Rate of Absorption of Carbon Dioxide Effect of Concentration and Viscosity of Caustic Solutions LAURENB. HITCHCOCK, University of
Virginia, Charlottesville, Va.
The initial (steady state) rate of absorption of 7 normal and of potassium hydroxide u p to 14 carbon dioxide by caustic solutions is studied as a normal at 30” C . The viscosily of these hydroxide function of hydroxide concentration and liquid vis- solutions, hitherto unknown oner lhis concentration cosity. Consideration of previous work based on range at 30” C., is determined. Equations of the the theory of the multizone liquid film suggests lhat: form gicen above are obtained which satisfactorily reproduce the experimental results over the entire range, using the term ea=as the viscosity function f(z) in that equation. The quantity ci is assumed for the where d V / A d e represenls initial current density, present to be constant at 0.06 gram equivalent per ci and c8 are interfacial and main-body concentra- liter. tions, respectively, of the’ reactants, and z is viscosity. The results are in agreement with qualitalive The initial (steady state) rate of absorption of conclusions already reported by a number of indepure carbon dioxide at a pressure of one atmosphere pendent observers but not previously correlated, and is measured f o r solutions of sodium hydroxide u p to point to a predominantly diffusional mechanism. N THE absorption of a gas by a liquid, factors are involved common to all types of heterogeneous reactions such as distillation, crystallization, solution, etc., where phase boundaries separate the reactants and fluid phases. Such processes are further distinguished by the presence of an interfacial layer where viscous motion exists, approaching stagnation a t the interface itself and depending almost entirely upon diffusion as a means of material transfer. These interfacial layers have frequently been designated as films, although this name suggests a definite thickness, and to some readers, as pointed out by Taylor (15),a dimension of molecular proportions. Actually the thickness, or depth, of the interfacial layer in which viscous motion persists is ordinarily unknown; it is supposed to vary greatly under various circumstances and probably lies between 0.01 and 0.5 mm. in general. The concept of this interfacial layer or “film” is extremely valuable, however, in spite of the fact that the exact thickness may not be known as a rule, for it calls attention to the importance of diffusion in heterogeneous reactions and to variables upon which diffusion may depend. Various types of heterogeneous reactions have been studied, but in the majority of cases so many variables are involved that it is impossible to deduce any one theory of the reaction mechanism to the exclusion of others. Payne and Dodge ( I S ) indicate the extent to which complicating factors may enter in the case of gas absorption by a liquid containing a solute which reacts irreversibly in two stages. Eight chemical reactions are suggested as possible though not all independent. Diffusion of dissolved gas and of the several ions require consideration. Observed absorption rates, they point out, are probably the net result of the combination of various diffusion rates and chemical reaction rates, and an attempt to postulate any definite mechanism seems impracticable in the light of the data available. In the more general case, however, where no irreversible chemical reaction is involved, a physical diffusion theory goes far toward interpreting results. The LeTvis and Whitman (8) “two-film” theory states that the rate of gas absorption per unit area of interface, dW/Ade, is proportional to the concentration difference of solute gas (1) between the main body of gas and the main body of liquid, (2) the main
body of gas and the interface, or (3) the interface and the main body of liquid. The proportionality constants are designated, respectively, as over-all coefficients, gas film coefficients, or liquid film coefficients. I n each case the concentration difference must be expressed in consistent units. The over-all coefficient may be evaluated in terms of liquid concentrations or partial pressures; in either case the expression of one concentration is required in terms of its equivalent in the desired units with the aid of equilibrium data :
That a simple equation of this form should fit all cases of gas absorption is unreasonable to expect. That it does apply satisfactorily to certain cases where conditions render some of the possible variables inoperative, indicates its usefulness and invites further development. Hanks and McAdams (3) have already shown, for example, that in the case of a very soluble gas the gas film coefficient, k ~ really , includes the factors of diffusivity, film thickness, and partial pressure of inert gas from which the soluble gas is being absorbed. Since film thickness 2, exactly as in heat transmission, is a function of the Reynolds’ number for a given shape factor, the gas film coefficient is revealed in this case as:
Similarly it has long been recognized that the liquid film coefficient contains the diffusion coefficient and the film thickness: AL = dL/XL
(2.4)
Other investigators have found that the simple Lewis and Whitman equation could be broken down into various components according to conditions. I n view of the excellent bibliographies contained in the recent papers referred to above, it is unnecessary to duplicate at this time the review of the subject of gas absorption. The case of gas absorption by a liquid containing a solute which reacts irreversibly, either in one or in two stages, how-
1158
November, 1934
ISDUSTRIAL AND ENGINEERING CHEMISTRY
ever, presents certain novel aspects for which the Lewis and mi^^^^ t,heory seems at first glance to be inadequate, as a process is of suggested by P a w and Dodge. That considerable industrial importance, as exemplified by the absorption of carbon dioxide by a&aline solutions, is evidented by the papers in recent years devoted to this subject, of which those by Hart% Baker, and Purcell (41, Weber and Nilsson ( I G ) , Whitman and Davis ( I 7 ) , and Williamson and Mathews (18) are representative, in addition to the work of payne and ~~d~~ already ~ ~ n,ith the ~ more purely theoretical aspects of the rate of absorption of carbon d i o x i d e b y alkaline solutions, papers by Davis VAC and Crandall ( I A ) ,Ledig and Weaver ( 7 ) , Hatta ( 5 ) , and Mitsukuri (11) have served to arouse general interest in the fundamental aspects of the problem as well. Weber a n d K i l s s o n (16) and Hatta ( 5 ) a p p l i e d t h e Lewis and Whitman two-film theory to the absorption of carbon dioxide from air by caustic solution, visualizing a m u l t i z o n e l i q u i d film in which rapid chemical reactions occur. H a t t a proposed the equation:
1159
suggested by Payne and Dodge report apparent "delay" reactions between carbon dioxide and potassium hydroxide, between the solution of carbon dioxide and the appearance of bicarbonate ions, etc., and certainly seem to justify the raising of the general question by them. Careful consideration of the references cited indicate that experimental conditions were not comparable with and there is no those used by Davis and CrandalL specific evidence that the chemical reactions discussed in those references are involved in the batch absorption of carbon dioxide by solutions of caustic soda or caustic potash at constant temperature and under constant gas pressure. As PaYne and Dodge knows exactly what chemical reactions l bepoint i out, ~no one may taking place, and~ the rates of unknown reactions must remain equally obscure.
n
where CY is a factor r e l a t i n g diffusion rates of the various ions and H i s the s o l u b i l i t y coefficient.
X
FIGURE 1.
While Equation 3 was applied to the progress of a batch absorption in the presence of inert gas, it reduces to
(AE ) = kL(CYC8 + Ci) d 0 initial
(4)
for the initial rate of absorption of a pure gas. This was later used by Davis and Crandall who assumed a to be unity. The term dW/Ade is the instantaneous rate of absorption in grams of carbon dioxide absorbed per minute per sq. cm. of interface; Ci and c s are the concentrations a t the interface and in the main body of the solution, respectively, expressed in gram equivalents per liter; k~ is the liquid film absorption coefficient. Davis and Crandall found that the relationship appears to be valid for dilute alkaline solutions but falls down a t concentrations above 1.O normal. These investigators recognized that Equation 4 is only qualitative and cite four factors which it neglects : (1) The time required for the completion of the chemical reactions which are assumed t o be instantaneous with respect t o diffusion rates; ( 2 ) the effects of the reaction products on the solubility of the gas; (3) the effects of the reaction products on the effective thickness of the stationary film and o n the diffusivities of the solutes; (4)the heat produced in the film by the reactions. (It must also flow away by diffusion since it is assumed convection plays a minor part in the film.)
BATCH
ABSORPTIONAppAR.4TUS
(2) The effects of the reaction products on the solubility of the gas cannot be important, because the interface concentration ci is itself relatively unimportant in Equation 4, except for a very
dilute solution. (3) Since only initial absorption rates are under consideration here. the film thickness and its diffusivity should not be materially affected by the presence of reaction- products at the start. (4) The effect of heat produced in the film by the reactions may be minimized by an efficient thermostat and design of apparatus. Since the thickness of the liquid film should depend on the Reynolds number, the following equation suggests itself:
The coefficient k~ of Equation 4 is replaced by a term including the diffusivity &, the typical dimension D , a power function of the Reynolds number ( D U S / z ) , and a proportionality constant k. In a given apparatus a t given liquid velocity, D and U mould be constant and could be included in a new constant k'. The density is a function of concentration and therefore of viscosity. The term d~ (S/z)" may therefore be expressed as some function of viscosity, and Equation 5 takes the following form:
Consider these points in the order listed: (1) That inorganic chemical reactions of the type here involved should have rates of the same order of magnitude as the diffusional rates would seem inconsistent with experience. The literature dealing with the rates of some of the chemical reactions
One further modification may still be made in Equation 6, for Davis and Crandall indicated that the interfacial concentration of carbon dioxide, ci, was some decreasing function of solute concentration cs. The quantity cz is relatively
1160
I I I A L A N D E ~ ( : I N E l ~ : i < I N GC N I S M I S T I < Y
Vol. 26, No. 11
small, Iiowever, arid f i x the presciit it will be assuiiied as coil. :Lgit:Ltioii wit~hoiit.produeinp dcform;rtion of the aurfaco. The stant, lyie < ) f l)ri,i,ary interest is if C(lliation (i I"."l>"ller and rod 1 L K l 1 m 1 ed o n :I Fhxft of 16-g:~gsstsirrless i..b 5u . b.th .antially correct, the deviatioiis of Daris aid Cran- stecl wire cnclused i n :L pi tube filled with a column of merdall's Equation 4 at conccntrations above 1.0 noniial are to a riseosity 8s sliggested by payne Dodge. EXPEIUM EN'PPL bf'ORK
Siiice the point a t issue is the relationship het.weeri the rate of absorption of carlmn dioxide, the conceiit~mtiimof sodium or potassium hydroxide, a n d t h e viscosit,y of the aqueoiis soliitioii, it must Ire emliliasiaed that the rate of absorption i n which i n t e r e s t centers at this tirue is tlie rate into a carhonate-free soliition. I n a s m u c h as carlionate is it product of t h e a b s o r p t i o n process, the rate of absorption under discussion becomes tlie initial rate into pure hydroxide solutions. A sharp distinction m i s t be drawn, however, between t h e instantaneous initial rate as stiidied by 1,edig and Weaver, which has an abnormal value for a fraction of a second aftcr carbon dioxide first encounters fresh solution, and vrliat rvill he called hereafter the "practical" initial rate, as defined by Ilavis and Crandall by the term "steady state." In tlie lmguage of the Lewis and Whitinan theory, coiiceiitrrition gradients have been establidied when the practical initial rate of absorption is realized. Briefly, the experiinental procediire developed is to me:rsure the time intervals over which successive incremer1t.s of pure gas are absorbed by stirred caustic solutions of known eoncentration and surface area illidor constant prmsure and constant temperature. Each increment is then plotted a t the middle of its time interval with respect to a contiiiuoiis tirue axis, and the points are found to fall on a straight line of negative slope. This line is extrapolated to zero time to give the practical initial rate of absorption. A series of experiments is performed in which the caustic concentration is varied progressively, yielding a ciirve (Figure F) wiiiclr shows initial rate as a function of initial hydroxide concciitration. These results were obtained at a tenlperatiire of 30'' C . Viscosity data for caustic solutions were available only over a very limited temperature and concentration range. A second phase of the experimental work therefore has been t.he determination of these values a t tlie temperatures and concentrations required by the present investig:itioii. APPARATUS. Figure 1 is a sketch of the apparatus, and Figure 2 sliows the essential portions:
,vhiell oliemtcs t.lre hoistj,,g N , tl,rough supersensitive mcreury rehy The net, height d the mercury column in E i s adjusted so t.hci,t,, nhen contact is made i n t h e left r mi, t h e h e i g h t , corrected t o 0" C., will corres ond to
B pressurc up760 mm. or other desired operating pressure in the apparatus, regwdless of the existing barometer. Motor N operxt,es the hoisting winch, P, through gear train 0 adjusted t o rneet the rate of abmrption in n givcn experimerrt with t h e least praeticablo deviatian. Motor L provides a n independent drive for winch P to provide rapid lowering of holbs ?.I" and M C when rech:rrging tho gw b"rcts.
hlnnomntcr 1) c i i ni,1es t h e op"n,t,"r t u follow thc ire in the absorption chamber during cvtmizitiuu thio, tie 120' three-brrrnch sicrpcock, Curbon dioxide gas is exp:mded from a cylinder through n preheisting coil t,o IL roservoii, U , irnmorsed in
X.
a sepamle i,he,mout:~t(not. .;howxi) in order t,o sup ly the gas at suhstarrtinlly opcmting tem eltilure iihen needo$. For certain sources of gns, it was d e s i A e to pass the gas through dryiug tuhes, Q , filled with trihydrite to remove traces
t%iA
of muistiire and oil. Absorpt,ion tuhe A and gas burets B and C were enclosed in i d c r jrmkets thningh \iliieh wtiler circulated in series with B thermostat, J . This was rniitrolled by n mercury thermoregulator of tho hydl.ogrn-s,tmosphere-Pealed t.ype mid supersensitive relay so ihzt Imth temperature measured hy thermometer l'* :It :L point close to the liquid surface was muintained within 0.02" Ttre,nri,mirtera u s 4 \vere cdibinted by the 13uretiu of Standnrda n t the st,art.of t,his invostig;rt.ion,sed ice points wem checked lociilly. They WPPC grtrdusted in 0.1" C. and read with a lens
c.
tion of t.hese intcrvals with respect to a continuous time reoord snt,isfactory double stopwatch system is clock simultaneously with the making of a t,hn cathetometer is also difficult. An acAn absorpbion tube, A , consisting of a glass cylindcr ( ~ ~ j > i x o x i -cessory was thrreforc developed for which t,he name "photomately 5 x 28 em.) is supplied Ivith carbon dioxide f n m grndnchmnogrnph" is proposed, and which may prow iiseiul in other ated burets B and C by meroury displrrcement. Thr buret,s invest.ig:Ltions as it is simple, accurate within the human revary in capacity nocording to the rate at which gas is rcquircd nctian .t i m e of appmrimntely 0.2 second, m d relatively inby a given solution. C is 100 cc., B is 300 CP., :ind it t.hird ~ X J I ~ ~ S L Y AP . motion picturc eamcra, 2, is mounted vert,ically brirct (not shown) has u capacity of one liter, gmdo:ittxl in 200- and rigidly i n n framework over an accurate timeniece. Y . and ce. increments. Absorption chamber A is provided witli ti stirrer, P,opw~ted from below so that during nn experimerit the liquid in the duhe ~~~.~~ ... m y be agititted without disturbing the surface in :my m y . I t onc jewel chronoketer, adjusted for 30 days by a wabchmakor Eo was iound that a speed of 80 r. p. m. gave definite over-:i/l uniform give R maximum variation of *I second in 24 hours. The 0~~~
~
~~~
~
I N D U S T R I A L A N D E N G I N E E R I - NG C H E M I S T R Y
1162
re lacement of the observers, failed to reveal any constant errors d e n check runs were made at the same concentrations.
TREATMENT OF RESULTS Results of a sample absorption experiment (the first with potassium hydroxide) are given in Table I and shown in Figure 4. The usefulness of the experimental method developed depends in part upon the fact that the rate of absorption into caustic solutions is a linear function of the time for the first 20 minutes and usually for the first hour. Successive increments of gas volume may therefore be plotted a t the mean of the initial and final time reading for the given increment without introducing error. Where the plot of rate against time develops more curvature, smaller volume increments may be taken, so that for all practical purposes one obtains in this way instantaneous absorption rates from a simple record of buret readings us. time.
For example, in charge A, Table I, time was recorded as 4 :08 :38 as the mercury level rose to point 50 on the gas buret; time was again read a t 4: 11: 15 a t point 70. Since zero time was 4:00:00, these time readings may be expressed for convenience as 8.63 and 11.25 minutes, respectively, or a A 0 of 2.62 minutes. The calibration curve shows a A V of 58.87 cc. between points 50 and 70. The interfacial area is 20.72 sq. cm.; hence AV/AAe = (58.87)/(20.72)(2.62) = 1.084 cc.
TABLEI. ABSORPTIONOF CARBONDIOXIDEBY POTASSIUM HYDROXIDE Run 29 Initial concn. (as total alkali) Initial concn. RE KOH Vol. of soln. taken, cc. Temp. O C. Sourcel of C O ~ Source of KOH Speed of stirring, r. p. m. Area of interface. 6q. cm.
GAS BGRET
Av
Cc.
..
10 20 30 50 70 90
~~
PHOTO-
POINT NO.
Dec. 3, 1932 0.870 N 0.861 N 400 30 Ohio Chemical Co. General Chemical Co. 80 20.72
... 30:35 30.13 59.69 58.87 58.94
CHRONOQRAPH
Hr.-min.-sec.
AY/AAe ,e Min. Cc./cm.2/min. CHARGE A
4-00-00 03-24 04-44 06-02 08-38 11-15 13-55
0. 3.40 4.73 6.03 8.63 11.25 13.92
Av.
... i:ioo
1.120 1.110 1.086 1.064 1.096
CHARGE B
10 20 30 50 70 90 Av.
30:35 30.13 59.69 58.87 58.94
4-16-42 18-09 19-34 22-30 25-17 28-14
60:i.S 59.69 58.87 58.94
4-31-16 34-26 37-35 40-42 43-57
30:35 30.13 59.69 58.87 58.94
4-46-55 48-40 50-25 53-54 57-18 5-00-50
30:35 30.13 59.69 58.87 58.94
5-04-20 06-16 08-03 11-56 15-48 19-49
30:35 30.13 59.69 58.87 58.94
5-23-03 25-16 27-19 31-38 35-56 40-23
83.05 85.27 87.32 91.63 95.93 100.38
90:k5 29.62 88.33 29.54
5-43-54 50-48 53-18 6-00-34 03-07
103.90 110.80 113.30 120.57 123.12
6-05-12
125.20
16.70 18.15 19.57 22.50 25.28 28.23
ea". Min.
.... ....
4.07 5.38 7.33 9.94 12.69
t e
c.
29: 97 30.00 29.99 29.97 29.98 29.95
7.86
....
1:009 1.023 0.984 1.023 0.965 1.001
17.43 18.86 21.04 23.89 26.76 26.60
0:9i3 0.916 0.912 0.874 0,906
32.85 36.01 39.14 42.33 37.58
0:%6 0.831 0.828 0.837 0.806 0.828
47.80 49.55 52.16 55.60 59.07 52.84
0:7i5 0.816 0.743 0.735 0.707 0.751
65.30 67.16 69.99 73.87 77.81 70.83
0:660 0.709 0.670 0.661 0.639 0.668
84.16 86.30 89.48 93.78 98.16 90.38
30.00 30.00 30.05 30.00 29.90
...
CHARGE C
10
30 50 70 90 Av.
31.27 34.43 37.58 40.70 43.95 CHARQE D
10 20 30
50 70 90 Av.
46.92 48.67 50.42 53.90 57.30 60.83 CHARQE E
10 20 30 50 70 90 AV.
64.33 66.27 68.05 71.93 75.80 79.82 CHARQE F
10 20 30 50 70 90 Av.
CHARGE G1
10
40
50 80 90 Av.
....
29.97 30.04 30.00 29.95 29.99
....
30.03 30.05 30.05 29.99 29 99 29 96
....
29.99 29.99 29:95 29.98 29.97
....
29.97 30.00 30.00 30.00 30.00 29.90
....
29.95 30.09 30.10 29.95 29.96
0:633 107.35 0.571 112.05 0.586 116.94 0.558 121.85 0.587 114.55 (Sample withdrawn)
Vol. 26, No. 11
PO
I
10
I
I
80
60
e,m~urEs
1
KM
I
I20
I
FIGURE4. EXPERIMENTAL POINTS(RUN 29) RATEOF ABSORPTIONOF CARBON DIOXIDE BY POTASSIUM HYDROXIDE (INITIALLY 0.870 N ) CS. TIMEAT 30" c. AND 1 ATMOSPHEREPRESSURE OF PUREGAS
FOR
per sq. cm. per minute. Since the instantaneous rate proves to be a linear function of the time, the arithmetical average of the rate a t 8.63 and a t 11.25 minutes is identical with the 11.25)/2 = 9.94 minutes, instantaneous rate a t (8.63 designated here as esv. I n this way the data in columns 5 and 6 of Table I are obtained; from them Figure 4 is prepared. As an aid in locating the "best" curve through these points, they are divided into groups and the mean of each group is determined by the method of averages with a calculating machine. The coordinates of these mean points are shown a t the end of each group in Table I. I n most cases these values represent the average of five readings and are therefore × as reliable as any one observation. They are plotted as double circles in Figure 4. The value of the rate at zero time, yo, is c a l c u l a t e d from the c o o r d i n a t e s of the first two (or more) meanpoints. I n this way it is in practice u n n e c e s s a r y to prepare Figure 4, and greater accuracy is obtained. One may, however, take the area under the curve in Figure 4 a t s u c c e s s i v e values of 8 to yield corresponding total volumes of carbon dioxide absorbed, when multiplied by interfacial area 20.72. These FIGURE5 . RATE OF ABSORPare readily converted to TION OF CARBON DIOXIDEBY POTASSIUM HYDROXIDE AS CARequivalents of caustic conBONATE ACCUMULATES, CALCUsumed by multiplying by LATED FROM FIGURE 4 a constant. The concenDotted line represents the location tration may thus be cal- of the initial rate curve into carbonatec u l a t e d a t a n y t i m e , free hydroxide. the rate of a b s o r p t i o n read from Figure 4 at the same value of t3, and the data secured as in Table I1 (shown in Figure 5 ) . A few measurements have been made a t 40' and 5' C. with both sodium and potassium hydroxides, but the work has not reached a point where a conclusion is justified as to the effect of temperature.
+
November. 1934
INDUSTRIAL AND EKGINEERIXG CHEMISTRY
1163
TABLE11. RATEOF ABSORPTION us. HYDROXIDE CONCENTRA-in the viscometer, which a t once yielded kinematic viscosities. TIOK (RTN 29)" The densities of the same solutions were determined simul9 v BVb C Y taneously, from which the absolute viscosity is calculated as .%fin. Cc . Equiualents/liter Cc./cm % / m i n . shown in Table IV. 1.163d 0.870e ... .. ..* 0 30 60 90 120 125.2
C
d
0 650 1190 1645 2028 2090
0.861C 0.730 0.622 0.530 0.453 0.441 0.435~
0 0.131 0.239 0.331 0.408 0.420
1.150 0.948 0.797 0.670 0.569 0.554
MEASUREMENTS AT 30" C. TABLEIV. VISCOSITY SOLN. NaOH NaOH NaOH NaOH KOH KOH
Calculated from Table I and Figure 4. (273)(2000) 2010 = (303) (400) (22410) By chemical analysis. Corrected for initial carbonate.
KOH
Since a t zero time the solution as prepared for use from e. P. reagents had 0.870 - 0.861 = 0.009 equivalent per liter of potassium carbonate, the value of yo = 1.150, calculated from Figure 4, is less than (dV/Ado) lmtlai into pure potassium hydroxide. Let this latter quantity be called yo' for brevity. By extrapolating Figure 4 to include under the curve the small area corresponding to 0.009 equivalent, the desired value of yo' is obtained. A sample calculation of yo' is as follows:
CONCN. Normality 1.084 3.023 4.592 6.057 3.526 7.165 10.221
....
TIME OF DENSITYI N V I S C O S I T Y O , FLUIDITY, FLOW, 9 VACUUM dzo E 1/ z Sec. Grams/cc. Centipoises Rhes X 10-% 0.9990 1.04070 1.001 258.9 1.546 0.6468 373.4 1.1128 0.4376 1.1661 2.285 519.4 3.329 0.3004 720.0 1.2116 1.223 0.8177 285.35 1.15358 0.4926 1,29230 2.030 418.8 3.263 0.3065 1.3968s 144.3 0.8004 1.2494 See citation 6 A 0.8954 1.117 See citation 12 0,8462 1.182 See citation 12
Hi0 KOH 1.00 KOH 0.50 a z = Kd&, where K = viscometer constant and d. (1 cc. air = 0.0012 gram).
= density in air
The viscometer was immersed in a well-stirred thermostat maintained a t 30.00' C. * 0.01'. The thermostat thermometer was calibrated by the Bureau of Standards and the ice point determined locally. Customary precautions were observed in cleaning and filling the viscometers. The mean of at least four determinations of the time of flow was found ( Ac) (nt) (0.870 - 0.861)(0.00691) = o.013 for each sample, and the average deviation of the mean was (Yo)(k)(A) = = (1.150)(2.010 X 10-4)(20.52) 0.05 per cent. Density determinations were made in pairs yo' = 1.163 and agreed within 0.02 per cent. The values reported are The slope, m, of the curve in Figure 4 is calculated from the reduced to weights in vacuum. Viscometer constants were determined by calibration with distilled water, specially first two "mean" points given in Table I: purified absolute ethyl alcohol solutions, and sucrose solutions prepared with material provided by the Bureau of Standards m = - - -1.096 - 1.001 - -o.oo691 7.86 - 21.60 for calorimetric work (Bureau of Standards standard sample 17, lot 502). Critical data for these liquids were taken from the I n similar fashion yo' is found for a wide range of concentra- work of Bingham and Jackson (1) and were used to detertions of potassium and sodium hydroxides a t 30' C., given mine viscometer constants. Viscosities of other reference in Table I11 and shown in Figure 6. liquids were then determined and found to agree with those given by the International Critical Tables within 0.3 per TABLE111. INITIAL ABSORPTION RATE O F CaRBON DIOXIDE BY Two ALKALISOLUTIONS AT 30" C. UNDER 1 ATMOSPHERE cent. The logarithms of viscosity recorded in Table IV were PRESSURE OF PUREGAS AND CONSTANT STIRRINGRATE OF 80 R. P. M. plotted against concentrations to yield the curves shown in -SODIUM HYDROXIDE--POTASSIUM HYDROXIDE Figure 7 , a method which gives more nearly a straight line RUN. No. yda co'b RUN. No. yUfa CO' than any other for this pair of substances. Values were then Cc./cm.2/min. Normaltty Cc./cm.l/min. Normality read a t round values of normality and are presented in Table V 1.152 0.870 9 0.695 0.6150 29 10 1.269 30 2.036 1.486 1.137 with the corresponding fluidities. 2.282 1.554 1.028 2.165 31 11
*"
-
12 15 16 17 18 19c 20 21 22 23 24 25 26 27 28 34 6d
2.898 2.640 2.326 2.850 0.04 1.554 0.548 0.583 1.151 1.146 1.636 1.635 0.682 1.510 0.598 0.375 2.10
3.542 5.806 2.165 3.995 0 2.201 0.5089 0.5089 1.0100 1.0100 1.500 1.500 0.5126 1.320 0.527 0.3097 6.742
32 33 35 51 53 54 55 56 57 58 59 60
18d
20d 22d 24d 26d 28d 30d
0.770 0.384 1.336 5.64 5.22 4.28 5.92 5.83 6.1:5 0 0.8448 5.85 4.088 2.050 2.700 1.377 0.673 3.302 3.332
0.504 0.301 1.025 6.179 5.060 3.526 10.221 7.165 8.667 13.928 11.910 8.752 3.070 1,430 1.843 1.020 0.508 2.310 2.467
Corrected for initial carbonate. b A convenient table for converting normality of NaOH aolutiona at 30' C. t o molality is found in a DaDer by Harned and Hecker ( $ A ) . CSpent liquor- from run 12 -cc?ntainhg 1.341 equivalents.per liter of N aiC;Vz. d Data of Lewis and Williams @ A ) . e Cryetale clogged surface and stopped absorption
__
VISCOSITYMEASUREMENTS A modified Ostwald viscometer of 10 cc. capacity was used in a procedure developed by Fenske (19) at Pennsylvania State College recently and kindly made available by him. The method is rapid and accurate and lends itself well to a wide range of viscosities by employing a series of viscometers differing solely in the size of the capillary. Carefully standardized solutions were prepared and times of flow determined
AND FLUIDITY OF Two ALKALIES AT 30" C . TABLEV. VISCOSITY AT ROUND VALUESOF THE NORMALITY
HYDROXIDE-
-$ODIU?d
Concn.
Log z
Normalzty
0 0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
....
-0.0793 -0.052 -0.0075 0.0385 0.0865 0.1360 0.1865 0.2385 0.2925 0.3485 0.405 0.460 0.5165
-POTASSIGM
2
+
Centa-
Rhes X
pomes
0,8004 0.832 0.887 0.983 1.093 1.220 1.368 1.536 1.732 1.961 2.231 2,541 2.884 3.285
Concn.
10-2 1.2494 1.2021 1.1272 1.0174 0.9152 0.8194 0.7311 0.6509 0.5774 0.5099 0.4482 0.3936 0.3467 0.3044
Log z
Normalitz, 0 0.5 .... 1.0 2.0 0.0045 3 . 0 0.059 4 . 0 0.1146 6 . 0 0.173 6 . 0 0.2335 7 . 0 0.2965 8 . 0 0.362 9 . 0 0.429 10.0 0.4976
.... ....
HYDROXIDE 7 z CenttRhes X 10-9 poises 1,2494 0.8004 0.8462 1.182 0.8954 1.117 1.010 0.9897 1.146 0.8730 1.302 0.7681 1.489 0.6714 1.712 0.5842 1.979 0.5052 2.301 0.4345 2.685 0.3724 3.145 0.3180
+
DISCUSSION OF RESULTS There is nothing novel about the proposal that viscosity may be an important factor in heterogeneous reactions. But limited viscosity data have in the past served to obscure the extent to which this factor may be of controlling importance. cs)/f(z), it seemed likely that I n the equation yo' = k'(ci the viscosity function might take an exponential form, in common with other phenomena following the so-called mass law, such as solubility and vapor pressure. Hovorka (6), for example, finds an exponential function involving the vis-
+
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cosity, vapor pressure, and density which yields a characteristic constant for many liquids. The viscosity function selected was therefore the most general form, ear. By plotting the smoothed data of Table I11 (Figure 6 ) against the viscs)/y0’] vs. z, it was cosity data of Figure 7 as log [(0.06 found that the points fell more nearly on a straight line than was the case with any other treatment. The y-intercept of
potassium carbonate in solution and a solid phase of KzC033/2 H 2 0 . I n the present experiments with sodium hydroxide, crystal formation was not observed a t the highest strength employed (6.7 normal) although Davis and Crandall report crystal formation in shallow stagnant layers a t concentrations as low as 1.3 normal. It is of interest to note that at the maxima in the two curves, approximately 4.5 A’ sodium hydroxide and 9.0 N potassium hydroxide, respectively, the two solutions have the same viscosity, while their absorption rates are 3.0 and 6.0 cc. per minute per sq. em., respectively. The constants in Equations 7 and 8 are in no sense “free” constants but are determined by the data themselves. On the other hand, taken individually they probably have little fundamental significance but would be profoundly affected by a change in the shape of the absorption tube, the design of the stirrer, and the rate of stirring. I n a d d i t i o n , t h e type of absorption apparatus convenient for experimental work of this nature has no industrial parallel. Nevertheless the measurements on sodium hydroxide and potassium hyFIGURE6. INITIALRATEOF ABSORPTION OF CARBON DIOXIDEBY POTAS- droxide were made under identical conditions SIUM HYDROXIDE AND SODIUM HYDROXIDE (CARBONATE-FREE) AT 30” C. in the same apparatus, whatever its shape A S A FUNCTION O F CONCENTRATION UNDER 1 ATXOSPHERE PRESSURE OF factor may be, and the data are t h e r e f o r e PUREGAS comparable. That the absorption rates apLiquid is stirred a t 80 r. p. m. Dotted lines indicate Lewis and Whitman theory with funcparently follow the same type of function for tion of viscosity, I, included. 0 = data of Lewis and Williams. both sodium and p o t a s s i u m h y d r o x i d e is the best line through these points is log k’ and the slope is a significant and would undoubtedly be the case whatever the log e. The following equations are obtained by this method: type of absorption equipment used.
+
For NaOH:
yo’ =
For KOH:
yo’ =
1.638(0.06 f
1.835(0.06
+
CS)
(7)
CQ)
e0.3846~
These equations are plotted on Figure 6 as dotted lines. Both are in fair agreement with the experimental data, the representation being somewhat better in the case of the sodium hydroxide. Equation 7 fits rather closely over the entire range except in the vicinity of 2.5 normal where the calculated curve falls below the experimental curve by a maximum of 5 per cent. Equation 8 falls below the experimental curve for potassium hydroxide by as much as 5 per cent in the vicinity of 3 normal and runs above the curve by 4 per cent around 7 normal. The limited data available at present for rates of absorption a t concentrations of potassium hydroxide above 7 normal are not conclusive. I n the case of potassium hydroxide solutions no crystal formation was visible in any part of the liquid phase throughout the course of any experiment until the initial concentration reached approximately 12 normal. I n contrast, Davis and Crandall observed crystal formation at concentrations as low as 9 normal. However, they used only 0.25 cc. of solution in a cell 0.2 cm. deep, with no stirring. Under such conditions the liquid soon became saturated and crystals were formed. In the author’s experiment the liquid layer was 21 cm. deep and the liquid was stirred a t SO r. p. m. Hence the rate of exhaustion of the alkali in the surface layer was much slower, and crystal formation was obtained only in the runs where the solution was initially so strong that the solubility of potassium carbonate was limited. Data in the International Critical Tables for the solubility of potassium carbonate in potassium hydroxide solutions a t 30” C. indicate that saturation is reached a t approximately 12 normal in potassium hydroxide with 2.50 per cent by weight of
COXPARISON WITH EARLIER RESULTS This is supported by the data of Ledig and Weaver (’7) on absorption of carbon dioxide bubbles in potassium and sodium hydroxides a t 25” C. using a radically different experimental method. Although their technic is designed primarily to show the high true initial rates of absorption while concentration gradients are being established, it is possible to estimate roughly what the practical initial rates are over a concentration range up to 12 normal or more. Davis and Crandall ( I A ) list these estimated values in a table expressing the rate of absorption a t any concentration as a ratio to the rate into pure water. These data are reproduced here in Table VI, together with similar calculations based on the present results. I n addition, at given concentrations, the ratio of the rate into potassium hydroxide to the rate into sodium hydroxide is shown, both for Ledig and Weaver and for the present results. Both sets of data for sodium hydroxide pass through a maximum a t approximately the same concentration. Ledig and Weaver’s data for potassium hydroxide seem to reach a constant ratio at about 9 normal, while the present investigation shows a n apparent maximum a t about 8 normal. The results of Wolf and Krause (20) are in qualitative agreement. They found in an Orsat type of apparatus that the rate of absorption of carbon dioxide by potassium hydroxide solutions rises rapidly up to 4 per cent by weight (0.74 normal), then more slowly with a maximum a t 28 per cent (6.32 normal). The rate for sodium hydroxide they state to be nearly the same as that for potassium hydroxide up to 4 per cent (1.045 N sodium hydroxide) (cf. Figure 6 and also Davis and Crandall, I A ) , beyond which the rate falls slowly. Hatta (6),working with mixtures of carbon dioxide and air, found that the rate depended profoundly on concentration and is considerably greater for potassium hydroxide
IYDUSTRIAL A N D EKGINEERING CHEMISTRY
November. 1934
solutions than for sodium hydroxide solutions. Masaki (10) studied 1.0 N potassium and sodium hydroxide solutions in an apparatus similar to that of Ledig and Weaver and found that between 11" and 25" C. a potassium hydroxide solution absorbs 1.14 times as fast as sodium hydroxide. (In the current investigation a t 30" C. the ratio is 1.22 for normal solutions.) Shchukarev and Bondareva ( 1 4 , working with very dilute solutions, report that the rate of solution of carbon dioxide in sodium hydroxide and ammonia solutions increases less rapidly than in direct proportion to the concentration of alkali, finally reaching a constant value.
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a change in temperature would not be expected to change anything except physical properties such as viscosity and density; the concentration as used in Equations 7 and 8 would remain constant. In such a case allowance for changed viscosity might be expected t o lead to complete agreement with the observed value. If, as is more reasonable t o suppose, the chemical characteristics of the system, such as activity, also
REFINEMENTS A possible improvement which suggests itself is the expression of the driving potential (a CS) in terms of activities rather than gross concentrations. There are two reasons for this proposal. First, the constants in Equations 7 and 8 differ by approximately 10 per cent. On a purely physical diffusion basis one would expect the equations to be the same; that is, for a given value of the viscosity, the rates of absorption per equivalent of driving potential should be equal. That these colistants are approximately the same may be regarded as an indication that "physical" factors (e. g., diffusivity, viscosity, density) account for most of the observed difference in rates of absorption for the two substances, since this difference is as much as FIGURE7 . VISCOSITY OF AQUEOUS SOLUTIONS OF SODIUM AND POTAS100 per cent of the rate into sodium hydroxide SIUM HYDROXIDES VS. CONCENTRATION at higher concentrations. Use of activities rather than concentrations might lead to a still closer agreement play a part, even though a minor one, complete agreement between the constants in the two equations. could not be secured until the variation of these chemical traits with temperature were known. Further study is being TABLE VI. RELATIVERATESOF ABSORPTIONOF CARBON given the subject. DIOXIDE BY ALKALIES With respect to the interfacial concentration of carbon (Referred to water at temperature, t ) dioxide, ci, as intimated earlier, this quantity must be a funcLEDIOAND WEAVER'BDATAAT tion of the solute concentration. Data are entirely lacking 25' C. (ACCORDING TO DAVIS A N D CR.4NDALL) HITCHCOCK'S DATAAT 30' C. on this point a t present, unless one wishes to make doubtful CONCN. NaOH KOH KOH/NaOH NaOH KOH KOH/NaOH approximations based on 2 A' potassium chloride. This MoZes/Ziter seemed fruitless when dealing with solutions up to 14 N po0 1.0 1.0 1.0 1.0 1.0 1.0 0.2 2.8 8.3 10.0 tassium hydroxide, and for the present c, has been taken con0.5 19.2 1:34 7:s 5.6 i.'25 23.3 1.0 9.0 36.2 1.22 11.0 45.9 1.21 stant a t its value in pure water a t 30" C. as reported by Davis 1.5 45.8 10.5 64.2 0.06 equivalent per liter. While and Crandall-namely, 2.0 1:45 11.0 1s:o 49.7 1:27 77.3 2.5 11.5 48.3 .. .. 86.6 .. ordinarily this is small in comparison with CS, the function 3.0 18 42.2 93.0 .. .. .. ii:o 3.3 .. 36.7 . . log [(ci 95.5 cs)/y0'] is sensitive to ci a t lowhydroxide concen4.5 19 .. .. .. .. 100 .. trations. It will be noted in Figure 6 that the agreement is 6.0 19 .. .... 5:5 6.6 .. .. excellent for both equations as the concentration of solute 7.5 . . 19 .. .. .. approaches zero. Further data may make possible a modifiThe second reason involves the effect of temperature upon cation in Equations 7 and 8 which will reduce the existing t o the vanishing point. rate of absorption. While data have been obtained in this deviation of a few per cent Other refinements are conceivable such as an allowance of laboratory a t 5" and 40" C. in the same apparatus, they are some sort for changing ionic mobility. Falkenhagen ( 2 ) available thus far only over the concentration range 0.5 to has shown from theoretical considerations, however, that the 1.5 normal for sodium and potassium hydroxides. If one uses Equations 7 and 8, obtained a t 30" C., substituting the viscosity of strong binary electrolytes is proportional, among viscosities a t 40" of the respective caustic solutions a t a con- other things, t o a complex function of the ion mobilities. centration of 1 normal, calculated rates of absorption1 are It appears that causes of viscosity are so interwoven with obtained which are 13 per cent below the observed rates for the other useful properties of the reactive solution that it is both sodium and potassium hydroxides. Hatta (j), using probable the function of viscosity as determined in this ina dimensionally similar apparatus stirred a t nearly the same vestigation includes the effect of related phenomena. Should the term (u CS) in Equation 6 developed by rate as in the present work, reports one experiment with a Hatta and by Davis and Crandall prove to be inadequate 98 per cent pure carbon dioxide gas a t 20" C. and a 2 N poto cover complex mechanisms and any chemical reaction tassium hydroxide solution from Tvhich it is possible to estirates which might be involved, then one would obtain from mate an initial rate of 2.15 cc. per sq. cm. per minute. Using Equation 8 and the viscosity of 2 N potassium hydroxide a t these data a viscosity function differing from the one emc5) was 20" C., a rate of 2.35 cc. per sq. cm. per minute is calculated. ployed herein, which assumed that the term (c, correct. I n other words, if a purely physical mechanism is involved Obviously the initial absorption rate could be expressed 1 Corrected for temperature effect on gas volume. empirically in terms of normality, viscosity, or density alone.
+
+
'
+
+
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INDUSTRIAL AND ENGINEERING CHEMISTRY
It seems better, however, to extend the applicability of the Hatta equation by introducing the viscosity term which would clearly affect the film thickness as in other unit operations. Whether viscosity is a factor of importance in other gas-liquid or liquid-solid systems encountered in heterogeneous reactions is a matter which is under investigation. PROGRESS OF A BATCH EXPERIMENT The present discussion is devoted, as stated a t the outset, to initial rates in pure hydroxide solutions. But the way in which absorption rate decreases during a given batch experiment, as hydroxide is converted to carbonate, is significant. Examination of Figure 5 where instantaneous absorption rate is plotted against concentration of residual hydroxide for a single experiment, reveals that, as absorption proceeds, the rate decreases more rapidly than can be accounted for simply by the decrease in hydroxide concentration.* Yet the reaction product, potassium carbonate, is itself reactive with carbon dioxide; that its presence serves actually to reduce the absorption rate below the value otherwise possible seems surprising. However, the viscosity of potassium carbonate solutions is considerably greater than that of potassium hydroxide solutions of the same normality, and presumably this effect outweighs the chemical potential which the carbonate would otherwise be expected to give. Practically no viscosity data for potassium carbonate and sodium carbonate are available a t present, nor for mixtures B-ith their respective hydroxides. That their mixtures do not follow the mixture rule even approximately a t higher concentrations is shown by preliminary experiments. Work is now in progress to secure this information.
SUMMARY The theory of the multizone liquid film has been applied by Hatta to the absorption of carbon dioxide from a mixture with air, employing potassium hydroxide solutions under batch conditions. Hatta's equation representing the progress of any given batch absorption is, for many important cases :
I n the initial absorption rate of pure gas the equation reduces to
which is substantially the equation found by Davis and Crandall to fit the data for both potassium and sodium hydroxides up to 1 or 2 normal. However, as concentration increases, the latter equation gives deviations from observed initial absorption rates which become rapidly larger. The viscosity of the absorbing solution appears to be a factor of importance, and earlier difficulties with the above equation are attributable to the fact that the effect of viscosity z on the coefficient JCL was neglected. The initial (steady state) rate of absorption of pure carbon dioxide a t one atmosphere has been measured for solutions of sodium hydroxide up to 7 normal and of potassium hydroxide up to 14 normal at 30" C. The viscosity of these hydroxide solutions, hitherto unknown over this concentration range at 30" C., has been determined. Equations in accord with the above theory are obtained 2 The curve of Figure 5 is definitely and uniformly curved and not a straight line. The procedure followed in obtaining this curve, as described above, requires i t t o have t h e form of a parabola initially. This may be readilv shown alnebraicallv. - . &s Dointed out in a Drivste communication from T. H. Chilton. H a t t a conaidered these curves t o be straight lines i n accordance with his theoretical equationa. I
~
which satisfactorily reproduce the experimental results over the entire range. The equation has the form, k'(0.06
+
CS)
ens
where C i is assumed for the present to be constant at 0.06 equivalent per liter. The constants in the equation are: SOLUTE NaOH
KOH
kl
1.638 1.835
a
0.4104 0.3840
The results are in agreement with qualitative conclusions already reached by a number of independent observers but not previously correlated. No crystal formation in the interface was observed until the solubility of potassium carbonate in potassium hydroxide was exceeded a t approximately 12 N potassium hydroxide. The equation may be used to estimate rates of absorption a t other temperatures than 30" C. if the viscosity is taken a t the desired temperature, but deviations from observed rates of absorption of the order of 10 per cent remain. It is suggested that the use of the activity in place of the concentration may reduce this difference.
ACKNOWLEDGMENT The writer is indebted to Warren K. Lewis and William P. Ryan for helpful suggestions. Certain of the data on rates of absorption were obtained by Hunter F. Lewis and John R. Williams (8A)while, in the measurement of viscosities and densities, assistance was rendered by John B. Hancock and S. D. Smiley. Figure 1 was prepared by Hunter F. Lewis.
XOMENCLATURE A = area of interface, sq. cm. B = constant for converting cc. COz gas at temp. t o K. to gram equivalents of alkali in "n" cc. of liquid = (273) (2000) (Ob)(22,410) c = liquid concn. in Equation 1 D = ty ical dimension of absorption apparatus H = sokbility coefficient K = over-all coefficient in Equation 1 P = partial pressure of gas s = density u = liquid velocity gas absorbed, cc. gas absorbed, grams a = exponent in equation f ( z ) = eas b = proportionality constant in equation x = bD(DUS/z)* c = concn., gram equivalents/liter (normality) a = diffusivity ; density, grams/cc. e = base of natural logarithms, 2.718.. , . f k'0 = = any function constant in equation k~ = k ' / j ( z ) k - individual film coefficient in Equation 1 m = initial slope of curve dV/Ade vs. e n = exponent on Reynolds' group 1 = temperature x = film thickness % = dV/Ade, cc./sq. cm./min. at zero time Yo' = dV/Ade, cc./sq. cm./min. at zero time, cor. for initial carbonate z = viscosity, centipoises f f = a factor relating diffusion rates of various ions A = an increment e = time d = fluidity, rhes X 10-2 Subscripts: B = inert gas
v= w=
Q = gas
interface (e. g., as used here, q = interface concn. of C02 gas, dissolved) L = liquid S = solute (e. g., NaOH, KOH, etc.)
i
=
LITERATURE CITED (1) Bingham and Jackson, Bur. Standards, Bull. 14,59 (1919). (1A) Davis and Crandall, J. Am. Chem. Soc., 52, 3757 (1930).
INDUSTRIAL AND ENGINEERING
November, 1934
(2) Falkenhagen, H., Physik. Z., 32, 745 (1931).
(3) Hanks and Mcildams, IND. ENQ.CHEM.,21, 1034 (1929). (3A) Harned and Hecker, J. Am. Chem. Soc., 55, 4842 (1933). (4) Harte, Baker, and Puroell, IND.ENQ. CHEM.,25, 528, 1128 (1933). (6) Hatta, S., Tech. Repts. Tihoku I m p . Univ., 8, 1 (1928); 10, 613,631 (1932). (6) Hovorka, F.,J. Am. Chem. Soc., 55, 4899 (1933). (6A) . , International Critical Tables, Vol. V, p . 10,McGraw-Hill Book Co.. New York. 1926. (7) Ledig and Weaver, J. Am. Chem. SOC.,46, 650 (1924); Ledig, IND.ENQ.CHEM.,16, 1251 (1924). (8) Lewis and Whitman, Ibid., 16, 1215 (1924). (SA) Lewis and Williams, Ch.E. thesis, Univ. Va., June, 1934. (9)‘Markham and Benton, J . Am. Chem. SOC.,53,497 (1931). (10) Masaki, J . Biochem. (Japan), 13, 211 (1931); 15,29 (1932). (11) Mitsukuri, Sci. Repts. T h k u Imp. Univ., 18,246 (1929). (12) Mohanlal and Dhar, Z . anorg. allgem. Chem., 174, 1-10 (1928).
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1167
(13) Payne and Dodge, IND. ENQ.CHEIM., 24, 630 (1932). (14) Shchukarev and Bondareva, Ukrain. Khem. Zhur., 7, Wiss. Teil. 1-11 (1932). (15) Taylor, H. J., “Treatise on Physical Chemistry,” Vol. 11, p. 1099, D.Van Nostrand Co., New York, 1931. (16) Weber and Nilsson, IND.ENQ.CHEM.,18, 1070 (1926). (17) Whitman and Davis, Ibid., 18, 264 (1926). (18) Williamson and Mathews, Ibid., 16, 1157 (1924). (19) Willihnganz, MoCluer, Fenske, and McGrew, Ibid., Anal. Ed., 6,231 (1934). (20) Wolf and Krause, Arch. Wtirmewirt, 10, 19 (1929). RECEIVED September 17, 1934. Presented as part of the Symposium on Diffusional Processes before the Division of Industrial and Engineering Chemistry a t the 88th Meeting of the American Chemical Society, Cleveland, Ohio, September 10 to 14, 1934. This paper is a contribution from the Department of Engineering and the Cobb Chemical Laboratory, University of Virginia.
Rate of Mixing of Gases in Closed Containers ALLENS. SMITH,Cryogenic Laboratory, U. S. Bureau of Mines, Amarillo, Texas
The theory of Loschmidt for the measurement of the diffusion coeficient has been applied to determine the time required to approximate complete mixing of a range of binary gas mixtures.
A
MONG the many chemical engineering processes in which gas diffusion is a controlling factor, the mixing of gases is relatively unimportant in plant design, although it must be considered and equipment provided to accomplish it. I n experimental work on the properties or reactions of gas mixtures, however, the subject is of greater importance, as the preparation of samples containing two or more constituents is often required. It is desirable to make up such samples under pressure, using the required proportions of the compressed pure gases, to reduce the labor of frequent preparation and analysis; samples after preparation are usually allowed to stand a few days and are then analyzed. Towend and Mandlekar (7), for example, made up mixtures of butane and air a t 20 to 30 atmospheres pressure, allowed them to stand for 18 hours, and then analyzed them before and after experiments. Campbell (1) prepared samples of methane and oxygen at normal pressure, and “homogeneity in the mixture before analysis was insured by allowing the gases to remain for at least 24 hours after mixing and shaking.” Carpenter and Fox (@, on the other hand, found that the diffusion of carbon dioxide into air is rapid and complete even under extreme conditions. The following pertinent questions are raised: Are repeated analyses of a sample necessary, and if not, what length of time is required for essentially complete mixing? The first question must be considered for a specific mixture. A single checked determination of the composition of a mixture would be representative of the gas, if it were made after mixing was complete, only when liquefaction, polymerization, or reaction either with the container or in the mixture did not occur. I n large vessels the phenomenon of thermal diffusion, which may cause stratification in a mixture and even partial separation, must also be considered. It has been pointed out recently by Keffler (4) that the composition of a gaseous mixture may change because of the effusion of the components more or less in accordance with Graham’s law. The importance of this effect will depend upon the relative
densities of the gases in the mixtures and the size of the orifice through which the gas is discharged. Having decided upon the possibility of the occurrence of these factors which retard or prevent the attainment of a uniform mixture, the second question, that of the time which should elapse between the preparation and analysis of a sample, follows. It does not seem desirable to determine experimentally the conditions by which a uniform mixture of gases may be obtained since the kinetic theory affords a method of calculation which, for all usual purposes, should be sufficiently exact. The reill be maxima under the conditions selected to simplify sults w the calculations.
THE DIFFUSIONCOEFFICIENT I n a cylinder of constant cross section in which diffusion occurs only along the axis of the cylinder, it is assumed that, if a concentration, c, exists a t a point, x, the concentration c dc will be present a t a distance x d x . A concentration gradient prevails between the two points which is equal to dc/dx and causes diffusion as long as i t exists. Fick’s law states that
+
+
de d n = - D A - dt
dx
where dn
amount of substance diffusing through area A in time 02 D = a proportionality constant depending upon the gases in question =
The application of integrated forms of Equation 1 requires a knowledge of the diffusion coefficient, D. Diffusivities may be obtained from the International Critical Tables or estimated by the empirical equation of Gilliland (S). Values of the diffusion coefficient have been reported, and calculation is facilitated for conditions a t 0” C. and 760 mm. pressure. I n changing the values of D12to different conditions of temperature and pressure, the following approximate equation has been used:
