RELATION B E T W E E N HEAT O F EVAPORATIOS AXD SURFACE TESSIOS
747
XOTE Oh' T H E RELATIOX B E T W E E S THE HEAT OF EVAPORATIOS ASD THE SURFACE TENSION O F LIQUIDS J. C. DE WIJS M r . Franckenstraat 10, iiijmegen, Holland
Received January 82, 1947
Some time ago the author proved (1)' a relation between the heat of evaporation of liquids and the temperature :
L = k ( T h - T)"
(1)
in which L is the total heat of evaporation a t the temperature T and Tkthe critical temperature, while k and m are two constants dependent on the nature of the substances under investigation; the value of m is about 0.4. This formula holds good for a great number of liquids between their melting point and their critical temperature, but not for the liquid gases. There is also a formula of the same form as mentioned above for another important property of liquids. I n connection with his thermodynamic theory of capillarity, van der Waals has given an exponential function for the surface tension, u , for liquids up to the critical point (4): u =
.4(Tk - T)"
(2)
in which A is a constant, dependent on the nature of the substances, while the exponent n should be the same for all sorts of liquids very near the critical temperature, namely, n = 1.5; in reality, for the so-called normal substances, n = 1.25 on the average a t lower temperatures, and for the liquid gases the value of n varies from 0.8 to 1.33. When we now combine these two formulas, it is obvious that this is possible in various ways. I n the first place we can eliminate the temperature, so that we obtain a relation between the heat of evaporation and the surface tension which is i n d e p e n d e n t of the temperature. From these two formulas it follows that:
and therefore
or if B = k/Apand p
=
m/n.
Other authors have also made use of the same formula, some with the same exponent for all substances, others with a special exponent for each one. 1
748
J. C. DE WIJS
In the second place we are able to combine both formulas in such a way that we get another relation between L and u, but one which is, in contrast to equation 3, dependent on the temperature. When we divide equation 1 by equation 2 we get:
L - -. L (Tk - T)" u A (Tk - T)"
-
or
4 = C(Tk - T)'
(4)
if C = k / A and q = m - n. TABLE 1 Ethyl alcohol
. D.
C = 274.2:
0.3922; p = -0.6071
L
-
~ 40 60 80 100 120 140
1m
Lb
Tx-T
T
_
i
_ 203.1 183.1 163.1 143.1 123.1 103.1 83.1
A
_
1
_ 20.20 18.43 16.61 14.67 12.68 10.59 8.45
-!
Observed
218.7
Equation 3
1 1
213.4 206.4 197.1 184.2 ] 171.1 ' 156.9
1 1
221.4 213.6 205.0 195.3 184.4 171.8 157.3
1 Equation 4 --l
A
Observed
+2.7
$0.2 -1.4 -1.8 +0.2 $0.7 $0.4 f0.4 +0.6
j j
' ~
10.82 11.58 12.43 13.44 14.52 16.15 18.57 22.35 29.22
I
10.89 11.59 12.42 13.49 14.76 16.44 18.75 22.19 27.97
$0.07
+0.01 -0.01 +0.05 +0.24 +0.29 +O. 18 -0.16 -1.25
Instead of combining the formula for the heat of evaporation with those of van der Waals for surface tension, one can also combine formula 1 with the formula of Eotvos for the molecular surface energy:
u V " ~= k(Ti, - T ) in which V 2 is the molecular volume a t temperature T. We then do not obtain the relations between L and u only, but instead of this, between L, u, and V (this is, however, not the theme of this paper). When we wish to examine formulas 3 and 4,it is obvious that this is possible only for substances for which the heat of evaporation as well as the surface tension are known in a wide range between the melting point and the critical temperature. The substances for which both series of data are known are, with the exception of the liquid gases, chiefly organic compounds (alcohols, esters, benzene, and some others). For most of them the data for L a n d u are known up to a few degrees below the critical point, but, because it is very difficult to obtain accurate values for u close to the critical temperature, we k d ourselves obliged to finish our calculations some tens of degrees below this point.
TABLE 2 Ethyl acetate
B = 31.14; C = 336.4; p
0.3340; p = -0.8073
I
L T
~
Tk-7
~
~~
80
liO.l
160
749
BETWEEN H E A T OF EV.4PORATIOS A S D SURFACE T E S S I O N
REL.4TIOK
90.1
16.32 13.98 11.i5 9.57 7.48 5 51
~
Observed
~
Equation 3
85,;s 82.15 7i.53 72.74 65.91 59 X i
86.34 81.98 7i.38 72.23 66.43 60 08
_
,
L ./
Observed
_
~
i~
-0 -0 -0 +0 +0
5 6 7 8 10
li
15 51 52 21
Xi5 599 600 812 86
1
1
Equation 1
'1
5 88i 6 621 7 567 8 890 10 95
_
~
+0 +O -0 TO
012 023 033 0% +0 09
1
L/m A
,
I
80 100 120 140 160 180 200 220 240 260
1
1 I
~
208 188 168 148 128 108 88 68 48 2s
5 5 5 5
20.28
95.45
5
5 5 5 5 5
3.41 1.75
1
Obseried 1 Equation 4
:quatian 1
54.11 43.82
94.28 90.93 87. 36 83.10 78.75 73.84 68.52 62.27 54.54 44.44
4.706 5.071 5.511 6.158 6.993
+1.17 -0.48 +0.78 +0.28 -0.19 -0.78 -0.29 $0.03 +0.43 +0.62
1
4.627 5.039 5.538 6.170 6.987 8,060 9.601 11.95 16.04 25.0
~
1
1
9.598 ~
15.87 25.04
i
-0.079 -0.032 +0.027 +0.012 -0.006 -0.095 +0 ,003 $0.10 +0.17
+0.16
TABLE 4 SUBSTANCE
Methyl alcohol .. . . . . . . ,. ..... . . . . .. .. . . .. Ethyl alcohol , . , . . . , . , . . . . . . . . . . . . . . . . . . . Methyl formate.. . , . . . , . . . . . . , . . . . . . . . . . . . . . . . Ethyl acetate . . , . . . . . . . , . , . . . . . . . . . . . . . . , . . . . . . . .Acetic acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . .
I
P
0.38 0.39 0.33 0.33 0.20
I
4
-0.87 -0.85 -0.63 -0.61 -0.83 -0.81 -0.91 -0.85
For the heat of evaporation there are available the measurements of Sydney Young and coworkers as published by Mills (2), while for the surface tension the data are taken from the well-known investigations of Ramsay and Shields (3).
750
J. C. DE WIJS
For testing the formulas \re have calculated for several substances the constants B , C, p , and q and with these the values of I, in equation 3 and of L/uin equation 4, Tyhich have been compared with the observed values. It is obvious that the values of the constants are dependent on the nature of the substances in general and, so far as R and C are concerned, also on the units in which the heat of evaporation and the surface tension are expressed. In tables 1-3 L is given in calories per gram and u in dynes per centimeter. I t appears from these tables that in all cases the agreement between the observed and calculated values for L and L / u , respectively, is very satisfactory. Most of the differences (A) for both equations arenot higher than 1 per cent, which is of about the same order as the accuracy of formulas 1 and 2. Owing to the fact that near the critical point' it is difficult to obtain exact values of u , the differences in this region are somen-hat greater, more especially for equation 4. Besides the esamples given above, about the same agreement for the formulas has been found for other substances, such as ether, carbon tetrachloride, methyl alcohol, acetic acid, and methyl formate. As follows from the deductions of the two equations, the values for the exponents p and q are given by those of m and n. With the average value for m = 0.4 and n = 1.25, we have p = m/n = 0.32 and q = m.- n = -0.85. table 4 shows, this holds good for the normal substances, but not for those, such as the alcohols and acetic acid, which are comples in the liquid state, as is mostly found for empirical relations. SUMMARY
Two empirical relations betwxn the heat of evaporation and the surface tension of liquids are given. REFERENCES
(1) DE WIJS, J. C . : Rec. trav. chim. 62, 459 (1943). ( 2 ) MILLS,J. E.: J. Am. Chem. SOC.31, 1099 (1909). W., AND SHIELDS, J.: Z.physik. Chem. 12, 433 (1893). (3) RAMSAY, (4) V A N DER WALLS, J. D.: Z. physik. Chem. 13, 716 (1894).
