DIFFUSION OF SOLUTIONS IN THE SYSTEM H~P04-Ca(HzP04)z-H20
A similar treatment, assuming a parent and an additional subspecies H of concentration y [I310 with
AH
=
[Hlh[Xl
gives the equation
217
+
AA = b[FIo[P(g(A) - !(A>> r(h(A) - !(A)>] (A6) It can be seen that if p, g(A), y, and h(A) are all appreciable, then only extremely fortuitous circumstances would give a constant shape for the differential spectrum.
Diffusion at 25” of Solutions in the System Phosphoric Acid-Monocalcium Phosphatewater’
by 0. W. Edwards, R. L. Dum, J. D. Hatfield, E. 0. Huffman, and K. L. Elmore DCiswn of Chemid Dmelopment, Tennessee Vdley Authority, Wa7sson Dam, Alabama (Received August $3, 1966)
Isothermal diffusion measurements at 25’ were made with a Gouy Musiometer on 18 solutions in the ternary system H3P04-Ca(H904)2-Hz0. The solutions were 0.95 to 4.65 M in &PO4 and 0.045 to 1.0 M in Ca(H2P04)2. The four volume-fixed diffusion coefficients were calculated on a digital computer by a least-squares method based on published theoretical equations, and the results were in good agreement with those calculated by the area method of Fujita and Gosting. The ranges of values of the Musion coefficients (cm.2/sec., X106) were [subscript 1 refers to HaP04; 2 refers to Ca(HJ?O&]: Dll, 12.3 to 6.3; 0 1 2 , -7.1 to f5.2; D22, 7.1 to 2.4; and 0 2 1 , -1.8 to -0.05. The large values of the cross-term coefficient DIZa t certain concentrations showed strong interactions between the flows of the solutes.
Diffusion coefficients for 0.036 to 16 M phosphoric acid solutions, determined by the Gouy method, have been reported.2 I n a continuing study of transport in phosphate systems, measurements were made of the four relevant diffusion coefficients for each of 18 solutions in the system HsP04-Ca(HzP04)z-H20at 25’. Several methods, which differ in accuracy and general applicability, for calculating diffusion coefficients for three-component systems from Gouy data were described by Gosting, et aLa4 A new general method for calculating diffusion coefficients from Gouy data is described in a companion paperlSand this “least-squares” method was used in the present study. The results are compared with those calculated by the area method.8
Measurements Apparatus. The two-lens Gouy diffusiometer, water bath, cell holder, and diffusion cells were those used in (1) Presented in part before the Division of Physical Chemistry, 145th National Meeting of the American Chemical Society, New York, N. Y., Sept. 8-13, 1963; Abstracts, p. 34-T. (2) 0.W.Edwards and E. 0. Huffman, J . Phys. Chem., 63, 1830 (1959). (3) D. F. Akeley and L. J. Gosting, J . Am. Chem. SOC.,75, 5685 (1953). (4)R.L. Baldwin, P . J. Dunlop, and L. J. Gosting, ibid., 77, 5235 (1956). (5) P. J. Dunlop and L. J. Gosting, ibid., 77, 5238 (1965). (6) H. Fujita and L . J. Gosting, ibid., 78, 1099 (1956). (7) P. J. Dunlop, J . Phys. Chem., 61, 994 (1957). (8) H.Fujita and L. J. Gosting, ibid., 64, 1256 (1960).
Volume 70, Number 1 January 1966
EDWARDS, DUNN,€€ATFIELD, HUFFMAN, AND ELMORE
218
Table I: Compositions and Refractive Index Data on Solution Pairs
Run no.
mole/1000 ml.
,-Composition,"
C1
-Density, TOP
C*
ACi
ACz
E O h
g./ml.-
Bottom soln.
Refractive fraction,b at
Refractive index derivative, 1000 ml./ mole, X104 R1 Ra
Total fringes, J,
1 m HIPO~
41.02 42.02
0.95006 0.95065
0.04500 0.04498
0.13713 -0.00239
0.00003 0.03551
1.05064 1.05122
1.05752 1.05705
0.99945 -0.01801
41.03 42.03
0.95048 0.95039
0.04500 0.04494
0.27446 -0.00344
-0.00008 0.07103
1.04404 1.04815
1.06102 1,05995
43.02 44.02
0.93662 0.93646
0.26990 0.26986
0.27689 0.00030
0.00029 0.06755
1.08437 1.08548
43.05 44.04 44.50
0.93637 0.93662 0.93734
0.26999 0.26993 0.26961
0.13787 -0.00053 0.04584
0.00015 0.03371 0.02191
45.03 46.03 46.50
0.92147 0.92203 0.92319
0.49990 0.49985 0.49928
0.15033 -0.00060 0.04978
47.00 48.00
0,90776 0.90657
0.75013 0.74914
0.23903 0.00207
80.609
305.38
50.68 48.84
-0.01294
81.049
309.08
102.11 98.79
1,09802 1.09682
0.99577 0.00109
75.314
305.74
96.02 94.80
1.08802 1,08846 1.08820
1.09472 1.09402 1.09411
-0.00583
75.218
305.38
47.63 46.83 46.82
0.00017 0.03594 0.02383
1.12489 1.12565 1.12578
1,13206 1,13147 1.13184
- 0,00633
71.220
295.38
49.11 48.25 48.83
-0.00084 0.07162
1.16312 1.16260
1.17428 1.17426
1.01487 0.00688
67.765
282.65
73.16 93.44
78.323
286.43
52.15 47.96
0.99890
0.99694 0.34094 0.99742 0.33575
2.12 m HePO4 1.03 2.03
1.91742 1.91733
0.04504 0,04501
- 0.00220
5.03 6.03 7.07
1.88662 1.89154 1.88744
0.26998 0.26996 0.27005
-0.00037
14.03 20.02 19.04
1.85537 1.85465 1,85387
21.01 24.01 9.01 12.01
0.04031 0.00001
1.09950 1.09937
1.10594 1.10599
-0.01512 0.99644
0.03769 0.01256 0.02599
1.13478 1.13487 1.13463
1.14086 1.14140 1.14113
-0.00362
0.09524 0.04864
0.66246 0.31810
74.433
286.47
49.27 49.24 50.17
0.49994 0.50000 0.50003
0,15272 0.05008 0.09829
0.00006 0.02997 0.01523
1.16975 1.16979 1.16976
1.17700 1.17701 1.17684
1.00014 0.29500 0.61693
70.213
280.64
49.07 54.83 51.57
1.81784 1.81789
0.74980 0.74978
0,29931 -0.00069
-0.00012 0.08945
1.20399 1.20403
1.21792 1.21796
1.00147 -0.00019
67.457
269.05
92.43 110.13
1.78041 1.78037
0.99988 0.99986
0.23952 -0.00142
-0.00058 0.06936
1.24260 1.24293
1.25351 1.25332
1.00995 -0.00506
64.166
261.04
69.77 82.59
75.382
262.64
93.47 91.91
0.13350
4 m HZ04 30.01 29.01
3.32824 3.34269
0.04374 0.04378
0.02893 0,26596
-0.00001
1.16634 1.16667
1.17879 1.17972
-0.00199 1.00002
30.03 33.01 34.01
3.34221 3.28283 3.28358
0.04378 0,26749 0.26750
-0.00100 0.28024 -0.00097
0.06939 -0.00010 0.07196
1.16785 1.19842 1.19965
1.17866 1.21193 1.21075
-0.00303 1.00130 -0.00372
72.531
263.90
83.38 93.05 86.72
33.03 34.03
3.28334 3.28215
0.26978 0.26980
0.14071 -0.00110
0.00005 0.03982
1.20215 1.20228
1.20906 1.20830
0.99867 -0.00768
72.757
262.49
47.08 47.63
35.01 36.01
3.27385 3.27626
0.49570 0.49602
0,30027 0.00292
0.00020 0.08976
1.23289 1.23323
1.24712 1.24713
0.99751 0.00860
69.135
259.27
95.39 107.60
37.02 38.02
3.15418 3.15340
0.74918 0.74910
-0.00206
0.29850
-0.00007 0.07355
1.26626 1.26752
1.28007 1.27867
1.00091 -0.00728
65.587
254.09
89.68 85.06
39.02 40.01
3.10060 3.08567
0.99832 0.99976
0.18707 -0.00060
0.00180 0.04989
1.30400 1.30416
1.31283 1.31167
0.96348 -0.00308
62.616
245.27
55.73 55.93
The Journal of Physicccl Chmislry
0.06949
DIFFUSIONOF SOLUTIONS IN THE SYSTEM HZO4-(H2PO4)~Ca-H20
219
Table I (Continued)
Run no.
----Composition,a
-Density, TOP aoln.
mole/1000 ml.
Cl
C?
49.01 50.00
4.63111 4.63134
0.04500 0.04501
0.27956 0.00094
0.00000 0.09001
55.01 56.00
4,54591 4.54415
0.26985 0.26987
0.30131 0.00170
- 0.00020
57.01 58.00
4.45698 4.45565
0.49973 0.49984
0.30193
- 0.00114
-0.00001
59.00 60.00
4.35793 4.35813
0.74960 0.74957
0.29865 -0.00134
a
ACi
ACz
0.08999 0.09000
- 0.00020 0.06960
Subscript 1denotes H3P0h; 2 denotes Ca(H2P0&.
g./ml.-
Bottom soln.
Refractive ai
Refractive index derivative, 1000 ml./ mole, XlO* R1 Ri
Total fringes, JUl
6 m HaPo4 1.22947 1.22943
1.24296 1.24312
1.00005 0.00304
72.671
248.22
93.12 102.73
1.25881 1.25924
1.27317 1.27278
1.00238 0.00535
69.949
245.84
96.38 101.96
1.28926 1.28954
1.30345 1.30299
1.00016 -0.00351
66.962
242.45
92.66 99.67
1.32205 1.32380
1.33578 1,33978
-0.00522
64.093
237.05
87.53 75.24
1.00242
‘Calculated from Ri by eq. 46, ref. 6.
the previous study.2 The optical lever arm (‘W’ distance) was unchanged from 232.25 cm. The four cells had thicknesses (“a” distances) of 2.5027,2.5031,2.5035, and 2.5078 cm. The Gouy patterns were measured with a Gaertner linear comparator, Model M1201-30B. Preparation of Solutions. Distilled water from the laboratory supply was passed through a mixed-bed ionexchanger; the effluent had a maximum specific conductance of 2 X ohm-’ cm.-’. Stock solutions of phosphoric acid were prepared from the triply crystallized hemihydrate. Stock three-component solutions were prepared by dissolving reagent monocalcium phosphate monohydrate in the acid solutions; the salt was spectroscopically pure, but its ratio CaO :PzO5 was slightly high. The Pz05and CaO contents of each stock solution were determined by precise For each diffusion run, two solutions that would provide suitable concentration gradients of the solutes were prepared from weighed amounts of a three-component stock solution, phosphoric acid stock solution, water, and specially purified monocalcium phosphate monoh ~ d r a t e . ~Weights ~ ? ~ ~ of the components were calculated from the desired molarities and the density of the average solution (top and bottom) of the three-component system.13 As a check on the composition of each solution, its density was determined in duplicate at 25’ in single-stem pycnometers; duplicate determinations agreed to within 1 part in 10,000. All weights were corrected for buoyancy of air. Procedure. Each arm of the cell was filled with the appropriate solution, the assembly was equilibrated at 25 f 0.05’ in the bath, and the usual photographs were taken to correct for optical errors of the cell. The junction was made, and the boundary was sharpened in the usual m a ~ m e by r ~siphoning ~ ~ ~ the solutions
through a single-tip stainless steel needle for 1hr. during which about 100 ml. of solution was withdrawn. Five photographs of horizontal Rayleigh patterns were taken at intervals during the sharpening period for use in determining F , the fractional part of a fringe.16 (The symbols used are those of Gosting, et aL3-9 Immediately after the sharpening period, 11 Gouy patterns were recorded over the next 2 hr. The whole number of Gouy fringes I was determined from the difference in the refractive indexes of the two initial solutions as measured with a high-precision Rayleigh interferometer (Zeiss No. 321005). The “fringe-skipping” error, inherent in the Rayleigh instrument, was eliminated by making measurements in several Zeiss cells, 0.5 to 2.0 cm. thick. The average starting-time correction16At was 21 sec. ; the average deviation from the average was 9.6 sec. Displacements of 16 fringe minima of each Gouy pattern were measured with the comparator; the fringe minima included those required for the C, extrapolation3 and, beyond those, 6 additional fringe minima distributed over values off({) up to 0.9. The composition and refractive index data are shown in Table I.
(9) R. L. Dunn and J. D. Hatfield, J. Phys. Chem., 69,4361 (1965). (IO) C.H. Perrin, J . Asaoc. Ofic.Agr. Chemists, 41,758 (1958). (11) H. H. Willard and A. W. Boldyreff, J . A m . Chem. Soc., 52, 1888 (1930). (12) W. L. Hill and S. B. Hendricks, I d . Eng. Chem., 28, 440 (1936). (13) A. J. Smith and E. 0. Huffman, Chem. Eng. Data Ser., 1, 99 (1966). (14) L. J. Gosting and M. S. Morris, J . Am. Chem. SOC.,71, 1998 (1949). (15) L. J. Gosting, ibid., 72, 4418 (1950). (16) L. G. Longsworth, ibid., 69, 2510 (1947).
Volume 70,Number 1 Januarg 1966
EDWARDS, DUNN,HATFIELD, HUFFMAN, AND ELMORE
220
~~
~
~
~
~~
~~
~
~~
~
Table II : Derived Quantities Rsquired for Calculating Diffusion Coefficientsa PA,
Run no.
cm.f/sec., x 106
41 .02 42.02
7.518 5.589
-57.3
0.4220 -0.2326
6.179
41.03 42.03
7.577 5.611
-58.2
0.4105 -0.2644
43.02 44.02
5.546 5.220
-13.1
0,0843 -0.0008
43.05 44.04 44.50
5.525 5.196 5.294
-13.3
45.03 46.03 46.50
4.595 4.440 4.513
-7.8
47.00 48.00
3.992 3.885
SA
-7.2
&,
8%
Areas under fringedeviation graph, X10'
x 106
x 106
10.191
7.874
5.292
39.88 -39.76
38.95 -39.76
6.278
10.281
7.926
5.253
37.75 -42.09
37.47 -44.05
5.223
12.388
5.830
5.217
30.65 -1.81
30.64 -0.31
0.0595 -0.0043 0.0085
5.233
17.734
5.954
5.135
40.74 -3.03 5.94
39.30 -2.95 5.77
0.0432 0.0023 0.0134
4.446
11.975
4.768
4.456
21.97 -3.74 64.90
20.46 1.11 64.77
3.877
13.013
4.165
3.897
21.22 0.50
21.23 1.65
-19.71 41.78
-19.94 41.06
r-
0.0321 0.0025
Qlb
QZC
2.12 m Hap01
- 0.0977
1.03 2.03
4.747 7.706
5.03 6.03 7.07
4.724 5.535 5.089
-52.6
14.03 20.02 19.04
5.232 4.513 4.817
-47.6
21.01 24.01
4.603 3.830
-44.7
0.2276 -0.0043
9.01 12.01
4.159 3.382
-52.6
-0.0009
-97.3
0.7768
4.984
8.909
8.034
4.651
4.886
10.500
6.638
4.595
-15.66 51.24 16.31
-15.31 52.09 16.01
4.335
11.205
5.962
4.165
92.40 24.09 54.67
90.59 24.05 54.68
3.857
9.784
5.203
3.804
82.76 -4.10
84.12 -4.08
3.384
8.397
4.715
3.387
91.31 -2.00
91.41 -0.37
29.33 25.51 9.91
31.35 25.13 9.16
- 0.0475 0.1944 0.0533 0.2390 0.0550 0.1321
0.2680
4 m HZ04 30.01 29.01 30.03
4.090 7.742 3.852
33.01 34.01
6.638 3.974
-112.9
33.03 34.03
6.541 3.950
35.01 36.01
- 150.23
0.1079 0.9406 0.0297
3.795
8.184
7.918
3.893
0.7526 0.0433
3.866
8.232
7.148
4.067
75.39 14.16
74.51 12.83
-111.4
0.7493 0.0724
3.768
8.188
7.084
4.110
79.79 23.41
79.23 22.00
5.846 3.744
-104.4
0.6538 0.0539
3.613
7.980
6.474
3.826
94.55 16.85
95.53 1V. 24
37.02 38.02
5.208 3,453
-99.1
0,6032 0.0645
3.305
7.524
5.848
3.593
105.99 22.33
106.47 21.86
39.02 40.01
4.654 3.110
-107.1
0.5707 0.0613
2.977
7.065
5.388
3.234
119.90 26.58
118.15 22.69
~~
The Journal of Physical Chemistry
DIFFUSION OF SOLUTIONS IN THE
SYSTEMH&P04-Ca(H2PO4)2-H20
22 1
Table II (Continued) 9 . 4 ,
Run no.
om.l/Rec.,
49.01 50.00
vu+
xin6
I
x 10'
vu-
I
x IO'
81,
82,
x 10'
x 10'
Areas under fringedeviation graph, X 104
SA
r-
7.747 3.329
-189.4
0.9770 0.2722
2.593
8.020
7.895
4.059
28.10 135.24
23.41 135.34
55.01 56.00
6.772 3.406
-158.0
0.9277 0.2662
2.730
7.421
7.075
3.962
56.80 110.41
52.15 107.79
57.01 58.00
6.074 3.242
-149.1
0.8650 0.2090
2.747
7.115
6.525
3.674
80.30 83.59
78.91 81.86
59.00 60.00
5.511 2.990
-151.2
0.8425 0.1806
2.602
6.584
5.951
3.335
85.58 68.77
84.17 69.15
Qzc
QIb
6 m HZ04
are available in AD1 Document No. 8666. 9~= reduced-height:area ratio; SA= slope of l / e Aus. al; r-, u+, Values of and U- are functions defined by eq. 51, 30a, and 31a of ref. 6; el and 02 are defined by eq. 25 and 26 of ref. 4. Observed values. Calculated from least-squares ( D c ~ ) ~ .
Calculations The four diffusion coefficients (D& for each set of and c2were calculated both solute concentrations by the "least-squares" methodg on an IBM 704 computer and by the area methods on an IBM 610 computer although some data, such as values for C,, p , e d f j 2 At, , DA,and QC,obtained from the IBM 704 computation were used in the area method. The extrapolation method6s6for determination of Ct has been considered valid only for systems in which values of I?+ and I?- are small or in which values of u-/u+ are near unity.8 I n the Appendix we present a different series expression that converges rapidly for all values of encountered in this study and from which were obtained valid extrapolations for e,. The intercept Cy, at 2, = 0 of the linear plot of Y j / e - r j 2 os. Z,1'8was determined by the method of least squares. The fringe displacements used in the extrapolation were those corresponding to the fringes j = 0 to 6, 10, 15, and 20.' Typical plots of the extrapolation are shown in Figure 1. I n general, more precise values of C, were obtained from patterns associated with longer diffusion times and from extrapolations based on the ten fringes used rather than on the seven lowest fringes. For a few compositions, approximate values of the main diffusion coefficients, DU and D22,were estimated by extrapolation of plots of the diffusion coefficients obtained at other compositions against concentration of the acid, cl. The crossdiffusion coeficients, DI2 and D21, for these compositions were then calculated from eq. 25 and 26 of ref. 4 where el and e2were experimental values.
cl
Discussion The Binary Systems HsPOrH-20 and Ca(HzP04)r
H20. Measurements of diffusion in solutions in the binary system HJ?04-H202showed that the boundaries were gaussian when the acid was more concentrated than 0.15 M. In further checks on the boundaries, measurements were made on a 1.08 M acid solution, and the interferograms were analyzed more extensively than in previous runs by measuring displacements of 16 fringe minima distributed over the range of values of f(f). The area under the relative fringe deviation curve was 3.5 X a result similar to that obtained by Wendt17for diffusion in sulfuric acid, which showed that the boundaries were gaussian within the precision of the measurements. Diffusion in solutions in the system Ca(H2P04)2-H20 cannot be measured because the salt hydrolyzes in water, deposits dicalcium phosphate, and yields a solution containing free phosphoric acid. The small fringe deviation for the binary system H&P04-H20showed, however, that the interferometer was suitable for measurement of diffusion in the threecomponent system. The Ternary System HaP04-Ca(H2P04)2-H20. Published flow equations and definitions of the four d 8 u sion coefficients that apply to three-component systems are based on cell-fixed or volume-fixed frames of referen~e.~-~J'The results reported here are based on a volume-fixed frame of reference (DZ.Jvbecause of slight (maximum 0.5%) changes in volume when some of the solutions are mixed. The differences between results based on the two frames of reference are small. Values of derived quantities required for calculating diffusion coefficients by the least-squares and area methods are shown in Table 11. The extensive tables ~~
~-
(17) R. P.Wendt, J. Phys. Chem., 66, 1279 (1962).
volume 70,Number 1
January 1966
EDWARDS, D u “ , HATFIELD, HUFFMAN, AND ELMORE
222
I
I
I
I
RUN 2.03, TIME = 7300 SEC.
-
1.30
1
0
I
2
I 4 z,2!3
I 6
1
S
Figure 1. ’l’ypical extrapolations for Ct.
of calculated and observed relative fringe deviations, Oj, that were supplied in the IBM 704 printout are available. l8 The fringe deviations resulted from interacting flows in the system and not from impurities; the components were free from contaminants, and sensitive tests of the solutions for h o d gshowed that the stainless steel needle was not corroded. The reliability of the values for the diffusion coefficients was assessed from the precision of the measured quantities from which they were computed. Values of fringe displacements Y , were reproducible within 5 X cm. for the broader fringes and within 2 X 10-4 cm. for closer fringes. The time and distance data of each run were tested by calculating values of Y,z/i, which should be constant for corresponding fringes of the patterns. There was no significant drift in this quantity in any run, and values of C t d also were constant for all interferograms of each run. Further, values of Jm were calculated for those sets of experiments in which more than two runs were made. For these sets, J m was calculated from least-squares values of R1and Rz and values of the concentration increments, ACl and AC2 (eq. 27 and 36 of ref. 5). The average deviation between calculated and observed values is 0.16 fringe. This uncertainty is an order of magnitude higher than the uncertainty reported for some binary21l 4 and ternarym systems. Concentration increments must be known to four significant figures when Jmis calculated; this precision may not have been attained because of in andyses of the stoc,k solution. Consequently, the error in Jm in the concentrations of the includes ponents. Calculated values of were from values of 6. On the r-, a+, and u- by eq. 55 of observed and calculated values of %A agreed within 0.5%. .~. I ” -
The standard errom for the four diffusion coefficients obtained from each set of runs were calculated by the The Journal of Physical C h i s t r y
rule of propagation of erromZ1 The rule was applied by taking the standard errors of %A and J, as 0.5% and 0.2 fringe, respectively, and obtaining the standard error of r- from the least-squares fit of Q vs. f(f). When only two rum were made, the errors in R,, l/u+, l/u-, all Ba and D,,were obtained by the rule of propagation of error using the known relations. When more than two rum were made, the errors in R1,l/a+, l/u-, and 6( were determined from the respective leastsquares fits described in the companion paper19and the errors in D$*were obtained from these errors by the rule of propagation of error. The average standard errors, AD,,, for runs in which the molality of HZO, was greater than 1.0 were: AD,,, h0.16 X ALh, k0.93 X lo+’; AD21, h0.04 X and A D 2 2 , k0.24 X 10“. For solutions in which the concentration of phosphoric acid was 1 m (except sets where cZ= 0.045), the diffusion coefficients were extrapolated from plots of the coefficients us. to = 0, at constant values of €2. As another measure of the precision of the diffusion coefficients, comparisons were made of the observed and calculated values of the relative fringe deviations. The IBM 704 program included a section that calculated the best fit of the experimental values of Oj with those calculated from theory (eq. A-8, ref. 6 and 9). Differences between calculated and observed relative fringe deviations range from 0 to 8 X and averaged 3 X the maximum uncertainty was about 6% of the value of Q,. When the diffusion coefficients were recalculated after addition of 5 X low4t o each maximum value of Q,, the recalculated diffusion coefficients d8ered from the original values by no more than ~ 0 . X2 10“ cm.Z/sec. The least-squares method of calculation yielded areas under the fringedeviation graph (Qz, Table 111) that deviated from the observed areas Q1 by an average of k1.2 X lo-‘. The “area” method, which is based on the observed areas Q1, gives slightly greater deviations in Q,,bsd - Qcslcd than does the least-squares method, in which this variance is minimized.
cl cl
(18) Material SUPpk”tary to this article has been deposited as Document No. 8666 with the AD1 Auxiliary Publication Project, Photoduplication Service, Library of Congress, Washington 26, D. c. A COPY may be secured by citing the Document number and by remitting $2.60 for photoprints or $1.76 for 35”. microam. Advance payment is required. Make checks or money orders payable to: Chief, Photoduplication Service, Library of Congress. (19) E. B. Sandell, “Colorimetric Determination of Traces of Metals,” 2nd Ed., Interscience Publishers, Inc., New York, N. Y., 1960. (20) G. Reinfelds and L. J. Gosting, J. Phgs. Chem., 68, 2464 (1964). (21) H. Margenm and G . M. Murphy, “The Mathematics of Physics and Chemistry,” 2nd Ed., D. Van Nostrand Co., Inc., Princeton, N. J., 1965, p. 615.
DIFFUSION OF SOLUTIONS IN THE SYSTEM H3P04-Ca(H2P04)2-H20
rQ
-0
-C2,
I
I
I
I
223
I
0.4 0.6 0.8 1.0 CONCN., M, OF Co(H2PO4Ip
0.2
us. Figure 2. Slopes of plots of l/a
LYI.
The error resulting from uncertainty in J , depends on the composition of the solution, as shown in Figure 2 in which slopes SA(of the linear relation l / f i vs. al) are plotted against concentration of Ca(H2P04)2 in several constant concentrations of H8PO4. The absolute values of the slopes vary directly with the acid concentration and inversely with the salt concentration in the range 0 to 0.75 M Ca(H2P04)2. An uncertainty in J , of 0.2 fringe produces an uncertainty of about 2 units in SA;when c1 = 0.95 and c z > 0.045, this uncertainty produces an uncertainty in SAof 15 to 28%, and reliable values for diffusion coefficients in such solutions cannot be calculated. At higher concentrations of acid, however, the uncertainty in SAis much less-2 to 4% in 2.12 m HiE'Od, 1 to 2% in 4 m, and 1.1 to 1.3% in 6 m. Wendt"] showed that SA depends on the composition of solutions in the system H2SO4-NdO4-H2O also and that low absolute values in SAlead to relatively large errors in values of the diffusion coefficients (Tables I1 and VII, ref. 17). It is concluded that, when l S A l > 50, the reliability of the values for the diffusion coefficients depends mainly on the precision with which J , is determined. All of the present methods yield unreliable results when ~SAI < 25; reliable results in this range could be obtained VoIum 70,Number 1 January 1966
EDWARDS, DUNN,HATFIELD, HUFFMAN, AND ELMORE
224
only if the precision with which J , is determined is improved by at least a factor of 10. Criteria for gravitational stability of diffusing ternary solutions have been derived by Wendt22and have been a,pplied by Reinfelds and Gosting.20 For the compositions listed in Table I, all values of the “sufficient but not necessary” criterion, sa, were positive. For experiment 12.01 and the experiments made in 4 and 6 m H3P04in which concentration gradients were primarily of monocalcium phosphate, the stability criterion > 0 was not met. However, there were no observable disturbances at the boundaries in these experiments, and the runs were not long enough to allow convective mixing, Le.
sa
2
IYI =
g
did not become large enough to render the solutions unstable. (See derivation of eq. 20 in ref. 22).
Diffusion Coefficients Values of the diffusion coefficients (DiJ calculated by both the least-squares and area methods are listed in Table 111. The results obtained by the two methods for solutions 4.0 and 6.0 m in H3P04 agree within about 1% and are about as reliable as values published for other ternary ~ y s t e m s . ~2J3 ~ ~Values ’~ for solutions 2.12 m in H3P04agree only within about 27& and those calculated for solutions 1.0 m in HaP04 are in even less agreement. Values for the main diffusion coefficients in most of the solutions 1.0 m in H3P04 were obtained by extrapolation of plots of diffusion coefficients at higher acid concentrations vs. concentration of a t each concentration of Ca(H2P04)2.The values of ]SA/for the dilute acid solutions were so small (about 10) that reliable values for (DiJV could not be calculated. The experimental diffusion coefficients generally satisfy the requirements of the flow equation^.^^^^^^ As C2approaches 0, DZlapproaches 0, and Dll satisfactorily approaches the value of the diffusion coefficient in HBPO4 solutions of the same concentration2 (Figure 3). The data are jnsufEcient, however, to show the approach of D12to 0 as C1approaches 0, as is also required by the flow equation. Extrapolation of DZ2to C2 = 0 (Figure 3) leads to in in 1.0 m H3P04, 5.0 X the values: 7.1 X 2.12m,3.8 x 10-6in4.0m,and2.5 X 10-6in6.0m. The approximate theoretical equations for ionic flows developed by GostingZ4are useful in interpreting the diffusion coefficients. Only the fint-order restrictions imposed on the four diffusion coefficients by the values of the limiting ionic mobilities and the requireTh.e Joumal of Phy%id Chemietrll
-
I .o,p
-4-
/
b
‘-r
& *RO
CONCN.,
I
/
a’
I
I
M, OF C a ( H 2 P O 4 I 2
Figure 3. Diffusion coefficientsfor H3P04 in the system HaPOrCa(HPOa)rHzO.
ment that the solution be everywhere electrically neutral are considered. It is assumed that the only ions present are H+, Ca2+,and H2P04-; their limiting ionic mobilities are, respectively, 349.82,26 59.50,26 and 33.26 For a system containing these ions in water, for which the experimental flow eq. 154 and 155 of ref. 21 are applicable, the “first-order theory” equations for the four diffusion coefficients can be written from eq. 168-172 of ref. 24. The Nernst limiting ionic diffu= 93.02 X 10“; = sion coefficients are: &t 15.834 X 10“; DHSO,- = 8.782 X These values substituted in the first-order theory equation yield calculated diffusion coefficients. From Table I11 it is seen that both the main-term coefficients Dll and DZ2are smaller than the limiting diffusion coefficients of their respective cations. Calculated values for the main diffusion coefficients are larger than the experimental values, but, except for D22 as a function of C1, the two sets of values are affected similarly by changes in concentration. (22) R. P. Wendt, J. Phys. Chem., 66, 1740 (1962). (23)I. J. O’Donnell and L. J. Gosting, “The Structure of Electrolytic Solutions,” W. J. Hamer, Ed., John Wiley and Sons, Inc., New York, N. Y.,1959,pp. 175-181. (24) L. J. Gosting, Advan. Protein. Chem., 11, 538 (1956). (25) D. A. MacInnes, “The Principles of Electrochemistry,” Reinhold Publishing Corp., New York, N. Y.,1939,p. 342. (26) C. M.Mason and J. B. Culvern, J. Am. Chem. SOC.,71, 2387 (1949).
DIFFUSION OF SOLUTIONS IN m SYSTEM H8P04a(HZOd)-Hz0
The relation between the calculated and the experimental values of the cross-term coefficients may be more significant. As with the main coefficients, the calculated values of the cross-term coefficients are larger in absolute magnitude than the experimental values, but the change of the experimental values with change in concentration seems especially important. The differences between the limiting diffusion coefficients of the respective cations and that of the anion H2P04- lead to large calculated valuea of D12and D21. Many of the experimental values of D12 are large, but smaller than the corresponding calculated values, whereas all the experimental values of Dzl are small; the experimental values of DI2change more with change in C1than do the calculated values. From the values given above, it is apparent that a of & + and &,*+ change in the effective value of &sor- with change in C1-as might be caused by a change in anion species with change in concentration-would result in a much larger change in D12 than in DZ1. This behavior of the cross-term coefficients-a striking feature of the experimental results-is shown in Figures 3 and 4. Change in anion mobility with change in C1 could explain this behavior, and several investigatorsz?28 report evidence for the existence of phosphate species that would be expected to be less mobile than H2P04-. It is noteworthy also that the absolute value of D12 as a function of C1is a minimum (zero) a t 3 m HsPOd, near the concentration (2.1 m) at which the time of re-
225
laxation in HsP04 solutions is a maximum2$-a harmonious relationship between interaction of solute flows and time of relaxation. The Onsager reciprocal relations have been shown to apply to diffusion of several ternary systems.@ For the system H8PO4-Ca(H2PO&-H20 the relations will be tested when the necesaary activities are available. These are forthcoming from vapor-pressure data and other measuremenkn The magnitude of the interaction of solute flows in the system HJ?OrCa(H2P04)rHz0 shows that where precise knowledge of the diffusional flows in multicomponent phosphate systems-such as soils and biological systems-is needed, the flows, or the pertinent diffusion coefficients, cannot be predicted reliably but must be measured. Acknowledgment. J. H. Christensen devised a method for eliminating the “fringe-skipping” error inherent in the Rayleigh interferometer.
Appendix Although the practice of obtaining C,for each Gouy pattern as the intercept a t 2, = 0 of a plot of Y,/e-fj’ is believed to be generally valid, it has been proved only for systems where values of I?+ and I?- are fairly small or where values of u-/rr+ are near unity.s For such systems, rapidly converging series expressions for fringe deviations have been developed (ref. 6, footnote 16; ref. 3, eq. 28-31). Here we derive a seriea expression for fringe deviations that converges rapidly for values of the parametens encountered within the system HaP04-Ca(H2P04)2H20. Values of I?+ and r- range from -1.3 to 1.3, and values of dZ/& fall as low as 0.5. An exact equation for the relative fringe deviation has been developed (eq. A-6 and A-8 of ref. 8).
The variable y must be expressed in terms of { so that a useful series can be developed. This expression ia obtained by expanding the numerator of the second
0.2 0.4 0.6 0.8 CONCN., M, OF Ca(H2P0& Figure 4. Diffusion coefficients for Ca(HSO& in the system HaPO,-Ca( HzPO4)rH*0.
1.0
(27) I(. L. Elmore, J. D. Hatfield, R. L. Dunn, and A. D. Jones, J . Phya. Chem., 69, 3520 (1965). (28) R. Ripan and C. Liteanu, A d . Rep. Popuhre Romiw, Bzcl. Stiint., A I , 387 (1949); M. Selveratnsm and M. Spiro, Trans. Faraday SOC.,61, 360 (1965); J. Zsako, Studk Univ. Victor Babes et Bolyai, 3, No.4, Ser. 1, No. 2, 69 (1958). (29) J. H. Christensen, A. J. Smith, R. B. Reed, and I(.L. Elmore, J. Chem. Eng. Data, in press. (30) P. J. Dunlop, J. Phys. Chem., 63, 612 (1959); P. J. Dunlop and L. J. Gosting, ibid., 63, 86 (1959); D. G. Miller, ibid., 63, 570 (1959); R. D. Wendt, ibid., 66, 1279 (1962); L. A. Woolf, D. G. Miller, and L. J. Gosting, J. Am. Chem. Soc., 84, 317 (1962).
Volum 70,Number I
Januaw 1866
EDWARDS, D m , HATFIELD, HUFFMAN, AND ELMORE
226
term of eq. A-1 as a Taylor series at y = making several substitutions. The expansion results in
r
6 and by
where p = l/Z/&+. Further, Taylor expansions of eq. A-3 lead to eq. 43 of ref. 5 and to eq. 72 of ref. 6, together with higher terms. (Note the substitutions in footnote 16 of ref. 6 to get eq. 72 of ref. 6.) The first three terms of eq. A-3 are
3=
where y1=
I
=
r+l/Q+e-'+m'
r
a
=
e-13
-
(I+ ~
n=l
--
6
I Jo
+ ~-&-e-*-m'
+
I n eq. A-2, replacing [ r + f ( G + y ) I'-f(fia)] by and in eq. A-1 replacing [ r + 6 + e - u t u 3 r - ~ e - c - u a by ] the right-hand side of eq. A-2, replacing the denominator of the second term of eq. A-1 yields by l / f i and y~by
f(r)
r
which on expansion become
+
V -Wr + f ( m A r-)
rf(4ZGr)ifl x
Taylor expansions of the bracketed terms show that eq. A 3 converges satisfactorily. These expansions lead to
If(s) - r+f(dQYAr>- r - f ( d Z G r ) l
-r+rn=2
=
+ (-l)fl~+n-i](l - p)" d B - r-(i - p ) ] ~ drn
f*[r-fl-l
[I
(A4
The result is the same aa that of Akeley and Gosting (ref. 3, eq. 28), but it is derived without restrictions on values of r-, I?+, and p. The series A-5 converged rapidly when values of rand I?+ were between - 1.3 and 1.3 and when values of p exceeded 0.5. These values included all those encountered in the system H3P04-Ca(HeP04)rH20. Finally, from eq. 28-31 of ref. 3, it is seen that the value of Crobtained as the intercept at 2, = 0 of a plot of Y,/e-r'388. 2 ;'' k valid for these systems.
