J. Phys. Chem. l W 3 , 87, 3407-3411 4
desorption of a sulfur compound other than SO2.
400.
Discussion and Conclusion
> W
a 0
b
100
200
300
500
400
1
800
700
600
-
TEMPERATURE ( " C )
Figure 6- TPD Curves 1 at 250 "C.
after sulfur dioxide adsorption on NiO a m p l e
> E
4J
VI 0 E
$ 40
.
0
a L
al W
u W a
3407
20 .
100
200
F W e 7. TPD Cur~eSOf 250 "C.
300
400
500
600
TEMPERATURE ( " C )
700
800
oxygen from NiO after oxygen adsorption at
ferent desorption maxima located at 150,300,570, and 750 "C. With sample 1, the results obtained are reported in ~i~~~~6. we can observe two so2desorption peaks (R = 0.4) at 150 and 300 OC, and a new Desk located at 710 "C with a ratio R = 1.6 which may be connected to the
The main point which appears from the different results is certainly the fact that only with sample 1, which contains all the absorbed oxygen species previously identified upon the surface, is there specific desorption of a new compound at 710 "C. If we compare this result to the TPD curves obtained from NiS04 decomposition, it seems reasonable to think that a similar energetic process controls the two phenomena which appear at the same temperature, and it can be deduced that sulfur dioxide reacts especially with oxygen species 0' in order to oxidize sulfur according to Sm. the reaction SW Such an interpretation could explain the difference in the calorimetric data for samples 1 and 2-5: the heat evolved by samples 2-5 concerns a SO2 adsorption only (by the fact that sample 5 contains none of the adsorbed oxygen species) while the heat evolved by sample 1 is representative of a SO2adsorption and a partial oxidation of sulfur. In order to investigate the part of gaseous oxygen, some complementary experiments were carried out at 250 "C with a gaseous mixture 02-SO2 (P(0,) = 100 mbar, P(S02) = 10 mbar) and with a clean sample, free of chemisorbed species. Under these conditions, the TPD results are similar to those obtained with samples 2-5, i.e., without a desorption peak located at 710 O C . Previous TPD experiments, performed to study oxygen adsorption at various temperatures, have shown that few chemisorbed oxygen species are created at 250 "C on a clean NiO surface (Figure 7). We can therefore conclude that the sulfur oxidation process involves necessarily an oxygen chemisorption step with the creation of weakly bound species 0'. In conclusion, thermal or gaseous initial treatment of nickel oxide appears as a very important parameter on SO2-NiO interactions and the knowledge of the different kinds of chemisorbed oxygen species on the solid can provide us with criteria for sulfur oxidation. Registry No. 02,7782-44-7; SOz, 7446-09-5; NiO, 1313-99-1.
-
Theory of the Surface Tension of Ionic Solutions in the Born-Green-Yvon Equatlon 1. L. Croxton and D. A. McQuarrle' Department of Ctmmlsfry, Unlversny of Celifornla, Davis, Cellfmla 95616 (Received: October 1, 1982; In Final Form: January 7, 1983)
The f i t member of the Bom-Green-Yvon (BGY)hierarchy is used to calculate the surface tension for a restricted primitive model of an electrolyte solution. The calculations extend the work of Onsager and Samaras and Buff and Stillinger to finite-size ions and higher concentrations. A comparison to experimental aqueous-solution surface tension data in the range 0.1-1.5 M is made.
Introduction An interesting specialization of double-layer theory is to a system of ions in contact with an uncharged wall separating regions of unlike dielectric constant. The electrostatic effect of the interface is described completely in terms of image potentials. Such a model may be used to represent an ionic solution/air interface or ionic solution/hydrocarbon interface. Experimental comparison is possible since the particle distribution functions obtained 0022-365418312087-3407$01.50/0
from a theory may be used to calculate the surface tension or interfacial tension of the system. A theory of the surface tension of ionic solutions is a particularly intriguing problem in itself since existing theories appear to be valid only in a concentration range for which experimental data are questionable. For example, the classic electrostatic theory of Onsager and Samaras' and the more general (1) L.Onsager and N. N.
T.Samaras, J. Chem. Phys., 2,528 (1934).
0 1983 American Chemical Society
3400
The Journal of Physlcal Chemistfy, Vol. 87, No. 18, 1983
theory of Buff and Stillinger,2 based on a linearized Kirkwood equation, are accurate only for very low concentrations where an experimental artifact, discovered by Jones and Ray: is dominant. However, it has been shown by Croxton, McQuarrie, Torrie, and Valleau* that the double-layer theory of Croxton and McQuarrie5 for this system, based on the Born-Green-Yvon (BGY) intergral hierarchy, yields a singlet distribution function g%) in good agreement with Monte Carlo data in the molar concentration range. Surface tensions calculated from the BGY theory should thus allow a meaningful comparison with experiment and hence a measure of the validity of the model used. This paper is a presentation of more extensive calculations in the BGY equation and a comparison to experimental aqueous solution surface tension data in the range 0.1-1.5 M. Related work on the calculation of the surface tension of electrolyte solutions appears in ref 6. Theory Consider a system of charged hard spheres contained in a half-space of uniform dielectric constant e. Let all the spheres have the same diameter, D , and let half of the particles bear a e+ charge and the other half bear a echarge, where e is the magnitude of the electronic charge. This is the familiar restricted primitive model of a 1-1 electrolyte. If we now place a hard planar wall at z = -D/2, the centers of the particles will be restricted to the halfspace z > 0. Let eo denote the uniform dielectric constant of the other half-space, z < -012. For such a model the total potential may be written as a s u m involving a singlet potential, ui(l)(rl),and a pair potential, u.J2)(rl,r2),where i and j deonte the species of particles 1 anJ 2, respectively. Furthermore, the singlet and pair potentials may be expressed as the sum of an electrostatic contribution and a short-range contribution. Thus, we have
ui(l)(i1) = uw"u(zl)+ u?l(z1)
(1)
uij(i1,i2)= uijHS(r) + uijel(r,rIM)
Croxton and McQuarrie e-BuHs(rl)
= H(r - D )
(7)
where u*(zl) represents a "soft" interaction of the ion with the wall and H is the Heaviside step function. We shall consider a form for u*(z) which crudely accounts for the necessary rearrangement of solvent structure as an ion approaches the interface. Consider a hydration sheath of thickness W surrounding each ion. If the ion is positioned at some z < W , a certain volume of this hydration sheath would extend beyond the hard wall at z = -012. We m u m e that u*(z) is proportional to the volume that must be displaced. From geometrical considerations we obtain
V = '/7rh2(3(D/2)+ 3W - h)
(8)
where h = max (0,W-z). Choosing the proportionality constant, a, to represent units of kT per H 2 0 molecule volume, we write u*(Z) = kTa7rh2(D/2
+ W - h/3)/(7rW/6)
(9)
where W is the diameter of one "spherical" H20molecule, taken to be 2.76 A. Under these assumptions of charge and size symmetry, the singlet density distribution function, g/l)(z),is the Same for both positive and negative ions, that is, gl(l) = gZ(l)= g(l). The double-layer theory of Croxton and McQuarrie5 then reduces to the single-integralequation (see Appendix 1) In g(')(z) = F ( z ) + 27rpS-dz2 [g'')(z2)- 1]K(z,z2) (10) 0
where F(z) = X -- A -2 ( 2 + Y2)
ah2(3
w
2TP + 6W - 2h) + -(m2 3
- 3m +
r
(2)
where rm is the distance of the jth ion to the image of the ith ion. The electrostatic terms, given by Buff and Stillinger,2 are "2
(3) (4) where qi is the charge of an ion of type i, X =
(e
- tO)/(t
+ to) r = ((xl - x 2 ) 2 + (yl- yJ2 +
(z1
- ZZ)~)''~
rIM= (r2+ (2z1+ D)(222 + D)1'/* The short-range potentials are defined by UW%1)
= u*(zJ + UHW(Z,)
(5)
= H(q)
(6)
e-Bu"W(z1)
(2)F. P. Buff and F. H. Stillinger,J. Chem. Phys., 25, 312 (1956). (3)G.Jones and W. A. Ray, Jr., J.Am. Chem. SOC.,69,187(1937);63, 288 (1941). (4)T.L. Croxton, D. A. McQuarrie, G . N. Patey, G . M. Torrie, and J. P. Valleau, Can. J. Chem., 69,1998 (1981). (5)T.L. Croxton and D. A. McQuarrie, Mol. Phys., 42, 141 (1981). (6)A. Bellemans, Phyaica, 30, 924 (1964);A. L. Nichols and L. R. Pratt, J.Chem. Phys., 76, 3782 (1982);B. Jancovici, J. Stat. Phya., 28, 43 (1982).
In Equations 10-12, ha(2)(r,rm) = 1/2[g12(2)+ g11(2)- 21 and hd(2)(r,r M) = 1/2[g12(2) - g11(2)],where gii(2)is approximated by gijh71,7d= gi+) exp(-X(qiqj/(EkTrIM)le-"wlwith gij(r) being the radial distribution function far from the wall and K the usual inverse Debye length. Other notation is defined by riM= rIMwhen r = 0, m = min ( z , l ) ,m2 = min (1,1z-z21),Ml = max (O,z,-l),A = q 2 / ( 2 e k T )where q lqll = 1q21,p = cD3 where c is the bulk concentration, k is the Boltzmann constant, and T i s the temperature. All lengths have been reduced by the ionic diameter D. We show in Appendix II that eq 10-12 yield the limiting form of Onsager and Samaras for low concentration. Our procedure is to evaluate numerically and store F ( z ) and K(z,z2)using bulk radial distribution functions gij(r) from the EXP(DH) theory of Olivares and M~Quarrie.~
The Journal of Physical Chemistty, Vol. 87, No. 18, 1983 3409
Surface Tension of Ionic Solutions
Eq 10 is then solved for g(%) by direct iteration. Computer time for iteration is insignificant relative to the time required for tabulation of K(z,z2)and F ( z ) and for calculation of the surface tension increment. The surface tension increment is given by the formally equivalent expressions2 AT1 =
Lm[‘Jdz) -
(13)
dZ
where the stress tensor components are given by ‘JT(z1) = r 2
-kT C i=l
2
pi(%,)
+ ‘/z i C j=l
Jdi2
X
C (mole/ liter)
2
2
-kT C
+ y2 C
i=l
ij=1
Jdi2
X
These equations are essentially those given by Buff and Stillinger,2 but uN is slightly altered by the fact that TIM depends upon the hard-sphere diameter by nature of the coordinate system used here. The numerical results presented below were obtained by using 121 points uniformly spaced over an interval of length L with L equal to about 5-D lengths. Similar results were obtained by using lower point densities and different interval lengths, indicating that our numerical representation of g%) is adequate. The infinite integrals involved in calculation of K(z,z2)and F ( z ) were evaluated as finite integrals with the upper limits ranging from 2L to 4L. An asymptotic tail was also added to the triple integral of F ( z ) which exactly cancels the asymptotic contribution of the first, ( z + 1/2)-1, term.
Results Plots of g%) vs. z are shown at several concentrations in ref 4. These plots show that the comparison of the BGY singlet distribution functions with Monte Carlo data is quite good up to 1.0 M. Figure 1 of this paper shows a comparison of the surface tension predictions of the BGY theory with the limiting law of Onsager and Samaras and the results of Buff and Stillinger. Most striking is the similarity of Ayl and AyI1 obtained from the BGY equation. The agreement of Ay obtained from the two thermodynamic routes is a measure of the internal consistency of the BGY theory. We believe that the Onsager-Samaras result falls below the BGY results because of neglect of higher order terms. The inconsistency of the Buff-Stillinger theory likely results from their linearization approximation and the larger values of Ay that they obtain are probably due to the point ion model which they use. This interpretation is supported by Figure 2, in which we ~~
(7)W.Olivares and D. A. MeQuarrie, Biophys. J., 15, 143 (1975).
Flgve 1. Surface tensbn (dynlcm) of an ionic solution in the restricted primitive model calculated from the Onsager-Samaras theory, the Buff-Stllllnger theory, and this work. The curves labeled by I and I 1 correspond to a calculation of Ay with eq 13 and 14, respectively. The diameter of the ions is 3.5 A, A = 1, and T = 298 K. 0.60 -
0.40 h
E
3
h U
,” 0.20 a
0
-
2
4
6
D(9 Flgure 2. Dependence of Ay (dyn/cm) on the diameter of the ions. The soli curve corresponds to a calculation of Ay with eq 13 and the dashed curve corresponds to the use of eq 14. The values of the parameters used are c = 0.5 M, T = 298.16 K, and y = 1. The difference between Ay’ and AT**Is small for 2.5 A I D I 4.5 A.
show the dependence of Ay on ionic diameter. The value of Ay increases for small and large values of D. Since the effect of ionic size is not great within the range characteristic of sodium and potassium halides, we have used the representative diameter of 3.5 A for all subsequent calculations. As did Buff and Stillinger, we have set X = 1 for convenience. The value of X is 0.975 if e = 78 and to = 1. Figure 3 shows how A y varies with A. Note that the value of Ay at X = 0.975 does not differ significantly from the value of Ay at X = 1.0. Figure 4 gives a comparison of this theory to the experimental data of Johanason and Eriksson read from Figure 3 of ref 8. As might be expected, the simple theory involving no short-ranged interaction with (8) K. Johnason and J. C. Eriksson,J . Colloid Interface Sci., 49, 469 (1974).
The Journal of Physical Chemistry, Vol. 87, No. 18, 1983
3410
0'30
T
Croxton and McQuarrie
Appendix I Derivation of Eq 1G12. We begin with the first member of the BGY hierarchy
which becomes
0
0.20
0.40
1.0
0.80
0.60
A
or since
Figure 3. Variation of Ay with A, which is equal to (e - eo)/(€ -t eo). The sdld curve corresponds to a calculation of Ay with eq 13 and the solM cwve corresponds to the use of eq 14, respectively. The values of the parameters used are c = 0.3 M, T = 298.16 K, and D = 2.5 A. The value of y = 0.975 for an air/water interface.
.; :a = o 2 0
With the definitions
1.20
.:
0
u(')(z)
F ( z ) = --
kT
+
h
E
;
0.80
3.
zi
and 0.40
(1-5)
0
0.20
0.60
c
1.0
we may write eq 10, which is computationally more convenient than eq 1-3. Using eq 2, 4, 7, and H'(x) = 6 ( x ) , we obtain
(mote/Iiter)
Figure 4. Comparison of Ay calculated from the BOY theory arid the experimental data in ref 8. The Ay calculated by eq 13 and 14 are essentially indistinguishable. The
and 0 represent data for KCI and
KI, respectively.
the wall yields poor agreement with experiment. A simple restricted primitive model of an electrolyte solution/air interface is an inadequate model to calculate surface tensions. If we allow the ions to interact with the interface through eq 9, then the agreement with experiment is substantially improved. Thus, we see that simple modification of the model to include short-range effects such as disruption of the ion's hydration sheath is capable of providing good agreement using physically reasonable parameters. The value of a,however, remains as an uncertain, adjustable parameter. Nevertheless, the BGY theory appears to provide an excellent approach to the surface tension of ionic solutions.
Acknowledgment. This work was supported by a grant from the National Institutes of Health, GM 26864-05.
Then
Surface Tension of Ionic Solutions
The Journal of Physical Chemistry, Vol. 87, No. 18, 1983
With notation defined in the text, we write eq 1-8.
3411
in F(z) which arise from hard-sphere interactions and may set a = 0. Thus, for low densities, we have
F(z) = -XA + 4pAimdz1J 2 2
dr',
half-space
(-
21
- 22
r3
+
Now if we set X = 1, substitute a Debye-Huckel radial distribution function into eq 11-1,and linearize the combined exponential, then we obtain
Equation 12 now follows easily. Similarly
F(z) = -- A + 8pA2Jmdz1J 2 %
half-space
dr2
Now
by symmetry. Similarly
Now since +fM,Zl+l dzz (zl- z2) = 0 unless z1 < 1 we have A
= -[1
- (1 + 2KZl)e-2u']
(11-7)
K2Zi2 1 1 3 /2[/3~
-z
+ y3]
if z < 1 = -Y6(m3 - 3m
+ 2)
If we substitute eq 11-7 into eq 11-4, then we obtain eventually
Equation 11 follows by substitution. Appendix I1 Extraction of a Limiting Expression for g(l)(z). Since the departure of g(l)from unity is first order in density, the integral term of eq 10 is second order in density and may be neglected. Furthermore, we may neglect all terms
or
in agreement with the theory of Onsager and Samaras.
