July, 1930
I N D VSTRIAL A N D ENGISEERILVG CHE-MIXTRY
The work is part of the research program on drying oils of Archer-Daniels-Midland Company and William 0. Goodrich Company, and affiliated companies. Acknowledgment is due these companies for permission to publish the results.
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Literature Cited ~
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~ 73 (1929) .
(3) Theis, Long, and ENG (4) Twitchell, I b d , 13, 606 (1921)
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al, 1244 (1929),
Vapor Pressure and Heat of Vaporization’ P. G. Nutting U.
s. GEOLOGICAL SURVEY, W A S H I N G T O S , D. C:
WIXG doubtlws to its industrial importance in plant design, a number of papers on the vapor pressure and latent heats of liquids have recently appeared in INDUSTRIAL ASD EKCXSEERISG CHEMISTRY (2, 3, 4 ) . Duhring’s rule has found wide application in the calculation of vapor pressures beyond the observed range, and recently Schultz (5) has developed a relation of similar form between the molal entropies of two liquids, yielding heats of vaporization and critical temperatures. A great many other empirical relations have been suggested, ranging from Trouton’s simple ML = 21 Tb t o the complicated formula of Cederberg. Any new formula must be both simple and exact to be useful. Thermodynamics leads no further than the exact relation
0
(1):
between internal thermal energy, E, mechanical work, TV, and vapor pressure, p . E is here identified with the internal latent heat, W with external latent heat, and p with pressure of saturated vapor. While Equation 1 cannot be salved in general terms, it is well adapted to the calculation of E when p is known through a range of temperatures. Another advantage is that it is dimensionless and therefore independent of the units employed. dW/dm is known exactly, if concentrations of vapor and liquid are known, from the Clapeyron equation: dW
RT
C,
dm = -p(G
calories - 1) gram
A check of the combination of Equations 1 and 2 in the form calories gram
(3)
with numerical data shows that it does give results as certain as the data, as it should. In setting u p empirical relations between L, p , and T for practical work, the forms of Equations 1 and 3 indicate the paramount importance of a relation between p and T which shall make d log p / d log T simple. Most such relations previously suggested are parabolic in form, exceedingly cumbersome to use, and afflicted with troublesome dimensions. The approximate energy line obtained by plotting log p against 1/T is nearly straight but not so convenient as the more curved line given by the plot of log p against log T . This can be very accurately represented by the hyperbola (1% 2-
+ a)(logP + b )
=
c
(4)
which is dimensionless throughout, since the constant a is the (negative) log of a temperature and b the log of a pressure 1 ReceiLed April 1 6 , 1930 Published by permission of t h e Director, U S Geological Survey.
+
that log T a = log T / A , etc. The derivative of Equation 4 is also dimensionless and very simple.
PO
(5)
Values of the constants a, b, and c for water, carbon tetrachloride, and toluene are tabulated below. All three are negative in each case and all are referred to natural logarithms. The pressure units are indicated, as are also the temperatures used in determining the constants. Constants of (log T T P Water Water Carbon tetrachloride Toluene
+ a) (logp + b ) = C
0, 50,100 mm. 100,200,300 a t m
-4 -4
a 19916 43976
-29 -19
b 73019 76527
-39 78249 -29 26610
100,180,260 mm. 110,214,320 a t m
-4 -4
19173 54265
-17 39190 -15 77465
-28 95956 - 2 2 19930
C
Pressures calculated from Equation 4 using these constants are in excellent agreement with those observed and will even stand for considerable extrapolation. Relation between Heat of Vaporization and Temperature
The other new formula is a relation between heat of vaporization and temperature, namely: or log
L = A ( T , - T)” = log A n log ( T c- T )
L
+
(6)
latent heat proportional t o a power of the temperature measured down from the critical temperature. There is a rational basis for this formula, since internal latent heat represents work done on a liquid by the cohesive force within it and T , - T represents an energy deficit from a temperature, T,, a t which the cohesive force is zero. The constant n is about for latent heats and 2/3 for surface tensions as might be anticipated. For carbon tetrachloride L = 6.128 ( T , - T)0.379 holds very well from 0” C. to the critical temperature 283.1. For water, L = 8.2674 ( T , - T)0.33357. For ethyl alcohol. n = 0.36; for COe, n = 0.424; for SO2,.n = 0.64. The linear relation indicated by Equation 6 is shown t o hold very well in all the cases tried. Log surface tension plotted against lag L is also a perfectly straight line for water, the only liquid for which a range of data are available. Critical temperatures may be calculated by Equation 6, but probably more easily by the Schultz formula relating differences in molal entropies. Literature Cited (1) Bridgman, “Thermodynamic Formulas,” p 20. (2) Cox, IWD. ENG CHEM, 16,592 (1923) (3) Davis, I b r d , 17, 735 (19251, 22, 360 (1930). (4) Krase a n d Goodman, Ibcd 22, 1 3 (1930). ( 5 ) Schultz, Ibzd , 21, 557 (1929) (6) White, I b r d , 22, 230 (1930)
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