SURFACE TENSIOX OF SATURATED VAPORS AND T H E EQUATIOK O F E o T V o S BY J. L. SHERESHEFSKY
In discussing phenomena of solid-liquid or liquid-liquid surfaces it is usually emphasized that the properties under consideration belong to neither of the phases making up the interface, but to the interface proper. Thus it is made clear that the tension in a benzene-water interface is characteristic, neither of the benzene, nor of the water, but of the new phase of the interface which is made up of the latter two. But there is generally a lack of definiteness when dealing with liquid surfaces in contact with their own vapors. The properties of such surfaces are usually ascribed to the liquids concerned, without giving due consideration to the effect that the respectiye vapor phases undoubtedly have on these properties. Van der Waals was the first to point out that the surface represents n third phase whose properties differ from those of the liquid or vapor. There is thus a complete parallelism between liquid-liquid and liquid-vapor interfaces. The surface tension, say, of benzene is therefore understood to refer neither to the liquid benzene, nor t? the benzene vapor, but to the benzene (liquid)-benzene (vapor) interface, and is also conceived as the resultant of two effects produced by the surfaces presented by the liquid and the vapor. The Surface Tension of Saturated Vapors The tension in a liquid-liquid interface as determined by the method of capillary rise is given by the expression U= $r h g (p1-p2) (1) where h is the height of rise, r the radius of the capillary, and pi and p z the densities of the immiscible liquids in equilibrium with each other. This expression also holds for the interface between a liquid and its vapor, pl. and p z being the respective densities. It is therefore evident that the tension of a surface formed by the contact of two fluid phases, whether they be two immiscible liquids, or a liquid and its vapor, is given by the difference of two terms, and thus is analogous tc Antonoff’s’ rule, which states that the interfacial tension between two liquids is equal to the difference of the surface tensions of the liquids in equilibrium with each other. Thus, if u is the relative, or interfacial tension, u1 the surface tension of liquid ( I ) in equilibrium with the vapor of liquid ( z ) , and u2 is the surface tension of liquid (2) in equilibrium with the vapor of ( I ) , Antonoff’s rule may be expressed by the equation u = u1 - u2 (2)
Phil. Mag., ( 6 ) 36, 377 (1918).
SURFACE TENSION OF SATURATED VAPORS
1713
Applying this rule to an interface between a liquid and its vapor u represents the relative surface tension, ul the tension of the liquid surface proper, that is, of a hypothetical liquid surface whose vapor was momentarily removed or frozen, and upis the tension of the surface presented by the vapor. From this follows by analogy that u1 = $ rhgp,
(3)
and up =
8 rhgp2
(4)
A more rigorous derivation of these equations may be obtained from considerations of equation ( I ) as applied to liquid surfaces in contact with vapors. At temperatures far removed from the critical the vapor density term, p2, is usually omitted, since it is negligible with respect to the density of the liquid. As a result equation ( I ) becomes only approximate and assumes the more simplified form of u =
8 rhgpl
(5)
This expression increases in accuracy at lower temperatures, and becomes exact as the vapor pressure of the liquid approaches zero. At this state of the liquid the relative surface tension, U , becomes equal to the absolute tension, u1, of the liquid surface proper. Since this state also corresponds to the hypothetical liquid surface postulated above equation ( j) becomes identical with equation (3), which defines the absolute surface tension of a liquid. The surface tension of a vapor as defined in equation (4) may now be obtained by combining equation (3) with the expression for the relative surface tension of a liquid as given by equation ( I ) , and with equation ( 2 ) expressing Antonoff's rule. Entirely different consideration led S'. Barbulescu' to the same conclusions. The Surface Tension of Saturated Vapors and Temperature To relate the surface tension of saturated vapors to temperature the following two equations are utilized. KIT,V-' ( I - T/Tc)'.' and p1 - p? = K2pc ( I - T;'Tc)0.3 u =
Equation (6) was deduced by van der Waals from his theory of corresponding states, and later modified by S. Sugden.* Equation (7) was suggested by D. H. Goldhamme9 and corrected by S. Sugden'. Here T, and pc are the
*
Physik. Z., 31, 48 (1930). J. Chem. SOC., 125, 32 (1924). Z. physik. Chem., 71, 577 (1910). J. Chem. Soc., 130 11, 1780 (1927).
1714
J. L. SHERESHEFSKY
critical temperature and critical density respectively, V the molar volume of the liquid, and K1 and K2 constants which are approximately universal for all non-polar liquids, Combining these with 2quations ( I ) and (4), we obtain the expression
which relates the surface tension of saturated vapors to the density of the vapor, its critical constants and temperature. The constant A equals K1/K2 and is nearly the same for all non-associated substances, M is the molecular weight, and the other terms have the designated significance. To test the validity of this equation the surface tension of the saturated vapors of six liquids a t various temperatures were calculated and compared with observed values. The agreement as shown in Tables I t o VI inclusive is very good up to the critical temperature for all substances except carbon
TABLE I Surface Tension of Saturated Vapors Benzene
A
Mol. K t . = 78.05; T, = 5 6 1 . 5
= 1.39
,353 ,302
210
,140
,0036 ,0076 .0144 ,0249 ,042 I
240
,086 ,016
.2209
90 I20
'50 I 80
,247 '
280
I93
.0904 ,1640 ,262 ,357 .453 .483 .344
.07 I j
PC = 304 5
,093 5 .1640 ,260 ,360 ,456 ,498 ,340
3.43 0.00
0.76 0.84 0.66 3.11 1.16
TABLE I1 Surface Tension of Saturated Vapors Chlorobenzene
A t"C
Mol. Wt. = 112. j; T, = 6 3 2 . 2
= 1.39
I
I1 I
-T/T, ,332 ,283 ,236
.188
. I41 '093 ,062 ,042
I11 P2
,0054 ,0102
,0180 ,0301 '0494 ,0778 ,1075 ,1360
IV ui(0hs.)
pc = ,3654
v u2(calc.)
VI Percent Deviation
0.67 0.58 1.98 1.44 3.00 I .02
1.98 7.78
SURFACE TENBION O F SATURATED VAPORS
TABLE I11 Surface Tension of Saturated Vapors Ethyl Ether A = 1.39 I t"C
Mol. Wt. = 74.1;T, = 466.8
I1 I
20 SO
80 1 IO
140 170
'85
-T/T, ,373 .308 .245 'I79 . I14 .049 . 0 17
pe = 0.2625 VI
I11
IV
P2
ui(0bs.)
u2(calc.)
,00187
.0447
,0442
1.12
.00508 ,01155
'1037
,1013
2.32
,1880 .2880 .3610 .3280 .1960
'1875 .2870 .3610 .33I . I94
.02349 .04448 ,0873I .1320
V
Percent Deviation
0.27
0.35 0.00
0.91 1.02
TABLE IV Surface Tension of Saturated Vapors Carbon Tetrachloride A = 1.39 Mol. Wt. = 153.84;To = 556.2 I t "C
I1 I
90 I20 150
I80 2
-T/T, ,3475 .293 '239 .185
. I32
IO
.077 .0225
40 2 70 2
po = 0.5576
I11
IV
V
P2
rz(0bs.)
u2(calc.)
.0079 . or63
. IO10 . I770
.0302
,2683 ,3730 ,4660 .4300
.os25
,0879 ,1464 .27IO
.222
,0995 .176j ,2720
.3780 ,465 ,477 .298
VI Percent Deviation
1.49 0.28 I .38 1.34 0.22
10.9j 34.20
TABLE V Surface Tension of Saturated Vapors Methyl Formate A = 1.3~ I I1 t"C 50
80
I
-T/T, ,337 .275
I IO
,2125
140
. I52
170 2 00
.090 .028
Mol. Wt. = 60.04;T, = 487 I11 P2
'0043 .or05 .0216 .0412 .0763
. I524
IV uz(obs.)
.09s ' I93 .316 .438 ,497 .340
= 0.3489 IV Percent rl(calc.) Deviation ~e
V
,0955
. I943 .317
.447 ,516 .361
0.53 0.67 0.28 2.06 3.83 6.~8
1716
J . L. SHERESHEFSKY T.4BLE
1’1
Surface Tension of Saturated Vapors Ethyl Acetate
A
= 1.43
I1
I t”C 90
I
-TITc ,307
hlol. Wt. I11 P?
,0047 ,0103
’50
,248 I92
I80
‘34
IO
,077
, 07
2 40
.or9
,1500
I20
2
.
= 88.06;T, = j23.1
,0206
0388 I2
p c = 0.3077
IV
V
u*(ObS.)
u2(calc.)
,400
,0907 ,1643 ,260 ,346 ,396
,229
,237
.0918 I645 ,260 ,353 ’
VI
Percent Deviation 1.20 0 . I2
0.00
2.76 I 00
3.49
tetrachloride. In the latter case the divergence is most probably due to the doubtful data in the neighborhood of the critical temperature. The observed values in the fourth column of each table were taken from the data of Ramsay and Shields as recalculated and corrected by S. Sugden.’ It is to be observed that the surface tension of saturated vapors attains a maximum value near the critical temperature, the ratio of the temperature at which the maximum occurs to the critical temperature being the same for all substances, as was shown by the author in a preceding paper.*
The Law of Eotvos To express the relationship between surface tension of a liquid-vapor surface and temperature Eotros suggested, in analogy to the gas law P V = R T , the similar equation ~ ( h l / p l ) ”= ~ I
