Langmuir 1995,11, 2843-2844
2843
Thermodynamic Method for the Evaluation of the Pressure Coefficient of a Liquid-Gas Interface
measured in the 0-35 MPa pressure range using axisymmetric drop shape analysis.12
Ivan Vavruch
Consider a two-component system consisting of a n inert gas over a pure liquid and separated from it by a planar interface. This two-component,two-phase system has two degrees of freedom for which we choose the temperature T and the external pressure P. We define the enthalpy H of the whole system by H=U+PV (1) where U is the internal energy of the whole system and Vis its total volume. Differentiation of eq 1with respect to the total surface area A a t constant absolute temperature T , external pressure P , and number of moles n in the system yields
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Received September 16, 1994. In Final Form: April 18, 1995
Introduction Experimental determination of the change of surface tension ( y ) with pressure ( P )exerted by an inert gas over a liquid raises difficulties. They originate in principle from the fact that the extraneous effects of adsorption of the pressurizing gas on the liquid surface and of gas dissolution in the liquid are both enhanced with pressure. These effects cannot be separated out experimentally and a second component is thus introduced into the system. It can be shown on thermodynamic grounds1B2that the pressure coefficient aylaP is related in the binary system liquid-gas to the volume change of the whole system associated with the creation of new surface (see eq 8). Rice1 has argued that this change should be positive and the surface tension should therefore increase with pressure. Since, however, usually one cannot subject a liquid surface to an increased pressure without introducing some pressurizing gas into the system, the pressure coefficient becomes n e g a t i ~ e . l - ~ Surface tension of pure liquids was measured under pressures of a variety of gases up to about 10 MPa. In these experiments the coefficient ay/aP turned out to be in general negative and the amount of lowering of the surface tension depended upon the nature of the gas used. Only in the case of h e l i ~ m ~were - ~ small but positive pressure coefficients noted. Evidently, this gas is so inert that its interaction is small enough not to dominate even a t elevated pressures. More recently, Hills and H@iland3 investigated the pressure dependence of the surface tension of mercury using helium pressure up to 100 MPa and they obtained, indeed, positive pressure coefficients. However, when nitrogen or argon constituted the pressurizing gas, a significant decrease of the surface tension of mercury with increasing pressure was observed. To find the way out of the outlined problems, it is necessary to look for methods which make it possible to evaluate the pressure coefficient of y from observable properties of the substance. Such a method based on thermodynamics is described in this paper. It enables aylaP to be calculated for a single liquid, i.e., in the absence of the extraneous effects mentioned above. The situation is different for interfaces between condensed phases. A rigorous thermodynamic treatment of the pressure coefficient of interfacial tension for binary and ternary liquid-liquid systems has been developed especially by Good and Motomura.loJ Very recently, highquality data for the system n-decane-water have been (1) Rice, 0. K. J. Chem. Phys. 1947, 15, 333. (2) Adamson, A. W. The Physical Chemistry of Surfaces, 4th ed.; Wiley: New York, 1982. (3) Hills, G. J.; Hgiland, H. J . Colloid Interface Sci. 1984, 99, 463. (4) Hough, E. W.;Wood, B. B.; Rzasa, M. J. J . Phys. Chem. 1952,56, 996. (5) Slowinski, E. J.;Gates, E. E.;Waring, C. E. J.Phys. Chem. 1957, 61, 808. (6) Gielessen, J.; Schmatz, W. 2.Phys. Chem. (Munich) 1961, 27, 157. (7) Eriksson, J. C. Acta Chem. Scand. 1962, 16, 2199.
(8) Rusanov, A. I.; Kochurova, N. N.; Khabarov, V. N. Dokl. Akad. Nauk SSSR 1972,202, 380. (9) Massoudi, R.; King, A. D., Jr. J . Phys. Chem. 1974, 78, 2262. (10) Good, R. J . J.Colloid Interface Sci. 1982, 85, 128 and 141.
Theory
+
( a H / a ) T p , , = H" = ( a U / a ) T p , , P ( a V / a ) T p , , (2) The derivative (aH/aA)T,p,,represents the increment of the enthalpy of the system with respect to a change in its surface area a t constant T , P, and n. It can therefore be identified with the total surface enthalpy which we have denoted above by the symbol Ha and it may be evaluated from the well-known relation13
H"= y - TdyIdT (3) To evaluate the derivative (aUlaA)Tp,,in eq 2 we write the equation
(auia),,,= (au/av,p,,(av/a),p,n (4) The first term on the right refers to the work which is required to overcome the intermolecular forces of attraction when the volume, V, of the system is increased a t constant temperature, pressure, and number of moles. It has the dimension of pressure and it may be identified with the expression for the internal (cohesive) pressure, P1.14Equation 4 then becomes (aula),,,,= p i ( a v / a ) T , , (5) and combining eq 2 with eq 5 gives finally the relation
+
(aV/aA),p,,= H"/(P, P ) (6) At atmospheric pressure P = 0.1 MPa; hence Pi >> P (see Table 1)and the total pressure (Pi P ) in eq 6 can be approximated by the internal pressure, P,. The volume-surface area coefficient (aVIaA)Tp,, may be interpreted as a measure of the looseness of packing of molecules a t the surface relative to their packing in the bulk. Thus it is a n "excess-type" property which has the dimension of length but has no direct relationship to thickness of a n interface. It follows thermodynamically that the change in the total volume of the system on the formation of interface is closely related also to the pressure dependence of surface tension.lJ3 The change in Gibbs free energy G ofthe whole system considered by us can be expressed as dG = -S dT V d P y dA + p dn (7) where S is the entropy of the whole system, p is the chemical potential, and the meanings ofthe others symbols are the same as in the preceeding equations. Since dG is an exact differential, we may use the Euler theorem to obtain from eq 7 (ayiap),,,, = (8) where P i s the pressure of the pressurizing gas and where
+
+
+
(av/wT,,
(ll)Motomura, K. J. Colloid Interface Sci. 1986, 110, 294, and references therein. (12) Susnar, S. S.; Hamza, H. A,; Neumann, A. W. Colloids Surf. 1994, 89, 169. (13) Lewis, G. N.; Randall, M. Thermodynamics (revised by Pitzer, K. S., Brewer, L.), 2nd ed.; McGraw-Hill: New York, 1961. (14) Barton, A. F. M. J . Chem. Educ. 1971,48, 156.
0743-746319512411-2843$09.0010 0 1995 American Chemical Society
Notes
2844 Langmuir, Vol. 11, No. 7, 1995
it is assumed that the mass of the system is held constant during the extension of the surface.13 The change of the surface tension with pressure is thus equal to the volume change of the whole system associated with unit increase in surface area. Considering eq 6 and eq 8 gives the sought final expression
(ayiap),,,, = HTP,
+P)
(9)
Evidently, if the total surface enthalpy H8 of the liquid and its internal pressure Pi are known, the surface tension-pressure coefficient (aylaP)Tp,,a t the pressure P can be calculated from eq 9. Thus, we have succeeded to derive a n expression enabling us to evaluate the pressure coefficient of surface tension from physical constants of the liquid. If temperature and pressure are held constant and the partial differentials are replaced by the total ones, eq 6 can be given the simple form
P,av=PaA
(10) Thus, the volume work Pi dV done against attractive intermolecular forces in bringing liquid molecules from the bulk into the surface is equal to the energy quantity HsdA which is in connection with the simultaneous extension of the surface. This energetic balance is useful for a complementary explanation of relationship 6.
Results and Discussion The numerical values of the pressure coefficient of y calculated by eq 9 for 14 selected liquids at room temperature and atmospheric pressure are summarized in Table 1. An analysis of the probable errors in the data used for the calculation has shown that the absolute error J m-2 Pa-'. We note that 1J m-2 Pa-' = is -5 x 1 m. The coefficient aylaP varies in organic liquids in the normal liquid range relatively slightly and it becomes smaller in strongly interacting systems. It ranges a t room temperature from ca. 1.3 x lO-'O J m-2 Pa-' to ca. 2.3 x J m-2 Pa-l. Exceptions are the strongly hydrogen bonded water and the liquid metal mercury in which the pressure coefficients of surface tension are much higher. It is useful to compare the numerical values of (ayiaP)Tp,, in Table 1calculated by eq 9 with the apparent pressure coefficients of surface tension obtained experimentally using helium as pressure transmitting phase. Thus Gielessen and Schmatz6 found a t room temperature for n-hexane 0.3 x lowloJ m-2 Pa-' and for n-nonane 0.25 x J m-2 Pa-l. Rusanov and co-workerss obtained a t 20 "C and 0.1 MPa for carbon tetrachloride -0.36 x J m-2 Pa-' and for water 0.37 x J m-2 Pa-'. Massoudi and King9 found in the system water-helium a t 25 "C up to 8 MPa for the pressure coefficient of y , ayIaP = 0 , and the values for 12 other investigated gases were all negative. Of special interest arb the careful measurements of Hills and Hailand3who obtained in the system mercury-helium a t 25 "C and 0.1 MPa for the pressure coefficient of y 3.0 x J m-2 Pa-l. Obviously, in this last system the extraneous effects of surface adsorption and gas dissolution are very small and the value calculated by eq 9 in J m-2 Pa-l. It is thus close to the Table 1is 4.05 x value of Hills and Hailand. The contrary is true for water. In conclusionit has to be stressed that all the experimental data are usually considerably smaller than the numbers calculated under comparable conditions by eq 9. They may thus serve as a verification of eq 9. The pressure coefficient of y depends on temperature. In common liquids, the total surface enthalpy is nearly temperature independent, provided it is not too close to
Table 1. Numerical Values of Pressure Coefficient (aylaP)TA,,, for Selected Liquids at 26 "C and 0.1 MPaa
Hs/ liquid
mJmT2
Pi/ MPa
2-methylbutane n-octane n-dodecane cyclohexane toluene anisole 1-bromobutane ethylenediamine pyridine carbon disulfide ethanol 1,2-ethanediol water mercury
47.33 49.49 51.55 60.07 63.14 71.00 59.47 82.96 75.50 75.83 46.78 74.52 118.0 546.7b
202 271 294 3 19 349 430 325 601 480 381 283 502 168 135OC
lo'o(ay/aP)TA,nd/
J m-2 Pa-' 2.34 1.83 1.75 1.88 1.81 1.65 1.83 1.38 1.57 1.99 1.65 1.48 7.02 4.05
a Data from ref 15. Data from ref 16. Data from ref 17. Calculated by eq 9.
Table 2. Temperature Dependence of the Pressure Coefficient (aylaP)TA, for Carbon Tetrachloride, Methanol, and Water at 0.1 MPa liquid carbon tetrachloride
methanol
water
tempi Hal "C mJm-2
0 20 40 60 70 25 35 45 55 10 20 40 60 80 100
P,"l MPa
62.9" 362 62.9" 344 62.9" 323 62.9" 299 62.9" 285 4 5 . 5 " ~ ~287 45.5"!b 280 4 5 . 5 " ~ ~276 4 5 . 5 " ~ ~272 116.9ac 51.6 117.3"s' 131 118.9a*c 272 122.3'9' 392 125.0Q3c 493 127.8"~~576
101O(aylaP)Tp,ndl Jm-2Pa-1 1.74 1.83 1.95 2.10 2.21 1.58 1.62 1.65 1.67 22.65 8.95 4.37 3.12 2.53 2.22
a Data from ref 15. Data from ref 19. Data from ref 20. Calculated by eq 9.
the critical temperature. Since internal pressure decreases with increasing t e m p e r a t ~ r e , 'the ~ numerical value of the coefficient (ay/aP)Tp,,should increase from 9 if the temperature is raised. This is shown for carbon tetrachloride and for methanol a t atmospheric pressure in Table 2. The numerical values ofthe pressure coefficient in these two liquids increase systematically if the temperature is raised. The contrary is true for water, however, and its very anomalous behavior may be explained by the cleavage ofhydrogen bonds in the system, due to enhanced thermal motion with increasing temperature.l8 We note that a t the boiling point of water, the coefficient 8ylaP finally equals to 2.22 x J m-2 Pa-' and it thus becomes a t room temperature comparable with the values for other liquids. Finally we note that a t atmospheric pressure the coefficient for the liquified argon surface tension in helium is a t 87 K -2 x J m-2 It is thus comparable with the values for a general class of liquids, and it increases if temperature is raised. The values are (in J m-2 Pa-l) 2.10 a t 105 K and 3.26 a t 140 K. LA9407440 (15)Vavruch, I. J . Colloid Interface Sci. 1993,156,258. (16) Jasper, J. J . J.Phys. Chem. Ref. Data 1972,1, 841. (17)Weast, R. C., Ed. Handbook of Chemistry and Physics, 64th ed.; CRC Press: Cleveland, OH, 1983/1984. (18)Leyendekkers, J. W. J.Phys. Chem. 1983,87,3327. (19)Korosi, G.;Kovbts, E. sz. J . Chem. Eng. Data 1981,26,323. (20)Riddick,J. A.;Bunger,B.; Sakano,T. K. TechniquesofChemistry, Vol. II. Organic Solvents, 4th ed.; Wiley: New York, 1986.
