THE SOLUBILITIES OF GASES AND SURFACE TENSION' H. H. UHLIG DiLision of Industrial Cooperation, Massachusetts Institute of Technology, Cambridge, .lf assachusetls ReceiLed June 22. 1937
The lack of adequate theory and of sufficiently exact experimental data has in the past discouraged attempts to correlate gas solubility data. Hildebrand (3) emphasized the lack of consistent data and Tamniann (11) pointed out t h a t certain deviations froni calculated relations, using an integrated thermodynamic expression for the solubility, could be ascribed to experimental error. Taniniann proposed the following expression derived from the Clau&x-Clapeyron equation: d 111 _ _ _y --- _-111 _ y dT T
q
R T?
where y is the Ostwald coefficient of solubility or the ratio of concentration of gas in the solvent to the concentration of gas in equilibrium with the solvent, q'RT? iq a heat of d u t i o i i correction term over and above a work term in y / T , and R and T ha1-e the uwal meaning. I n cases in which q 1 dlny could be neglected it was shown that tlie average value of - . __ for ln y d T several gaces in a variety of solvents was given by the reciprocal of the abqolute temperature T. Horiuti (4) shon-ed that, reasoning froni thermodynamics, one obtains the equation:
d_ ln _ _y - - _-_L, __ dT RT2
a ill r1 dT
where V1 is the molal volume of colute and L , is the so-called internal heat of solution equal to 4H - RT. The internal heat of solution is in this instance equal, therefore, to - ACT, the change in energy of the system. Sisskind and Iiasarnowslry (9), treating the solubility of argon in various solvents, shon-ed similarly to Taniniaiiii that the energy of solution could be divided into two terms. The first is a work term proportional to 4 7 ~ ~ where ~ 7 , r is the radius of the solute molecule and u is the surface tenzjon of tlie solvent. The second is a molecular solvent-solute interacPresented before the Division of Physical and Inorganic Chemistry a t tlie Sinety-third IIeeting of the .imerican Chemical Society, held a t Chapel Hill, S o r t h Carolina, April, 1937. 1313 T H E J O U R S . A L OF PHTSICAL C H E M I S T R T , Y O L .
41, S O . 9
1216
H. H. UHLIG
tion term. Despite the proximity of solvent and solute molecules, the authors attempted to evaluate this term using London’s expression for the attractive force between two non-polar molecules. I n the present paper an expression for the solubility is derived starting with premises differing from those of either Tammann or Horiuti but similar to those of Sisskind and Kasarnomsky, which can be reduced to the thermodynamic equation. The derived equation expresses the solubility of a gas as a function of its molecular radius and the surface tension of the solvent. It is employed successfully in correlating the data for the solubilities of several gases reported in the literature, and in addition sheds light on the physical processes which accompany the solution of one substance by another. DERIVATION O F AN EXPRESSIOS FOR T H E SOLUBILITY O F A GAS
If we assume a spherical gas molecule of radius r to enter the solvent, a spherical cavity is produced in the solvent of essentially the same radius. A certain amount of work is done in producing this cavity, since any increase in surface of a liquid is associated with a definite energy change. This energy change is, in its simplest terms, given by the increase in area of the cavity multiplied by the surface tension of the solvent, or 4rr2u, where u is the surface tension in ergs per square centimeter. I n calculating this effect in this simple manner we are using a macroscopic concept, namely, surface tension at molecular dimensions, which introduces, of course, an approximation. In a comparison with experiment, however, it is shown later that deviations arising from this effect alone are apparently not large even in the cases of gases of smallest dimensions. In addition to the surface tension term there are energy terms which arise because of interaction of solvent and solute molecules and which account for repulsive or attractive forces at the intermolecular distances characteristic of molecular separation in liquids. The nature and magnitude of the latter forces depend on the specific properties of the solute and solvent molecules which, in general, present a complex picture too difficult to resolve using the present-day theories of molecular interaction. d qualitative picture is possible, however; for example, interaction would be expected to be large and positive between a gas and solvent molecule constituted of many dipole groups, with or without a resultant moment, and relatively small or of opposite sign for an elementary gas dissolved in a non-polar solvent. If we group all energies of the latter category under the designation “interaction energies” and call these E , then Au, the change in energy of the system in transferring a molecule from the solvent phase to a low pressure gas phase, is given by: Au = 4rr% - E (2)
SOLCBILITIES O F GASES A S D SURFACE TENSION
1217
In the above expression the energies attending interaction between solute molecules and gas-phase solvent molecules are neglected, since at any but appreciably high solvent vapor pressures these are small. Using the Maxwell-Boltzmann distribution theorem, it is well known that the equilibrium concentrations of solute molecules in the two phases can be calculated as follows:
where c’ is the concentration of gas molecules in the solvent, c is the concentration of gas molecules in the gas phase, Au is the energy difference in transferring a molecule from the solvent to the gas phase, k is Boltzmann’s constant, and T is the absolute temperature. But C‘
- --Y
C
(4)
by definition, where y is the Ostwald coefficient of solubility. The quantity y in the region where Henry’s law is obeyed is independent of the pressure. It is possible, therefore, to extend calculated values of y at limiting low pressures to practical pressures for which experimental data exist. In expression 2, Au is actually the free energy, not the total energy, change. In substituting a free energy change for Au in the MaxwellBoltzmann equation this equation then becomes analogous to the thermodynamic equation expressing the free energy change of transfer of a perfect solute from one concentration to another concentration ( 6 ) . Substituting expression 2 for Au and y for c’/c we can express equation 3 as
Expression 5 indicates that if the interaction energy E remains constant, then the greater the radius of the solute molecule and the surface tension of the solvent, the smaller will be the solubility y. If there are gases which interact but little with the surrounding solvent molecules and do not tend to become solvated, we should expect that E for these gases would be small. We should also expect that variations in the solvent-solute interaction in going from one solvent to another would be inappreciable and hence the major portion of the solubility would depend on the magnitude of the term 47rr‘u. It is readily seen, therefore, that for gases of this kind, if log y, the logarithm of the solubility, is plotted with U , the surface tension of the solvent for which values of y correspond, a straight line should result whose slope would be proportional to the square of the gas molecular
1218
H. H. UHLIG
radius and whose intercept would be a measure of E. Gases normally expected to fit into this category would be the elementary gases hydrogen, oxygen, nitrogen, and the inert gases, although evidence is presented later that interaction characterized by repulsive forces is appreciable for hydrogen, helium, and neon. J. Horiuti (5) a few years ago published accurate solubilities of the first three gases in carbon tetrachloride, acetone, benzene, ethyl ether, chlorobenzene, and methyl acetate. His data at 20°C., pIotted with surface tension data taken from the International Critical Tables at the same temperature, are shown in figure 1. A straight
FIG.1. Relation betveen t h e !ogarithm of gas solubility and surface tension of solvelit
line can be dran-n through the data for each gas. From the slope n-e can calculate the molecular radii of the solutes. The values obtained are cni. for oxygen, and 1.38 X 1.43 x 10-8 e m . for nitrogen, 1.36 X em. for hydrogen, which are in good agreement with the values 1.57, 1.45, and 1.38 X 10-8 e m . , respectively, calculated from Tan der Waals' gas constants as given in Landolt-Bornstein's tables. Recently Maxted and Moon (8) published data on the solubility of hydrogen in ethyl alcohol, ethyl acetate, benzene, chloroform, and acetic acid at various temperatures. Their data, when treated as Horiuti's data, give a molecular radius for Hz of 1.29 A.T. as compared with 1.38 AT=.
SOLUBILITIES O F GASES AND SURF-4CE TEXSION
1219
The intercept for Horiuti’s data for nitrogen is very nearly zero, for oxygen slightly positive, but for hydrogen is definitely negative. This negative intercept can be interpreted as an absorption of energy by the system when hydrogen dissolves, which serves to diminish the solubility of the gas as compared with other gases and accounts in part, according t o the rule of Le Chatelier, for the increase in solubility of hydrogen with increase in temperature. This negative energy E for hydrogen can also be associated n-ith repulsive forces at close range acting between hydrogen and the solvent molecules which increase the effective radii of the cavities surrounding the hydrogen molecules. It will be noticed that the radius of hydrogen calculated froni the slope is larger than the value usually assigned to H in the crystal lattice. For oxygen the energy E is positive, the forces attractire, and the radius of O2 more nearly corresponds to the radius giren by crystal lattice dimensions of 0. It is interesting to observe for the three gases that relatively small deviations from linearity, recalling experimental difficulties of the measurements, show that interaction a-ith large organic solvent molecules is approximately the same in each instance. It is also interesting to observe that the snialler the surface tension of the solvent, the greater is the capacity to dissolve gases. The qualitative relation between surface tension and gas solubility was o b w v e d in the early part of this century by Skirroiy (10) and by Christoff (1). Their qualitative obqervations become definitive in equation 5 , according to TI-hich the reciprocal relation betn-een surface tension and solubility holds for those solvents and solutes for which E is either small or only slightly variable in going from one solvent to another. The success which attended the plot of the data for oxygen, nitrogen, and hydrogen prompted similar treatment for other gases. In figure 1 Horiuti’s data for methane are plotted; in figure 2 data for carbon monoxide and ethylene. Methane and carbon monoxide show a regular linear arrangement with positive intercepts. Despite a small pernianent moment for carbon monoxide, interaction with solvents is small, as shown by the small value of E . The larger positive intercept for methane can be interpreted as appreciable interaction or solvation, which may also account for more pronounced scattering of the data. Khen the interaction term E becomes appreciable, it is reasonable to suppose that specific interaction effects m-ith one solvent, as compared n-ith another, n-ill then become noticeable. I n the case of ethylene, for example, the regularity of the data plotted as in figure 2 is not ss good as for the gases previously considered. Interaction lvith the solrent is distinctly more pronounced, solubility is larger, and specific effects are apparent. To assist in determining the slope, the solubility of ethylene in water obtained from Landolt-Bornstein’s tables was used. From the slope a radius for ethylene of 1.51 x 10-8 em. and an intercept of 1.4 are calculated. The large positive intercept indi-
1220
H. H. UHLIG
cates attractive forces between solute and solvent, and the calculated value of the radius should approximate crystal lattice dimensions. This is found to be the case. Horiuti’s data for acetylene and ethane behave much as those for ethylene, the acetylene data showing the largest scattering. The water solubility data for the gases mentioned above found in Landolt-Bornstein’s tables do not in every case fall on the line which best fits Horiuti’s data. Good agreement exists for oxygen and methane, but, not for carbon monoxide, nitrogen, and hydrogen. No explanation is apparent.
FIG.2. Relation between the logarithm of gas solubility and surface tension of solvent
In table 1 values are given for the slopes
*
E
and intercepts ___ dg 2.303kT of the data plotted in figures 1 and 2 , the radii of the gas molecules calculated from the slopes, and comparative radii obtained from other sources such as from compressibility data (van der Waals) or viscosity data taken from Landolt-Bornstein’s tables or crystal lattice dimensions. Constants derived for helium, neon, and argon are also included, the gas solubility data being taken from Lannung’s (7) measurements in methyl alcohol, ethyl alcohol, acetone, benzene, and water. Lannung’s data for
SOLUBILITIES O F GASES A S D SURFACE TENSION
1221
organic solvents alone are not sufficiently regular to define a singular slope, but by including his values for water, the surface tension of which is approximately three tinies that of the organic solvents, the slopes are readily determined. ,111 the derived constants are taken from measurenients at 20°C. Considering the simplicity of the theory, the radii calculated froni the solubility measurements are in good agreement with radii calculated from other types of physical measurements. The gases with smallest solubility shon- the lowest yalues for the intercept and vice versa. I t is interesting to observe that the intercepts are fairly linear with the boiling points of the solutes. From equation 5 certain deductions can be made with regard to the temperature coefficient of the solubility of gases. It will be observed TABLE 1 Radii of gas solutes from solubility r X 108 om. GAS
Clalcu-
lated
Hr . . . . . . . . . . . . . . . . . . . . . . . . . . 0 2 .........................
co.. . . . . . . . . . . . . . . . . . . . . . . .
CH?.. . . . . . . . . . . . . . . . . . . . . . . He.. . . . . . . . . . . . . . . . . . . . . . . . . S e ,. . . . . . . . . . . . . . . . . . . . . . . . . A .......................... C2H.j.. . . . . . . . . . . . . . . . . . . . . . .
0 0 0 0 0 0 0 0
0259 0252 0263 0254 0114 0122 0183 0313 1 ~
-0 40 0 08 0 037 0 48 -1 20 -1 07 -0 12 1 40
Observed
~1.43 1.38 1 1.36 1.40 1.37 0.92 0.95 1.16 1.51
1 ~
~
1.57 (van d e r Kaals) 1 38 (van der Kaals) 1.45 (van der Kaals) 1.15 (band spectra) 1 . 4 (atomic radii) 1.OO (viscosity) 1.17 (viscosity) 1.43 (viscosity) 1 . 5 (atomic radii)
that if 4nr2u > E , the solubility of a gas will increase in temperature, while if 47rr?u < E , the solubility will decrease in temperature. Tamniann (11) generalized similarly on the sign of the temperature coefficient of solubility in d i i c h he stated that if y > 1 the temperature coefficient will be positive, and that if y < 1, it will be negative. This rule seems to hold fairly generally except for water. K i t h a knowledge of values of E through plots of logy with u i t is possible to compute the temperature coefficient of solubility. This has been done and calculated values have been compared with the observed coefficients recorded by Horiuti and Lannung. Instead of computing the coefficients
d 1/T
which, referring to equation 5 , is equivalent to -45rr'u
+E
2.303 k
1222
H.
H. CHLIG
+
values of the free energy of solution, -4ar% E , have been computed in calories per mole. To arrive at values for - 4 a h E , from surface tension and temperature dependence of solubilities, the following relations were employed :
=
2.303 1;
+
(1 log y d 1!T ~
d 1% Y is where I is the intercept in the plotted data of log y with u and ___ du the slope. The quantity
*
is taken from the observatioiis of Horiuti d l T (4);Lannung’s (7) published values of this quantity from his own data are used for helium, neon, and argon. The values are given in table 2 for the solvents benzene and acetone. These solrenti were chosen a t random,-any of the other organic qolvents show similar correspondence. By and large, agreement in the ralues for the free energy of solution - 4ar% E calculated from surface tension and temperature depencleiice of solubility is as good as the simplified theory n-arrants. For many of the gases listed in table 2, the calculated values are well within the experimental error. llention has already been made of the difficult technique and large experimental error in measuring gas solubilities. Methane values sholy the largest deviation between observed aiid calculated values, owing probably to the specific interaction with solvent which was assumed to be the cause of the somewhat scattered data for methane in figure 1. I n this connection, although the calculated values are not recorded in table 2 , water appears t o be an anomalous solvent. From Horiuti’s data, extending 01-er a considerable temperature range, it is possible to calculate the temperature dependence of the solute radius r and the solvent-solute interaction E . Surface tension data were obtained by making use of a compilation of surface tension data by -4. Ferguson and S. Kennedy (2) aiid by reference to the International Critical Tables. Slopes and intercepts of the logarithms of the solubility plotted with corresponding surface tension data for the solveiitq n-ere obtained by the method of least squares. I n table 3 value. are li.tec1 for r and E for nitrogen and carbon monoxide at three temperatures. The solute radius is constant for nitrogen and very nearly so for carbon nioiioside over a temperature range of 4OoC. The interaction energy E , shown previously to be approximately the same for all solutes at the sanie temperature, is shown to decrease with increase in temperature. T-alues of E actually
+
1223
SOLUBILITIES O F GASES A S D SURFACE TENSIOK
represent the differences in energy of interaction of solute--solrent molecules in the liquid phase as compared with the gas phase of the solvent. As the temperature increases near the critical temperature of tlic solvent, the distinguishing differences between gas and liquid tend to diasppear, TABLE 2 Jlolal j r e e e n w g y of solutions f r o m siirJacc tension data Calories per mole a t 20°C. +E
-471%
i
GAS
4rrr?a
'
E Calculated
1
Observed
Solvent : benzene
K2.. . . . . . . . . . . . . . . . . . . . . . . . . . . . , CO... . . . . . . . . . . . . . . . . . . . . . . . . . I CHa. . . . . . . . . . . . . . . . . . . . . . . . . . . H~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . , He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S e............................. , A , .. . . . . . . . . . . . . . . . . . . . . . . . . . . .
-1135 -1030 -984 -1002 -442 -472 -708 -998
' '
0 2.
............................
I
1 ,
i
~
'
1
-53.7 67.1 645 -537 -1610 -1439 -161 110
-1189 -963 -339 - 1539 -2Q52 -1911 - 869 -889
- 1390
- 989 - 807 -154 - 1362 - 1972 - 1827 - 7-13 -i o 9
- 980 -640 91.5 - 1640 -3300 -2000 - 800 - 64G
- 990
- 92 - 1840 -2850 -2700 - 1000 -810
Solvent :acetone AT2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
-935 - 847
co
....................... CHa. . . . . . . . . . . . . . . . . . . . . . . . . H,. ........................... He, . . . . . . . . . . . . . . . . . . . . . . . . . . . . S e........................... A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0: . . . . . . . . . . . . . . . . . . . . .
-53.7 67.1 654 - 537 -1610 - 1.439 - 161 110
-808 -825 -362 -388 -582 -819
TABLE 3 Efect of t e m p w a t u r e on gas i a d i ! i s axd "E" r X 10s em.
I ~
T
IX
E IN C I L O R I E S P E R
"C.
0 20
-10
~.
'
K2
1 .43 1.45 1.43
' ' I
MOLE ~~
CO
S?
l.3G 1.39 1,41
-10.3 - 89 -210
______
~
I
CO
~ _ _ _ ~
127 51 - 23
until at the critical temperature or above one phase is possible and differences no longer exist. I t appears reasonable therefore that values of E go through a minimum with increasing temperature and that at the critical teniperature their values become zero. This is illustrated in figure 3.
1224
H. H. UHLIG
Values of E in calories per mole for nitrogen and methane in chlorobenzene are plotted with the absolute temperature. The dotted line represents an extrapolation of the experimental data in accordance with the expected trend. At the critical temperature of chlorobenzene (359"C.), the values of E for both gases are shown to be zero.
FIG.3. Relation between solvent-solute interaction energy and absolute temperature SUMbIARY
Considering the energy change in transferring a solute molecule of radius r t o a solvent of surface tension u, an expression as follows, making use of the Maxwell-Boltzmaiin distribution theorem, is derived for the solubility of a gas : 111 y
+
-4~7,~u E
= ____
kT
E is called the interaction energy of solute and solvent, and y is the ratio of concentration of solute in the solvent to t h a t in the gas phase. For several gases, including oxygen, hydrogen, nitrogen, carbon monoxide, and methane, E is found to be approximately constant for any one gas in a
SOLUBILITIES O F GASES Ah-D SURFACE T E N S I O S
1225
variety of solvents. From a plot of log y with surface tension of the solvents (for any gas), a straight line is obtained from the slope of which can be calculated the solute molecular radius. The radii calculated in this manner are in good agreement with radii calculated from physical measurements of other types. T7alues of E, the interaction energy, are determined from the intercept. I n general, if the solubility of a gas is small, E is negative; if large, positive. From values of E so determined, the free energies of solution of gases can be calculated. These calculated values compare favorably with values obtained from the temperature dependence of gas solubility for most solvents, the notable exception being water. Least square treatment of Horiuti’s data for nitrogen and carbon nionoxide data over a 40°C. range of temperature show that the solute niolecular radii determined from the surface tension plots tend to be constant. The interaction energies decrease with increase in temperature. Because the interaction term E is the difference in the solute-solvent interaction of gas and liquid phases, its value at the critical temperature must be zero. It is shown t h a t E passes through a minimum at some temperature lower than the critical temperature of the solvent. REFERESCES
(1) CHRISTOFF, A . : Z. physik. Chem. 41, 139 (1902). A , , A N D KENSEDY,S.: Trans. Faraday SOC.32, 1174 (1936). (2) FERGUSON, (3) HILDEBRAND, J. H. : Solubility of Son-electrolytes. Reinhold Publishing Corp., S e w York (1936). (4) HORIUTI,J.: Sei. Papers Inst. Phys. Chem. Research (Tokyo) 17, 222 (1931). ( 5 ) HORIUTI,J.: Sci. Papers Inst. Phys Chem. Research (Tokyo) 17, 126 (1931). (6) LANGYUIR, I . : Colloid Symposium Monograph 1936, p. 48. (7) LANNUNG, A . : J. Am. Chem. SOC. 62, 68 (1930). (8) MAXTED, F. B., A N D MOON,C. H . : Trans. Faraday SOC.32, 772 (1936). (9) SISSKISD,B., A X D K.~SARSOWSKY, I.: anorg. allgem. Chem. 214, 385 (1933). (10) SKIRROW, F . : Z. physik. Chem. 41, 139 (1902). G.: Z. anorg. allgem. Chem. 168, 17 (1926); 194, 159 (1930). (11) TAMMANS,
z.
