THE DIFFUSION OF COPPER SULFATE I N AQUEOUS SOLUTIONS OF SULFURIC ACID A. F. W. COLE AND A. R. GORDON Chemistry Department, University of Toronto, Toronto, Canada Received August, 19%
For an electrochemical problem now being investigated in this laboratory, it was necessary to know the diffusion constant of copper sulfate in aqueous solution in the presence of varying amounts of sulfuric acid. A search of the literature showed that the plaints of other investigators as to the meagerness and uncertainty of the diffusion data would have been justified in the case of copper sulfate also. We have therefore determined the diffusion constant of copper sulfate in aqueous solutions of sulfuric acid, and the results are recorded here. The method was that originally developed by Northrup and Anson (4) and since that time used extensively by McBain and his associates (3); since it has already been discussed adequately in the literature, no lengthy description is needed. In brief, it consists in enclosing in a glass cell of known volume a solution of the material whose diffusion constant is desired; the base of the cell is a sintered glass diaphragm whose pores are sufficiently small to prevent streaming of the solution as a whole, and at the same time are large enough to permit diffusion to take place unhindered. The cell is suspended with the diaphragm just dipping below the surface of an equal volume of pure solvent, and after allowing diffusion to proceed for a suitable length of time, the amount of solute that has diffused through the diaphragm is determined; from this it is possible to compute the diffusion constant, provided the cell has previously been calibrated with a substance of known diffusion constant. It has been found desirable, after filling the cell with solution, to carry out a preliminary diffusion to ensure a more or less linear drop of concentration in the diaphragm before transferring the cell to the sample of solvent into which the fkal diffusion is to take place. Under theseconditions, it can easily be shown, on the assumption that Fick's diffusion law is obeyed, that to a first approximation
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A. E”. W. COLE AND A. R. GORDON
where co is the initial concentration inside the cell in equivalents per liter, c’ and c” are the concentrations of the inner and outer solutions after the diffusion has proceeded for t seconds, A and I are the effective crosssectional area and thickness of the diaphragm, and V is the volume of both the inner and outer solutions. Hence
Bkt = log (c’
+ c”)
- log (c’
-
c”)
(2)
where @, the “cell constant,” is 28/2.3026 IV, and can be determined once and for all for the cell by diffusion measurements with a substance of known k. The use of equations 1 and 2 involves the assumption (in addition to that implicit in Fick’s law) that the concentrations of the inner and outer solutions are changing so slowly that a steady state is set up in the diaphragm, Le., that there is a linear drop of concentration throughout the diaphragm at all stages of the experiment. Actually, as Barnes (1) has shown, this is not necessarily the case, but he finds from a rigorous solution of the problem, that to a high order of approximation the only change introduced in equations 1 and 2 is that p becomes (2A/2.3026IV) X (1 - X/6), where X is the ratio of the volume of the liquid in the diaphragm to that in the cel!. Since this revised cell constant depends only on the geometry of the apparatus and is determined automatically by the calibration, no error is introduced by this cause in equation 2. The assumption of Fick’s law, however, implying that k is independent of c, is a more serious matter. If k does depend on the concentration, the concentration gradient bc/bz will vary from top to bottom of the diffusion layer even when a steady state has been attained in the diaphragm. However, on the assumption that k = ko (1 - ac), which fortunately covers the case of copper sulfate, it is easy to allow for this. At any cross section of the diaphragm, the total amount diffusing in a time dt is -kA(bc/bx)dt, which, for a steady state in the diaphragm, is constant from the upper surface of the diffusion layer (x = 0) to the lower surface (x = I ) . Assume an “effective” diffusion constant k’, independent of c, which will cause the same amount of solute to diffuse for the same total drop in concentration across the diaphragm. Then
Simplifying, multiplying through by dz, and integrating between the limits = 0 and x = 1,
x
DIFFUSION OF COPPER SULFATE
735
Hence
k’ = ko
- koa(c’
+ c”)/2
+
but (c’ c”) is constant throughout the experiment. Therefore the “effective” diffusion constant, k‘, computed by equation 2 is the true (differential) diffusion constant for the average of the inside and outside concentrations, or (what is the same thing) for one-half the initial inside concentration. The cells used were similar to those employed by McBain,-of Jena glass with G4 pore size and of about 100-cc. capacity. Their volumes were determined by weighing the cell with liquid just to the top of the diaphragm, and then weighing again when full; the volumes so determined could be checked to 0.1 cc. The liquid in the outer compartments (small beakers of 125-cc. capacity) was measured by specially calibrated pipets. During a measurement, each cell with its attendant beaker was suspended in an individual glass air bath immersed in a water thermostat electrically controlled to f0.02”C. The cells and beakers were carried by a bridge which had no connection with the thermostat and which was supported on sponge rubber blocks to minimize vibration; each cell was hung from a wire suspension to ensure that the diaphragm was horizontal, and this suspension and the support for the beaker forming the outside compartment were attached to ebonite rods to avoid any transfer of heat from the outside air to the solutions.‘ The top of each air bath was closed with a slotted fiber disk covered with absorbent cotton, and as an additional precaution against evaporation from the outside compartment, a few cubic centimeters of a solution roughly isotonic with that in the outer compartment were placed in the bottom of each air bath. The cell constants were determined by calibration at 20°C. with N/10 potassium chloride solution, whose diffusion constant has been found by Cohen and Bruins (2) to be 1.448 cm.2 per day = 1.676 X 10-6 cm.* per second; successive calibrations for a cell gave values of B differing in general by less than one fifth of 1per cent. The acid copper sulfate solutions with which the cells were initially filled were made up from recrystallized B.D.H. analytical reagent copper sulfate, a stock solution of C.P. sulfuric acid, and boiled conductivity water; the sulfuric acid solution which was to be placed in the outer compartment was always made up to the same volume concentration of acid as the solution to be used in the cell. After a preliminary diffusion to set up a steady 1 We have found that to obtain consistent results it was necessary to eliminate vibration as far 88 possible, and also to control temperature somewhat more carefully than has been suggested by other investigators; a fluctuating temperature produces a pumping action of the solution in the cell which tends to break up the diffusion layer in the diaphragm.
T E S~ O U ~ A LOF PBYSICAL CHDYLIIYI.RY,VOL.
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A. F. W. COLE A N D A. R. GORDON
state in the diaphragm (usually with copper sulfate solutions this required ten to twelve hours), the cell and beaker (now containing a fresh sample of TABLE 1 Data for run N o . 176 Acid concentration = 3.6 equivalents per liter; T t = 125.62 hours = 4.522 X lo6 see.
1
CELL
Cell Cell Cell Cell
I1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Equivalent H 8 0 4 per liter . . . . . . . . . . . . . . . . . T"C . . . . . . . . . . . . . . . . . . . 10e.ko. . . . . . . . . . . . . . . . . 106.koa. . . . . . . . . . . . . . . .
I
0
0.0777 0.0826 0.0549 0.0654
0 0.005 0.10 18 18 10 6.28 6.28 4.97 2.56 2.56 2.001
1
c'
0.5208 0.5162 0.5496 0.5379
0.10 18 6.28 1.64
= 18.00"C.;
e*
0.1016 0.1072 0.0763 0.0883
0.10 0 . 5 1.0 18 18 25 7.81 6.24 6.01 2.70 0.94 0.821
1
kX108
4.89 4.90 4.88 4.87
3 . 6 7.6 18 18 5.02 3.66 0.32 0.12
I 0.005 N X
0
0.1 N 0.5 N 1.0 N
a
7.8 N
A 4.0
3.6 N
a*' 360
L
F - 0 2
c
e 015
m Equivdenh
o(l
C"S0,
o(5 per /me.
016
07
FIQ.1. The diffusion of copper sulfate in aqueous solutions of sulfuric acid a t 18°C.
the acid solution) were left in the thermostat for from five to eight days; at the end of this time, the cell was removed from the beaker, and both solutions were analyzed electrolytically for copper. As an example, the
DIFFUSION OF COPPER SULFATE
737
data for one such run, selected at random, are summarized in table 1. Thus from the table, the diffusion constant of copper sulfate in a solution containing in 1 liter 3.6 equivalents of sulfuric acid and 0.31 equivalent of copper sulfate is 4.89 X per second. The results of a large number of such measurements, all at 18’C., are shown in figure 1; each point is the average of four measurements similar to those recorded in table 1. It is evident from the graph that for a given acid concentration and for the range of copper concentrations used, k is a linear function of c to a close approximation. Table 2 gives the values of ko and of koa for various strengths of acid, and also for various temperatures for one acid concentration. It is evident from the table that for solutions extremely dilute in copper sulfate the diffusion constant decreases in a more or less linear manner with increasing acid concentration. Since the forces retarding diffusion are usually assumed to be proportional to the viscosity, one would expect that the product k07 for a given temperature would be roughly constant, and this is very approximately the case.2 I n the more concentrated solutions, however, increase in acid concentration tends to decrease the dependence of k on the copper concentration, and at the moment there does not seem to be any plausible explanation of this. We have also carried out a few experiments to measure any diffusion of the sulfuric acid during an experiment. Naturally this will only be prominent if the acid concentration is relatively small, and the difficulties of analysis (barium sulfate precipitation in the presence of a copper salt) are considerable. However, our results show, as might be expected, a definite diffusion of the acid from the outer solution to the inner, i.e., against the copper sulfate; thus in one experiment (CO = 1.6312, c’ = 1.3200, c” = 0.3122 equivalent of copper sulfate per liter) with 0.1 N sulfuric acid as solvent, it.was found after one hundred and ninety-two hours diffusion that the inner solution was 0.014 equivalent of sulfuric acid per liter stronger than the outer. On the other hand, with relatively high concentrations of acid (1 N or more), there is evidence of a slight but definite diffusion of acid with the copper; this effect is now being investigated in this laboratory. REFERENCES (1) B A R N E s : Physics 6,4 (1934). (2) CORENAND BRUINS:Z. physik. Chem. 103,349 (1923);113, 157 (1924). (3) MCBAINAND LIU: J. Am. Chem. SOC.63,59 (1931). MCBAINAND DAWSON: J. Am. Chem. SOC.66, 52, 1021 (1934); Proc. Roy. SOC. London 148A,32 (1935). (4)NORTRRUP AND ANSON: J. Gen. Physiol. 12,543 (1929). The product k07 increases about 15 per cent when the acid conrentration increases fromO.l Nto7.6 N .
.
