1407
THECALCULATION OF COHESIVE AND ADHESIVE ENERGIES mum value of the dipole moment observed for odimethoxybenxene in the two solutions which showed a levelling was still considerably smaller than the value calculated for free rotation. Examination of molecular scale models shows that certain rotational positions are restricted 13terically. The cis-cis position in which both methoxy carbons are adjacent and coplanar with the benzene ring is most severely restricted. I n addition, other positions in which the methoxy carbons
are coplanar with the ring are somewhat restricted by the ring hydrogens. Reduction of the contribution of these rotational positions may account for the difference between the observed maximum dipole moment and the calculated free rotation value.
Acknowledgment. The authors wish to acknowledge the contribution of Villanova University in providing financial support for carrying out this study.
The Calculation of Cohesive and Adhesive Energies from Intermolecular Forces at a Surface
by J. F. Padday and N. D. Uffindell Research Laboratories, Kodak Ltd., Wealdstone, Harrow, Middlesex, England
(Received November 16, 1967)
Surface tensions of the n-alkanes and interfacial tensions between the %-alkanesand water have been calculated. The calculations use a modified form of the Moelwyn-Hughes’ equation for the dispersion interaction between two particles, the integration method of Hamaker to derive the total interaction across a plane surface, the geometric mean relationship of Good and Girifalco for the interaction of two dissimilar phases, and an assumption that the entropy of surface formation equals the difference between the interaction energy so calculated and the total internal energy of surface formation. The calculated suiface tensions of the nalkanes are compared with and agree well with experimentally determined values; also, some of their calculated interfacial-tension, contact-angle, and spreading-coefficient measurements with water all agree with the corresponding experimental values. For other systems, calculations are limited to the contribution of the dispersion forces to the total interaction of the system.
Introduction
WlZ = Wlzd
The work of cohesion, W,, and the work of adhesion, W,, are defined by Young1 and Duprb2 by the equations
w,
= 2YL
w* = YL + Ys + YSL
(1)
(2) W, and W, are both surface free energy terms of an idealized system and in practice their values have been obtained by substituting appropriate surface and interfacial tensionfi into eq 1 and Z3 Such substitution is unjustified for almost all real solid-liquid systems because the surface tension is likely to vary from one part of the surface to another and because some elastic deformation of the bulk solid is inevitable. These calculations are confined to pure liquids and to lowenergy hydrocarbon surfaces where such effects, although present, are unlikely to produce significant errors. The interaction energy W I ~arising from physical or van der Waals forces between two particles is given by
+ WIZD + WIZK
(3)
where W12d is the dispersion contribution and W1ZD and WIZK the respective Debye and Keesom contributions to the total interaction energy. In this treatment, the calculations are restricted to nonpolar systems in which the Debye and Keesom forces are zero. The dispersion interaction energy, W12d, between two particles is the decrease in their potential energy arising from only dispersion forces when bringing them from an infinite distance of separation to a distance T apart and is given by Moelwyn-Hughes4 as (1) T. Young, “Collected Works,” Vol. 1, G. Peacock, Ed., John Murray, London, 1855. (2) A. Dupr6, “Theorie Mecanique de la Chaleur,” Gauthier-Villars, Paris, 1869,p 369. (3) N.K. Adam, “The Physics and Chemistry of Surfaces,” 3rd ed, Oxford University Press, London, 1941,p 2; F. 0. Koenig, 2.Elektrochem., 57, 361 (1953); R. E. Johnson, J. Phys. Chem., 63, 1656 (1959). (4) E. A. Moelwyn-Hughes, “Physical Chemistry,” 2nd ed, Pergamon Press Ltd., London, 1961,p 392.
Volume 76,Number 6 M a y 1968
1408
J. F. PADDAY AND N. D. UFFINDELL (4)
obtained by replacing the vo of Slater and Kirkwood’s5 expression with v&/’ where vo is given by
The algebraic sum of these two energies, Wmd and Q (the quantity of heat entering the system to obtain equilibrium), divided by 2 gives the increase in internal energy, Ed,for surface formation
+Q
Ed = Wmd
e 2 ?r (mcyo) ‘1’
2
Moelwyn-Hughes found S , the effective number of dispersion electrons, to be smaller than 2. We have taken the value of S in these calculations to be the valency or valencies of the atom or atoms involved. To obtain the total interaction arising from dispersion forces at the surface of a macroscopic system, we calculate the decrease in potential energy, Wmd, of unit area of a system in which two semiinfinite and parallel surfaces of a liquid, separated by a large or infinite distance, tire brought together to a distance at which the surface region is indistinguishable from the bulk liquid. Wmdis thus obtained by summing the energy change of every pair interaction acting across the semiinfinite surfaces, using eq 4 to obtain the coefficient, p, of l/r6 for the interaction of a single pair. To obtain Wmd,eq 4 was integrated according to the method of Hamaker6 (see Overbeek’) to give
(5) where ,811 is the coefficient of l/r6 of eq 4 and A is the Hamaker constant. This method of calculation is preferred to that of FowkesJswho integrated London’s0 equation for surface volume elements only. To obtain the total interaction energy of a system in which the semiinfinite surfaces of two different and mutually insoluble materials are brought together, the p11 coefficient of eq 5 is replaced by the coefficient for the interaction between two liquids (or phases) using the geometric mean relationship of van Laarlo and Good and Girifalco“ P12
= (PllP22)1/2
(6)
Relationship between the Total Interaction Potential and Surface Tension The total interaction energy derived by eq 5 is the increase in potential energy of the system when two unit areas of liquid surface are formed from bulk liquid. It represents the work done on the system. During the process of separation no account has been taken of the equilibration of the surface region. It has been assumed that the bulk structure of the liquid phase is maintained right up to the surface and is retained a t all stages of surface formation. The structure of the surface region is unlikely to be the same as the bulk due to the imbalance of attractive forces; therefore, further energy must enter or leave the system as the freshly formed surfaces approach equilibrium. The Journal of Phgsical Chemistry
(7)
where
Ed - TS
= y
(8) Thus to obtain the surface tension, y, from Wmdit is necessary to know Q (a quantity almost impossible to measure) and 8, the entropy of surface formation. It is implicit in Fowked8 work that the quantities Q/2 and TS are numerically equal and cancel each other, but no clear reason for this has been given. In this work we have found, like Fowkes, that the potential energy term, Wmd/2, equals the surface tension for a suprising number of systems over relatively wide temperature ranges and we will, therefore, make the same assumption. A second assumption to be made is that the calculated value of the surface free energy, i.e., the surface tension, applies to solid as well as liquid surfaces. Using this assumption it is now possible to calculate such properties as contact angle, spreading coefficient, and work of adhesion according to the relationships given below. However, to avoid writing out all the components of the coefficient l/rIl2 in the expression for Wmd,the Hamaker constant, ALL,will be used, as in eq 5 , for the interaction between two like liquid par~ that between two unlike particles of ticles and A L for liquid and solid.
Relationships between the Hamaker Constant and Wetting Properties The energy of interaction, Wmd, arising from dispersion forces between two semiinfinite plane surfaces of a liquid, has been derived above in eq 5 . The relationships between this interaction energy and wetting properties may be summarized as follows. (i) Surface Tension of a Pure Liquid, y ~ .Within the assumption, eq 5 , 7, and 8 may be written as (9)
(5) J. C. Slater and J. G. Kirkwood, Phys. Rev., 37, 682 (1931). (6) H.C. Hamaker, Physica, 4, 1068 (1937). (7) J. Th. G. Overbeek, “Colloid Science,” Vol. I, H . R.Kruyt, Ed., Elsevier Publishing Co., The Netherlands, 1952,Chapter 6. (8) F. M.Fowkes, Ind. Eng. Chem., 56 (12),40 (1964);Advances in Chemistry Series, No. 43,American Chemical Society, Washington, D. C., 1964,p 99. (9) F. London, Z.Phys.,?63, 246 (1930); Trans. Faraday Soc., 33, 8 (1937). (10) J. J. van Laar, “Die Thermodynamik einheitlicher Stoffe und binBrer Geraische,” Noordhoff, Denmark, 1936. (11) R. J. Good and L. A. Girifalco, J. Phys. Chem., 64, 661 (1960).
THECALCULATION
1409
COHESIVE AND ADHESIVE ENERGIES
OF
where
so=
I n practice lboth NLL and rLLare obtained from the density, p L , of the liquid for the temperature at which ALL is required. The symbols on the right-hand side of eq 10 are all identified in the Glossary. (ii) Interfacial Tension between Two Liquids or a Solid and a Liquid, 71. If mutual insolubility is assumed, the interfacial tension may be expressed using eq 6 and 9 to give
-2yL
+2
ASSALL
(v) Wetting Energy or Adhesion Tension, We. The wetting energy is defined by the equation
We
=
YL COS
(18)
Therefore ASSALL
(vi) Work of Adhesion, W,. is defined by
ASSALL
6
w,
= YL(C0S
The work of adhesion
e + 1)
(20)
Theref ore This equation has also been used for water in contact with a nonpolar surfaces in which case
where ALL^ is the dispersion contribution to the total energy of water and YL is the true experimental surface tension. The terms ASS and r s s although designated to describe the properties of a solid surface may, of course, be those of a second pure liquid. Because YSS and r L L are often not very different, an approximation is sometimes useds Ylr = Y S
+
YL
- 2(YSdYLd)'/*
(13)
where 7sd and Y L are ~ the hypothetical surface tensions the solid and liquid would possess if only dispersion forces acted a t their respective surfaces. (iii) Conlact Angb, 6. Combining the Young equation ys
-
TI =
YL COS
6
w,= 2
ASSALL ( 2 4 4 2( rss
'/a
(21)
42
Equations 15, 17, and 19 all contain the work of adhesion. I n order to simplify calculations the expression for the work of adhesion may be replaced by the geometric mean of the dispersion contribution to the surface tension without serious error, as in the derivation of eq 13. These relationships have been used to calculate the surface properties of systems which rely on dispersion forces alone. These relationships may also be used to calculate the surface properties of systems in which one component is polar by estimating the contribution due to dispersion forces to obtain the work of adhesion and using the experimental value of the surface tension for the total of all contributions. To use the above equations it is necessary to calculate first the Hamaker constants from values of a0,8, 2, and N I L The value of N I I equals (l/rll3) and is obtained from p , the density, according to
(14)
with eq 10 and 11gives cos6 = --1
[
+2
A WALL
:
'/a
YL
(15)
(24~)~("~
When this equation is used for a system including a polar liquid or water, ALL refers to the dispersion contribution ALL^ and YL is the total surface tension of the liquid. (iv) Spreading Coeflcient, So. The spreading coefficient is definled by
so = YS - YI Therefore, from eq 10, 11, and 16
YL
(16)
For large molecules the interparticle distance rll will vary for each part of the molecule; therefore, in the subsequent calculations with the n-alkanes we have calculated the Hamaker constants on the basis that CH2 or CHs groups are the basic volume elements or particles.
Calculation of the Surface Tension of n-Alkanes The surface tensions of the n-alkanes at 20" calculated using eq 9 are given in Table I. To illustrate the difference between taking the CH2 unit and the whole molecule as the volume element, n-hexane is taken as an example. The values of the various terms in eq 9 Volume 72, Number 6 M a y 1968
1410
J. F. PADDAY AND N. D. UFFINDELL
Table I: Surface Tensions of the n-Alkanes, C%HZ,+~ -(a
=
0.35, b = 1.83)*-
(20°),=
g/ml
n
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 30
-(a
= 0.425, b
3
YL
P
0.144a 5.4 0.286d 11.4 0.430d 17.6 0.567d 23.5 0.626 26.1 0.660 27.7 0.684 28.8 0.704 29.8 0.718 30.3 0.730 30.8 0.741 31.4 0.751 31.8 0.757 32.2 0.765 32.5 0.769 32.7 0.775 32.9 0.778 33.1 0.777” 33.1 0.777’ 33.1 0.77@ 33.2 0.765 33.4
YL
1
102“~, oma
dn, oma
(oalcd), ergs/om2
lOZ4a0,
5.70 4.44 3.84 3.49 3 37 3.30 3.26 3.22 3.21 3.19 3.17 3.15 3.14 3.13 3.13 3.12 3.11 3.11 3.11 3.11 3.11
2.53 4.36 6.19 8.02 9.85 11.70 13.50 15.35 17.20 19.00 20.85 22.70 24.50 26.30 28.15 30.00 31.80 33.65 35.50 37.30 55.60
2.53 2.18 2.06 2.01 1.97 1.95 1.93 1.92 1.91 1.90 1.895 1.89 1.885 1.88 1.875 1.875 1.87 1.87 1.87 1.865 1.865
0.34 2.12 6.16 12.7 16.4 19.0 20.6 22.5 23.2 24.1 25.3 26.2 26.9 27.5 27.6 28.2 28.6 28.6 28.6 28.7 29.0
2.62 4.39 6.16 7.93 9.70 11.45 13.25 15.00 16.75 18.55 20.30 22.05 23.85 25.65 27.40 29.15 30.95 32.70 34.45 36.25 53.95
TLL,
10-nlN~~
1.77)’-
oma
adn,
oma
2.62 2.20 2.08 1.98 1.94 1.91 1.895 1.875 186 1.855 1.845 1.84 1.835 1.83 1.825 1.82 1.82 1.815 1.815 1.81 1.80
(calcd), ergs/cmz
ergs/cmaMeasd Ref
--YL,
0.38 2.15 6.25 12.4 16.1 16.0 h 18.4 18.4-19.24i-k 20.1 20.4 1, h 21.7 21.5-21.8 h , j , k 22.3 22.9 h, j 23.3 23.9 h, 1 24.4 24.7 h, 1 h 25.2 25.4 25.8 25.9 m 26.2 25.6-26.7 i, 1, n 26.6 27.0 27.6 i,j 27.5 27.4 27.4 27.4 27.5
with water, erga/cm*Calcd Measd Ref
--?I
68.0 62.1 57.2 54.3 53.7 53.6 50.2-51.1 1, O, p 53.6 50.2-52.6 I, 0, p 53.7 50.8-51.681, O-(I 53.7 53.8 51.2-52.3 1, p 54.0 54.1 52.78 q 54.2 54.3 52.2-54.2 1, n, p 54.3 54.4 53.77 q 54.5 54.5 54.5 54.5 54.5
a “Handbook of Chemistry and Physics,” 37th ed, Chemical Rubber Publishing Co., Cleveland, Ohio, 1955-1956. * H. H. Landolt and R. Bornstein, “Zahlenwerte and Funktionen,” Vol. 1, Springer-Verlag, Heidelberg, Germany, 1950-1951. Th. G. Scholte and F. C. deVos, Rec. Trav. Chim., 72, 625 (1953). Liquid density a t 20’. Liquid density a t 28’. Liquid density a t 32’. Liquid density a t 37’. 0. R. Quayle, ef al., J . Amer. Chem. Soc., 66, 938 (1944). ’ A. I.Vogel, J . Chem. Soc., 133 (1946); 616 (1948). F. M. Fowkes, J . Phys. Chem., 67, 2538 (1963). W. D.Harkins and E. H. Grafton, J . Amer. Chem. Soc., 42, 2534 (1920). ‘See E. G. Shafrin and W. A. Zisman, J. Phys. Chem., 66, 740 (1962). A. Weissberger, “Techniques of Organic Chemistry,” ref 11. Vol. 1, 3rd ed, Interscience Publishers, Inc., New York, N. Y., 1959,p 763. W. D.Harkins, et al., J. Amer. Chem. Soc., 42, 700 (1920). J. A. Krynitsky and W. D. Garrett, J . Colloid Sci., 18,893(1963). ‘ R.Aveyard and D. A. Haydon, Trans. Faraday Soc.,
’
61, 2255 (1965).
Table I1 : The Constants Involved in the Calculation of the Contributions of Dispersion Forces to the Surface Tensions of n-Hexane and Water
Term
NLL
z
S 010
rLL
YLd
(ZOO)
Relationship
Volume element CH2 in n-hexane
CeHlz in n-hexane
NAp/mOl W t Outer electrons Valency electrons Ref 9,13,14 (No)
4.62X loz1 38 38 11.80X om3 6.01X cm
27.7 X 1021 6
Eq 8
14.9ergs/cm2
18.7ergs/cm2
and 10 are set out in Table I1 for n-hexane and CH2 taken as the volume element. The calculated values of y~ of 14.9 and 18.7 ergs/cm2 compare with experimental values between 18.4 and 19.24 ergs/cm2 given in the literature cited in Table I. This method of calculation is suitable for all the alkanes for which the polarizabilities are known. Since yL is dependent on aO8/’,it is important to have good experimental values of the polarizability. In order to obtain unknown values by interpolation, known values T h e Journal of Physical Chemistry
6
1.97x 10-24 cm3 3.3X lo-* cm
>
HzO in water
3.34x 1021 8
4 1.48X cm3 3,114X lo-* cm 19.2ergs/cmz
of molecular polarizability of the n-alkanes were plotted as a function of the number, n, of carbon atoms in the alkane chain, and a straight line was obtained. The 2a, where b is the polarizability of equation a0 = nb the CH2 group and a b that of the CHI group, fitted these data. Hence the polarizability of any alkane was calculated by inserting the requisite value of n. Two literature source^^^^^^ have been used for values of polarizability of the alkanes and these are shown plotted in Figure 1. These two sets of data have been
+
+
1411
THECALCULATION O F COHESIVE AND ADHESIVEENERGIES 71 = 18.7
+ 72.8 2(18.7 X 19.2)”* = 53.6 ergs/cm2
Similar calculations have been performed for other alkanes with water, as shown in Table I. Use of eq 13 instead of the more accurate eq 11 introduces no appreciable error because, as seen in Table 11, the interparticle distances for water and CH2 in n-hexane are very similar. The calculated value of 53.6 ergs/cm2 is compared with the experimental value of 51.1 ergs/cm2.
Calculation of the Interfacial Tension between %-Hexaneand Mercury
0
5
10
Fowkes* gives the experimental value of y~~ for mercury as 200 ergs/cm2. Calculation of this value was not attempted because ao, the polarizability, is unknown. The interfacial tension is calculated as before using eq 13.
Number of carbon atoms in alkane.
Figure 1. The total molecular polarizabilities of n-alkanes as a function of the number of carbon atoms: (A) 2a = 0.70, b = 1.83 (aee ref 12); (B) 2a = 0.85, b = 1.77 (see ref 13).
used independently to calculate, by means of eq 9, two sets of surface-tension data. These are shown in Table I and plotted in Figure 2 together with the experimental values taken from the literature. The surface tension of the gases methane, ethane, propane, and butane are those calculated for liquids under some 40-50 atm of pressure a t 20”. The liquid densities of these four alkanes were obtained by plotting the densities of the higher alkanes a t 20” against the number, n, of carbon atoms in the alkane molecule and extrapolation to the lower values of n.
Calculation of the Dispersion-Force Contribution to the Surface Tension of Water Substituting; the values given in the last column of Table I1 into eq 9 and 10, taking the water molecule as the volume element, one obtains a value of 19.2 ergs/cm2 for the dispersion contribution to the total surface tension. Thus dispersion forces contribute 26% of the true surface tension of water. We shall to distinguish this value from the total surface use 71,~ tension y~ of t,he liquid. From the value of y~~ of water and on the basis that polar liquids interact principally through dispersion forces with nonpolar substances, the wetting properties between nonpolar compounds and water were calculated.
Calculation of the Interfacial Tension between n-Hexane and Water Using eq 111 or 13 the interfacial tension between n-hexane and water has been calculated. Using the latter equation
30
25
20 N*
8
7;; 13
6 15
I
2 rn 0
10
5
I
.i 0 0
5
10
15
20
Number of carbon atoms in alkane.
Figure 2. The calculated and measured surface energies of n-alkanes at 20’ as a function of the number of carbon atoms: , experimental values; - - -, calculated values from a. values of ref 13; - - * - calculated values from a0 values of ref 12.
-
(12) “Handbook of Chemistry and Physics,” 37th ed, Chemical Rubber Publishing Co., Cleveland, Ohio, 1955-1956. (13) H. H.Landolt and R. Rornstein “Zahlenwerte and Funktionen,” Vol. 1, Springer-Verlag, Heidelberg, Germany, 1950-1951.
Volume 78, Number 6 May 1968
1412
J. F. PADDAY AND N. D. UFFINDELL
71 = 18.7
+ 484 - 2(18.7 X 200)'/' = 380 ergs/cm2
This compares well with the observed value14 of 378 ergs/cm2
Calculation of the Spreading Coefficient of Water on Triacontane According to eq 16 and the simplified eq 13
8,
= -2(72.8)
+ 2(19.2~ss)'/'
7 5 s is the calculated surface tension of triacontane shown in Table I. Two values are given, the mean of which is 28.2 ergs/cm2. Inserting this value in the above calculations gives X, = -99 ergs/cm2. The measured S, on paraffin wax is -99 ergs/cm2.
Calculation of the Angle of Contact of Water on Triacontane COS^
=
-1
X 19.2)'/* + 2 (28.2 72.8
= -0.36
e
= 110"
This compares with the measured contact angle of water on paraffin wax, 110-115°.15
Calculation of the Work of Adhesion of Water on Triacontane W , = 2(28.2 X 19.2)'12= 46.6 ergs/cm2 1
Using the more precise eq 21 gives 45.9 ergs/cm2.
The Effect of the Presence of the Vapor Phase In the derivation of eq 9 it was supposed that the bulk liquid surface was ruptured to produce two unit areas of semiinfinite liquid surface. In practice such a process involves the replacement of each semiinfinite surface of liquid with a corresponding semiinfinite surface of vapor. The effective surface tension of the vapor has been calculated using eq 9 and 10, wherein all quantities except NLL are the same as for the liquid. Using eq 11 with the substitution of Ass by Avv, the Hamaker constant of the vapor phase, and T S S by TLL, it being assumed that the distance of closest approach in the vapor is equal to that in the liquid, then
and the simpler expression of eq 9 used without serious error. At 20" the liquid and vapor densities of n-hexane are 0.6595 and 0.0006 g/ml; thus the difference between N L and ~ (NL - N v ) ~ is only 0.2%. As the critical temperature is approached, NV approaches NL and YLV tends toward zero.
Discussion The L ~ n d o n ,the ~ Slater-Kirkwood,6 the Xeugebauer,16 the Moelwyn-Hughe~,~ and the Lifshitz1'118 formulas have all been used to calculate the fl coefficient for dispersion interaction between two particles.1*719 The Lifshitz formula used by Gregory was for fully retarded forces applied at large distances (T, 2000 d) and is not applicable to the wetting system here because the unretarded forces acting over molecular dimensions are thought to predominate. Of the formulas applicable to these surface systems, the London expression gives the lowest value, Seugebauer's the next, then Slater-Kirkwood's, and finally Moelwyn-Hughes', giving the largest value for PII. Strictly, London's equation should be applied to interaction of hydrogen-like atoms having one orbital electron. The equation of Moelwyn-Hughes used in these calculations takes account of all outer shell electrons and also, unlike the other equations, the difference between their average frequency of oscillation and the frequency of each electron in isolation. The integration of the pair potential by the method of Hamaker assumes that the radial distribution function of volume elements is a discontinuous stepped function as shown in Figure 3, curve a for water at 20". This is compared with the radial distribution function given by Narten, Danford, and Levy20 (curve b) ob-
-J
2
0
0
1
2
i ,IKu, 3
4
,
,
,
,
5
b
7
8
~
9
1
I
0
Figure 3. Radial distribution function of water at room temperature: (a) assumed from Hamaker summation, 20'; (b) from X-ray diffraction of waterz0 at 25'.
Substituting values for ALLand Avv from eq 10 in terms of the common value PU (i.e., PLL), eq 23 becomes YLV
=
PLL
-(NL - Nv)2 2 4 r ~ ~ ~
At temperatures well below the critical point, Nv is so small compared with N L that it may be neglected The Journal of Physical Chemistry
(14) W. D. Harkins, "The Physical Chemistry of Surface Films," Reinold Publishing Corp., New York, N. Y . , 1952. (15) R . B. Ray and F. E. Bartell, J . Colloid Sci., 8, 214 (1953). (16) T. Neugebauer, 2. Phys., 107, 785 (1937). (17) I. E. Dzyaloshinskii, E. M. Lifshitz, and L. P. Pitaevskii, Soviet Phys. J E T P , 37, 229 (1959). (18) J . Gregory, Discussions Faraday SOC., 42, 168 (1966). (19) R. H. Ottewill and J . N . Shaw, ibid., 42, 154 (1966). (20) A. H. Narten, M.D. Danford, and H. A. Levy, ibid., 43, 97 (1967).
THECALCULATION OF
COHESIVE AND
ADHESIVEENERGIES
1413
of TS a t room temperature (-26 ergs/cm2) is, in fact, tained from X-ray diffraction data. The latter curve compensated for in the method of calculation we have refers to 25" but the authors show that the temperature used. coefficient is sufficiently small to allow comparison To summarize, it was found that the contribution of of the radial distribution a t 25" with that a t 20". dispersion forces to the total interaction between two The step of thLe Hamaker curve lies to the right-hand particles could be calculated. Difficulties in finding a side of the peak of the X-ray diffraction curve and this suitable value for the ionization potential have been would have the effect of producing a low value for the avoided by using the equation of Moelwyn-Hughes. calculated dispersion contribution to the syrface Only polarizabilities and densities are then required tension. If the value for rLL is taken as 2.76 A, the and these are known with certainty. The integration length of the hydrogen bond, then the dispersion conof pair potentials for a bulk system followed Hamaker's tribution to the surface tension of water is increased method and the total energy of suface formation was from 19.2 to 24.4 ergs/cm2. A similar comparison of equated to the surface tension on the basis that the radial distribution functions of the n-alkanes is not entropy term cancels with the energy required to bring possible became of lack of data. a freshly formed surface to equilibrium. The surface In calculating the wetting properties of water a t the tensions of the n-alkanes so calculated agree within a surface of a n-alkane we have assumed that the Debye few per cent of the measured values. The surface contribution to the work of adhesion is negligible. free energy of a n-alkane that is solid at 20" was calThe calculated interfacial tensions of Table I are all culated and agreed well with values derived from expergreater than t,he measured values and this could be iment either directly or indirectly, mutatus mutandis, explained either by our calculated value of the disfrom the angle of contact, the work of adhesion, or other persion contribution to the surface tension of water wetting properties. We have also attempted to clarify being too low or by the Debye forces being significant. the assumptions made in the calculation of surface The extent of the Debye interaction of water dipoles tension implicit in the work of Good and Girifalco, with CH2 groups of a n-alkane is proportional to aop2, Fowkes, and the present authors. where a0 referir t o the polarizability of CH2 groups and p refers to the dipole moment of water. The constant Glossary obtained is 5% of the dispersion value given by Moelwyn-Hughes' formula and represents about 2.3 ergs/ A Hamaker constant (= + N # ) ; area cm2. Electronic charge (=4.80 X e esu) Planck's constant ( = 6.62 X 10-27 erg/sec) Moelwyn-HughesZ1 derived an expression for W M ~ h Electronic mass (=9.11 X lo-** g) m that took into account Born repulsion and equated it to Avogardro's number ( = 6.02 X 1023mol-') NA the internal energy of surface formation. His expresNumber of molecules or volume elements per cubic NLL sion for WM' is proportional to the inverse square of the centimeter interparticle distance but is not suitable for calculating P Pressure Intermolecular distance or distance between volume TLL surface tension at other temperatures than 0°K. elements Fowked8 expression for surface tension was a different Entropy; effective number of dispersion electrons, x function of iinterparticle distance obtained (erronetaken as valency ously we believe) by summing the potential energy of Spreading coefficient S O surface molecules only and ignoring bulk-bulk attracTemperature (OK) T V Volume tion. The expression for WM' used here thus seemed Work of adhesion W, to be more adequate. Work of cohesion WO The assumption that the work or potential energy Wetting energy; adhesion tension W, term WM' of eq 5 equals the surface free energy (and Interaction energy between isolated pairs of moleWI2 hence surface tension of a pure liquid) involves the cules Interaction energy between macroscopic surfaces WM arbitrary assuinption that Q / 2 of eq 7 equals TS of Z Number of outer-shell electrons eq 8, where S is the surface entropy. The main justiStatic polarizability ff0 fication in this assumption lies in the ability of eq 9 to Dispersion constant (coefficient of r-6) P predict, in the same way as Moelwyn-Hughes,21 the Surface tension; force required to stretch unit s u p Y surface tension the liquid would possess if it were at face E Surface energy; energy required to create unit new 0°K. surface Using eq 9, ithe surface tension of n-decane at 0°K is Density P calculated to be 44 f 4 ergs/cm2, using the density data 6' Angle of contact between two phases of TimmermarqZ2which is somewhat lower than the Proper frequency of an isolated electron vo extrapolated experimental value of 50 st 2 ergs/cm2. At O'K, TS must be zero and Q / 2 is likely to be small (21) E. A. Moelwyn-Hughes, J . Colloid Sci., 11, 501 (1956). or zero, hence i;ome agreement is expected. The agree(22) J. Timmermans, "Physico-chemical Constants of Pure Organic ment, although not good, suggests that the large value Compounds," Vol. 2, Elsevier Publishing Co., London, 1965, p 41. Volume '78, Number 6 M a y 1968
1414 c1
d L or LL
P. 0. MASLOV Dipole moment Dispersion contribution (superscript) Liquid phase (subscript)
S or SS
Solid phase (subscript)
I m
Interface (subscript) For a macroscopic system (subscript)
The Semiempirical Method for the Calculation of Some Parameters for High-Resolution Nuclear Magnetic Resonance Spectra by P. G.Maslov A . I . Gerzen State Pedagogical Institute, Leningrad, U.S. S. R .
(Received October 98, 1966)
In the present paper, a useful semiempirical method for calculating magnetic shielding constants, IT, chemical shifts, 6, the polarizabilities, a,the Langevin magnetism, Xd, Van Vleck paramagnetism, xP,and other molecular parameters is offered. In the work, it is shown that at least part of the known difficulties, connected with the calculations of the above mentioned parameters, can be removed by means of using an equation of weighted averages. Comparative data leave no doubt as to the effectiveness and rather high precision of the proposed method, of the order of O.l-l.O%.
The study of nmr spectra at the present time has acquired great scientific and practical significance.1-20 However, one entire category of questions on the theory of nmr spectra still awaits r e ~ o l u t i o n . ~ -For ~ example, there is still no development of the theory of nmr spectra for nuclei with spin I 2 1 available. I n the calculation of the case of nuclei with spin I = nmr spectra is often difficult because of the lack of the necessary parameters: the magnetic shielding constant, u, of nuclei in molecules, the chemical shifts, and ~ t h e r s . l - ~Approximate quantum-mechanical metho ~ s , ~ - unfortunately, ~ J ~ J ~ are not available at a level sufficient to permit theoreticians to calculate, in important cases, the nmr parameters with sufficient precision to have practical importance. I n such situations, the development of new methods, even if they are approximate or semiempirical, but which are not constrained with the deficiencies of the quantum-mechanical methods, becomes urgent. I n the present paper, a useful semiempirical method for calculating magnetic shielding constants, u, chemical shifts, 6,the polarizabilities, a,the Langevin magnetism, Xd, Van Vleck paramagnetism, xB,and other molecular parameters is offered. The susceptibility is given by eq 1
x
= ‘/dXzz
+ xuv + X Z J
=
where xt (i = 1, 2, 3) are the components of the molecuThe Journal of Physical Chemistry
lar magnetic susceptibility, x, averaged along all directions, M is the molecular weight, d is the density of the substance, and xv is the volume magnetic sus ceptibility.
(1) J. W. Emsley, J. Feeney, and L. H. Sutcliffe, “High Resolution Nuclear Magnetic Resonance Spectroscopy,” Pergamon Press, Oxford, 1965. (2) A. Losche, “Kerninduktion,” Veb Deutscher Verlag der Wissenschaften, Berlin, 1957. (3) J. A. Pople, W. G . Schneider, and H . J. Bernstein, “HighResolution Nuclear Magnetic Resonance,” McGraw-Hill Book Co., Inc., New York, N. Y., 1959. (4) E. R. Andrew, “Nuclear Magnetic Resonance,” Cambridge IJniversity Press, London, 1955. (5) R . E. Richards, Advan. Spectr., 2, 101 (1961). (6) G. H. Townes and A. L. Schawlow, “Microwave Spectroscopy,” McGraw-Hill Book Co., Inc., New York, N. Y., 1955. (7) R. H. Bible, “Interpretation of NMR Spectra, An Empirical Approach,” Plenum Press, New York, N. Y., 1965. (8) M. A. Eliashevich, “Atomic and Molecular Spectroscopy,” State Physical and Mathematical Publishing, Moscow, 1962. (9) P. Pascal, A. Pacault, and J. Hoarau, Compt. Rend., 233, 1078 (1951). (10) P. Pacault, Ezperientia, X , 41 (1954). (11) P. G . Maslov, Dissertation, V. I. Lenin State Pedagogical Institute, Moscow, 1953. (12) I. Mizoguchi and M. Inoue, “Nuclear Magnetic Relaxation in Magnetite,” Tokyo, 1966. (13) J. M. Anderson and J. D. Baldeschwieler, J. Chem. Phys., 40, 3241 (1964). (14) J. M. Anderson, J . Chem. Educ., 42, 363 (1965). (15) R . H. Herber, Ann, Rev. Phys. Chem., 17, 261 (1966). (16) M . T. Jones and W. D . Phillips, ibid., 17, 323 (1966). (17) J. S. Waugh, Ed., “Advances in Magnetic Resonances,” Academic Press, New York, N. Y . , 1965.
