Changes in Volume on Mixing Solutions P. W. PARSONS AND F. J. ESTRADA U. S. Industrial Chemicals, Inc., New York, N. Y.
or expansion when one comFor practical operations that involve ponent is added, and to comusually occur when solushrinkage and expansion of solutions on pute the amount by a simple tions are mixed; these mixing, a simple and convenient method of differencing of the function changes are generally shrinkcalculation has been devised. By combinand a multiplication. ages rather than expansions, ing readily obtainable data-the percentThe theoretical significance and, when highly polar solvents of these volume changes is of no such as water are involved, the age composition of a solution, its density, concern here ( I ) , nor is any atlargest changes occur; these and the densities of the pure componentstempt made to fit empirical or three facts are well known. into a function called the "shrinkage factheoretical mathematical equaThe methods developed by tor", the amounts of actual volume change tions to the data. Thus, the Lewis and Randall (2) and can be computed quickly. treatment is entirely rigorous, others, including the concept as no assumptions are made of the partial molal volume, Derivation and examples of the method other than the experimentally are useful and often required in are shown, with special reference to ethdetermined data themselves as thermodynamic calculations. anol-water solutions. The fact that cercompiled principally from the By these methods the changes tain of these solutions actually expand on International Critical Tables. in volume which occur when The change in volume that dilution has not been generally recognized infinitesimally small amounts occurs on mixing water and of the constituents are added hitherto alcohol can be conveniently can be computed. However, The relation to the established concepts found from Figure 1. they are laborious and do not of apparent volume and partial volume is conveniently give the volume EXAMPLE I. 190-proof alcodiscussed. changes when finite quantities hol (95 per cent by volume of of tvhe constituents are added, ethanol at 60" F. or 0.95 volume fraction of ethanol) is diluted auantities that will make large with water to 100 proof (0.50 volume fraction of ethanol). By ihanges in the percentage- composition of the solution. definition, the proof of alcohol (more accurately speaking, the Therefore, these methods are cumbersome for computing per cent of proof) divided by 2 gives the per cent by volume of the deviations from additivity Of VOhm.eS in most operations ethanol at 60" F., and divided by 200 gives the volume fraction in the laboratory and in the industrial plant. of ethanol at 60' F. The shrinkage factor corresponding to a volume fraction of By employing a function of the percentage composition of a 0.50 is 0.075, and that corresponding to a volume fraction of solution, its density, and the densities of the pure oomponents, 0.95 is 0.012. The difference, 0.063, multiplied by 0.95, which will be called the "shrinkage factor", it is possible to the volume fraction at the beginning gives o , which ~ is~ ascertain by simple inspection whether there is contraction the shrinkage per volume of 190-proof ethanol used (for example,
HANGES in volume
C
.
FIGURE1. VOLUME FRACTION-SHRINKAQE FACTOR FIGURE 2. WEIGHTFRACTIONSHRINKAGE FACTOR FOR FOR SYSTEM A = ETHANOL, B = WATERA T 60" F. SYSTEM A = ETHANOL, B = WATERAT 20' C. (15.6' C.)
949
~
INDUSTRIAL AND ENGINEERING CHEMISTRY
950
-
FACTORS FOR VOLUME AND WEIOHT FRACTIONS AT VARIOUS TEMPERATURES TABLE I. SHRINKAGE
-
Volume Fractions of A Ethanol, E Water a t 60' F. fA P/PB $A 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.32 0.30 0.28 0.20 0.10 0.05 0.02 0.00
Vol. 34, No. 8
0.79389 0.83382 0.86380 0.88999 0.91344 0.93426 0.95178 0.96296 0.96534 0.96760 0.97596 0.98661 0.99282 0.99703 1.00000
0.00000 0.02147 0.03586 0.04895 0.06184 0.07463 0.08556 0.09036 0.09058 0.09039 0.08591 0.07221 0.06251 0.05761
Weight Fractions of A = Ethanol, B = Water a t 10' C.
FA
P
1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.25 0.20 0.15 0.10 0.05 0.00
0.79784 0.82654 0.85197 0.87602 0.89927 0.92162 0.94238 0,95977 0.96665 0.97252 0.97800 0.98393 0.99098 0.99973
Weight Fractions of A = Ethanol, E = Water a t 40° C.
FA
P
*A
1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.25 0.20 0.15 0.10 0.05 0.00
0.77203 0.80028 0.82578 0.85025 0.87417 0.89750 0,91992 0.94055 0.94991 0.95856 0.96670 0.97475 0.98311 0.99225
0.00000 0.01887 0.03353 0.04704 0,06060 0,07468 0.08938 0.10281 0.10780 0.11038 0.10988 0.10658 0.10008
VA 0.00000 0.02023 0.03626 0.05131 0.06688 0.08355 0.10093 0.11428 0.11619 0.11316 0.10498 0.09251 0.07711
FA
P
PA
1.00 0.92 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.24 0.20 0.10 0.05 0.02 0.00
0.78934 0.81142 0.81795 0.84344 0.86766 0.891 13 0.91384 0.93518 0.95382 0.96312 0.96864 0.98187 0.98938 0,99454 0.99823
0.00000 0.01556 0.01978 0.03530 0.04974 0.06444 0.08009 0.09626 0.10961 0.11294 0.11206 0.09821 0.08591 0.07861
Weight Fractions of A = Ethanol, B = Water a t 20' C. FB P PB 1.00 0.90 0.80 0.70 0.60 0.60 0.40 0.30 0.20 0.10 0.05 0.00
0,99823 0.98187 0,96864 0.95382 0.93518 0.91384 0.89113 0.86766 0.84344 0.81795 0.80424 0,78934
if 100 gallons of 190-proof ethanol were taken the shrinkage would be 6 gallons). On the other hand, if the difference is multiplied by 0.50 (the volume fraction of ethanol after dilution), the shrinkage per volume of 100-proof alcohol produced is obtained, which is 0.032. To make this dilution from 190 to 100 proof, enough water must be added per volume of 190 proof to bring the total volume after dilution to 0.95/0.50, or 1.90 volumes. Since the shrinkage was 0.060 per volume of 190-proof alcohol taken, the amount of water added would have to be 1.90 - 1.00 0,060 or 0.960 volume (190 - 100 6.0 = 96.0 gallons for the case where 100 gallons of 190-proof alcohol were taken).
+
Weight Fractions of B = Water a t 20' C.
A = Ethanol,
+
Inspection of Figure 1 shows that, when alcohol of 60 proof is diluted with water, there will be expansion instead of contraction. Such an expansion also occurs with certain other monohydric alcohols (methyl, n-propyl, and isopropyl) although for these other alcohols the change from contraction to expansion occurs a t somewhat different percentage compositions. The use of volume percentages, which is general in the alcohol ,industry, is open to at least three serious objections that do not apply to weight percentages or weight fractions :
0.00000 0.01091 0.02802 0.04698 0.06417 0.08009 0.09667 0,11606 0.14119 0.17799 0.20429
A
Weight Fraotions of Ethanol, B = Water a t 30' C . FA P PA
3
1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.25 0.20 0.15 0.10 0.05 0.00
0.78075 0.80922 0.83473 0,85908 0.88278 0.90580 0.92770 0,94741 0,95607 0.96396 0.97133 0.97878 0.98670 0.99668
0.00000 0.01935 0.03442 0.04834 0.06240 0.07716 0.09250 0.10591 0.11004 0,11118 0.10861 0.09368 0.10278
Weight Fractions of A = Sulfuric Acid,B = Water at20'C.
FA
P
PA
1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.08 0.06 0.04 0.02 0.00
1.8305 1,8144 1,7272 1.6105 1,4983 1.3951 1.3028 1.2185 1.1394 1.0661 1.0522 1.0385 1.0250 1.0118 0.9982
0.0000 0.0452 0.0730 0.0886 0.1018 0.1145 0.1301 0.1483 0.1652 0.1825 0.1872 0.1928 0.1996 0.2178
liters per kg. of 100-proof alcohol produced. On the other hand, multiplying the difference 0.077 by 0.924 gives 0.071, which is the shrinkage in liters per kg. of 190-proof alcohol taken. The symbols used in the derivation of the shrinkage function are as follows: p
F f z
P
density; number subscripts denote particular solutions and letter subscripts denote ure constituents = weight fraction; letter sufscripts show particular constituents = volume fraction; letter subscripts show particular constituents = weight of solution taken = shrinkage factor for weight fractions; letter subscript denotes constituent, the quantity of which is held constant = shrinkage factor for volume fractions; letter subscript denotes constituent, the quantity of which is held constant
=
Taking 21, weights of solution 1 containing FA^ weight fraction of constituent A , add (z2 - zl)weights of constituent B t o give z2 weights of solution 2, containing FA^ weight
1. The volume percentage of a given sample varies with the temperature and with the external pressure. For instance, 100proof alcohol contains 60.00 volume per cent of ethanol at 60" F. (15.6" C.) but 50.12 per cent at 77" F. (25" C.). 2. The sum of the volume percentages of the two constituents seldom is 100. Thus, at 60" F. 190-proof alcohol contains 95.00 volume per cent of ethanol and 6.18 per cent of water, or a total of 101.18 per cent. 3. When definite quantities of solutions of different concentrations are mixed, it is difficult to compute accurately the percentage composition of the resulting mixture.
For these reasons and because weight percentages are more widely used, the determination of volume changes when weight percentages or weight fractions are given will be discussed. EXAMPLE 11. A quantity of alcohol, weight fraction of ethanol 0.924 (190 proof), is diluted with water to an ethanol weight fraction of 0.425 (100 proof). On Figure 2 the rending of the shrinkage factor corresponding to a weight fraction of 0.924, which is 0.015, is subtracted from that corresponding to a weight fraction of 0.425, which is 0.092, to give 0.077. Multiplying this difference by 0.425 gives 0.033, which is the shrinkage in
fl
- JXrioksye
F u c t o r , i!/kers/fiAyrqm
FIGURE 3. WEIGHTFRACTIOX-SHRINKAGE FACTOR FOR SYSTEM A = ETHASOL, B = WATER AT 20' C.
INDUSTRIAL AND ENGINEERING CHEMISTRY
August, 1942
95 1
+
weight fraction of A to give (zl x i ) weights of solution 3 containing FA^= ( ~ ~ F ~ ~ + x ~ ~ weight ~ ~ ) fraction / ( z ~ of+ Az . ~ ) EXAMPLE 111. One kilogram of alcohol, weight fraction of ethanol 0.924 (190 proof), is added to 5 kg. of alcohol, weight fraction of ethanol 0.121 (30 proof). The wei ht fraction of the resulting mixture can be at once computed from the formula:
and the shrinkage from the formula, Shrinkage =
XiF~i('@.as
-
*AI)
+ XzFa2
(*A3
-
*Ad
\EA being taken from Figure 2 as follows: shrinkage = 1(0.924)(0.113 - 0.102)
+
5(0.121)(0.113 - 0.015) = 0.0695 liter
FACTOR FOR FIGURE4. WEIGHTFRACTION-SHRINKAGE SYSTEMA = ETHANOL,B = WATERAT VARIOUS TEMPERATURES fraction of A . Since the quantity of A is unchanged, x~F= A ~X z F a 2 . The volume of B so added is (x2 xi)/pB so the total volume of solution 2 if no shrinkage occurred would be ( a l p l ) (x2 q)/pB. But the volume of sohtion 2 actually is z z / p z ; so, on making the substitution xz = x ~ F A ~ /and F Arearranging, ~ the shrinkage is: 1 -- 1 +-- 1 2+"21-!?=xlFAl P1 PB PZ ( F A Z P B F A ~ P ZF A I P ~
-
+
-
-
If * A is chosen such that *A
=
1 FAPB
-'
I +c
FAP
where C is any constant, it is apparent that the shrinbage can be found by forming the difference \Ens - \EA,and multiplying by FA^ and xl. Moreover, since X ~ F A= I X ~ A Z the , shrinkage can also be found by multiplying the difference of \EA by Fa2 and x 2 . The value of C selected was ( l / p A ) (l/PB)l
so
This choice makes !PA = 0 when F A = 1. Because weight fraction F is dimensionless, the dimensions of \zr itself are the reciprocal of those of density. I n all the examples given here the units of density are grams per milliliter, so the units of \E are milliliters per gram or liters per kilogram. Had the density been taken in units of pounds per gallon, then the units of \k would have been gallons per pound. (To convert from milliliters per gram to gallons per pound, divide by 8.3452.) As would be expected, \EA,in which the quantity of constituent A is held constant, is different from \ k ~ in , which the quantity of constituent B is held constant. Figure 3 shows the difference for the system A = ethanol, B = water. Nevertheless, either \EA or * B can be used to solve most problems by means of the following general equation with properly interchanged subscripts: shrinkage
=
ZIFAI( I n s
-
*AI)
+ rzFa2
(*AS
I n computing * A it is found that as F A approaches zero, \EA approaches the indeterminate form O/O, which means that more and more accurate densities are needed for the weight fractions approaching zero to give * A the same degree of accuracy. This loss of accuracy in computing \EA in this region is offset by a reduced need for accuracy, since the values of \EA will be multiplied by smaller and smaller values of F A in computing the shrinkage. The shrinkage factor is related in a simple manner to a function which has a certain vogue, the "apparent volume" usually denoted by @. If the difference between the actual volume of a solution and the sum of the volumes of the two pure constituents A and B is ascribed wholly to constituent A (by expansion or contraction of A ) , then @ A is defined thus: @ A = - -1FA PFA PBFA where the units of @ A are the reciprocal of those of density. By comparison with the expression for \EA,it is seen that@A = ( l / p ~ ) - \Ea. The apparent volume can also be used for computing the shrinkage by substituting - @ A for !PA in the equations. The fact that a function such as @ A , in which is made an assumption concerning the division of the volume change of the solution as between the two components, can be used to compute precisely the volume change on mixing, is quite unexpected. @ A is frequently given as the apparent molal volume, in which case the molecular weight must be taken into account. The partial volumes may be readily calculated from * A by measuring the slope of the curve \EA us. F A and using the equation which follows.
- '@Az)
This equation shows the shrinkage when X I weights of solution 1 containing FA^ weight fraction of A are mixed with x 2 weights of solution 2 containing FA^
FIQURE5. WEIQHTFRACTION-SHRINKAQE FACTOR FOR SYSTEM A = SULFURIC ACID,B = WATERAT 20" C.
INDUSTRIAL AND ENGINEERING CHEMISTRY
952
where TJB = partial molal volume of B W E = molecular weight of B. By reasoning similar to that above, the shrinkage factor for volume fractions is found to be
I n this case the units are liters per liter, gallons per gallon, etc. Figure 4, which shows \kAfor ethanol-water mixtures a t various temperatures, indicates that a t higher temperatures the amount of the contraction becomes reduced and the expansion a t low ethanol concentrations tends to disappear.
Vol. 34, No. 8
Figure 5 shows Q 4 for sulfuric acid-water solutions. When the initial solution that is to be diluted with water has a round weight fraction, it becomes unnecessary to take two readings of \kA and to form the difference. The chart is entered on the appropriate curve (that corresponding to the initial weight fraction), and the curve is followed to the weight fraction corresponding to the solution after dilution. This value of \EA is then multiplied by the weight of solution and the weight fraction as before.
Literature Cited (1) Baxter. G. P.. and others, J. Am. Chem. SOC.,33, 901, 922 (1911): 38, 70 (1916). (2) Lexis and Randall, "Thermodynamics and the Free Energy of
Chemical Substances", Chap. IV, New York, McGraw-Hill Book Co.. 1923.
Correlating Gas Solubilities and Partial Pressure Data DONALD F. OTHMER AND ROBERT E. WHITE' Polytechnic Institute, Brooklyn, N. Y.
F FUNDAMENTAL importance in the study of any gas-liquid absorption or desorption process are the gas-liquid equilibrium or solubility data. A survey of the literature for these data reveals that much is still to be desired in the quality, quantity, and form of data presentation. Since a t least four variables (temperature, concentration in the liquid phase, concentration in the gas phase, and total pressure) must always be evaluated, it is evident that many data are required to define a system. Although experimenters have worked many years with various types of apparatus to obtain such data, these data are incomplete for practically every system. However, a variety of methods does exist for expressing gas solubility. Bunsen coefficient 01 is defined as the volume of gas (reduced to 0" C. and 760 mm.) dissolved in a unit volume of solvent a t the temperature of the experiment when the partial pressure of the gas (excluding the vapor pressure of the solvent) is 760 mm. (Data are expressed as corrected "volume in volume,'). Experimental results often are recorded in terms of this coefficient without maintaining the specified conditions. I n some cases the partial pressure of the gas has not been maintained a t 760 mm., due either to neglect of the vapor pressure effect of the solvent or otherwise. The Ostwald solubility expression, I , represents the ratio of volume of gas dissolved a t any pressure and temperature to volume of absorbing liquid. This expression differs from the Bunsen absorption coefficient in that the volume of dissolved gas is not reduced to 0" C. and 760 mm. The solubility is therefore the volume of gas dissolved in unit volume of solvent under conditions of the experiment (volume in volume). Similar to the Bunsen coefficient is the Kuenen coefficient which gives the volume of gas, measured a t standard conditions, dissolved in one gram of the solvent a t the temperature of the experiment, when the partial pressure of the gas is 760 mm. (volume in weight).
0
1
Present address, York Ice Machinery Corporation, York, Penna.
The Raoult coefficient expresses the solubility as grams of gas dissolved in 100 cc. of solvent a t the temperature of the experiment, when the partial pressure of the gas is 760 mm. (weight in volume). There seems to be no commonly used name for the solubility expressions in terms of the other combination of measurements, such as weight of gas dissolved in weight of solvent, or moles of gas in a mole of solvent. These latter two ratios are by far the most useful in any engineering calculations. Furthermore, gas solubility is often expressed in terms of the Henry law constant, H , which relates the concentration of dissolved gas to the partial pressure or concentration of gas in the gaseous phase. (This has been used as a measure of the gas dissolved, even in those cases where the law is specifically indicated as inapplicable.) The Henry law constant varies with temperature and with the nature of gas and solvent; its numerical value depends on the units in which the pressure and concentration are expressed. Henry's law is applicable over a wide pressure range with reasonable accuracy for many systems. On the other hand, many systems deviate widely from this relation; in general, deviations increase with increase of partial pressure of the gas and with decrease of temperature. Gases of low solubility, such as inert gases, and gases that do not change molecular form in solution usually follow Henry's law closely. Gases that combine with or dissociate in the solvent show large deviations. A particular handicap has been the fact that the variation of the Henry law constant with temperature is not linear.. Markham and Kobe ( 5 ) ,in a review of gas solubility determinations and data, mention the shortcomings of the methods for expressing gas solubility but made no attempt to suggest more useful methods of expression or correlation. The International Critical Tables contain considerable gas solubility data expressed in terms of the Bunsen coefficient and also the Henry law constant. Here, in particular, there has been a confusion of several coefficients,due to variation of the pressures.
