Data for equilibrium 'constant K , used in distillation and gas absorption design calculations, are correlated by a logarithmic plot against vapor pressure o f a reference substance at the same temperatures to give substantially straight lines. A nom ogram has been prepared based on this correlation which gives K values for the hydrocarbons below nonane at different pressures and temperatures in a form which is useful when constant K values are to be compared. A second nom ogram is also presented which is particularly useful for obtaining K values for hydrocar
bons in this range at the sam e temperature and pressure as would he necessary, for example, in plate-to-platc cal
culations o f a distilling column handling hydrocarbons.
I
N PRE VIO U S articles (J, 2) a logarithm plot of vapor or related pressures of a compound against the pressures exerted by a reference substance at the same temperature was used to correlate data of vapor
p r e s s u r e s , latent heats, heats of chemical reaction, g a s s o l u b i l i t i e s , heats of solution, adsorption equilib
rium pressures, heats of adsorption, and other prop
erties. It was also shown that the c o r r e s p o n d i n g logarithmic plot of reduced pressures at the same re
duced temperatures added somewhat to the precision of the plot and the accuracy of data obtained therefrom.
It appeared that the same plot might be a useful tool for correlating values of the so-called equilibrium con
stant, K , used in the design of distillation, absorption, and related systems involv
ing particularly hydrocar
bons, such as petroleum fractions.
This equilibrium constant is usually derived, as by Sherwood (4), from Raoult’s law
p = P-x (1)
D O N ALD F . O T H M E R Polytechnic Institute, Brooklyn, N. Y.
and D alton’s law,
P = (2)
which, when combined with Avogadro’ s law, give the equilibrium relation:
p = P ,x = P y (3)
where, at a given temperature, p is the partial pressure of the hydrocarbon under discussion in the gas phase; P . is the vapor pressure of the pure component; P is the total pressure;
x and y represent the mole fractions of the particular hydrocar
bon in the liquid and the gas phase, respectively; and S P
repre-T e m p e r a l u r e ° F .
Figure 1. Equilibrium Constant K Values o f M ethane (Solid Lines) and o f Propane (Dashed Lines) at Constant Pressures, Plotted as Straight Lines against Vapor Pressures
of W ater and Corresponding Temperatures on Log Paper
670 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y Vol. 36, No. 7
Figure 2. Noraogrum for Determining Equilibrium Constant K
A M tr a i^ h t l i n e t h r o u g h t h e t e m p e r a t u r e s c a l e a n d t h e p o i n t r e p r e s e n t i n g t h e p a r t i c u l a r c o m p o u n d a n d p r e s s u r e o n t h e c e n t r a l g r i d g iv e s K. v a l u e o n t h e l e f t - h a n d s c a l e .
seats the sum of the partial pressures of the various components.
More usually fugacities are substituted for the pressures and the above equation is written:
used as the reference substance in all subsequent work; it follows that if straight lines are ob
tained when plotted against water, straight lines would also be obtained if plotted against axis at the appropriate values corresponding to the vapor pressures of the material used as a reference, in this case water; ordinates corre
sponding to these temperatures are erected for the data tabulated. The temperatures are 60°, 80°, 100°, 150°, 200°, 300°, and 400° F .; there
fore, the corresponding seven ordinates were located and drawn at the respective vapor pres
sures of water. plotted on the temperature ordinates.
5. Points representing the same pressure
sponding to the total pressure, and K is the so-called equilibrium constant which is thus defined. Sherwood (4) gives tabulated values of K for the lower hydrocarbons which have been calcu
lated from the fugacity curves for these materials. Other data are available also from other sources (e.g., Robinson and Gilli
land, 8), extending the limits both as to temperature and pressure for these values of K.
L O G A R I T H M I C P L O T O F K
This ratio of the composition in the vapor phase to that in the liquid phase is extremely important in design w ork; in an at at the same temperatures gave a substantially straight line in the temperature range to be used. Because of the ease of working
on logarithmic paper; if the slope is a constant, the line is straight..
Thus:
log K = a log P , + & ( 8 )
where a and b are constants which depend, among other things, on the particular total pressure in question. Equation 8 is based on an assumption of Raoult’s law and the gas laws. Where these do not hold, fugacities have to be substituted for pressures; and a similar equation would result which would hold over a wide range.
The present correlation is most useful in the temperature and.
July, 1944 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y 671 drawn for the temperature scale, calibrated on a logarithmic scale of pressures with temperatures corresponding to the vapor pres
sures of water. This was done as for the X axis o f Figure 1, and values of temperature increased downward.
3. Several values of K for methane were taken from the tables at the same pressure and different temperatures. Lines were drawn between the K values on the left-hand scale and the corresponding temperature values on the right-hand scale.
These lines intersected in a common point between the two
scales. v
4. Other values o f K for methane and corresponding tempera
tures were taken at another pressure, and an additional point was likewise found.
5. These steps were repeated for each o f the six pressures for methane, and then at all six pressures for each of the other hydro
carbons, whose values are tabulated by Sherwood (-•{).
6. Lines connected all of the points for each compound, and give greater precision in use. Although the intersections deter
mining the points on the grid showed slight variations, which at the same temperature and pressure, another nomogram may be made. Figure 3 was constructed graphically starting with the linear relation shown in Figure 1 between log K and log Pv at the same temperature and the additional relation; at constant temperatures there is a roughly proportional variation of tabu
lated values of K with the absolute pressure (as indicated by Equation 5 or the gas laws). This representation gives a logical presentation of the three variables, pressure, temperature, and K ,
672 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y Vol. 36, No. 7 since K is usually regarded as the dependent variable; in this
chart it falls on lines between those representing the other two.
As a first step in constructing Figure 3, plots of K against the vapor pressure of water at the same temperature were drawn as isobars, as indicated in Figure 1 for all of Sherwood’s data (4), covering the hydrocarbons up to n-octane. (Because o f the fact that in most cases the relation o f K to pressure changes be
tween 25 and 50 atmospheres, data at 50 atmospheres were not used.) These plots were simply made as all of the data are tabu
lated at the same six temperatures. The six ordinates, corre
sponding to the vapor pressures o f water at the six temperatures, were drawn on identical sheets of logarithmic paper. The values of K were then plotted to give the isobars for each compound. drawn at the left-hand margin of the sheet and calibrated logarith
mically throughout the range o f pressures from 0.5 to 25 at
mospheres. These values increased upward. The temperature scale, a parallel line, was then drawn a convenient distance to the right and calibrated identically with the calibrations on the X axes o f the plots o f log K vs. log P , just mentioned. These tem
peratures increased downward. The original sheet o f Figure 1 (for methane) was then superimposed on the sheet for the nom o
gram so that the X axis (the temperature axis) coincided with the temperature line o f the nomogram. A ny given value of K on Figure 1 was now a vertical line, since Figure 1 had been ro
tated 90° in superimposing its X axis on the temperature scale.
This line o f constant K intersected the isobars at different points.
Thus, in the insert sketch of Figure 3, the vertical line on the K plot intersects three isobars at as many different temperatures.
The points for the K scale for methane were then located mechanically by the following steps:
1. The intersections o f the values of a constant K line with the isobars are projected to the left until they intersect the temperature scale: the resulting points indicate the temperatures corresponding, respectively, to the pressures where K has the points o f pressure are used. As a mechanical aid in construction, pins are driven into the drawing board through each of these points on the pressure scale. The straight edge is always pressed against one o f them in drawing the line to locate the K points. This could be done automatically with one hand to locate the one point on the line, while the other hand was moving the other end o f the straight edge to the temperature point on at the higher temperatures and pressures. These calibrations were not considered; and the lines were discontinued at values of K where the original data could not be represented on the nomogram.
7. It is possible to construct a grid work of these intermediate scales by connecting points o f constant K on each of the scales each particular plate. B y connecting the points corresponding to the operating conditions on the pressure and temperature scales with a straight line, the intersections of this line with the intermediate scales representing each individual component may be read by this one setting to give the respective values of K for each o f the components at the conditions o f temperature and pressure existing on the plate. Here, again, as in Figure 2, the representation of the unsaturated hydrocarbons is somewhat dif
ferent from that for the saturated compounds.