyzed by a cation-exchange method and by volumetric determination of calcium ions with EDTA. The stock solution of sodium bromide was analyzed by drying to constant weight and by volumetric determination of bromide ions with silver nitrate. The results obtained by the two different methods in each case agreed to within f0.06%. All solutions were prepared by using distilled water that was passed through a N A N 0 Pure (Barnstead) ion-exchange apparatus.
Cell Arrangement The cell used consisted of a membrane type Ca ion selective electrode (Coraing Model 476041), a N a ion selective electrode (HNU: ISE 45-1 1-00), and a solid-state bromide ion selective electrode (Corning, Model 476128) immersed in a mixture of NaBr and CaBr, solutions in a double-walled cell. The solutions were stirred magnetically at a constant rate and the temperature was maintained at 25 f 0.05 "C. The cell may be represented as N a or C a ion s e l e c t i v e electrode
1
N a B r (") CaBrz ( m 2 )
1
Br ion selective electrode
where m, and m2 are the molalities of sodium bromide and calcium bromide solutions, respectively. On leave from The Department of Chemistry, Osmania University, Hyderabad 500 007. India.
0022-3654/87/2091-4796$01 SO10
The potentials were measured with a dual electrometer amplifier (DEA) which was primarily designed to measure the potential of ion selective electrodes (ISE) and reads to 0.01 mV. The amplifier has two high-impedance inputs so that the potential of the Na and Ca electrodes can be monitored with respect to the Br electrode. Potential readings were recorded at equilibrium in the test solution and an Apple I1 Plus computer was used for averaging the potential. The potentials were observed to be stable to within f 0 . 5 mV for periods of time from 1/2 to 1 h. All the electrodes were conditioned by allowing them to stand in a mixture of 0.01 m NaBr and 0.01 m CaBr, while not in use. Before studying a new ionic strength, the electrodes were calibrated by measuring the potentials of Ca electrode vs. Br electrode in pure CaBr, and Na electrode vs. Br electrode in NaBr solutions. In each case the potentials were plotted against the logarithm of the corresponding activities by taking the activity coefficients of the pure NaBr and CaBrz solutions at 25 OC from NBS data.I9 The selectivity coefficients were determined by the separate solution Ifiethod. Thus the selectivity coefficient of the Ca ion selective electrode toward Na' ions (K") were determined by measuring the potential of Ca ion selective electrode first in a series of pure CaBr, solutions and then in a series of pure NaBr solutions. The potentials of the Ca electrode vs. Br electrode in NaBr-CaBr2 mixtures, in pure CaBr2, and in pure NaBr are described by the eq 1, 2, and 3, respectively.
(1) Pitzer, K. S . J . Phys. Chem. 1973, 7 7 , 268. (2) Pitzer, K. S.; Mayorga, G. J . Phys. Chem. 1973, 7 7 , 2300. (3) Pitzer, K. S.; Kim, J. J. J. Am. Chem. SOC.1974, 96, 5701. (4) Pitzer, K. S. J . Solufion Chem. 1975, 4, 249. (5) Pitzer, K. S.; Mayorga, G. J . Solution Chem. 1974, 3, 539. (6) Ananthaswamy, J.; Atkinson, G. J . Solufion Chem. 1982, 11, S09. (7) Hanna, T. M.; Atkinson, G. J . Phys. Chem. 1985, 89, 4884. (8) Roy, R. N.; Swensson, E. E. J . Solution Chem. 1975, 4, 431. (9) Roy, R. N.; Gibbons J. J.; Krueger, C.; White, T. J . Chem. Soc.. Faraday Trans. 1 1976, 72, 2197. (IO) Roy, R. N.; Gibbons, J. J.; Snelling, R.; Moeller, J.; White, T. J. Phys. Chem. 1977, 81, 391. ( 1 1 ) Roy, R. N.; Gibbons, J. J.; Trower, J. K.; Lee, G.A. J . Solution Chem. 1980, 9, 535. (12) Roy, R. N.; Gibbons, J. J.; Bliss, D. P. Jr.; Casebolt, R. G.; Baker, B. K. J. Solurion Chem. 1980, 9, 911. ( 1 3) Khw, K. H.; Chan, C. Y.; Lim, T. K. J . Chem. Soc., Faraday Trans. 1 1978, 7 4 , 837. (14) Khm, K. H.; Chan, C. Y . ;Lim, T. K . J . Solution Chem. 1977,6,651. (15) Khw, K. H.; Lim, T. K.; Chan, C. Y. J . Solution Chem. 1978, 7, 291. (16) Boyd, G. E. J . SoLution Chem. 1977, 6 , 135. ( 1 7 ) Macaskill, J . B.; White, D. R. Jr.; Robinson, R. A,: Bates, R. G. J . Solution Chem. 1978 7 , 339. (18) White, D. R. Jr.; Robinson, R. A,; Bates, R. G. J . Solution Chem. 1980, 9, 457. (19) Goldberg, R. N.; Nuttall, R. L. J . Phys. Chem. Ref Data 1978, 7 , 263.
0 1987 American Chemical Society
The Journal of Physical Chemistry, Vol. 91, No. 18, 1987 4191
Thermodynamics of Concentrated Electrolyte Mixtures ECaBr2(purc)
ENaBr(pure)
= EO
= EO
+
k
In (aCauBr?
K“ =
exp[2(ENam(pure)- Eo)/k’l (mNaB17NaBr)~
=-
eXP[2(ENaBr(pure)-
From a calibration run with Ca ion selective electrode, Eo in eq 2 was determined and K”was calculated from eq 4. Following the same procedure the selectivity coefficient of the N a ion selective electrode toward Ca ions (K’) was determined. The relevant equations for the N a ion selective electrode are = Eo’
ENaBr(pure) ECaBrz(pure)
+ k‘ In (aNaaBr + K’a~a]’~ag,) (5) = Eo’ + k’ In (aNaaBr)
= Eo’
k’ In (K’aCa”zaBr)
YNaBr
(6) (7)
and K’ was calculated from eq 8. The selectivity coefficient values are very much dependent on the method used for the determination, the concentration level of the primary as well as the interfering ion, and the nature of the electrode membrane. The variation in the selectivity coefficient values has been explained as being due to the changing environment of the ion in solution and to the mechanism of electrode response.20 Thus Gadzekpo and Christian2’ reported the selectivity coefficient (K’) of the N a ion selective electrode (Beckman Instruments) as 1.8 X 3.2 X and 7.1 X by mixed solution, separate solution, and matched potential methods, respectively. Kiselev et aLZzobtained values of 6.8 X and 1.9 X by an automated method and a manual method (in accordance with the recommendations of IUPAC), respectively. We obtained an average value of 4.0 X over the ionic strength range studied. Pungor et summarized the selectivity coefficient values (K”) for various types of calcium ion selective electrodes, determined by different methods and under different conditions of measurement. The values reported varies from -0.27 to 10.7 X W e obtained an average value of 2.6 X over the ionic strength range studied.
Results and Discussion The potentials of Na and Ca ISE vs. the Br ISE were measured at ionic strengths ranging from 0.1 to 7.5 m. A titration procedure was used to alter the composition of the mixture, so that the cell was undisturbed during a series of readings. At each ionic strength two sets of potential measurements were made to test the reproducibility and accuracy of the measurements. In the first set, potentials were measured by taking pure CaBrz solution and using NaBr as titrant and in the second set taking pure NaBr solution and using CaBrz as the titrant. A good overlap was observed between the two sets of readings. The experimental activity coefficients of both NaBr and CaBr, were determined by solving
log
YCaBr2
YNaBr
log
YCaBr2
YNaBr
I = 0.1 0.9005 0.8354 0.7002 0.6002
-0.1045 -0.1040 -0.1034 -0.1033
0.5005 0.4990 0.4014 0.3005
0.9003 0.7998 0.7015 0.5994
-0.1321 -0.1362 -0.1396 -0.1427
0.4988 0.5820 0.4003 0.3004
(4)
aNaBr
ENaBr-CaBq
log
(3)
K” as
1
1
TABLE I: Mean Activity Coefficients of NaBr in NaBr-CaBr2 Mixtures at 25 OC YCaBr2
k
+2 In (K/’aNaZaB?)
Equation 3 can be rearranged to give
(2)
0.1944 0.00903
-0.1035 -0.1035 -0.1039 -0.1046
-0.1054 -0.1063
I = 0.5 0.1996 -0.1523 0.09752 -0.1547
-0.1454 -0.1432 -0.1477 -0.1500
0.9010 -0.1619 0.8012 -0.1610 0.697 1 -0.1604
I = 1.0 0.5927 -0.1600 0.4922 -0.1599 0.3961 -0.1600
0.8894 0.7868 0.6878 0.5875
0.4909 0.4963 0.3974 0.2947
0.2885 0.2016 0.1024
-0.1604 -0.1610 -0.1619
0.1940 0.09697
-0.1073 -0.1 111
I = 2.5 -0.1283 -0.1163 -0.1084 -0.1038
-0.1019 -0.1019 -0.1021 -0.1041
I = 5.0 0.9018 -0.009901 0.7341 0.01220 0.6983 0.01549 0.5983 0.02280
0.4990 0.4987 0.2942 0.3997
0.8997 0.8000 0.7001 0.6003
0.5004 0.4652 0.4003 0.3006
0.02724 0.02731 0.03149 0.02979
I = 7.5 0.1814 0.1845 0.1896 0.1959
0.1241 0.1426 0.1582 0.1710
0.2942 0.1946 0.09278
0.03100 0.03150 0.03210
0.2012 0.09996
0.2006 0.2041
1
1
0.2
0.0
Figure 1. log
?NaBr
0.4
0.6
1.0
0.8
1er2
vs. Yc,BrZ: (+) I = 1; (*) I = 2.5; (0)I = 5.
0.21
=o f3
-0.2
-0.3
-0.41
0. 0
1
1
I
0. 2
i
1
0. 4
1
0.6
’
1
1
0. B
I
1.0
~
(20) Bard, A. J.; Faulkner, L.R. Electrochemical Methods: Fundamentals and Applications; Wiley: New York, 1980. (21) Gadzekpo, V. P. Y.; Christian, G. D. Anal. Chim. Acta 1984, 164, 279. (22) Kiselev, G. G.; Mezhburd, T. A,; Petrukhin, 0. M.; Avdeeva, E. N.; Trofimova, E. V . Zh. Anal. Khim. 1985, 40, 8 8 . (23) Pungor, E.; Toth, K.; Hrabeczy-Pall, A. Pure Appl. Chem. 1979, 51, 1913.
yXaEr
Figure 2. log
yCaBrZ vs. YNaBr:
(+) I = 1; (*) I = 2.5; (0)I = 5.
eq 1 and 5 by an iterative procedure. The results are given in Tables I and 11. Equation 9 is a form of Harned’s equationz4which can be used
4790
The Journal of Physical Chemistry, Vol. 91, No. 18, 1987
TABLE II: Mean Activity Coefficients of CaBr, in NaBr-CaBr, Mixtures at 25 O C
0.09949 0.1646 0.2998 0.3998
-0.1998 -0.1969 -0.1919 -0.1892
I = 0.1 0.4996 -0.1876 0.501 1 -0.1876 0.5986 -0.1872 0.6995 -0.1881
0.09975 0.2002 0.2985 0.4006
-0.2841 -0.2794 -0.2750 -0.2709
I = 0.5 0.5012 -0.2673 0.41 80 -0.2702 0.5997 -0.2642 0.6996 -0.2615
0.09899 0.1988 0.3029
-0.2995 -0.2939 -0.2895
I = 1.0 0.4073 -0.2860 0.5078 -0.2829 0.6039 -0.2797
0.1 107 0.2132 0.3122 0.4125
-0.2443 -0.2371 -0.2310 -0.2253
I = 2.5 0.5091 -0.2201 0.5038 -0.2204 0.6026 -0.2151 0.7053 -0.2094
0.09820 -0.03910 0.2659 -0.01031 0.3017 -0.00070 0.4017 0.02090
I = 0.5010 0.501 3 0.6003 0.7058
5.0 0.04396 0.04402 0.06098 0.08267
0.1003 0.2001 0.2999 0.3997
I = 0.4996 0.5348 0.5997 0.6994
7.5 0.2882 0.2992 0.3213 0.3594
0.2077 0.2201 0.2376 0.2603
TABLE IV Harned Coefficients for CaBr, in NaBr-CaBr, Mixtures at 25 O C I log YEo %A PEA RMSD
0.8006 0.9010
-0.1906 -0.1946
0.8004 0.9025
-0.2594 -0.2579
0.7115 0.7985 0.8977
-0.2751 -0.2701 -0.2627
0.8060 0.9030
-0.2034 -0.1969
0.1 0.5 1.0 2.5 5.0 7.5
log Y A O -0.1078 -0.1559 -0.1630 -0.1205 0.02605 0.2034
0.1 0.5 1.0 2.5 5.0 7.5
0.8054 0.9072
rL
0.004046 0.511 2 0.021 15 -0.1103 -0.03539 -0.04759
4.733 -0.8092 -0.002955 0.1242 0.021 56 0.01245
0.1082 0.1412
0.7988 0.9000
0.4029 0.4514
In
YB
IO4 (1.242
X
(1.746 X IO-' (1.699 X (1.740 X lo-' (3.989 X lo-,) lo-) (6.909 X lo-,) and $
= p + 2mBr(QNaBr + ECNaBr) + "amBrBhaBr
+ + "amBrCNaBr + mCamBrCCaBr2 + mCa(20NaCa + mBr$NaCaBr) + "amCaEO'NaCa (l l )
YNa
In
=
YCa
4p+ 2mBr(BCaBrl
+ ECCaBr2) + + "amBrB'NaBr) -t 2(mCamBrCCaBr2
4(mCamBrB'CaBr2 mNamBrCNaBr)
+ mNa(20NaCa + mBr$NaCaBr)
+
+ 4mNamCaE0'NaCa (12)
RMSD 8.797 X lo-' 2.822 X IOw4 3.924 X 10" 1.183 X 1.687 X 6.851 X
In
= p + 2mNa(BNaBr + E C N a B r )
YBr
ECCaBr2)
+ 2mCa(BCaBr2 + + m N a m B r B h a B r + mCamBrB'CaBrl + "amBrCNaBr m C a m B r C C a B r 2 + mNamCa$NaCaBr + "amCaEO'NaCa
+ ( l 3,
where
p
=
+ bZi/2) + ( 2 / b ) In (1 + bZ'/2)] (14)
-A+[Z1l2/(l
A, = 0.39145 at 25 OC and b = 1.2
log
?Ao
-
aABYB
- PABYB2
(9)
= log
YBO
-
aBAYA
- PBAYA2
(10)
and for CaBr, (ye) log
RMSD" 3.811 X 8.778 X 1.143 X 4.475 X 5.549 X 2.284 X
"The RMSD values in parentheses refer to the existing values. = 0.07 and rL = -0.007.
E =
YA =
3.288 X 1.163 X 1.432 X 5.765 X 3.410 x 10-3 1.734 X
and rL Values Obtained from y, S- of NaBr so
to conveniently express the experimental activity coefficients of NaBr in the mixture (yA) log
0.05998 0.02323 -0.01337 0.002495 -11.0soo4 -0.2580
mCamBrB'CaBr2
0.01182 -0.01254 0.01242 0.1134 0.1002 0.1081
-0.01441 -0.01481 -0.01243 -0.09369 -0.05262 -0.009941
-0.06743 -0.05624 -0.02886 -0.06058 -0.1675 -0.04706
mixture, Equations for the activity coefficients of Na, Ca, and Br and other relevant equations for the system are given below.
PAE
AB
-0.2063 -0.2896 -0.3010 -0.2503 -0.05589 0.2003
TABLE V I
TABLE 111: Harned Coefficients for NaBr in NaBr-CaBr, Mixtures at 25 OC I 0.1 0.5 1.0 2.5 5.0 7.5
Usha et al.
where yAoand yeoare the activity coefficients of NaBr and CaBr, in their pure solutions at the same ionic strength as the mixture. Y A = mNaer/("aBr + 3mCaBr2) and YB = 3 m C a B r 2 / ( m N a B r + 3mcaer2) are the ionic strength fractions of NaBr and CaBr,, respectively. The plots of log y vs. Y at a few representative ionic strengths are shown in Figures 1 and 2. The experimental data were fitted to eq 9 and I O . The values DAB,aBA, and PBAtogether with their root mean square of aAB, deviations obtained from an analysis by eq 9 and I O are listed in Tables 111 and IV. The fact that the quadratic terms are needed to fit the experimental data indicates the importance of cation-cation and ternary interactions. Pitzer's recent treatment of electrolyte solutions is used to derive expressions for the activity coefficients of NaBr and CaBr, in the
CIZclmc = ClZalma C a
41
=
(15)
1
s41+ EO,(l)
The first term, sOO, is a constant and second, EO,(l), depends upon ionic strength and charge of the ions i and j. The other terms have their usual significance. The quantities B M X , B'MX,and Cwxare properties of single electrolytes and are given by B,,
=
pgf, + 2p&f,(1 - (1
+ aZ'i2)exp(-aZ'12)) /a2Z
( 1 6)
+ (1 + aZ112+ 0 . 5 ~ ~exp(-aZ'12))/a2Z2 ~1)
BIMX = 2Pf,&(-1
(17) CMX
=
~X/(21zMzXl"2)
(18)
YYaBr
=
(YNaYBr)'"
(19)
YCaBr2
=
(YCa?'B?)"3
(20)
The parameters @@), @ ( I ) , and C, for the pure salts used for the calculation are as follows: PgiB,
= 0.09730
P~;B,= 0.27910 (24) Harned, H. S.; Owen, B. B. The Physical Chemistry of Electrolyte Solutions, 3rd ed.; Reinhold: New York, 1958; Chapter 14.
%Zn,lz,l
=
QaBr
= 0.001 16
pi^^^
= 0.3562
@gia)Br2 = 1.872 QaBr2
= 0.005291
4799
J . Phys. Chem. 1987, 91, 4799-4801
g
0.
TABLE VI: I
0.
0.1 0.5 1.0 2.5 5.0 7.5
0.
and J. Values Obtained from y* of CaBrl so $ RMSD"
-3.4784 0.1639 0.3500 0.1593 0.3541 0.5425
80.11 0.5913 -0.3501 -0.08433 -0.1013 -0.01237
2.086 X (2.837 X 8.364 X (3.655 X 5.101 X (2.390 X 1.838 X lo-' (4.598 X 8.194 X lo-' (4.961 X 6.107 X lo-' (9.419 X
OThe RMSD values in parentheses refer to the existing values. = 0.07 and $ = -0.007. -0.31
1
2. 0
0.0
1
1
4.0
8.0
1
I 0.1 0.5 1.0
Figure 3. log y" vs. I: (+) NaBr and (*) CaBrz.
-0.15
-
t
0
.
2
~
IO-') lo-') IO-') lo-')
and $
TABLE VII: Trace Activity Coefficients for NaBr and CaBr, in NaBr-CaBr2 Mixtures
1
6. 0
lo-')
I
log Y k B r -0.1052 -0.1286 -0.1630
log yEaBr2 -0.1989 -0.2566 -0.2588
2.5 5.0 7.5
log Y&aBr
log YZaBr2
-0.1402 -0.02153 0.1052
-0.1922 0.1617 0.5054
values. If we use the weighted averages for and $ with being weighted proportional to I and $ proportional to 12, we get different values from the 7 N a B r and 7CaBr2 data. NaBr (Table V) gives = -0.0321 and $ = 0.0209 while CaBr, (Table VI) gives = 0.3808 and $ = -0.0356. The experimental and calculated yNaBr at I = 2.5 m are plotted in Figure 4 by using the three different sets of (%, $) described above. From the system studied in our laboratory, we conclude that and $ are also distinct functions of ionic strength. We are unable to explain this behavior due to the lack of sufficient lit~ ~ mixing parameters ~ ~ well as several ~ erature on these as other~ systems. To facilitate a more detailed comparison of the effect of one electrolyte on the other in the mixture, the trace activity coefficients of both NaBr and CaBr2 were calculated by using the following equations.
which can be obtained from eq 9 and 10 by substituting YB = 1 and YA = 1, respectively. These trace activity coefficients of NaBr and CaBrz as a function of ionic strength are given in Table VI1 and are plotted in Figure 3. ( 2 5 ) Rogers,
P. S. Z. Ph.D. Dissertation, University of California,
Berkeley, 1980.
Acknowledgment. We acknowledge the support of this research by a consortium of 11 petroleum production and service companies. Registry No. NaBr, 7647-15-6; CaBrz, 7789-41-5.
Umbrella Sampling: Avoiding Possible Artifacts and Statistical Biases Stephen C. Harvey* and M. Prabhakaran Department of Biochemistry, University of Alabama at Birmingham, Birmingham, Alabama 35294 (Received: January 15, 1987) The umbrella sampling procedure for Monte Carlo or molecular dynamics simulations allows the examination of structural and energetic questions in regions of conformational space that are not near the minimum-energy structure. This paper discusses biases that can occur in the determination of thermodynamic parameters in umbrella sampling simulations in multidimensionalconformational space. Two tests are described for determining whether such biases exist in a given simulation, and criteria for selecting appropriate umbrella sampling potentials are suggested.
Introduction Among the most popular computer modeling methods for examining molecular structure are the Monte Carlo'-2 (MC) and (1) Valleau, J. P.; Whittington, S. G. In Stafistical Mechanics, Parr A : Equilibrium Techniques; Berne, B. J . , Ed.; Plenum: New York, 1977; pp 137-168.
0022-3654/87/2091-4799$01.50/0
molecular dynamics3" (MD) algorithms. Both of these allow one to survey conformational space in the neighborhood of a particular (2) Valleau, J. P.; Torrie, G. M. In Statistical Mechanics, Part A : Equilibrium Techniques; Berne, B. J., Ed.; Plenum: New York, 1977; pp 169-194. (3) McCammon, J. A,; Karplus, M. Annu. Reu. Phys. Chem. 1980,31,29.
0 1987 American Chemical Society
~
