SURFACE TENSION AND VAPOR PRESSURE
1163
correction was made for the heat of vaporization a t 25’C. in arriving at the above reported value. REFERENCES (1) MILES,C. B.,AND HUNT, H.: J. Phys. Chem. 46, 1346 (1941). (2) ROBSINI, F. D.: Bur. Standards J. Research 6, 1 (1931).
SURFACE TENSIOK AND VAPOR PRESSURE LUIGI Z . POLLARA Department of Chemistry, Siena CoZEege, Loudonville, New York Received January 10, 194.8
In order that a molecule escape from a liquid into the saturated vapor it must have sufficient energy to overcome the energy barrier set up by the surface forces. Consequently the number of molecules per unit volume in the vapor must be a function of the height of the barrier. In terms of measurable quantities this amounts to saying that the vapor pressure must be some function of the surface tension. The following empirical relationship between the vapor pressure and the surface tension is proposed, Tlogp = -ay
(,“f d)liai-b ~
where a and b are constants, M is the molecular weight, D and d are the densities of the liquid and the vapor, respectively, y is the surface tension, and 2’ and p have their usual meaning.
(D - d )
wa
Graphs of T log p vs. y
~
are shown in figure 1. The plots are
remarkably straight, even near the critical point, thus bearing out the fact that a and b are constants.
!!
Since a t the critical point y __ becomes zero, the constant b assumes (D d)l” the value Tolog p,. Therefore we can write as a final equation:
T log p =
-ay
(Sd)li3 + T, log p s
(1)
Equation 1 contains some useful implications, two of which are given here and others of which are to be discussed in a paper to follow.
1164
LUIOI 2. POLLARA
If equation 1 is combined with Katayama’s modification of the l%tvos equation,
a useful result is obtsjned:
T log p
FIQ.1. Plot of T log p uersw
= -k(Tc
y
- T) + Tolog p :
: curve 1,
ether; curve 2, methyl formate;
curve 3, ethyl acetate; curve 4, carbon tetrachloride.
The constant k can be evaluated at the boiling point:
Substituting this value in equation 2 and solving for log p : log p = log Pc (&)(I
-):
(3)
This is an expression for the vapor pressure in terms of the critical constants and the boiling point. Values of the vapor presaure calculated for several liquids, using equation 3, are compared with the experimental values in table 2.
1165
SURFACE TENSION AND VAPOR PRESSURE
TABLE 1 Data for j i g w e 1
Ether *K.
dm.
323 353 413 443 458
1.68 3.94 14.6 24.6 31.1
13.7 10.3 4.00 1.42 0.4
0.6713 0.6286 0.4936 0.3785 0.2698
72.6 210 480 618 684
315 246 113 47.9 16.9
Carbon tetrachloride
363 393 423 453 483
1.46 3.15 6.00 10.4 16.8
18.1 15.0 11.5 8.50 5.67
1.4475 1.3740 1.2914 1.1945 1.0687
59.8 196 329 461 592
415 347 277 217 156
323 413 473
1.47 4.61 17.8 48.5
20.5 16.0 7.63 0.93
0.9251 0.8698 0.7156 0.4131
53.7 234 516 797
331 269 146 22.7
363
1.50
15.7 12.1 5.70 0.50
63 .a 209 494 777
344
353
Ethyl acetate
453 513
0.8065
0.7580 0.6265 0.3278
288
154 20.8
Differentiating equation 3 with respect to temperature,
and comparing this with the Clapeyron-Clausius equation, d-l n p-- AH dT RP
an expression for Trouton’s ratio is obtained in terms of the critical constants and the boiling point:
Values of AH/TB have been calculated by equation 4 and compared with experimental values in table 3. Although it was originally stated that equation 1 is empirical, its form is certainly suggested by the following semi-thermodynamical reasoning.
1166
LUIGI 2. POLLARA
Since the free energy per molecule in the surface phase is equal to that in the vapor phase, we have Pu
= PO
TABLE 2 Comparison of ezperimental and calculated values of the vapor pressure
Carbon tetrachloride (b.p., 350°K.) 'K.
353 373 423 473 523
OM.
1
Methyl alcohol (b.p., 338°K.)
atm.
1.10 1.92 6.00 14.4 29.5
OK.
1.09 1.93 6.17 14.0 33.0
353 373, 413 453 493
I1
Ethyl ether (b.p., 308'K.) ~
323 353 373 423 453
1.68 3.94 6.40 17.6 28.8
atm.
atm.
1.76 3.45 10.6 26.4 56.2
1.71 3.34 10.3 26.0 56.2
Water (b.p., 373OK.)
-
~
~~
1.62 3.79 6.18 17.2 28.5
0.121 4.67 15.3 39.2 84.8
323 423 473 523 573
~
~
1
I
0.139
4.44 14.6 38.4 86.7
TABLE 3 Comparison of ezperimental and calculated values of Trouton's ratio AE TB
Tc-TB TO
LIQUID
ULPSEPIYENTAL
AH_ _
TB :c*LCuumn) -L_
Benzene.. . . . . . . . . . . . . . . . . . . . . . . . . . , , . Ethyl ether.. . , . . , . . . . . . , , . . . . . , . . . . . Ethyl formate. . . . . . . . . . . . . . . . . . . . . , . Carbon tetrachloride Argon, . . . . , . . . . . . , . . . . . . . . . . . . . . . . . . , Hydrogen. . . . . . . . . . , . . . . . . . . . . . . . . , . , Hydrogen bromide. . , . . . . . . . . , . . . . . . , Ethyl chloride. . . . . . . . . . . . . . . . . . . . . . , Methyl alcohol.. . . , , . . . . , , . . . . . , . . . . . Ethyl alcohol. . . . . , . . . . . . . . . . . . , . . . . . Water
0.372 0.341 0.357 0.360 0.424 0.394 0.433 0.381 0.341 0.319 0.424
1.68 1.55 1.67 1.65 1.68 1.11 1.92 1.72 1.90 1.80 2.34
B.8 20.2
20.7
22.0 20.4 17.3 10.9 20.4 20.9 24.9 26.8 26.1
21.4 20.9 18.1 12.8 20.2 20.6 25.4 25.7 25.2
20.6
The free energy in the vapor phase is given by the well-known equation: PO
+
= P ~ T ) k~ In
P
(5)
The free energy per molecule in the surface phase is given by the equation P~
=
+ Ar
where A is the area occupied by a molecule.
(6)
1167
SURFACE TEKSION AND VAPOR PRESSURE
The quantity A is somewhat arbitrary, since it depends on the position the bounding surface is given in the heterogeneous region between the vapor and the liquid. Subtracting equation 6 from equation 5 we have
+
kT In P = AY ( d T ) - &TI) (7) In order to make further progress it becomes necessary to make some assumptions as to the nature of the surface phase. As a first rough approximation we assume the surface layer as one molecule thick with the density of the liquid phase.
(g)
2/3
Using these assumptions the area per molecule becomes 2\r213
Com-
bining this with equation 7
where a absorbs all constants including the conversion factors.
To determine the nature of the term
-1& ( T )
assumptions, we can plot T In p vs. ( g f y .
- p ; ( T ) ) under the previous
The plots are essentially straight
lines except in the neighborhood of the critical temperature. We can therefore write
In order to get slightly better agreement with the experimental data and also
(g)
2/3
to have the value of b closer to T , In p,, the quantity
is replaced by
(x) , D-d zia
even though the latter cannot represent the area occupied by a
molecule.at the critical point. SUMMARY
An equation relating the vapor pressure to the surface tension is proposed:
From equation 1 an equation for the vapor pressure in terms of the critical constants and the boiling point is given as
Trouton's ratio in the following form is also given: A T, H = 2.303 R logP.(T.--T,) TO
