October 1955
INDUSTRIAL AND ENGINEERING CHEMISTRY
in bench scale data which indicate serious contamination, whereas in actual operation the concentration of contaminants does not exceed that which may be rendered ineffective b y the ever-present deposit of residual coke. LITERATURE CITED
(1) Alexander. Proc. Am. Petroleum Inst.. 27 (111).51 (1947). (2j Alexander and Shimp, Natl. Petroleum ’lVews, 36 ( 3 i ) , R-537 (1
944).
2157
(3) Berkhimer, Macwa, and Leum, Proc. Am. Petroleum. Inst., 27
(111),90 (1947). (4) Diakel Corp., “Physical, Chemical, and Catalytic Testing of Diakel Powdered Cracking Catalyst,” June 7, 1943. (5) Shankland and Schmitkons, Proc. Am. Petroleum Inst.,27 (III), 57 (1947). (6) Voorhies, IND. ENG.CHEM.,37, 318 (1945). RECEIVED for review October 15, 1954. ACCEPTED April 21, 1955. Division of Petroleum Chemistry, 126th Meeting ACS, New York, N . Y., 1954.
Fractional Precipitation or Crystallization- Systems EFFICIENCY OF FRACTIONATION E. F. JOY1
AND JOHN H. PAYNE, J R . ~ Mound Laboratory, Monsanto Chemical Co., Miamisburg, Ohio
T
HE precipitation of barium-radium chromate mixtures from homogeneous solution is a suitable method for concentrating radium because of the high distribution coefficient obtained and the ease with which the fraction precipitated can be controlled (6). Concentrating the radium present as a microcomponent requires repeated fractional precipitation in a systematic scheme. This paper deals with the method of selecting the most efficient fractionation scheme for a binary mixture of this general type. T o determine the “efficiency” of a fractionation step some quantitative measure of the separation achieved must be designated as a basis on which the “efficiency” is to be calculated. Consider the separation of B from a mixture with A where B is present in small concentrations. When the mixture is split into two fractions in which B is enriched in one fraction and depleted in the other, the amount of separation which has been achieved is related to the amount of B recovered in the enriched fraction and to the degree of enrichment which it has undergone. At the same time, the amount of B which is now in the depleted fraction must be discounted, taking into account the degree of depletion. The problem is one of balancing these two opposing factors to obtain a maximum separation. This problem has been solved by a thermodynamic approach involving the entropy of mixing. The entropy change for one fractionation step was found to be a function of the product of the yield of B and logarithm of the enrichment of B, summed up for the enriched and depleted fractions. The conditions for obtaining a maximum negative entropy change corresponding t o maximum separation have been determined, The “efficiency” of separation of a fractionation step is designated as the ratio of the entropy change for the fractionation step t o the maximum obtainable entropy change for the system being considered. The entropy change depends only on the initial and final states of the system and is independent of the mechanism of the separation. The “efficiency” of separation as used is concerned only with measuring the amount of separation and is not concerned with the means of making the separation. I n a continued fractionation, the mixing of fractions of differing composition results in an increase of entropy and this loss in efficiency of separation should be avoided. I t is important, therefore, to have a point in the scheme where the composition of the starting material is duplicated so that fresh additions can be made without loss of efficiency. Previous workers ( I , 7) have stated this need and have solved the problem for specific cases in an Present address, J. T. Baker Chemical Co., Phillipsburg, N. J. a Prevent address, Monsanto Chemical Co., Research Dept., Inorganic Chemicals Division, Dayton, Ohio. 1
empirical way. A more systematic approach t o the general case has been made and the mathematical relationships for obtaining repeating compositions have been developed. The efficiencies of fractionation of several workable systems were calculated from their entropy of separation and the one chosen was the simplest method for separating radium from barium. THEORY OF SYSTEMS WITH REPEATING FRACTIONS
A typical triangular fractionation scheme (8) is shown in Figure 1. Individual fractions are represented by the circles. The fractions are designated by a horizontal row number and a diagonal column letter, Consider the separation of two components, A and B, by fractional crystallization or fractional precipitation. If a constant fraction z of Component A is precipitated a t each operation and fractions are combined in the usual triangular fractionation scheme, the fraction of the original A which mill appear a t any point in the scheme is given by the terms of the binomial expansion [2
+ (1 - 211” = 1
(1)
as shown in Figure 1. I n similar fashion for Component B , where a different but constant fraction y is precipitated a t each step, the distribution of B in the various fractions will be given by the expansion [Y
+ (1 -
!/)In
=
1
(2)
Fractionation schemes are simplified when fractions of repeating composition recur a t regular intervals. The first fraction which can have the same composition as the original mixture is fraction 2b. I n this case every fraction will have the same composition as the fraction vertically above it in the scheme. The composition of fraction 2b will be the same as the original if the fraction of A which reaches that point in the scheme is the same as the fraction of B which also reaches that point, Le.,
241
- 2) = 2y(l - y)
(3)
This equation is satisfied by two solutions 2 = y
x = l - y
(4)
The first solution, z = y, is a trivial solution. Although the concentration of the 2b fraction repeats the original concentra-
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INDUSTRIAL AND ENGINEERING CHEMISTRY
tion, the end fractions also have the same composition and no separation of the two components is obtained. Therefore, z = y is the equation for the line-of-no-separation. This equation is plotted in Figure 2. The line x = y divides the diagram into two sections. Points above the line-of-no-separation represent enrichment of Component B in the precipitate. The enrichment ( 4 ) , expressed by the ratio of y t o z, is greater than 1.0 for any points in this region. Below the line, depletion of B, or enrichment of A , occurs in the precipitates.
Figure 1. Triangular fractionation s c h e m e
and 5d. The solutions for the equations of the operating lines are given in Table I. The operating lines plotted in Figure 2 are general for the fractionation of any pair of salts in which a constant fraction of each salt can be precipitated or crystallized. They represent all possible combinations of z and y which repeat the original composition within the first 5 rows of fractions. Repeating compositions in fractions beyond the fifth row ( n > 5 ) are not considered since in practice such systems become too complicated. The relative amounts of the two salts which will precipitate in any specific case is a function of the relative solubilities of the two salts, their isomorphous properties, and the fractionation conditions. I n the chloride fractional crystalIization of radium and barium mixtures, approximately 50% of the barium and 80% of the radium is crystallized a t each crystallization step ( 1 ) . These values, plotted in Figure 2 letting z equal the fraction of barium and y the fraction of radium crystallized, give the point 0.5, 0.8. This point falls close t o the operating curve for a repeating composition in fraction 3b, which is the fraction t o which original material is added in chloride crystallization systems in practice. When barium-radium bromide mixtures are fractionally crystallized, 33.3% of the barium will carry 83.1% of the radium a t each crystallization step ( 7 ) . The point in Figure 2 obtained from these values falls on the operating curve for fraction 5c. I n practice, original material is added t o this fraction. For a number of salts, including the barium-radium chromate mixtures (6), the reIationship between the amounts of the two components precipitated follows fairly closely the logarithmic equations proposed by Doerner and Hoskins ( 2 ) log (fraction of radium left in solution) = A log (fraction of barium left in solution)
+
The second solution, z y = 1, is of interest because it gives repeating compositions for fractions in the same vertical columns and a t the same time gives enrichment in the end fractions (Column a of the triangular scheme) of (y/z)m. This equation, also plotted in Figure 2, might be termed the operating line for a repeating composition in fraction '2b. The next fraction in the triangular scheme which can repeat the original composition is fraction 3b. This fraction will be the same as the original providing zZ(1
-
2) =
p(1 - y)
one solution of which is again z = y. the equation t o y2
a quadratic in y.
-
(1 - z)y
-
Dividing by y
(5)
- zreduces
(1 - z)2 = 0
(6)
2)2
where z and y equal the fraction of barium and radium precipitated, respectively. The value of the constant, h, depends on the precipitation conditions. For the purpose of illustration, a value of 7.213 was chosen for h, which approximates the value obtained for the fractional chromate precipitations a t 0' C. This Doerner
Table I.
(7)
- Z) =
y3(1
- y)
(8)
Values which satisfy this fourth degree equation were obtained by making the simplifying substitutions, y = cos2 e and 1 y = sin2 e. From these values an operating line for a repeating composition in fraction 4b was plotted in Figure 2. I n a similar manner, operating lines were obtained for repeating compositions in fractions 5b and 5c. Operating lines for Fractions 4d, 5d, and 5e also can be plotted since a complementary relationship exists between the points on curves 4 b and 44 5b and 5e, and 5c
-
4b
Y
5
The negative solution has no physical meaning since negative fractions of a component cannot be precipitated. The positive solution, plotted in Figure 2 , is the operating line for a repeating composition in fraction 3b. .4n operating line for fraction 3c also can be plotted, since a complementary relationship holds between the points on curves 3b and 3c. For a point x, y on curve 3b, the corresponding point on curve 3c will be (1 - z), (1 - Y). Fraction 4b will have the same composition as the original if z3(1
Solutions of Equations for Operating Lines
3b
+ 4(1 - z)z
(9)
Using the notation given in Figure 2 ,
This equation has two solutions
(1 - z) f d(1Y = 2
Vol. 47, No. 10
2:
5,
5b I4
5
Y 0.99420 0.9836 0.9638 0.9310 0.8798 0,7428 0.6655 0,4228
X
2/
0.30 0.25 0.98784 0.10 0.97026 0.99083 0.10 0.9685 0.40 0.20 0.9188 0,9478 0.35 0.25 0.9561 0 . 5 0 0.30 0.8543 0 . 4 0 0.9265 0.30 0.40 0.7794 0.60 0.875 0.9196 0.50 0.40 0.6947 0.8678 0.50 0.60 0.70 0.809 0.50 0.4941 0.7959 0.70 0.85 0.729 0.70 0.60 0.80 0,3740 0.632 0.6951 0.80 0.90 0.70 0,2310 0.5123 0,5419 0.90 0.98 0.90 0.80 .. .. ..... 0.354 0.4196 0.95 ..... 0.90 .. .. ..... 0,2443 0,2995 0.98 ..... 0.95 Solutions for operating lines for fractions 3c, 4d, 5e, and 5d were obtained from these solutions by complementary relationships g = 1 x and x = 1 Y.
-
-
Table 11. Repeating Fraction 5e 4d 3c 5d
a
Operating Data €or Various Fractionation Schemes, X = 7.213 A Precipitated, %" ( z x 100) 6.9 8.8
12.2 15.1
Yield of B Precipitated, 7%" (Y x 100) 40.2 48.4 60.5 60.1
80.0 20.0 2b 88.3 25.9 5C 92.6 30.2 3b 9 6.3 3 7 . 2 4b 97.9 42.5 5b Values determined graphically from Figure 2.
Enrichment of B / Fractionation Step (%//XI
2.f33
:4 .:5E8
4.00 3.41 3.06 2.59 2.30
INDUSTRIAL AND ENGINEERING CHEMISTRY
October 1955
2159
the net change in entropy would be zero. If the separation is not made reversibly the external system would undergo a n even greater increase in entropy so that the net change in entropy would be positive. The entropy of separation thus represents the minimum increase in entropy which the external system must undergo to effect the separation and it should be a quantitative measure of the separation which has been achieved. Equation 12 is valid for ideal solutions. Mixtures of barium and radium salts may be considered as nearly ideal solid solutions, because the elements are isomorphous, almost identical in chemical behavior, and radium is usually present in the mixture only as a microcomponent. E
-7.213 7
0
0.1
0.2 0.3
0.4 0.5
0.6
0.7 0.8 0.9
X (FRACTION OF A PRECIPITATED)
Figure 2.
6
Operating curves and points for systems with repeating composition
and Hoskins curve is plotted in Figure 2. The intersections of this curve with the operating curves are operating points for fractionation systems having fractions of regularly repeating composition. Schemes having repeating fractions could be built up b y precipitating the various percentages of Component A in accordance with the values of z a t these intersections. These are given in Table 11. For example, to obtain a repeating composition in fraction 5~~ one would precipitate 6.9% of Component A and would obtain a 40.2% yield of B in the precipitate, giving an enrichment of 5.84 for B in the precipitate. The enrichment of Component B for a single precipitation equal to y/z is greatest for small values of z,reaching a maximum value of X as z approaches zero (Figure 3). If it is desired to obtain B in high purity in a few steps, an operating point on the lower part of the Doerner and Hoskins curve, corresponding to smaller values of z,would be used. Under these conditions the recovery of B is comparatively low. By operating a t points higher on the distribution curve, higher yields of B are obtained, but at a somewhat lower enrichment. The maximum efficiency of separation will occur a t some intermediate point which can be determined from a thermodynamic consideration based on the entropy change on separating the two components.
5 \
\
I-
z
W
5
1
0
4
K
z
92 >IX
3
2
I
OA
? .0.9
0:2 0'.3 014 0'5 0'.6 0'.7 0'.8 X (FRACTION OF A PRECIPITATED)
Figure 3. Variation of enrichment with fraction precipitated for system showing logarithmic distribution
For a binary mixture of A moles of Component il and B moles of Component B , the entropy of separation into the pure components is
ENTROPY OF SEPARATION
The entropy of a system is a measure of the degree of randomness of the system. The spontaneous process of allowing two pure materials to mix results in an increase of entropy. For ideal solutions the increase in entropy is given b y the entropy of mixing (S), AS,,,
=
-XN,R In X , ,
=
- AS,,,
=
Z N , R In
B + B R In ,4___ +B
where A = A +B
X*
(11)
where iV%and X i are the number of moles and the mole fraction, respectively, of each component and R is the molar gas constant. For the reverse process of separating a mixture into its pure components, the entropy decrease should be a quantitative measure of the separation achieved. AS,,,
A A + B
A S = A R I n __-
X,
(12)
The entropy change depends only on the initial and final state and is not concerned with the mechanism of the separation. Horn-ever, if the separation were carried out reversibly, the external system would undergo an increase in entropy exactly equal to the decrease in entropy of the internal system so that
If A is large compared with B , two simplifying approximations A 1 = In can be made. I n the first term, In A + B B,A, where B / A is small compared with one. The approximation 1 In ___ In ( 1 - B/A) - B / A can be made. I n the 1 B/A second term, B can be neglected in comparison with A in the B A B. This gives denominator of In ~
+
+
+
B A
AS = - B R f B R l n -
(14)
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47,No. 10
Equation 19 becomes
Equation 20 shows a relationship between the entropy of fractionation and the concepts of yield and enrichment. The first term in the bracket represents yield, y, multiplied by the logarithm of the enrichment, y/x, for Component B in the precipitate; and the second term has a similar meaning for the solution. For any separation the terms will always be opposite in sign, positive for enrichment of B in the fraction, but negative for depletion. The conditions for maximum negative entropy change can be obtained by differentiating this equation with respect t o x and setting the derivative equal to zero. This gives
02
0:l
d.5
0.3 0:4
6.6
6.1 O h
0:s
1.0
X (FRACTION OF A PRECIPITATED)
Figure 4. Maximum entropy of separation for systems showing logarithmic distribution
At A : B = 100: 1 the error in using the approximate equation is
