EngFering pocess development
T. M. REED’ 111
AND
H. S.
MYERS
THE PENNSYLVANIA STATE COLLEGE, STATE COLLEGE, PA,
M
ULTIPLATE vapor-liquid equilibrium stills have been built and used in thepast few years with satisfactory remlts (1,g). Such stills may be used, for instance, to obtain an wtimate of the relative volatility in systems where difficulties are encountered in attaining accurate differences in composition of the liquid and vapor in single theoretical plate stills. Analytical difficulties might arise when the relative volatility is too close to unity to allow a reasonable composition change in one plate. Again a difference in some convenient physical property of the components may be too small to allow accurate analysis unless the changes in composition are allowed to accumulate over several equilibrium plates. There is, however, an unattractive feature associated with the data from such stills. A liquid and a vapor sample are taken several equilibrium plates apart in a multiplate equilibrium still. Under these circumstances the conventional liquid-vapor equilibrium curve for one theoretical plate is not immediately obtainable. The composition of the liquid in equilibrium with the vapor sample withdrawn and the composition of the vapor in equilibrium with the liquid sample withdrawn are not known because these samples are separated by more than one equilibrium plate. Any relative volatility calculated using the measured compositions must necessarily be an average over the composition range between the samples withdrawn. If the relative volatility is conetant over the whole composition range, the difficulty of matching a liquid composition with a vapor composition in equilibrium does not arise. For example, it was found (1)in an equilibrium still of two theoretical plates that the syst,em n-decane-trans-Decalin exhibits a t each of several total pressures a constant relative volatility over the whole composition range. From the value of the relative volatility at each pressure the corresponding equilibrium curve for one theoretical plate may be calculated by means of the expression
Y/(l
- Y ) = aX/(l - X )
reasonable accuracy. When such an approximation is inadequate, the conventional X-Y diagram for liquid-vapor equilibrium may be constructed from multiplate equilibrium still data by the methods discussed in this paper. ”HE TEREESAMPLE METHOD
The first method involves initially the sampling of the equilib rium still a t at least three separate equilibrium plates. The sample points must be so located with respect to one another that the number of equilibrium plates between successive sample points is a t least some minimum number dictated by the composition difference desired for analytical purposes. It is shown in a later section of this paper that only certain combinations of sampling point locations will yield the one-plate equilibrium curve by the three-sample method.
3-%ATE
0 X I LlQUIO COMPOSITION
Figure 1. Three-Sample fer Obtaining the True Liquid Equilibrium Curve Multiplate Equilibrium
(1)
where CY = relative volatility, X = mole fraction of the more volatile component in the liquid phaue, and Y = mole fraction of the more volatile component in the vapor phase in equilibrium with the liquid phase. From the rearrangement of the above relationship into Y = CYX/(l ax)
Method VaporUsing a Still
This method is illustrated with reference to Figure 1. The experimental data in this hypothetical case were obtained by sampling the liquid phase a t the ‘‘bottom” equilibrium plate-Le., the plate least rich in the more vohtile component; the vapor (or condensed vapor) of the third equilibrium plate from the bottom; and the vapor of the fifth equilibrium plate from the bottom. The five-plate “equilibrium” curve is the locus of experimental points, each of which has BS abscissa a bottomplate liquid composition and as ordinate the corresponding vapor composition from the fifth equilibrium plate. Each point located on the three-plate curve has as abscissa a bottom plate liquid composition and as ordinate the corresponding vapor composition from the third equilibrium plate. To obtain the complete curves,
x+
the vapor compositions in equilibrium with a series of liquid compositions from X = 0 to X = 1.0 may be calculated and plotted as the conventional liquid-vapor equilibriam diagram. If the change in composition over the several equilibrium plates iu the multiplate still is very small-i.e., when the relative volatility is near unity-the relative volatility calculated directly from the data may be assigned to the average composition with 1
‘tQUILI0RIUM”CU
Present address, University of Florida, Grtinesville, Fla.
914
INDUSTRIAL AND ENGINEERING CHEMISTRY
April 1952
the bottom plate composition is changed by addition of one of the components t o the still. The one-plate equilibrium diagram is then obtained by graphical subtraction in the following manner. Beginning a t some point A on the X = Y-line of Figure 1 the plates are stepped off in the conventional McCabe-Thiele method for each of the multiplate curves. One step on the three-plate curve equals three theoretical plates; one step on the five-plate curve represents five theoretical plates. On Figure 1, the first step on the five-plate curve ends a t point B . The &st two steps on the three-plate curve end a t point C. The step from A to B represents five theoretical plates and the steps from A to C represent six theoretical plates. The step from B to C then is the difference in liquid composition on adjacent theoretical plates. The intersection, D, of the horizontal line through C and the vertical line through B gives a point on the oneplate curve-i.e., a true equilibrium point. Similarly, point G is located as the difference between two fiveplate steps (10 theoretical plates) and three three-plate steps (9 theoretical plates) to give another point on the true equilibrium curve. This process can be extended and again repeated by beginning a t another point A t o give more points on the one-plate curve. In this manner the usual equilibrium X - Y diagram can be constructed from multiplate data obtained a t increments greater than one plate.
915
-
abscissa of A’, but rather at a liquid composition n 1 plates above the abscissa of A’, that is, a t point B‘. BD then represents one plate change beginning a t B‘. Thua, B D layed off along B’E beginning a t B’ gives a point D’ on the one-plate equilibrium curve. By an identical argument the points I , K , P , etc., are located on the one-plate ourve. The (n- 1)-plate curve is determined by the points, B , F , Q, R, S, eta., on Figure 2.
*/I 0
mRnoN OILUTE
C A L W L A ~ E O BY SOLUTION EOUATIONS
Figure 2. Two-Sample Obtaining the True Liquid Equilibrium Curve Multiplate Equilibrium
for THE TWO-SAMPLE METHOD
The second method of constructing the one-plate curve from multiplate measurements depends upon an accurate calculation or measiirement of a t least one point on the one-plate equilibrium curve. Figure 2 illustrates the general method wherein samples are taken n plates apart and an m-plate equilibrium curve is plotted. A single point A , or better a small portion 0 to A of the one-plate curve is obtained by some suitable method valid accurately a t low concentrations. Either end of the X scale may be used for this calculation or measurement. Beginning a t the abscissa of point A (or of any point on the calculated part of the true equilibrium eurve) on the X Y-line, the “plates” of the n-plate curve are stepped off. This is the series of steps represented by the dotdash lines on Figure 2. The step formed by the lengths of the vertical and horizontal lines between point A and the X = Y-line represents the change in composition over one theoretical plate. Beginning another series of n-plate steps a t the point A‘ of the horizontal projection of point A on the X = Y-line, a second series of steps (the dmhed lines) is obtained which differs from the first series by one theoretical equilibrium plate. In each n-plate step, e.g., HCB’, there are two intersections of the dashed and the dot-dash lines. One of these intersections in each step-that one on the vertical dashed line-gives a point on the (n-1)-plate curve. The other intersection in the same step-that one on the horizontal dashed line-gives a point on the one-plate curve or the true equilibrium curve. By starting the two series of n-plate steps a t other calculated, measured, or constructed points on the one-plate curve additional points on the one-plate curve may be constructed in the same manner. The argument showing that the intersections of the steps on the horizontal dashed lines give the one-plate curve follows. On Figure2 the distance AH is the change in vapor composition over one plate. Since HC is the change over n plates, HC - H A = AC is the change over ( n - 1)plates. However, the change AC begins one plate above H a t A’, so that the actual abscissa of the change AC is that corresponding t o the abscissa of A’. The length AC layed off from A’ along A D gives a point B which is a change of ( n - 1) plates. (This operation amounts to projecting AC on A‘D t o give A’B.) Since, then, A’B represents n- 1and A’D represents n plates, A’D - A’B = BD represents the change over one plate-Le., n-(n-1) = 1 plate. Thechange BD, however, does not occur a t a liquid composition represented by the
-
t
X. LlOUlD COMPOSITION
Method VaporUsing a Still
In addition to depending upon the reliability of the experimental data, the accuracy of the two-sample method is determined by the accuracy of the calculated point or few points required to translate the data into a one-plate curve. The accuracy of the results of both methods are, of course, dependent upon the accuracy of the mechanical construction used. It would be possible to obtain the one-plate curve by purely algebraic means only if the equations of the multiplate curves were known. SELECTION OF SAMPLING POINTS IN THE THREE-SAMPLE METHOD
The relative location of the three points a t which samples are withdrawn from the multiplate equilibrium still may not be chosen arbitrarily. Some combinations of sampling points locations will not yield the one-plate X Y curve by this method. The bottom-plate sample may be denoted by 0, the intermediately located sample by M , and the top-plate sample by N . The top plate is the sample richest in the more volatile constituent. These samples are each separated by a known number of theoretical equilibrium plates. The condition that the data yield the one-plate or true equilibrium curve is expressed by the relationship
-
N
-jM
= f l
(2)
in which M and N are the plate numbers as defined above, and j is an integer, such as 1, 2, 3, etc. The j value is a number which allows the useful combinations of M and N to be calculated. This relationship implicitly means that it must be possible with any given combination of N and M t o arrive a t a difference of one theoretical plate (thus, f1) by using in Equation 2 some integral multiple of the intermediate plate number M . For example, if the minimum difference between any two samples should be fixed a t 2 theoretical plates or greater, then M - N cannot be less than 2 and the M-plate sample must be a t least the second-theoretical-plate sample. Thus, making PA = 2, Equation 2 can be solved for N corresponding to various values of j . Then, by calculating the difference in theoretical plates between N and M for each j , the permissible combinations of M and N can be determined as those which give the required or a greater difference.
INDUSTRIAL AND ENGINEERING CHEMISTRY
916
This is illustrated in the following table for the above example, M = 2 and the minimum theoretical plate difference 6 = 2 ,M = 2 1
N 3 1
2
5
3
3
3
7 5
4
9
Etc.
7
6
1 1 3
1 5
3 7 5
This table shows that for this example where the minimum 6 is supposed to be 2, the following combinations (0,M , AT)of sampling points are permissible in so far as the table has been carried (0 refers to the bottom sample): 0, 2, 5 ; 0, 2, 7 ; and 0, 2, 9. -4 similar calculation using the intermediate sample plate as M = 3 and 6 = 2 gives the following permissible combination of sampling points for j = 2 and 3: 0, 3, 5; 0, 3, 7 ; 0, 3, 8 ; 0, 3, 10. Again, taking the intermediate sample at the third theoretical equilibrium plate from the bottom sample ( M = 3),and requiring the minimum 6 to be equal to 3 instead of 2, the following combinations are permissible for j = 2 or 3: 0 , 3, 7 ; 0, 3, 8; 0, 3, 10.
Higher values of j greater than 3 required more than 8 theoretical plates in the multiplate still. An examination of the permissible groups 0, M , N above reveals that the numbers in a group are prime with respect to one another. That is, M and N in any one permissible group do not contain a common integer factor, and M is not divisible by N to give an integer, and vice versa. For example, the groups 0, 2, 4 will not yield the one-plate curve because 2 is a factor of 4. Similarly, for 0, 2, 6; 0, 3, 6; and 0, 4, 6. In this last instance, although 4 and 6 are not integer multiples of one another, they both do contain the common factor 2. This fact makes the group 0,4,6 unsatisfactory. The plate number six, the sixth theoretical plate from the bottom sample, has an integer factor in common with all numbers between 0 and 10. Thus, a multiplate still of less than eleven theoretical plates should not be sampled a t the sixth theoretical plate when the data are to be treated by the three-sample method. LITERATURE CITED
(1) Fenske, M. R., Myers, H. S., and Quiggle, D., IND.ENQ.CHEM., 42, 649 (1950). (2) Herring, J. P., M.S. thesis, Pennsylvania State College, 1948. RBCEIVED for review January 25, 1851.
ACCEPTED November lQ, 1951.
of Ions in Aaueous
Engineering p ","c:s -, development
Vol. 44, No. 4
I I
Solution with Glass STUDIES WITH RADIOACTIVE TRACERS ARTHUR 0. L O N G AND JOHN E. WILLARD UNIVERSITY
OF W I S C O N S I N , M A D I S O N , WIS.
I
T HAS been shown earlier (11)that radioactive tracers afford a sensitive and relatively rapid means for studying the sorption of ions from solution by surfaces. This exploratory work on techniques and on the sorption and desorption of sodium and cesium ions on glass has now been extended. Particular emphasis has been placed on experiments designed to determine whether the sorption process is an ion exchange process uncomplicated by other processes. I n this work, a8 in the previous investigations, the specimens have been glass squares, 1 inch on a side, cut from soda-lime-silica microscope slides (11). Except where otherwise specified all of the specimens were scrubbed under water, rinsed with carbon tetrachloride, and moderately flamed before use. The radioactive-counting techniques were similar to those previously described (11). Results are given in terms of monolayers sorbed, the monolayer being arbitrarily defined as the number of atoms required' to cover the macrosurface area of the sample if each ion covers an area equal to the square of its ionic diameter. SIMULTANEOUS SORPTION AND DESORPTION O F SODIUM IONS
I n order to observe the rates of simultaneous sorption and desorption of sodium ions, glass specimens were tagged by immersion in a solution of sodium ions tagged with Na2*(15-hour halflife), rinsed, and immersed in a solution of the same concentration tagged with Na22 (3-year half-life). The amount of NaZ4sorbed just prior to immersion in the Na22 solution mas determined by drying and counting one of the samples a t this time. The sum of the Na22 sorbed and the Na24 remaining after immersion in the
Na22 solution was determined from the total count due to the two isotopes when each sample was removed from the Na22 solution. The NaZ2was determined from the residual count after the Na24 had decayed. Corrections were made for Na24 decay and for the long-lived impurity (11) which was found on the Ns24 control sample and was assumed to remain in equal quantity on the samples immersed in the Na22 solutions. The results of such teste made for different times of immersion, different temperatures, and different pH's suggest a t first sight that the rates of simultaneous sorption and desorption of sodium ion are qualitatively similar (Figure 1). In each case the initial rate of desorption at 72" 6. appears to be greater than the corresponding rate of sorption. After this initial interval the sum of the monolayers represented by the sorbing and desorbing curves remains approximately constant. This constancy does not, however, indicate that sodium ions were entering and leaving the surface at equal rates. The plotted values for monolayers of Na24 are calculated on the assumption that the specific activity (disintegrationsper minute per milligram of sodium) of this isotope during desorption was the same as before it was sorbed, Ghereas it quite certainly had been decreased by inactive sodium in the glass. If such dilution occurred, the rate of departure of sodium from the glass was greater than indicated by Figure 1 and significantly greater than the rate of sorption, indicating a continual net loss of sodium from the glass. LEACHING O F NEUTRON-IRRADIATED GLASS SAMPLE§
I t may be calculated that a 1.63-gram glass square of the type used in this work, bombarded with a thermal neutron flux of 1012
