Correlatin
s
an roperties
elate
Vapor compositions, equilibrium constants ( K ) , activity coefficients (y), and relative volatilities (a)as functions of pressure and temperature are plotted directly to give straight lines by the method previously suggested (8) for vapor pressures. A minimum of data may thus be used to define a whole system, arid the consistency of experimental data may be evaluated readily. A new and simpler plot and the corresponding equations are presented for partial pressures, vapor compositions, equilibrium constants, activities, and relative volatilities as direct logarithmic functions of the total pressure of the system.
This method i s convenient for correlating experimenta! data which are usually taken at constant total pressures, The slopes of lines on both types of plots are functions of latent heals and partial heats of solution of the components. Thus, experimental data on heats, vapor pressures, vapor compositions, and activieies, however determined, may be used for mutual cross checking, correction, and filling in where incomplete. Activities of SOlUtiOnb, only the solvent of which i s volatile, may be correlated. Furthermore, critical pressures and temperatures may be used (9) for either, especially near the critical range.
DONALD F. OTHMER AND R0GER GILMONT Polytechnic Institute, Brooklyn, N. Y.
V
APOR pressure and latent heats have been correlated by a simple plot or the use of the equation, log P = (LIL’) log P‘
+c
where a t the same temperatures, P and P’ are vapor pressures and L and L’molal latent heats, respectively, of two compounds (the latter in each case being that of a standard or reference substance) and C is a constant. Log P’ really serves as the temperature variable and is obtained directly from vapor pressure data of the standard substance. L/L‘ is nearly independent of temperature. A logarithmic plot gives a line which is substantially gives latent heat data a t any straight, the slope of which (L/L’) temperature for the compound in question from that of the reference substance. This vapor pressure plot was extended (9) to be used with reduced pressures a t reduced temperatures in order to increasi? the precision in use. Further applications were made to gas solubilities and partial pressures ( l a ) ,and to the pressures of adsorbed materials from adsorbents (11). A recent application of the method correlated equilibrium constants of hydrocarbons by means of a nomogram (10). The method has now been extended to systems of two or more volatile components; in addition to temperature and pressure variables, the variables of compositions of both the liquid and vagor phases have been included; the partial heats of solution calculated by the method satisfactorily correlate with thermal data. PARTIAL AND TOTAL PRESSURES OF BINARY SOLUTIONS
PLOTTED AGAINST VAFORPRESSURES OF REFERENCE SUBIn most cases (8) the equation is not needed and vapor pressures are plotted on logarithmic graph paper (or logarithms are plotted on ordinary graph paper) by three steps: (a) Corresponding temperatures and vapor pressurcs of the reference substance are read from a table; temperatures are indicated on the X axis a t appropriate values of pressures, with ordinates erected accordingly. ( b ) Pressure is plotted from the logarithmic scale of the Y axis on the respective temperature ordinates. (The STANCE.
same prcssure unit does not have to be used on both the X and ’iaxis, since there is a constant ratio between any two units; thib ratio would merely move the line up or down on the plot without changing its form or slope.) (c) Points so obtained are connected by a line, usually straight. When applied to solutions of two or more volatile liquids, it IF convenient but not necessary to use one of the liquids as the reference material. Figure 1 shows the total vapor pressure of various solutions of constant composition of the system acetic acidwater from other data (4); here, as in the other logarithmic plots, data are plotted at corresponding tempcratures. Figures 2A and 3 show partial pressures of mater (reference substance) and acetic acid. (Some slight curvature of the total pressure lines and of the lines of some other functions should result theoretically from addition of the partial pressure lines near the middle range of liquid composition when the molal latent heats of the two component$ are widely different. In the usual case, as in Figure 1, a plot doel; not show this very small deviation which may, for practical purposes. be neglected). PLOTTED AGAINST TOTAL PRESSURES OF SYBTEM. Another simple method of plotting which is useful in handling experimental data taken a t constant total pressure is shown in Figure 2B. The partial pressure of water from acetic acid solutions ir plotted against the total pressure existing on the system, The slopes of the lines of Figures 2 A and 1 are constant and are equal, respectively, to d log p J d log p ; and t o d log P / d log p i . The quotient, d log p , / d log P , must also be constant and is the value of the slopes of the lines of Figure 2B; hence these lines of partial pressure directly plotted against total pressure must be straight, a9 they are shown to be. VAPOR COMPOSITION
The vapor composition VI, percentage of more volatile component in the vapor in equilibrium with the liquid of composition 2,. is defined by Dalton’s law: VI
= p l / P or log y1 = log pl
- log P
INDUSTRIAL AND ENGINEERING CHEMISTRY
September, 1944
TernReralwe.
859
of the slopes of lines of partial pressure and of the lines of total pressure for the same liquid composition. That these slopes we constant (i.e., straight lines are obtained for the vapor composition plot) is shown in Figure 4 for aqueous acetic acid solutiom. This algebraic derivation t o explain the previously observed straightneas of the lines was suggested by Buffington (3). A plot of vapor compositions against total pressures also gives straight lines, as shown in Figure 5 where the data of Keyes (6) are included, This new plot against total pressures is particula& useful in correlating experimental data, which are usually taken at constant total pressures and are thus plotted directly om a standard sheet of logarithmic paper.
"C,
EQUILIBRIUM CONSTANTS, RELATIVE VOLATILITIES, ACTIVITY COEFFICIENTS
200
100
Vapor
600
400
1000 1500
Presswe Water, rnrn.Hg
Figure 1. Total Pressures of Aqueous Acetic . Pressures of Water at Acid Solutions ~ s Vapor Same Temperatures Lines of constant composition,indicated in mole % water
At a given liqu',d composition, this may be differentiated with respect to log p i (the logarithm of the vapor pressure of pure reference substance at the same temperature) : d log YI d log p:
dlog pi d log p?
dlog P d log p t
The last term is the slope of the lines in the plot of total presawes (Figure 11, and the middle term is the slope in the partial prassure plot (Figure 2A). Both terms on the right are constant dnce the lines of these slopes are straight; the left-hand term must, therefore, also be cohstant. It represents the slopes of lines on P log plot of vapor composition, and is equal to the difference
50
60
5 ,
By similar treatment of algebraic quantities representing the slopes of lines, other properties give straight lines, both on the earlier plot against a reference substance and on the new direct plot against t o t d pressures. While these functions may be considered for both components, only the more volatile are included in Table I, which also gives the slopes of the lines. The equilibrium constant K = y/x may be plotted more readily than y since, as y approaches zero, the plot extends indefinitely while K approaches a aonstant. The K plots have the same slopes as the corresponding y plots; and the slopes of both may vary somewhat if y shows very large variations with pressure and temperature (at constant x) which it usually does not. plots of the relative volatility OL = K1/& = ylxz/xyz do not have these disadvantages, since this function does not approach eero, and the very slight curvature of the lines is not present. A useful plot is that of activity or activity coefficient, yi pi/p:xl. Figure 6 presents the activities of water and acetic acid. Because of the small range of values, the vertical scale is expanded considerably by plotting logarit,hms on the ordinary vertical ruling of semilog paper; and this expansion could be carried as far as might be desirable t o accentuate the spread of the lines or their slope.
Tern perot ure , "c. 70
00
90
Temperature,
a
200 500 1000 1500 Vapor Pressure Water, rnnrHg
100
OC.
I
Figure 2. Partial Pressures of Water from Acetic Acid Solutions us. Vapor Pressures o f Water ( A ) at Same Tern eratures and us. Total Pressures of System ( B ) at Same d?emperatures
too
200 500 1000 1500 Vapor Pressure Woler, mmHg .
Figure 3. Partial Pressures of Acetic Acid from Aqueous Solutions us. Vapor Pressures of Water at Same Temperatures
860
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY TABLEI. Loo PLOTSFOR CONSTANT VALUESOF x (ISOSTERES)
Vol. 36, No. 9
APPLICATION TO HYDROCHLORIC ACID-WATER
The first twelve of these functions give straight lines on logarithmic paper when plotted against the total pressures on the system or the vapor pressures of a reference substance a t the same temperatures. E / T us. pressure gives straight lines on semilog paper, and E vs. temperature gives straight lines on standard paper. The slopes of these lines are the heat ratios indicated in the last column. Only equations and plots for one component are indicated, those for a second or subsequent volatile component are similar; similar equations and plots may be made using constant values of vapor composition where desired.
In the application of these methods t c i g a solubilities (12), the plots of partial pressures of hydrogen chloride over aqueow mlution8 were straight lines. Similarly the plot of total pressures gives substantially straight lines, ae does the plot of partial pressures of water Figwe 8 shoas that the lines for vapor composition are straight within the accuracy of experimental data. A maximum C.B.M. ~tppears which is reflected in the intersections of the lines of total pressure. The C.B.M. ipJ also indicated in Figure 8, where a line of given weight per cent of hydrogen chloride (in the liquid) intersects the abscissa of the corresponding mole per cent of water (in the vapor). The temperature corresponding to the constant boiling point is the ordinate. Akerlof and Teare (a)carefully determined tlir activities of water in hydrochloric acid solutions by measurements of electromotive force rather than that of pressure. The lines representing their data (Figure 9) are straight which corroborates this method of correlation PARTIAL HEATS OF SOLUTIOh
The slope of a line in a vapor pressure plot the ratio of the heat quantity in converting t o the vapor state one mole of the giveD material compared t o the molal latent heat at the given temperature of the reference substance. The heat required to vaporize one mole of one coniponent ( L ) from a solution ie equal to the latent heat of the pure component ( L O ) , plus its partial heat of solution (11) which is equal to and opposite in sign to the heat evolved in dissolving one mole in B volume of solution so large that itr composition will not change. From the equation8 of Table I the slopes of the lines of successive plots are shown constant by interrelation with the slopes of lines of previous straightline plots. Furthermore, the slopes are shown successively to be ratios of heat, quantities which may also be shown to be constant. 1s equal to
APPLICATION TO CHLOROFORM-ETHANOL
The careful measurements of Scatchard and Raymond (15) on the system chloroform-ethanol are used to illustrate the application of some of the correlation plots developed above. Vapor and liquid compositions were given for isothermal distillation a t three temperatures instead of the more usual isobaric distillation. Plots were constructed on Figure 7 against ethanol for total pressure (P),equilibrium constants ( & = yl/xl and K z = yz/zJ, and relitive volatility ( a = KJKZ). Straight lines result in each case, even though there is a constant boiling mixture (C.B.M.) and considerable deviation from ideality. The curved line in the relative volatility plot represents a constant total-pressure line of 300 mm. of mercury; its shape (nhich mould be straight if Raoult's law held) illustrates the irregularity of the system. The presence of the minimum C.B.?YI. (maximum total vapor pressure) may also be noted. On the plot of total pressures some of the lines intersect the line for lOOy0chloroform; in others K , = Kz = 01 = 1for the C.B.M. These plots offer a simple means for obtaining the change of C.B.M. with temperature and pressure. Composition lines can be extrapolated in the a or K plots until they intersect the line for which 01 = K = 1, giving the corresponding temperature; the prr3sure is then obtained from thr plot of total pressures.
Ternveraiure, "C
100
200 Vapor
300
Pressure
500 800 I000 W a t e r , mmHg
Figure 4. Composition of Vapors over ,4queaus Acetic Acid Solutions CS. Vapor Pressures of Water a t Same Temperatures
INDUSTRIAL AND ENGINEERING CHEMISTRY
September, 1944
861
uncertainty was also determined. The length of the arrows indicates the uncertainties involved in the lower direction only, since it appears that these values are too high. From Figure 9 the slopes of the lines of activities were determined by the method of least squares to give the ratio directly. Finally, the values of the ratio as calculated from the partial heats of solution (IS) and the integral heat of solution (14) were plotted. Excellent agreement is shown in Figure 10 among the data obtained by these independent types of measurement and backgrounds of theory. OTHER APPLICATIONS
IO
100
1,"i I\:,),]
200
300 400 Total Pressure, mm.Hg
5
600 800 1000
Figure 5. Compositions of Vapors over Aqueous Acetic Acid Solutions us. Total Pressures of System at Same Temperatiires The squares represent data of Keyem (6)
The lines of both Figures 4 and 5 are nearly horizontal, an indication that the vapor composition does not change greatly with a moderate change of pressure (the normal case). In the particular case where L: = Li and Raoult's law holds, the slopes reduce to zero; and the lines are horizontal. The vapor composition (2,v) diagrams are then identical at all pressures; and all of the lines of partial pressure and of total pressure (including those of the pure components) are parallel a t 45" on a log plot such as Figure 1, since the molal latent heats are equal for the pure substances and every possible mixture at the same temperature. Furthermore, the heats of solution would be zero, as is necessarily true whenever Raoult's law holds. (The constancy of the molal latent heat is assumed in the familiar McCabe-Thiele method of calculation of distilling columns, which thus assumes that the 2, y diagrams are identical at all pressures and the vapor composition lines on the log plot are all horizontal.) As indicated above, the slopes of the equilibrium constant ( K ) plots are the same as the corresponding vapor composition plots; hence the heat terms are also the same. The relative volatility function is useful because its equation is free of the term for the total latent heat of the mixture. The dopes of the lines on its respective plots are equal to the differences of the slopes of the vapor composition lines for the two components, and also to the differences of the slopes of the partial pressure lines. Even more interesting is the plot for activities or activity coefficients; the slopes are the same for both and give directly the ratios of heats of solution t o latent heats. Since latent heats are usually known (or may be determined by means of the basic log plots or other methods quite accurately), the heats of solution may thus be obtained from this plot. When desirable, the slopes may be increased for more accurate reading by increasing the scale of the Y axis, If Raoult's law holds, the lines all have zero slope and coincide with the X axis. If it does not hold, deviations are indicated by the heat of solution; on the plot of activity coefficients, this shows up directly by the relative slopes of the lines. HEAT QUANTITIES IN HYDROGEN CHLORIDE-WATER
This system was selected to test the correlation of heats since data are available for partial pressures ( 6 ) , vapor compositions (7), activity coefficient (g), and heats of solution (13,14), as functions of temperature and liquid composition. Figure 10 ehows the ratio of partial heat of solution to the heat of vaporiaation of pure water determined from these three sources. The slopes of the lines in a partial pressure plot for water ( 5 ) were calculated by the method of least squares from which the range of
Numerous other systems, some of three or more components, have been examined, and the results have corroborated the results above. A few studies of the three systems discussed above were selecled because they illustrate the mutual correlation of: (a) usual vapor composition data at constant pressures (wateracetic acid and hydrogen chloride-water) ; (b) vapor composition data taken a t constant temperature (chloroform-ethanol) ; (c) activity coefficients bbtained from e.m.f. measurements (hydrogen chloride-water) ; and ( d ) heats of solution directly obtained (water-acetic acid and hydrogen chloride-water) Furthermore, the systems represent the three usual types of binary mixturesLe., no C.B.M. (water-acetic acid), minimum C.B.M. (chloroform-ethanol), and maximum C.B.M. (hydrochloric acid-water). One of the most useful features of these plots is the ease of working back and forth between constant pressure and constant temperature conditions, depending on the method of determining or using the data.
.
5 IO
.-
E
20
2
70
a
100
I00
200 Vapor
300 Pressure
500
700
I066
Water, rnm Hg
u .u4r z >
LIU
c
50 60 90
5 .oz
a
80
inn
Vapor
' g
Pressure Water, mrnHg
Figure 6. Logarithms of Activity Coefficients of Water in Acetic Acid Solutions and of Acetic Acid in Aqueous Soiutions against Vapor Pressures of Water at Same Temperatures
INDUSTRIAL A N D ENGINEERING CHEMISTRY
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I00
150
200
300
Vol. 36, No.
calrulations f o r disl illiiig columns. Et, IF more advantageous to WE 21 as the par. ameter in the plot of log P us. log p y I ana it is necessary to have the specific heats
