Simple Equations for Vapor and Liquid Densities R. R. DREISBACH AND R. S . SPENCER The Daw Chemical Company, Midland, Mich.
Methods have been developed for calculating the orthobaric vapor density of compounds accurately to about 4 atmospheres by means of a modified Antoine equation where only the molecular weight and the boiling point at any pressure are known and for calculating liquid densities of compounds over the same range using law of rectilinear diameters, where boiling point at any pressure and liquid densities at two temperatures are known.
I
N a previous paper the authors ( g ) gave formulas for the determination of the rate of change of boiling point, with pressure at any pressure and temperature up to the critical point. As a further development of this work, new methods for calculating vapor and liquid density have been worked out. Antoine ( 1 ) apparently was the first worker to represent the relationship between the density of a saturated vapor and the temperature by means of an equation. Later, Schmidt (6) proposed a similar method for representing the product of saturated vapor and liquid densities. Antoine's equation was: log v
=
K
+ 947 5A
where 8 was the temperature plus a constant varying from 219 to 246 depending on the substance. This relationship was never developed to the state of general usefulness, probably because the Antoine equation for vapor pressure has been little used. The infrequent use of the Antoine equation probably stems from the fact that its accuracy has been underest,imated. The Thomson (6) paper and the authors' ( 2 ) paper have given considerable data t,o prove the high accuracy of this equation for vapor pressure. In the present, investigation the authors were able to obtain the parameters of the vapor density equation from the gas laws themselves and the plot of the vapor density on the Cox chart and, in consequence, this expression becomes of great, value oM-ing to it's accuracy and the ease with which the parameters are obtainable from the molecular weight and the A and B values of the Antoine equat,ion for the vapor pressure of the compound. If the equation for the perfect gas law is expressed in terms of millimeters, grams, milliliters, and degrees Centigrade, then the expression PV = RT becomes:
tm
gr when t,he expression A from the Aiitoirie vapor pressure equation is substituted for its equivalent log,,P. d, = vapor density a t t o C. in grams per ml., M = molecular weight, A and B = constants of the Antoine equation for vapor pressure, R = gas constant = 62,363 m1.-mm. per C., and t = temperature in O C. O
As this equation is based on the perfect gas laws it is accurate only at temperatures at which the vapor of the product obeys
this gas law. This condition seems t80hold very well a t reduced temperatures ( T R )of about 0.5-i.e., (3) where tL = critical temperature in C. However, the equivalent reduced pressure ( P R )seems to be more nearly a corresponding state than the reduced temperature ( T R ) . As a starting point n-hexane was used and the reduced pressure ( P R )corresponding to a reduced temperature ( T R )of 0.5 was determined by means of the Antoine equation for vapor pressure. This gave a reduced pressure, P R , of 0.00072. For all other compounds, then, the temperature used for evaluating d, was the temperature corresponding to a P R of 0.00072. The value used for the pressure mould be the critical pressure times 0.00072. This value is then changed into the corresponding temperature by means of Ihe Antoine equation for vapor pressure. The authors now arrive, probably, at the real reason why this method has not been used previously, except in a very narrow range. It is the fact that the values obtained for the vapor density by means of Equation 2 are valid only over the narrow range where the compound rigidly obeyed the gas laws. There is a way, however, of removing this narrow limitation A plot of accurately determined vapor density on the Cox chari ( 2 ) is a straight line, when the ordinate log scale reads 0.00001 gram per ml. for the first cycle and increases by a value of 10 for each successive cycle and the abscissa temperature scale is left intact. These values of d, plotted against temperature will lie on a straight line to about 10 atmospheres, but beyond this the plot curves upward and to the left of the continuation of the straight line. These line8 in the hydrocarbons of a Cox chart family tend to meet in a small circle, the so-called infinite point. Since the vapor density values of a compound plot as a straight line, a simple equation in two parameters will express it. Such an equation is: O
D*
logd,
=
A*-
D '
~
t
+ 230
which has the same form as the Antoine equatiorl for the vapor pressure. The relation between the constants A* and H * and the constants of the vapor pressure equation will be discussed below. This equation gives values that are accurate and eliminates the errors due to deviation from the gas laws, for it is based on experimental data. To evaluate the two parameters A* and B* in Equat,ion 4 it is necessary only to have the value of log d, a t two temperatures. As seen from Equation 3, log d, is easily evaluated a t any temperature where the gas laws are obeyed by the vapor of a compound. By trial it was found that values of d, obtained at a temperature corresponding to a reduced pressure ( P R ) of 0.00072-and a t this temperature plus 10" C . for the second value of dv-gave values of A * and B* which check the experimental values with excellent precision up to about 4 atmospheres and with decreasing precisinn up to about 10 atmospheres.
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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
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The method described above for obtaining A * and B* is disadvantageous in that the use of a small temperature interval decreases the accuracy of obtaining these constants. It was pointed out by Thomson ( 7 ) that A* and B* can be obtained in a very simple manner as a function of a single temperature, tl, a t which the vapor obeys the ideal gas laws.
It is shown easily from the preceding equations that:
TABLEI.
Vol. 41, No, 7
DATA USED FOR CALCULATING VAPOR DENSITIES 0.42%; A * = log iLI 1 log R log 7'' X. - B - H * / t 230) ChloioComliound Pentane Hexane Heptane Hrndene benzene Boiling point, C. 36.074 68.742 98.428 80.103 131.70 A , value 6.82858 6.93882 7.04427 7.04623 7.18402 B , value 1050.4 1212.3 1367.4 1291.4 1556.8 Tc, K. 470.4 508.0 540.0 562.7 632.2 B B* 66.3 75.0 83.5 84.0 101.0 B 984.1 1137.3 1283.9 1207.4 1455.8 A* 1.7521 1.31889 1.45069 1.34130 1.57147 t , O -41.8 -16.5 -1.5 +0.33 39.4
(B
- B*
= 84.14
+
+
-
+
I -
-
c.
If t P - tl is less than 30" C., i t can be shown empirically that B - B* is close to a h e a r function of tz - ti for each t,, 80 that:
B
- B" =
where
e
e = 84.14
and
f
=
0.205
ndion
- ti)
f f ( t 2
+ 0.00035 ti
P, Atmos-
Compound
=
84.14
+ 0.428 ti,
pheree
( 71
which is valid t o *0.1 fioni 0 " to 200" dr ocltrboii separations In the following derivation, redefinition of components for calculating p , and is essential for arriving at a u~liversalrelationship among the separation selectivities and in a few instances, results in selrctivities which are reciprocals of the selectivities obtained by the usual definitions. I n many cases no changr in U , 6, or y reaultr. The definition ~ n arrived s at bv x
consideration of the tR-0 equilibrium liquid phases involvcd ili liquid-liquid extraction and t,heir equilibrium vapor. I n such H t,hree-phase system the liquid extract, phase with the ve,por phase constitutes a simple vapor-liquid extraction system which must be related thermodynamically to the simple liquid-liquid extraction system formed by the two liquid phases. The components A and B to be separated are chosen such that the following relation exists between their activities, a , as solutw a t infinite dilution in the solvent a t t.he temperature involvcd:
> a,H
aA
(1)
which, where the vapors behave as perfect gases is sirnpl?,
for equiniolar solutions of A and B in the solvent a t concentratioils There Henry's law applies. The term vapor-liquid is used herein in the broadest sense embracing the processes often referred to ns azeotr opic and extractive distillation which differ only in that the proximity of the volatilities of the feed and solvent used in azeotropic distillation results in formation of a constant boiling mixture between the solvent and one or more feed components. The only assumption made in the following derivation is that the vapors behave as perfect gases. Under conditions such that the. deviations are appreciable, fugacities may be used directlv. DEFINING
a,
8, AND
y IN
TERMS OF ACTIVITY
F O R 4 L I Q W ~AND D A FT.4P0R P H ~xuLEQT'ILIBRII v The. wntional volatilitv of e given voniponent i i drfinrtl v =
the relati.vr volatlhtv ( C Y ) ot
c,
{YJII-
(2)
A\
co1~lpoll~~~t* PE
& . = % " ' =s/ A VB 1LT R
