ACTIVITYCOEFFICIENTS FOR WATER-UREA SYSTEMS
1831
Activity Coefficients for the Systems Water-Urea and Water-Urea-Sucrose at 25 from Isopiestic Measurements1
by H. David Ellerton and Peter J. Dunlop Department of Physieal and Inorganic Chemistry, University of Adelaide, Adelaide, South Australia (Received November 10,1066)
Isopiestic vapor pressure measurements are reported for the systems HzO-urea and HzOurea-sucrose at 25”. Osmotic and activity coefficients have been calculated for the system HzO-urea, and the deviations from ideality have been discussed in terms of association of urea. Activity coefficients have also been calculated for the system HzO-urea-sucrose, and the results for relatively dilute solutions of this system are discussed in terms of association of urea, hydration of sucrose, and binding of urea monomer to the hydrated sucrose. A method is indicated for obtaining the activity of the solvent in this ternary system and hence the molal activity coefficients of the solutes by utilizing the experimental information from both binary and ternary systems.
The isopiestic vapor pressure method is currently being used in t,his laboratory to measure solute activity coefficients, which form part of the experimental data necessary to test current theories of ternary diffusion. This technique has been used elsewhere to study several aqueous ternary systems containing two electrolyte^,^-^ to study others containing an electrolyte and a nonelectrolyte,s-ll and also to study one system containing two nonelectrolytes. l 2 For the system HPO-sucrosemannitol, Robinson and StokeslZ reported that each solute was “salted out” by the other, whereas in the system H2O-urea-NaC1, Bower and Robinsonlo showed that a t moderate solute concentrations a very small “salting out” effect was observed, although a t lower and higher concentrations the effect was one of “salting in.” I n this work it has been found that, up to 7 m urea and 4.3 m sucrose, each solute component is “salted in” by the other. The deviations from ideality in the HzOsucrose-mannitol system were discussed12 in terms of hydration of both solutes, whereas in the system HzOurea-sucrose we discuss the behavior in terms of hydration of the sucrose, association of the urea, and binding of the urea monomer to the hydrated sucrose.
Theory Notation. y1, y2 = molal activity coefficients of solutes 1 and 2 in a ternary solution containing components 1 and 2 with molalities ml and m2, respectively.
$1, y ~ ”= molal osmotic coefficient and activity coefficient, respectively, of a binary solution containing only solute 1 at molality ml. 42, yzo = molal osmotic coefficient and activity coefficient, respectively, of a binary solution containing only solute 2 at molality m2. WLR, (PR = molality and molal osmotic coefficient, respectively, of the reference solution. a. = activity of the solvent. Mo = molecular weight of solvent. wo = 1000/M,.
(1) This investigation was supported in part by a grant from the Colonial Sugar Refining Co. Ltd. of Australia. (2) R. A. Robinson, J . Phys. Chem., 65, 662 (1961). (3) R. A. Robinson and V. E. Bower, J . Res. Sat?. Bur. Std., A69, 19, 365 (1965). (4) M. S. Stakhanova, V. A. Vasilev, and Y. A. Epikhin, Russ. J . Phys. Chem., 37, 182 (1963). (5) M. M. Shul’ts, L. L. Makarov, A. N. Norinichev, and S. TuZhen’, ibid., 37, 652 (1963). (6) A. N. Kirgintsev and A. V. Luk’yanov, ibid., 37, 1501 (1963); 38, 702, 867 (1964). (7) L. L. Makarov, Yu. G. Vlasov, and R. Kopunets, ibid., 37, 1495 (1963); 38, 1055, 1297 (1964). (8) F. J. Kelly, R. A. Robinson, and R. H. Stokes, J . Phys. Chem., 65, 1958 (1961). (9) R. A. Robinson and R. H. Stokes, ibid., 66, 506 (1962). (IO) V. E. Bower and R. A. Robinson, ibid., 67,1524 (1963). (11) V. E. Bower and R. A. Robinson, ibid., 67, 1540 (1963); J . Res. Nat2. Bur. Std., A69, 131 (1965). (12) R. A. Robinson and R. H. Stokes, J . Phys. Chem., 65, 1954 (1961).
Volume 70. hTumber6 June 1966
H. DAVIDELLERTON AND PETERJ. DUNLOP
1832
We assume that the activity coefficients of two nonelectrolytes in a ternary solution may be represented by the Taylor series m
In
71
m
A,,m12m23
=
(AOO= 0)
(1)
1=0 j = o
and
A
"
In
72
BIjmmlZm23
=
(BOO = 0)
(2)
2=0 j=o
where the A,, and B j j are not independent and are functions of the derivatives. of In y1 and In yz, respectively, at ml = m2 = 0. Our aim is to find a method for evaluating the A,, and the B,, from isopiestic vapor One pressure measurements of ternary systems. method has already been reported by Robinson and Stokes,12 but their derivation is somewhat different from that given here. We first expand eq 1 to include all terms of fourth order
+
In y1 = A I O ~ I A01m2 -I- A20m12 +Allmlmz
+
+ A30m13 + A21ml2m2 + Alzm1mz2 + Ao3mz3 + A40m14 + A31m13in2 + A22m1~m2~ + A13m1mz3 + Ao4mz4 A02m22
(3)
For ternary systems it is found that further terms are generally not required to represent the experimental data. The coefficients of those terms in eq 3 which contain only powers of ml can be obtained from isopiestic measurements for the two-component system.13 Thus the coefficients in the expression In
=
71'
Aloml
+ A20m12 + A30m13 +
(4)
may be derived from the osmotic coefficients, 41. With the aid of eq 4, eq 3 becomes In
+ A01m + A l l m ~ m+~ A o m 2 + + A21ml2m2 + A m l m ~+~ A31ml3m2 + A22m12m~2 + A13m1mz3 + A04m2~ (5)
y1 =
In
Explicit relations between the At, and the Bt, may be obtained by equating the corresponding coefficients in eq 2 when it is expanded to include terms of fourth order. The coefficients AI, in eq 5 and 7 may be related to an experimental quantity A defined byg
71'
Ao3m23
The relations between the A,, and the B,, may be obtained by differentiating eq 5 with respect to m2 and using the cross-diff erentiation relation
[XCI~, [TI,
b In
b In YI
72
=
(6)
to give, after integration
+ + + + 3A03mm2 + (A31/4h4) + (2A22/3)m13m2 + (3A13/2)m1~m~~ + 4A04mn3
- mlh
- m2h
(8)
which may be obtained directly from isopiestic vapor pressure measurements by equilibrating a binary reference solution, R, with a ternary solution containing the solutes 1 and 2. V R is the number of species given in solution by 1 mole of the reference solute. The derivation utilizes the Gibbs-Duhem relation for both two- and three-component systems. After some manipulations and one integration, one obtains'O
+ Allml + 2A02m + + (3/2)Ai2?nim2+ 3A03mz2 + A 3 m 3 + (4/3)A22m12m~ + 2A13m1m2 + 4A04m3
A/(mlmd = A01 A21m12
(9)
The A/(mlmz) can be measured directly and the coefficients A,, in eq 9 can be found by the method of least squares. Thus the concentration dependencies of y1 and y2 for ternary systems (eq 5 and 7) can be obtained if 71' and yz' have also been measured as functions of ml and m2, respectively.
Experimental Section Materials. Urea was obtained from British Drug Houses, and the Analar grade material was once recrystallized from doubly distilled water. Samples which had been once and twice recrystallized were shown by the isopiestic method14*to be identical. Sucrose was BDH microanalytical grade and was used without further purification. Sodium chloride (reference solute) was from the same sample as described previ0us1y.l~ The solutions were prepared gravimetrically, and the weights were corrected to vacuum15 assuming the density of the stainless steel weights to be 7.76 g/cc. Doubly distilled demineralized water was used as solvent. The densities of solid sodium chloride, urea, and sucrose for vacuum corrections were taken as 2.165, 1.335, and 1.588 g/cc, respectively, while the corresponding molecular weights used were 58.443, 60.056, and 342.303, respectively. (13) H. D. Ellerton, G. Reinfelds, D. E. Mulcahy, and P. J. Dunlop,
+
J . Phzle. Chem., 68, 398 (1964).
Aolml (A11/2)m12 In YZ = In 7 2 ' 2A02mlm2 (A21/3)m13 A12m12m2 -t
The Journal of Physical Chemistry
E VRmR4R
(7)
(14) R. A. Robinson and R. H. Stokes, "Electrolyte Solutions," Butterworth and Co. (Publishers) Ltd., London, 1959: (a) pp 177181; (b) Chapter 1. (15) For the method of applying vacuum corrections, see Mettler News (1-22)E, 1961, p 81. This publication is available from Mettler agents.
ACTIVITYCOEFFICIENTS FOR WATER-UREA SYSTEMS
~~
~~
~~
1833
~~
Table I : Isopiestic Solutions of Urea and Sodium Chloride "aC1
0.2054 0,2101 0.4071 0.4137 0.6100 0.6192 0.8094 0.8365 1.0280 1.0372 1.4238 1.4374 1.9905 2.0015 2.4309 2.4509 2.4848 a
0.9909 0.9895 0.9835 0.9838 0.9765 0.9763 0.9682 0.9658 0.9613 0.9607 0.9493 0.9486 0.9333 0.9345 0,9228 0.9220 0.9206
0.1093 0.1117 0.2168 0.2204 0.3234 0.3282 0.4258 0.4397 0.5361 0.5405 0.7294 0.7357 0.9932 0.9997 1.1901 1.1984 1.2125
2.5035 2.9434 2.9922 3.8354 3.8590 4.9354 4.9638 5.3814 5.4766 5.8135 5.8638 6.0428 6.0930 6.3962 6.4421 6.8203 6.8760
0.9202 0.9105 0.909g 0.8937 0.8931 0.8757 0.8757 0.8691 0.8676 0.8634 0.8627 0.8601 0.8596 0.8557 0.8555 0.8505 0.8502
1.2207 1.4081 1.4290 1.7660 1.7748 2.1754 2.1865 2,3322 2.3648 2.4805 2.4973 2.5564 2 I5737 2.6725 2.6883 2.8086 2.8272
m1
61
7.1813 7.2449 7.9257 8.0696 9.8320 9.9048 12.0781 12.1676 14.0050 14.1703 15.6122 16.0436 18.1145 18.3441 19.8828 20.0270"
0.8469 0,8460 0.8387 0.8374 0.8236 0.8235 0.8093 0.8088 0.7991 0.7981 0.7905 0.7886 0.7819 0.7802 0.7753 0,7750
"aCl
2,9233 2,9426 3.1496 3.1930 3.7008 3.7223 4.2955 4.3181 4.7678 4.8061 5.1340 5.2299 5.6852 5.7288 6.0482 6.0783
Saturated solution.
Apparatus. The apparatus used was the same as that which has been described previously. l 3 About 5 ml of the solution whose solvent activity we wished to determine was placed in each of three or four cylindrical silver dishes, which were approximately 3.5 cm in diameter and 2 cm in height. A sihilar quantity of the reference solution was placed in another set of silver dishes. The dishes, with lids removed, were placed on a flat, silver-plated copper block contained in a vacuum desiccator and evacuated to a pressure of approximat,ely 25 mm. The evacuation process was generally carried out in stages for as long as 24 hr for newly prepared solutions to avoid splattering, although shorter periods were used for solutions that were being evacuated for the second or third time. The solutions were then allowed to equilibrate under vacuum for 5 or 6 days in a large thermostat both at 25", controlled to approximately f0.002", l7 before being reweighed. Each dish contained three stainless steel ball bearings to assist equilibration while the desiccator was being rocked in the thermostat bath. Generally, several solute compositions, containing urea and sucrose in varying proportions, were equilibrated with the same reference solution. Samples of each solution were present in triplicate dishes, and the sodium chloride reference solute was present either in triplicate or quadruplicate. A set of dishes containing urea only accompanied most runs. Thus, a total of 21 or 22 dishes were usually equilibrated together at the one time. The concentrations of unknown and reference solutions were prepared initially
so as to be fairly close to the anticipated equilibrium value. For ternary solutions containing a total solute concentration of approximately 1 m or less, some difficulty was experienced in attaining equilibrium, owing to bacterial activity. To overcome this problem, extra precautions were taken in cleaning and sterilizing the apparatus, and the water used as solvent for the solutions was distilled once more. The solutions were accepted as being at equilibrium if the molalities of a particular set differed by not more than 0.1%, but, as an additional precaution, the desiccator containing the solutions was reevacuated and the solutions were allowed to equilibrate for a further 3 days and were then reweighed. In some cases, the solutions were concentrated further by pumping under vacuum for a prolonged period of time. They were then allowed to reequilibrate for 6 days, after which the solutions were reweighed. A further check weighing was made after another 3-day equilibration.
Results The System HzO-Urea. Isopiestic data for the system HzO-urea have been reported previously by Scatchard, Hamer, and Wood.'* It was decided, how(16) Using atomic weights compiled in International Union of Pure and Applied Chemistry, Information Bulletin No. 14b, 1961. (17) Thermometers were calibrated by National Standards Laboratory, Commonwealth Scientific and Industrial Research Organisation, Sydney, N.S.W., Australia. (18) G. Scatchard, W. J. Hamer, and 5. E. Wood, J . Am. Chem. Soc., 60, 3061 (1938).
Volume 70, Number 6 June 1966
1834
H. DAVIDELLERTON AND PETER J. DUNLOP
Table 11: The System Urea-Sucrose-Water I
An -
r
-
ml mz
ml
0.1635 0.1663 0.2711 0.2749 0.3747 0.3807 0.4573 0.4650 0.1456 0.3386 0.1443 0.3354 0.5482 0.7473 0.9224 0.7552 0.9320 0.2500 0.6092 1.0149 1.7810 0.2515 0.6134 1.0231 1,4745 1.7927 0.2960 0.7292 1.2291 1.7873 2.1783 0.2982 0.7348 1.2392 1,8022 2.1961 0.4552 0.9598 1.9423 1.2840 2.5049 0.4621 0.9746 1.3028 1.9727 2.5464 0.4116 1.0345 1.7838 2.6527 3 2611 0.4159 1,0463 1.8054 2.6880 3.3045 ~
m2
0.3500 0.3561 0.2493 0.2528 0.1459 0.1483 0.06315 0.06422 0.8355 0.6791 0.8277 0.6728 0.4976 0.2925 0.1227 0.2956 0.1240 1.4587 1,2075 0.9011 0.2214 1.4671 1.2159 0.9084 0.5257 0.2229 1,7270 1,4452 1,0913 0.6372 0.2708 1,7395 1.4565 1.1003 0.6425 0.2730 1.9574 1.6455 0.9443 1,4307 0.4571 1.9872 1,6710 1.4517 0.9591 0.4647 2.4013 2.0505 1,5838 0.9458 0.4055 2.4264 2.0738 1,6030 0.9584 0.4109
Expt
Calcd
mR
%
mi
m2
0.11737 0.11629 0.11266 0.11523 0.12288 0.11725 0.11656 0.11299 0,10884 0.11058 0.11464 0.11531 0.11296 0.10819 0.11420 0.10567 0.10772 0,10566 0,09669 0.09552 0.09911 0,10304 0.09763 0.09776 0.09546 0.09611 0,09499 0.09185 0.09254 0.09155 0,09402 0.09812 0.09345 0,09356 0,09225 0.09479 0.08877 0.08607 0.08410 0,08535 0.08667 0.09207 0,08643 0.08359 0.08239 0,08501 0.08800 0,08128 0.07818 0.07595 0.07847 0.08775 0.08133 0.07792 0.07605 0.07798
0.11740 0.11726 0.11755 0.11744 0.11785 0.11772 0.11812 0.11799 0.109g1 0,10955 0.11005 0.10969 0,10947 0.11008 0.11058 0.10992 0.11042 0.09977 0.09813 0.09700 0.09774 0.09965 0.09796 0.09678 0.09670 0.09758 0.09586 0.09344 0.09158 0,09112 0.09214 0.09569 0.09322 0.09132 0.09086 0.091go 0.09112 0,08801 0.08485 0.08648 0.08546 0,09070 0.08752 0,08599 0.08429 0.08488 0.08727 0.08264 0.07863 0.07682 0.07815 0.08699 0.08226 0.07816 0.07627 0.07763
0.2793 0.2843 0.2790 0.2828 0.2764 0.2810 0.2759 0.2806 0.5558
0.00 -0.01 -0.06 -0.02 0.05 0.00 0.00 -0.02 -0.01 0.02 0.05 0.12 0.09 -0.04 0.04 -0.09 -0.03 0.11 -0.05 -0.07 0.02 0.06 -0.01 0.04 -0.05 -0.03
3.7417 0.6117 2.4134 4.0646 0.6147 1.4691 2.4286 3.2774 4.0885 4.5572 0.6539 2.6051 3.5329 4.4253 4.9365 0.6630 1.5932 2.6498 3.5971 4.5031 0.6947 1,6728 2.7900 3.8001 4.7783 0.6993 1.6845 2.8110 3.8311 4.8197 0.6400 2 I9908 4,4204 5.4515 0.6443 1.7163 3.0144 4.4574 5.4960 0.6689 3.1427 4.6648 5.7678 0.6723 3.1641 4.6994 5.8093 0.7020 1.8794 6.1464 0.7066 1.8923 6.1951 0.7297 1.9588 6.4667
0.1378 3.0066 2.0333 0.8637 3.0214 2.5875 2.0461 1.4897 0.8688 0.4456 3.2141 2.1948 1,6059 0.9403 0.4827 3.2589 2.8060 2.2324 1.6351 0.9569 3.4149 2.9461 2.3506 1,7273 1.0153 3.4374 2.9669 2.3683 1.7415 1,0241 3,5475 2.3638 1.4567 0.6247 3.5712 3.0673 2.3824 1.4689 0.6298 3.7076 2.4838 1.5373 0.6609 3.7263 2.5007 1.5487 0.6657 3.8907 3.3590 0.7043 3.9164 3.3820 0.7099 4.0448 3.5008 0.7410
The Journal of Physical Chemistry
0.5501 0.5544 0.5429 0.5486 0.9931
0.9997
1.1900
1.1984
1.4080
1.4289
1.6887
-0.01 -0.07 0.05 0.02 0.04 0.05 0.01 0.13 0.07 0.07 -0.07 -0.11 -0.05 -0.07 0.05 0.04 -0.06 -0.16 -0.13 0.00 0.02 -0.08 -0.03 -0.06
0.01 1.7074
0.02 -0.06 -0.02 -0.01 0.02
-
Expt
0.08301 0.07857 0.06581 0.06266 0.07842 0.07059 0.06622 0.06243 0.06206 0.06474 0.07625 0.06329 0.05976 0.05968 0,06169 0,07690 0.06902 0.06411 0.06051 0.05939 0.07671 0,06697 0.06139 0.05764 0.05754 0.07663 0.06683 0.06121 0.05753 0.05751 0,07516 0,05870 0.05517 0.05643 0.07542 0.06494 0.05875 0.05504 0,05566 0.07587 0,05690 0,05305 0.05406 0.07397 0.05686 0,05299 0.05340 0.07341 0.06221 0.05178 0.07335 0,06196 0.05074 0,07055 0,06046 0.04891
mR
%
1.7747 2.1753
0.08 -0.05 -0.12 -0.11 -0.05 -0.16 -0.04 -0.15 -0.14
2.1864
-0.03 2.3321
2.3647
2.4805
2.4972
-0.07 -0.01 -0.04 -0.03 -0.01 -0.03 -0.04 0.18 0.15 0.01 0.01 -0.06 0.14 0.15 0.12 0.02
-0.05
2.5564
2.5736
2.6724
2.6882
2,8086
2.8272
2.9233
0.17 0.19 0.16 -0.05 0.06 0.13 0.00 -0.03 -0.14 0.12 0.17 -0.01 0.02 0.15 0.22 0.02
-0.05 0.19 0.26 0.00 -0.03 -0.08 0.07 -0.03 -0.08 0.02 -0.13 -0.12 0.03
ACTIVITY COEFFICIENTS FOR WATER-UREA SYSTEMS
1835
Table I1 (Continued) Aa
-
-
r
-
7
mimz
ml
mz
Expt
0,6069 1.3371 2.1204 2.7396 3.1214 3.4380 3.5872 3.7205 0.6099 1.3439 2.1322 2.7560 3.1411 3.4592 3.6071
2.4073 1,9962 1.4915 1.0366 0.7179 0.4246 0.2790 0.1370 2.4194 2.0064 1.4999 1.0428 0.7225 0.4273 0.2806
0.08169 0.07894 0.07421 0.07326 0.07317 0,07424 0.07894 0.08454 0.08229 0.07877 0.07415 0.07334 0.07362 0.07461 0.07753
(1
A' -
-
mlmz
mR
%
ml
mz
1.7660
-0.12 -0.04 -0.15 -0.11 -0.12 -0.07 0.06 0.10 -0.09 -0.03 -0.13 -0.09 -0.07 -0.05 0.02
0.7350 1.9740 6.5251 0.2083 4.0646 0.2100 4.6496 0.2410 5.5874 0.2430 5.6473 0.2843 6.9686 0.2869 7.0566
4.0737 3.5280 0.7477 3.1172 0,2007 3.1431 0.2027 3.6067 0.2435 3.6375 0.2461 4.2556 0.3037 4.2933 0.3076
Calcd
1.7747
Expt
Calod
mR
%
0,07320 0.06126 0.04812 0.08431 0.06880 0.08412 0.06840 0.08110 0.06064 0.08092 0.06019 0.07823 0.05156 0.07811 0.05108
2.9426
-0.06 -0.01 0.08 -0.01 -0.03 0.07 -0.06 0.03 -0.03
2.1197 2.1370 2.4757 2.4967 2,9416 2.9686
0.10 -0.04 0.04 -0.10 0.03 -0.05
The fourth and fifth decimals were retained to minimize "rounding off" errors in the least squaring of the data,
ever, to make a completely new set of measurements concurrently with the work on the three-component system. All experiments were performed either in triplicate or quadruplicate, and the technique of prolonged pumping to concentrate the solutions as described above was found particularly useful at the higher urea concentrations. The molality of a saturated urea solution was found to be 20.027, which may be compared with a value of 20.007 obtained by Scatchard, et a1.18 Table I gives the molalities of urea solutions in isopiestic equilibrium with sodium chloride solutions. In this paper, 0, 1, and 2 are used to designate water, urea, and sucrose, respectively. The osmotic coefficients of urea solution up to 7 m (the region covered by the three-component work) can be represented by the equationlg
+
-
41 = 1 - 0.043702~~1 0 . 0 0 6 3 4 8 ~ ~ 1 ~ 0.000695m13 4-0.000034m14
(ml
< 7)
(10)
with an average deviation of 10.047%. The osmotic coefficients over the complete concentration range are best represented by the extended equation
b1 = 1 - 0.043384m1 0.006122m120.000703mi3 0.000052mi4 0.0000020m15 (m
+
< 20)
(11)
with an average deviation of ~0.054%. The corresponding activity coefficients of urea may be obtained13 from the relation
where the Ei are the coefficients of the m12terms in eq 11, and hence In y1'
=
+ 0 . 0 0 9 1 8 3 ~ ~-1 ~ 0.000937mi3 + 0.000065~~1~ -
-0.086768m1
0.0000024m16 (13) Since this work was commenced, Bower and Robinsonlo have published a limited amount of data for the HzO-urea system. Their results are substantially in agreement with this work, particularly at lower concentrations, although at higher concentrations their osmotic coefficients for some points show a scatter of as much as *0.6% from the data presented here. The System H2O-Sucrose. Accurate isopiestic data for the HzO-sucrose system have been published recently by Robinson and Stokes,12 and their eq 5 for the relationship of osmotic coefficient to molality was used throughout this work
+ 0.07028~2+ 0 . 0 1 8 4 7 ~ ~-2 ~ 0.004045m23+ 0.000228m24 ( m < 5.8)
42 = 1
(14)
The System H2O- Urea-Sucrose. The equilibrium molalities of urea and sucrose mixtures and the reference sodium chloride solution are given in Table 11. Also given in this table are values of the experimental quantity (Almlmz), which is derived from eq 8. Values of 4~ were interpolated from data already published,*O while 61 and 42 were obtained from eq 11 and 13, respectively. Column 4 of Table I1 gives the values of (19) An IBM 1620 computer was used t o least square the data. (20) See ref 14, Appendix 8.3, p 476.
Volume 70, Number 6 June 1966
H. DAVIDELLERTON AND PETER J. DUNLOP
1836
A / m l m 2 calculated from the least-square eq 17. The last column gives the percentage error in the molality of the reference solution which would account for the difference between the experimental and computed values of A / m l m . This percentage error is defined8 by
where mR(ca1cd)
=
(ml$l
+ m2$2 +
(16)
Aoaled)/YR$R
Acalcd can be derived from column 4. $R is assumed to remain constant for small changes in mR, and when sodium chloride solutions are used as reference solute, Y R = 2. To fit the values of A/mlm~,derived experimentally from eq 8, to a power series in ml and m2 with the form of eq 9, four different polynomials were tried, involving terms up to either squares of cubes in 1721 and m2, both with and without cross-terms present.21 The series that best represented the data was
A
-- -
m1m2
u+u+uz
-0.12597 -I- 0.015004ml -I- 0.018236m2 Uz
0.000710m12 - 0.000500mlm2 - 0.001848m22
< 7.0, m2 < 4.3)
(ml
(17)
*
with an average deviation of 0.0015. Surprisingly, the cubic series with cross-terms gave the greatest average deviation, while the two series without crossterms had average deviations between these two. An equation involving the first powers only in ml and m222 was also found to represent the data quite well. Activity coefficients for the ternary system may be determined from a knowledge of the values of the coefficientsof eq 17. Thus from eq 5,9, and 17, we obtain In
y1 =
In yl0 - 0.12597m2 f 0.015004m1m2
+
0.009118m22- 0.000616m~~ 0.00071Om1~?n~ - 0.000333m1m2~ (18) and from eq 7,9, and 17 In yz
=
N H groups. Using the activity data of Scatchard, et uZ.,~* and the heat of dilution of Gucker and P i ~ k a r d , ~ ~ Schellman obtained the enthalpy of formation of an amide hydrogen bond in waterz3 and later extended the work to discuss the stability of hydrogen bonding in peptides. 25 Kresheck and Scheraga,26in an extension of Schellman’s work, have discussed the temperature dependence of the enthalpy of formation of the amide hydrogen bond, and have pointed out that it may be possible to account for the nonideal behavior of aqueous urea solutions by mechanisms other than dimerization, trimerization, etc. Nevertheless, T’so, Melvin, and Olson,27 in a binary study of aqueous purine and nucleoside solutions, consider that it is reasonable to assume that the large lowering of osmotic and activity coefficients in those systems may be ascribed to association. Stokes2* has also discussed association of urea in connection with binary tracer and diffusion coefficients. If it is assumed that urea forms a series of dimers, trimers, etc., in water, we may write
+
In 7 2 ’ - 0.12597m1 0.007502m12 -k 0.018235m1m~- 0.001848mlm~~
-
0.000237ml3 - 0.000333m12m2 (19)
Discussion The System H2O-Urea. The deviations from ideality of the water activity of urea solutions have been discussed previously by S~hellman.2~The negative heat of dilution of the polar urea molecule was attributed to intermolecular hydrogen bonding between CO and The Journal of Physical Chemistry
+ U +UB,etc.
(20)
...... so that mu, mu,, . . , , etc., are equilibrium concentrations of monomer, dimer, etc. If we further assume, as our data suggest, that at low concentrations dimerization is the important step, then the equilibrium constant, K1, for the mole fraction28 concentration scale may be written
where N U and NU,are the equilibrium mole fractions of monomer and dimer, respectively. (21) This was determined by the method of least squares using a CDC 3600 computer belonging to the Commonwealth Scientific and Industrial Research Organisation, Canberra. Double precision was used to minimize errors due to “rounding off .” (22)We are indebted to Dr. R. A. Robinson of the National Bureau of Standards, Washington, D. C., for arranging for this computation to be made, and also for a trial computation of a second-order polynomial. (23) J. A. Schellman, Compt. Rend. trav. lab. Carlsberg, Ser. Chim., 29, 223 (1955). (24) F.Gucker and W. Pickard, J . Am. Chem. SOC.,62, 1464 (1940). (25) J. A. Schellman, Compt. Rend. Trav. Lab. Carlsberg, Ser. Chim., 29.230 (1955). (26)G. C. Kresheck and H. A. Scheraga, J . Phys. Chem., 69, 1704 (1965). (27) P.0. P.Ts’o, I. S. Melvin, and A. C. Olson, J . Am. Chem. Soc., 85, 1289 (1963). (28) R. H.Stokes, J . Phys. Chem., 69, 4012 (1965).
ACTIVITYCOEFFICIENTS FOR WATER-UREA SYSTEMS
1837
The total molality (monomer and dimer), (m&, of urea in soluticq may be computed from the relation
these phenomena would describe the solvent activities of the ternary solutions. However, using the parameters K1 and h, it was not possible to compute a. values which agreed with the data in Table 11. Kow, inspection of Table I1 and use of eq 9 indicates that the two solutes are “salted in” by one another. We therefore postulate that the urea monomer is bound to the hydrated sucrose (hS)
(ml)T
=
+
( 4 ~ 1 1)ml -
wo
+ ( 4 ~ +1 l)(ml + 2 w ~ m 1
+ 2(4K1
+ 1)
(22) Sow, if dimerization is regarded as the cause for the nonideality of dilute aqueous urea solutions, then the corresponding water activities, ao, are given by a0 = [1
+ (ml)T/aOl-’
(23) and hence the corresponding osmotic coefficient is given by
41 =
( a 0 4In [1
+ (ml)T/WO]
(24)
Using eq 22 and 24, it was found that, up to 2 m, a value of K 1 = 1.78 gave agreement with the experimental results in Table I to better than 0.1%. This agrees with Stokes,28who has recently reported that, the data of Scatchard, Hamer, and Wood and of Bower and Robinsonlo are consistent with K1 = 1.8. Thus it seems reasonable to interpret nonideality of dilute aqueous urea solutions in terms of dimerization of the solute. However, as has been pointed out by Stokes12* this does not prove that association actually occurs. The System H20Urea-Sucrose. The deviations from ideality of the water activity of H20-sucrose solutions have been discussed previously by ScatchardZ9and more recently by Robinson and Stokes12 in terms of a hydration number, h, the average number of moles of water bound to 1 mole of sucrose (component 2). The latter authors derived the relation
and by expanding the logarithmic factor obtained a value of h = 4.61 by considering the linear term in m2. By solving eq 25 numerically, we find that the water activity of sucrose solutions up to 2 m may be represented, within the error of measurement, by ii hydration number of 4.86 f 0.05. At higher concentrations the value of h becomes lower, but even for a near-saturated 6 nz solution, a hydration number of 3.6 will reproduce the data. Now if association of urea and hydration of sucrose can be used to describe the deviations from ideality of the activity of water in the two-component systems, it might be expected that a simple combination of
U
+ hS +UhS
(26)
with an equilibrium constant K 2given by
On this basis it was possible to compute the water activities of the ternary solutions in Table I1 in terms of the three parameters K1, Ka, and h. Equations 21 and 27 have to be solved for KZwith the restrictions
+ mchs -k 2mv, = mhs + muhs
ml = mu m2
(284 (28b)
where mu, m v , mhs, and muhS are the equilibrium molalities of urea monomer, urea dimer, hydrated sucrose, and the hydrated urea-sucrose complex, respectively. Explicit solutions to the equations were not obtained. Equation 21 was first solved for mu, in terms of mu, ma, K1, and h. Then eq 27 was solved for a series of K2 for certain assumed values of mu. The corresponding values of mu and KZwere then used to compute a series of a. values for various compositions of the system. It was thus possible to choose corresponding values of mu and KZwhich best represented each composition of the ternary system. All computations were made with a CDC3200 electronic computer. I n this way it was found that a value of Kz = 13.0 would reproduce the experimental water activities with an average deviation in a. of ltO.00005 up to ml = m2 = 1.0. By assuming a reasonable error of 0.2% in the product ( w R ~ R ) , the estimated average deviation for the same concentration range is ltO.00004. Thus in this relatively dilute range of concentrations only three parameters, &, Kz, and h, are necessary to represent the data in Table 11. It would seem reasonable to suggest that use of further parameters should enable agreement between experimental and calculated values to be obtained for more concentrated solutions. (29) G. Scatchard, J . Am. Chem. Soc., 43, 2406 (1921).
Volume 70, Number 6 June 1966
