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A new method for the calculation of free surface energies of liquids and solids

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A new method for the calculation of free surface energies of liquids and solids

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Content A HEW METHOD FOR THE CALCULATIOH OF FREE SURFACE EHERCIES OF LIQUIDS AHD SOLIDS A Thesis Presented to the Faculty of the Department of Chemistry The University of Southern California In Partial Fulfillment of the Requirements for the Degree Master of Science in Chemistry fey Edward Herman Strisower October 1945 UMI.Number: EP41556 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI EP41556 Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 TABLE OP CORTERTS Page Chapter I - statement of the Problem 1 7 ^ ° Chapter II - Review of the Literature 2 * f Chapter III- Theoretical Part . A '.IP® Derivation of Surface Tension Equation 13 Chapter IV - Experimental Part Statement of the Problem 28 Description of the Apparatus 31 Purification of Samples 37 Calibrations 39 Surface Tension Measurments 41 Vapor pressure Measurments 47 Chapter V - Conclusion 30 Chapter VI - Bibliography 31 1 CHAPTER I The Problem In the past numerous empirical relations have been developed expressing the surface tension of liquids as a function of temperature, vapor pressure and density. From a theoretical viewpoint it seemed desirable to develop such an equation from thermodynamic considerations. This has been the purpose of this paper* The problem is to develop an equation expressing sur face tension in terms of such easily available experimental quantities as vapor pressures and densities and to keep the treatment sufficiently general to allow application of the resultant equation to solids and liquids. CHAPTER II 2 Review of the Literature on Surface Tension Theory It is a well known faot that the surface of a liquid tends to adjust itself so as to expose the minimum surface area. This is due to an unbalanced force of attraction ex erted upon the molecules in the surface and tending to pull them inward. This phenomenon is at the root of the property of surface tension of a liquid, where surface tension can be identified thermodynamically as a free surface energy P. Since surface tensions are ordinarily measured under conditions of constant temperature and pressure and since volume changes are negligible, P may be replaced by the Helmholtz free energy A. This definition of surface tension in terms of thermody namic functions is very helpful in investigations concerning the theoretical foundations of many empirical equations relat ing surface tension to temperature, density and other variables. Many experimental methods for the determination of the surface tension of liquids have been devised"*". Of these the capillary rise method, the drop weight method and the wire ring method are the most common. With the aid of these methods I horsey, Bur. Standards Sci. Paper, 21, 562 (1926) A general survey of experimental methods. 3 the surface tension of many liquids has been determined over a wide range of temperature* The data so obtained has been analyzed and various empirical equations for the temperature dependence of surface tension have been developed. The Oldest and best known of these equations is the Efitvos Equa tion^ developed in 1886 where o;p( M, k are surface tension liquid density, molecular weight and an empirical constant r(f)% = K fU -T j respectively. This relation was later modified by Ramsay and Shields^ to read *(f ) % 5 K(T‘-r -d) where d is a constant equal to six. Also well known is Mac- leod's4 equation relating surface tension to liquid and vapor density. ^ i f i y - c C is a constant independent of temperature, D and d are liquid g and vapor densities respectively. H. Katayama developed an equation taking into account the density of the vapor above the liquid as well as the liquid density. His equation reads and reproduces experimental data more accurately than most E EStvos, Wied. Ann., 87, 456 (1886) 3 Ramsay Shields, Phil. Trans., A 184, 647 (1893) 4 Maoleod, Trans. Farad. Soc., 19, 38 (1923) 5 H. Katayama, Tohoku Imp. Univ. So* Reports, (1), (4), 373 (1916) 4 other equations which were developed previously. The impor tance of the vapor phase and the concept of an interface with properties different from either the liquid or gas phase was emphasized by Shereshefsky^. He developed an equation contain ing critical, liquid and vapor densities and the critical where aT& is ah empirioal constant. Shereshefsky showed that his equation represented the temperature variation of the surface tension for several organio liquids (benzene, ohloro- benzene, ethyl ether, methyl formate, ethyl acetate) more closely than did even Katayama*s equation. It is a curious fact that Shereshefsky*s equation fails for the case of car bon tetrachloride. While all the above equations are capable of represent ing experimental data, they contain empirical constants and are therefore unsatisfactory from a theoretical point of view until these constants can be justified by theoretical con siderations. Two main avenues of attack appeared possible to overcome this short coming. The first consisted in the theo retical derivation of equations similar in form to the old well known empirical equations - comparison between the two would then throw some light on the significance of the era 's T.X. Shereshefsky, <T. Phys, Chem., 35, 1712 (1931) 5 pirical constants contained in these old equations. The second method is hased on a strictly deductive approach, starting with some plausible model of a liquid and predict ing the properties to be expected from such a model. This latter method was followed by Margenau7, l.S. Kassel®, the former by Einstein9, Fowler10, and Lennard-Jones11. It may be noted at this point that the above division into two dis tinct methods of approach is, of course, highly artificial since the treatment of each investigator invariably contains elements of both methods. It has been used merely as a de vice of classification; the same end might have been accom plished by differentiation in terms of statistical or non- statistical methods employed. Einstein*s and Margenau*s treatment would then be classified as non-statistical, the others as statistical. It now remains to examine the more important ideas of these investigators in the treatment of the problem and to see how closely experimental results can be predicted by their respective methods. Margenau calculated the work necessary to separate two 7 Margenau, Phys. Rev., 3£, 365 (1931) 8 L.S. Kassel and M. Muskat, Phys. Rev., 40, 627 (1932) 9 Einstein, A.D. Physik, 34, 165 (1911) 10 R.H. Fowler, Proc. Roy. Soc., A 159, 229 (1937) 11 J.E. Lennard-Jones and J. Corner, Trans. Farad. Soc., 36, 1156 (1940) 6 halves of a liquid column by integrating the force (expressed as a function of distance) necessary in this separation. He then equated this work W to TJ, V= -b J /l where U is the total surface energy of the liquid, related to the free surface energy "by the well known equation V ' - a- r (!fj - p(t?J where or is the free surface energy and is identical with surface tension as ordinarily measured. The use of the relation V--W/l has been questioned11 as it appears that Margenau has calculated the free surfaoe energy rather than TJ, the total surface energy of the system. However, the values obtained from Margenau*s final equation V - - 1 ] d x (l) where n is the average number of molecules, E(r) the mutual potential energy of two molecules a distance r apart, x 0 the distance between the two liquid halves, are in fair agreement with experiment (see table I) for He, He, A, Hg, Og, and Gig. Q This agreement has been criticized by Kassel as being due to merely a fortunate ohoice of temperatures. Kassel has calculated the mutual potential energy be tween two halves of a liquid, obtaining using as potential field ° £(?)- j° r , B(z) = fo r £* where f(z) is a distribution function, z the distance between two molecules, R0 the molecular radius. Concerning f(z) two possible assumptions were investigated f ( z ) = 0 j o r 2<r£0 j f ( z ) - e The corresponding values of U then take the form 7/ = I M u * e v - d-2 (2) (3) Experimental values do not agree with the values calculated from either equation, but appear to lie closer to the values calculated from equation (2) as shown in table I. It thus appears that the distribution function assuming uniform dis~ tribution is based on a fundamentally more correct model and that the Boltzmann distribution while holding for the case of a very dilute gas is not applicable to a liquid. This is another point of criticism of Margenau*s treatment pointed out by lennard-Jones^. Table I Element XJ calculated Eq.l Eq.2 Eq. 3 U experimental T, *K irexperime. He 0,78 0,78 7.10® 0.6 4 0*12 He 20.0 5.88 26.3 15.0 26 5.15 A 50.0 17.76 145.1 35.0 87 12.5 % 26.0 12.63 55.8 25.8 80 18,27 Cl2 100.0 45.11 6688.0 73.0 233 29,2 °2 21.0 14,21 40.4 35.0 80 15,7 8 Attempts to deduce the old empirical equations from theoretical considerations date from 1911 when Einstein^ showed the relationship existing between the free surface energy and the heat of vaporization and on the basis of a simple liquid model was able to deduce the EJJtvos equation* His considerations led to a value of k, the constant in the EStvos equation, which cheoked in the order of magnitude with the value obtained by substituting experimental values of surface tension into the EStvos equation and solving for K. In a paper which appeared in 1937 R.H. Fowler^ derived MaoleodTs equation and Sugdenls parachor by Quantum statisti cal methods. The scope of this treatment is best expressed in Fowler?s own words: "...It is sufficient to recognize that the quantity P defined by Kafr. p ft-Pa ' is in fact a constant (very nearly) for any given substance, independent of the temperature for a wide range, from the critical temperature downwards - and having recognized this fact, to attempt to derive it as a theorem in statistical mechanics applied to a reasonable model, and to give a for mula for P in terms of molecular diameters and (or) inter- molecular forces.." The results of this treatment are given in table II for the case of argon and mercury, showing good agreement with experiment considering the uncertainty with which, some of the fundamental constants involved in the cal culation are known. Table IX Element P calculated P observed A 52.5-54 H g 53 68 According to Fowler*s treatment and theory there is no ground for expeoting Macleod’s equation to hold over a wide temperature range. The fact that it does seems to be due to a cancellation of a number of subsidiary effects. Fowler*s treatment shows that the paraohor is a con stant near the critical temperature but did not apply to temperatures in the neighborhood of the melting point or boil ing point as was pointed out by lennard-Jones^. A treatment more general in this respect appeared in 1940 by Lennard-Jones11. The authors make use of a partition function for the liquid given by where $0 is the potential energy of the molecules when arranged at their mean positions in a face centered cubic lattioe of infinite extent, containing N molecules per volume JLffmk ( 10 TT. is the 'free volume* of an atom when all others are fixed in their mean positions; sharing of this free volume is taken into account by the termo< whose value equals zero for no sharing and one for complete sharing. The partition func tion has been discussed in earlier papers of the authors^2”14. The Helmholtz free energy A is given by A - -k7~&>gQ, and hence can easily be expressed in terms of the above partition function for the liquid. For the particular case of a liquid column of volume V, cross section l/E, surface S, and con taining H molecules, the free energy A is given by A , = A„+ & -Ho-k-T-NkT/tf V/ • where A0 is a term taking into account the free energy associa ted with the surface S. If the liquid column is now broken at its central cross section and the two halves are removed to an infinite distance in an isothermal, reversible process, the free energy of the resulting system will be given by At= A„+K -f/VfeTfrjpSLkJ] _ Afc(kT -^V~/Vsjkl 6 ° ^ ~ A / i \/ s where and 1 * ' are the average number of molecules per unit area of surface and the free volume of a surface molecule IE J.E. Lennard-Jones and A.F, Devonshire, Proc. Roy. Soc. A163, 53 (193?) 13 Ibid., 165, 1 (1938) 14 Ibid., 169, 317 (1939) respectively. is the potential energy of the N molecules in their equilibrium positions in the infinitely separated halves of the column, a quantity which may be calculated by a lattice summation. Since in the above process of breaking the liquid column unit' area of surface was created, the differ ence Ag - A*}, should give the free energy increase due to this unit area of surfaoe, i.e. Ax - A, - b = Y o - & i- M KTfy V*/ | / * since / Y 0 ~j>o) an<i are Proportional to the surface den- -V3 sity of the atoms i.e. to V and since the temperature variation of Vf / /$ . is small and may therefore be neglected, the above equation reduces easily to a yV i = k(r,-T) which is 18tvofs law. K is a constant having a value of about 2.1 ergs/degree. A value close to this observed value was obtained by Madelung^® by assuming a plausible variation of the free volume, while Born and Courant^^ used the hypothesis of an elastic continuum as in the Debye theory of specific heats and obtained agreement with experiment within a few percent. A treatment of the total surface energy of crystals 17 appeared in 1943 in a.paper by R. Haul*' in which he related 15' "¥f Made lung, Physik. Z., 14t 729 (1931) 16 M. Born and R. Courant, Physik. Z., 14, 731 (1931) 17 R.B. Haul, Z. Physik. Chera. 53, 331 (1943) that quantity to the heat of sublimation by 12 A(i) ~ 1/ - f Ao'j at constant T where AciJ inner molar heat of sublimation, Agj the energy necessary to vaporize the atoms from the surface of the crystal into the surrounding space. A f l f j then represents a fraction of y | ( i ) gig©*1 by the ratio of the number of bonds to be broken per atom in the interior and the surface of the crystal, therefore k = t hv z where Z is the number of neighboring atoms in the interior of the crystal, 2* the number of neighboring atoms in the surface of the crystal. It then follows that I f = ^ ' A ( » ) . . , . which gives the total surface energy of’the crystal in terms of the heat of sublimation and fundamental crystallographio constants. Oh the basis of this theory Haul calculated the surface energies of the solid noble gases obtaining (fair agreement with experiment) the data shown in table III. 13 Table III Element T,°K Heat of Subl. cal/gm Total Surf 100 plane . Energy 111 plane Free 111 Xenon 0 3790 45.7 39.6 39.6 Krypton 0 2678 39.4 34.2 34.2 Argon 0 1847 29.6 25.6 25.6 He on 83.8 0 1690 26,2 22.7 14.3 24.6 471 10.6 9,2 6.7 The assumptions underlying Haul’s treatment may be briefly summarized as follows: 1) The measured heat of sublimation minus the volume work represents the energy expanded against the forces of attraotion between the atoms. 2) A perfectly ordered crystal lattice is assumed. This is known to be an approximation from an inves tigation by Hennard-Jones1® indicating distortion of the lattice near the surface, decreasing the distance of the uppermost layer by about 5$ of the normal distance between planes* 3) Only the effects of the nearest lattice neighbors are considered, an assumption which appears to be justified by an experimental investigation of ' 5 77E7 Hennard-Jones and Dent, Proc. Roy. Soc.,A121, 247 (1928) 14 S t r a n s k i ^ who showed that for the case of Cd of the bond energies are accounted for by the effects of the nearest neighbors. From this value it can be deduced that neglecting the effect of the more distant 'neighbors* affects the calcu lated surface energies by only about one per cent. In conclusion, the papers of gtrankkliand Kaishew20 may be mentioned; Kaishew's treatment relating surface energy to the work necessary to split a crystal leads to the same results as Haul's considerations outlined above. 15 CHAPTER III THEORETICAL PART The Derivation of the Surface Tension Equation In. the derivation of the surface tension equation the environment of a molecule in the interior of a liquid is con sidered first. Such a molecule is obviously completely sur rounded by other liquid molecules which exert attractive forces, such as Van der Waal forces, on each molecule in the interior of the liquid. These forces, termed bonds in the succeeding discussion, cannot be exerted with equal symmetry and in all directions upon a molecule in the surface, since such a molecule has lost part of its liquid environment. In terms of bonds the molecule will now possess only a fraction of its former number of bonds. • Because of this difference in the number of bonds, a molecule in the surface has a larger free energy content than, a molecule in the interior of the liquid. The increase in the free energy content of a molecule in the surface over one in the interior of the liquid is then the surface free energy of that particular molecule. Designating Fb as the free energy increase per mole of liquid associated with the dissolution of all bonds, f as the fraation of bonds broken in the liquid surface and K as the number of moles per square cm of surface, the free energy per square cm ^ is given by < r = K f - F t in 16 5?he problem now resolves itself into the evaluation of K, f and . £he term Fy, oan he calculated by means of the thermodynamic cycle shown below. - Meaning of symbols: s vapor pressure of the real liquid pe s density of the real liquid p> s pressure required to retain the density of the real liquid upon dissolution of all bonds p* s pressure of dilute ideal gas s density of dilute ideal gas 3?he free energy content of the actual liquid, Fg , can be considered as the sum of two terms, Ft and Fc , where Fb represents the free energy Increase due to bonding and FA the 17 free energy due to any other cause. F t - + Pc. The free energy content of the liquid under the pressure p*, Fg(, is then given by Ft, ~ Ft + Fc + aPJ and the free energy content of the ideal liquid under a pressure Pf by ' Fb f Fc+ aF, + AF'e But for the ideal liquid Fj, is zero by definition, whence ^ Hence calculating AFf f c from the cycle will give the free energy due to bonding. In the evaluation of dF*g the actual liquid under its vapor pressure Pj and density^ may be taken as the starting point. Compressing this liquid under a pressure P* will not appreciably affect its volume (though a correction may be applied here for a more accurate calculation) so that the free energy changeAF,to the state designated by P T, is g i v e n b y & F , = £ ( P ' - P j ) The next step involves the dissolution of all bonds resulting in an ideal liquid and is associated with a free energy change designated by AF*g. Releasing the pressure P* on the ideal liquid to the very low pressure P* of a dilute vapor results in a free energy change given by AF, = RT K Pi The pressure P* on this dilute, ideal gas is now in creased till it reaches the vapor pressure of the actual gas; this step is accompanied by a free energy change given by Since the real vapor is in equilibrium with the liquid, the free energy change from vapor to liquid is equal to zero. This step completes the cycle. It then follows from the cycle, that _ KTi».ffP* = '+ + Rr^ p*/p/ (l) whence, 4F't = n U - f / f - £(?'-?,). £2) The surface tension ft of a substance is equal to the decrease in free surface energy per square cm of surface formed. In terms of the theory this increase in free surface energy is largely due to the partial loss of liquid environ ment resulting in a partial non-bondedness of the molecules in the surface. To a first approximation we would expect a molecule in the surface of a liquid to have lost one half its liquid environment or one half of all its liquid bonds. Hence we can write where K is the number of molecules per square cm of liquid surface, the increase in free energy accompanying the dissolution of all bonds per molecule. To estimate the value of K a simple liquid structure was postulated by assuming a distribution of molecules similar to that in a face-centered cubic lattice. K was then calcu lated for three crystal planes; the details of this calcula tion are shown for the 100 plane. Each corner molecule in a unit face centered cubic lattice is shared by eight cubes, each face centered molecule by two cubes, giving a total of four molecules per unit cube. Then where p s liquid density £ - length of edge of unit cube | v j - molecular weight 2 - Avoga^dio's number Since there are two molecules per unit cube in a surface composed of 100 planes of such face centered unit cubes, the number of molecules per square cm of surface is given by 20 A similar calculation for the 110 and 111 planes gives, respectively r ^ ^ 1 X (0 9 tiiK wfoi/cjM - (o-'X 'b x (o substituting equation (2) in (S) gives J r = iV-iffhTt^v/f -n(?'-?3 ) W Equation (5) can be expected to apply only at tempera tures at which the effect of the vapor above the liquid is negligible; it should hold strictly for the (physically im possible) case of a liquid in contact with a vacuum. At higher temperatures the effect of the vapor becomes increasingly important and equation (5) may be modified as shown below to include the effect of the vapor on the liquid surface• By means of a cycle analogous to the one employed for the liquid, a term ean be evaluated having the same significance for the vapor as AF'e for the liquid. That means that gives the free energy increase per mole of vapor associated with the dissolution of all vapor bonds resulting in an ideal gas. AkF^ is given by 21 The molecules in the surface of the liquid may he imagined to be bonded one half to liquid molecules and one half to vapor molecules. The increase in free energy due to the dissolution of one half of all liquid bonds is given to this quantity. This is evident since vapor bonding must increase the free energy of liquid molecules. Hence Combining equations (3), (5), (6) and (?) it follows that Equation (8) gives the surface free energy of a liquid surface, but disregards the vapor surface in contact with it. To take into aeoount the mutual effect of the liquid surface on the vapor surface and vice versa the following procedure is taken. A vapor surface can be shown, by considerations similar to these for the liquid, to have a free energy per mole given W but the free energy increase ) must be added v % 3 (?) and gg Equation (10) is analogous to equation (8), involving the assumption that the constant K is the same for both liquid and vapor. This assumption has been made as the simplest assumption in the absence of any more definite information. At high temperatures close to the critical temperature K for the liquid will be close to Kf for the gas and K will equal K 1 at the oritioal temperature. At lower temperatures, on the other hand, the effect of the vapor becomes negligible under all conditions irrespeotive of whether or not K equals K'; the assumption that K equals K f therefore appears reasonably valid. The free energy at the interface is due to the free energy contributions of both liquid and vapor surfaces which have Just been calculated, i.e. The situation at the interface (liquid-vapor) existing always at the Surface' of any liquid is therefore given by + ISvxp.SU+j- <1:L) which in turn becomes upon performing the indicated substitu tions Equation (12) represents the surface tension of a liquid expressed as a function of liquid and vapor densities and vapor pressure; it takes into acoount the effect of the 23 vapor on the liquid surface and applies to the interface (liquid-vapor) rather than to the physically impossible liquid surface by itself. This equation has been checked against experimental data for a number of liquids. The procedure has been to fit K at one of the lower temperatures and then use this value of K over the entire temperature range. This was suggested by consideration of the assumptions that 1) the liquid would approximate any model most closely at low temperatures, and 2) that no transitions in molecular distribution would occur. Two independent checks were thus obtained, one for the a priori calculated value of K resulting flfom the assumption of a specific liquid model, the other for the temperature dependence of the surface tension predicted by equation (12). In general, good agreement with experiment was found in all oases on both counts, i.e. K and the temperature co efficient, even for the oase of acetone which is highly polar. This seems to indicate that orientation effects due to dipole interactions are relatively unimportant in the surface, since bonding was assumed to be uniform in the deduction of equation (12). In the oase of water the calculated temperature depen dence of the surface tension agrees with experiment, while K is lower than for all other liquids. This may be due to a liquid structure different from that of non-hydrogen bonded liquids. 24 The average value of K obtained for 11 widely different liquids (including acetone, mercury, and water) is 7*0 as compared with the theoretically deduced value of 6,6 for the 110 plane in a face centered cubic lattice, and 6,2 for the 111 plane. While this agreement is somewhat closer than that for each individual liquid, the fact that the cal culated value for K in each case oomea within plus or minus fifty percent, and often much closer, to the experimental value indioates that rough predictions about liquid behavior based on a simple liquid structure may be made. The temperature dependence of surface tension predioted by equation (12) is in fair agreement with experiment over a very wide range of temperature, as appears from table IV. Considering the uncertainty with which some of the experimental data is known, agreement on the average of well within 10 per cent for 11 widely different liquids indicates the general validity of the assumptions underlying the derivation of equation (12). Table IV Carbon Dioxide, T0 s 304.2° K K / i m 9 T, °K pe c P K*t. f 0 ys (9vUt/cc.) (flWoc) (d *jiA e s /c vu) 2.75 230 r.126 .024 8.99 13.9 13.9 5.9 4.6 2.7 1.4 0.5 .06 T, °K P t A (Job 8.99 / 230 r.i26 fgwt/oc.) .024 273 0.914 .096 34.3 293 0.772 .19 56.5 303 .598 .334 70.7 25 Table IV (aont*d) Carbon Tetrachloride, T = 556.2°K K/lxlo9 3,03 T/K £98 Pe (jWcc; 1.595 (a&s/ct.) ,0007 3 ,12 r “ V 26.8 26.8 373 1.434 .0103 1.915 17.7 17,3 473 1.189 1.0742 14.5 8.9 6.5 543 .67 .271 38.15 2.1 0,7 Isopentane, T s 460.9 3.75 243 .6662 ,000325 .078 20.1 20.1 221 .6862 .000137 .0303 21.0 22.8 258 .6525 ,000635 .162 18,7 18.3 271 .6400 .00113 .302 17.6 16.8 289 .6235 .00221 .631 16,2 15.1 298 ,6148 .00318 .935 14.5 14.0 , Chlorine, T e 417°K 3,4 273 1,468 .0128 3.65 21,7 21.7 203 1.646 .0006 .151 42.5 32.4 303 1,377 .030 8.6 17.2 16,7 323 1.31 .0486 14,1 14.5 13.4 403 .92 .258 61,4 3.2 1.6 26 1 fax(Oq 2.97 Tj°K 041 Pt .943 n-Butyl Chloride a pj (qiMt/tc) (cShwJ .000023 .005 j / » r*- 29.5 29.5 224 .962 .0000077 .00155 29.6 31.4 253 .930 .00005 .0115 25.8 28,0 268 .913 .00012 .0281 84.4 26.1 281 .899 .00022 .0561 23.0 24.5 297 .882 .00046 .123 21,6 22,5 t-Butyl Chloride 3.37 262 .872 ,00038 .091 22.4 22.4 251 .885 .000227 .0513 23.4 23.7 271 .864 .00055 .135 21.3 21.3 285 .849 .00096 ,248 19.9 19.8 295 .838 .00139 Neon .371 19.1 18.6 3.8 25 1.24 .005 .466 5.5 5,5 26 1,23 ,00625 • 666 5.3 5.2 27 1.28 .008 .914 5.2 4,8 28 1.20 .0105 Helium 1.247 4,9 4.5 4.5 2.3 .1469 ,00159 .0777 ,314 .314 3.5 ,1377 .0082 .454 .166 .177 4.2 .1260 .0159 1.000 .169 ,098 k/ W 5,8 2,76 2,04 Tatle IV (oont'd) Mercury T,°K ft 293 13.546 P & ifrrt/cc) .00000152 27 46sc ^ « ^ 465 473 13.113 .0227 394 434 626 12.747 Acetone ,934 346 393 293 .792 .00058 .243 23.7 23.7 313 .769 .00177 .555 21.1 21.2 333 .746 .003 1.14 19.3 18. 6 353 .719 ,004 Water 2.12 17.3 16.2 .. 273 1.0000 .00000485 .00603 75,6 75.6 373 1.0435 ,000598 1.000 61,0 58.9 CHAPTER IV EXPERIMENTAL PART Statement of the Problem The problem -was to build a capillary rise surface ten- tension of liquids in contact with their own vapor only, over a temperature range of approximately+50 X to -50 °0, For this purpose a closed vacuum system was used per mitting the removal of practically all gases except the vapor in equilibrium with the liquid being studied. The choice of liquids to be investigated was based upon the following theoretical considerations. In the deri vation of equation (IS) conformity of the vapor to the ideal gas laws was implied by writing . At the high pressure P* occurring in the oyole it appeared plausible that the volume occupied by the molecules them selves may be a non-negligible fraction of the total molar volume V. Hence a correction seemed in order to take into account the volume b of the molecule itself resulting in an equation of state of the form. sion apparatus suitable for the measurement of the surface - RT (14) and a free energy equation . fer _ j,jdip - ■ I " 29 Using equations (14 and (15) in the cycle resulted in a modified surface tension equation of greater complexity than equation (12). This equation, which had not heen de veloped till the experimental investigations were well under way, was expected to show significant dependence upon b showing the effect of molecular configuration of which b in turn is a function, b was assumed to be equal to the volume of rotation of the molecule and was calculated from direct measurements of molecular diameters on Hirshfelder . models. Significant differences between essentially spherical and linear molecules were anticipated since the correspond ing volumes of rotation showed appreciable variation. To check these expectations the surface tension of an essentially spherical molecule, t-butyl chloride and its linear iaomer, n-butyl chloride were determined. Isopentane served as another example of a linear molecule. The theory outlined above, however, could not be sat isfactorily extended to take care of non-mas3 point molecules of volume b within the time limit set for this extension. The experimental results of the surface tensions of n-butyl chloride, t-butyl chloride and isopentane therefore serve only as an additional check on the validity of equation (12) in much the same manner as the experimental data found in the literature. The surface tensions of these three 30 liquids have heretofore not been determined over any con siderable range of temperature. 31 Description of the Apparatus Surface tensions were determined by means of the cap illary rise method employing an apparatus similar to the one SI used by O.Maas and taking essentially the same precautions while not attempting the same precision* Care was particu larly taken in making the wide tube large enough to prevent a capillary rise in it* It was made 4cm in diameter which gg has been shown ample for this purpose from theoretical as 23 well as experimental investigations of previous workers* Pig. 1 represents the glass part of the system. To 3 wet the capillary before each measurement and to agitate the liquid in the wide tube so as to bring it into uniform and complete thermal equilibrium a glass plunger was used. This plunger could be raised and lowered by means of a pair of strong permanent magnets acting on the upper part of the stem of the plunger which contained small iron rods built in for that purpose* After stirring the liquid the plunger could be fixed in any desired position by means of a guide, bearing and lock nut. The wide tube is connected to the sample bottle into which the liquids to be studied were intro duced. The procedure then consisted in freezing the liquid 21 0. Mass and C.H. Wright, J. Am. Chem. Soc., 43, 1107 (1921) 22 lord Rayleigh, Proc. Roy. Soc., A92, 184 (1915) 23 T.W. Richards and L.B. Coombs, J. Am. Chem. Soc., 37, 1659 (1915) u • M PQ o j r l s ' h a t f t / t U 3S in the sample bottle by immersing it into a dry ioe acetone (or liquid nitrogen) bath. When all the material had solidified stopcock A was opened to a cold trap connected to a high-vac pump. The entire system was then evacuated keeping stopcooks B, C and 3> open. The pressure was then measured by the Maoleod gage. Stopoock A was then turned to shut off the trap from the system and the liquid in the sample bottle was allowed to melt. The process was repeated to remove all dissolved gases in the liquid. Stopcocks B and C were then closed and some of the vapor in the system was condensed into the cold trap. This created a pressure differential by means of which the liquid in the sample bottle was forced into the wide tube by merely opening stopcock C. The few drops of liquid remaining in the sample bottle were distilled over into the cold trap by allowing the system to stand for about half an hour with stopcooks A, B and C open. When all the liquid had thus been removed from the sample bottle, stopcock A was closed, B and C were left open and the appara tus was ready for use. To remove the liquid from the apparatus at the comple tion of a set of measurements, stopcocks B and C were closed and most of the vapor in the remaining parts of the system were condensed into the cold trap. Stopcock B was then opened which forced the liquid back into the sample bottle by the same process by which it had originally been forced into the S3 wide tube. The sample bottle was then removed from its ground glass ;joint and the liquid poured out, then returned to the system and with stopcooks B and C open all of the residual vapor was condensed into the cold trap, thereby oleaning the system and making it ready for the investigation of the next liquid. The glass part of the system was immersed into a bath to a depth indicated in figure 1. The bath consisted of a rectangular 16 gage sheet metal box provided with double plate-glass windows in the front and back. These windows were held in place by pressure exerted by two metal frames against the front (and identically at the back) part of the box. Rubber gaskets were used between all glass to metal surfaces to insure complete tightness. The metal frames in turn were held in place by heavy bolts in the manner shown in fig. 2. The space between the double windows could be partially evacuated to serve as thermal insulation; in practice however, it was found sufficient to introduce oalcium chloride as a drying agent to prevent condensation of water vapor on the inner surfaces of the double windows. The outer surface of the front window through which the measurements were made was kept under a continuous stream of air to prevent condensation of water vapor and fogging. A series of air ^ets distributed around the window was installed 34 for that purpose and found to work satisfactorily in all low temperature measurements* The sheet metal box was set into a larger wooden box so constructed as to allow space for a layer of crushed asbestos of about £ in width to act as thermal insulator. The top layers of the asbestos were held down by cork sheets. With all surfaces of the metal box thus insulated (except the top) the temperature could be kept sufficiently constant over the entire temperature range to allow ample time for all manipula tions involved in the measurements to be carried out carefully and within the stated temperature invariance. The bath was filled with a mixture of isopropyl alcohol and acetone as a bath liquid. This mixture was found to remain clear and was readily available at the time of this investigation. The temperature of the bath could be lowered by means of a dry-ioe-aoetone mixture contained in a tin can immersed in the bath. Within less than an hour the temperature of the bath could be lowered from room temperature to about -55 °C. A stirrer in the bath kept the liquid constantly agitated to insure uniform temperature throughout the bath. To raise the temperature quickly and conveniently an ordinary E50 watt immersion type electrical heater was in stalled in the bath. As soon as any desired temperature was registered by the thermocouple used for the temperature 35 measurements, further heating or cooling was stopped and the bath proved sufficiently well insulated to hold that particu lar temperature long enough to permit the measurement to be carried out within + 0.2 °C. Temperature measurements were made by means of a copper-constantan thermocouple oonneoted in the usual way to a potentiometer circuit* A factory-new type K-2 potentio meter and a very sensitive Leeds and Northrup type HS galvan ometer permitted the detection of 0.1 of a millionth of a volt corresponding to temperature variations far in excess of the claimed accuracy of + 0.2 0C. The hot function of the thermocouple was immersed into a glass vessel of approximately the same shape and dimensions as the 4cm wide tube connected to the capillary. This vessel was filled with the same liquid as the one being used in the capillary apparatus for measurements. It was placed as close as possible to the wide tube and approximately the same distance away from both heating and cooling elements as the wide tube. With this arrangement errors due to temperature lags through the glass walls of the capillary apparatus are compensated. The thermocouple was calibrated against a tenth degree oertifled Bureau of Standards mercury thermometer at temper atures above zero and against the freezing points of purified carbon tetrachloride and chloroform below zero. The data ob- 36 tallied in these calibrations is tabulated below (table I) and was used for a large scale plot of emf against temperature. A smooth curve of very slight curvature was thus obtained and used in all subsequent temperature measurements. At the conclusion of the surface tension measurements the thermo couple was recalibrated showing deviations much smaller than the experimental error. Table I Substance Freezing Pts. in °C E.M.F. in volts final calibration 0.000857 0.002224 0.000000 0.000960 extrapolated figure (except of Hg.0 where initial temperature is given) calibration water at carbon tet. chloroform water at 21.78 -22,95 -63,5 00.0 24,40 24 25 0.000852 0.000857 0.002223 0.000000 0,000960* IT4"' Xanftolt BBrnstein, Yol. 1, 377 (1923) 25 Ibid. 1st supplement, 198 (1923) 37 The Purification of the Samples a) Carbon tetrachloride. Eastman Tflfoite Label cp. grade carbon tetraohloride was fractionally recrystallized till further recrystalliza tions showed no change in the emf values at the freezing point. b) Chloroform. Commercial grade chloroform was shaken out with sulfuric acid repeatedly, washed with water, dried over calcium chloride, then fractionated in an 18 in vigreaux column collecting the middle portion at 60*95-61.00°C (unoor- reoted)• This middle portion was refractionated and the middle fraction again collected, coming over at 60,98-60.98°C (unoorreoted). Applying emergent stem and pressure corrections this temperature was calculated to correspond to 61.27*0 as the normal boiling point. The accepted value is 61.2°C.^® o) Ethyl Ether (used in calibration of capillary). Commercial anhydrous ether was dried over sodium wire, then fractionated in the same column, a middle portion collected at 34.2°C (uncorrected). This portion was then refractionated, middle out again collected at 34.21- 34.21°C. Applying emergent stem and pressure corrections this temperature was calculated to correspond to 34.27°0 38 as the normal "boiling point. The accepted value is 34.6°C^. d) Isopentane, A high grade sample of isopentane (guaranteed to he at least 99 mole percent pure) was obtained from the Philipps Go. and used as starting material. This isopentane was repeatedly fractionated in a silvered six foot fractionating column having an efficiency of 30 plates. As final sample a middle fraction coming over at 28.2-28.4°C (corrected) was colleoted and used in the surface tension measurement. The IOT tables^7 give 27.95°C as the normal boiling point. e) n-Butyl chloride. cp. n-butyl bromide was fractionated four times in the 18 in. vigreaux column and the portion coming over at 78.3g°C (corrected) was collected and used in the measurements. The accepted value of the normal boiling point is 78.50°C.£8 f) t-Butyl chloride. High grade, Eastman White label t-butyl chloride was repeated ly fractionated in an 18 in. vigreaux column and the portion coming over at 50.5^°C (corrected) collected and used in the measurements. ^5 EancTolt BBrnstein, Vol. 1, 366 (1923) 27 I.C.T. 3, 22d (1928) 28 J. Timmermans, J. Ghim. phys., 2£, 401 (1930) Calibrations 39 a) The Thermocouple, The emf values shown in table I were in all cases the mean of at least three separate measurements with a maxi mum deviation of £0.000002 volts corresponding to +0.05 degrees centigrade. An attempt was made to use powdered dry ice contained 29 in a dewar as a fixed point as described by R.B. Scott Successive runs, however, showed variations as large as +1.0°G, The difficulty was finally ascribed to the poor contact made by the thermocouple wire against the dry ice; slight pressure on the thermocouple was observed to result in emf variations corresponding to several degrees. It therefore appears that this fixed point is not de sirable for use with thermocouples, while it may be per fectly valid for use with platinum resistance thermo meters as used by Scott. b) Capillary Calibration. The capillary was calibrated directly in the usual way by means of a mercury thread®^. This calibration was cheeked by using ether as a standard in some surface tension measurements. The results are shown in table II. ^9 R.B. Scott, Temp. Measurement and Control Symp. p 212. Hat’l. Bureau of Standards. 30 T.W. Richards, J. Am. Chem. Soc., 37, 1657 (1915) 40 Table II Method Average radius of capillary in cm mercury thread 0.0318 ether 0.0317 In the mercury thread method the radius of the capillary was calculated from the formula \ f r s w /2 UcLc' where m is the weight and 1 the average length of the mercury thread used, d its density. In the ether calibration the surface tension equation £ g rp (h + - 5- r J was solved for ? using accepted values of gamma at various temperatures. In this way r in table III was calculated. Table III Temp. °0 he ight cm density gms/cc ‘ 4 calc cm 21.4 1.475 .712 .0318 -19.4 1.810 .758 .0316 —40 .2 1.960 .782 .0317 -49.1 2.030 .792 .0317 average .0317 Note: each horizontal row in table III is the result of averaging three measurements at the temperature in question. 41 Surface Tension Measurements of n-butyl chloride, t-butyl chloride and Isopentane Using the technique described above the surface ten sion data shown in table IV was obtained. In each oase several measurements were made at any given temperature to insure reproducibility, To prove that thermal equili brium had been attained before each measurement was taken, data were collected on both a rising and falling tempera ture trend of the system. Upon plotting this data it wag found that all points fell on the same smooth surface ten- si on-temperature curve, regardless of the temperature trend of the system. Essentially the.same technique was followed in the determination of n-butyl chloride and t-butyl chloride ex cept that in the latter oase liquid nitrogen was replaced by a solid carbon dioxide-acetone cooling mixture. Bata on n-butyl chloride and t-butyl chloride are shown in tables V and VI respectively. 4 2 Table IV - Surface Tension of Isopentane emf before measurement (volts) capillary emf after t,°C rise in cm measurement (volts) li<*. dens, gms/cc surf. t< dynes/ci .000609 1.540 .000613 15.7 .6235 15.1 614 1.540 615 15.7 .6235 15.1 619 1.535 621 15.8 .000080 1.645 .000092 2.1 95 1.645 102 2.5 .6355 16,4 105 1.645 111 2.6 .6355 16.4 .000563 1.825 .000544* -15.1 578 1.810 580 -15,4 579 1.790 571 -15.3 .6525 18.3 560 1.790 547 -14.7 .6525 18,3 .001164 2.080 .001169 -31.6 1150 1.985 1147 -31.0 .6662 1136 2.000 1139 —30 , 8 1140 1.930 1141 -30.8 1136 1.930 1141 i 03 o * CO 20.1 1145 1.925 1147 -31.0 20.1 1146 1.920 1142 -31.0 20.1 .001790 2.110 .001799 -50,3 1805 2.110 1814 -50.8 1827 2.120 1834 -51.4 1842 2.115 1846 -51,8 43 Table 17 (Cont'cL) emf before capillary emf after t, °Q llqt* 4ens. surf. tens, measurement rise in om measurement gms/oo dynes/cm (volts) (volts) 1848 8:.100 1852 -52.0 .6862 22.8 1854 2.115 1856 -52.1 22.8 1858 2.095 1858 -52.2 ,000070 1.675 .000066 - 1.9 65 1.675 62 - 1.8 16.8 60 1.675 58 - 1.7 .6400 16.8 57 1.670 54 - 1.6 16.8 ,000366 1.575 ,000373 9.5 374 1.580 374 9.6 15.6 373 1.58Q 372 9.6 .6280 15.6 371 1.585 369 9.5 15.6 000979 1.445 .000981 25.0 980 1.450 976 25.0 .6148 14.0 976 1.445 981 25.0 14.0 980 1.455 976 25.0 14.0 Measurements at lower region of capillary 000910 1.465 .000911 23.2 .6156 14.1{ 912 1,455 913 23.2 914 1.465 915 23.2 000431 1.765 .000435 -11.2 44 Table IV (Cont'd) emf before measurement (volts) capillary rise in cm emf after t ,°Q measurement (volts) 433 1.760 411 394 1.750 396 -10.6 398 1.750 398 -10.6 .001708 2.100 .001721 -47.6 1724 2.040 1726 -48.1 1725 2.080 1715 -48,0 t,°C liq* dens. surf, tens gms/ao dynes/cm .6475 .681 .682) ) av. ) ~ ~ 17 .8 17.8 22.4 22.1 The ahove data was used to make a large scale surface tension-temperature plot. The result was a smooth curye of slight curvature. The capillary rise was shown to be inde pendent of the particular region of the capillary in which the rise took place; this was proved by taking measurements in a lower region of the capillary and plotting the points thus obtained on the plot of the surface tension-temperature curve obtained from data taken in a higher region of the capillary. It was then found that all the data regardless of the region of the capillary in which it was taken fell on the same smooth curve. 45 Table 7 - Surfaoe Tension of n-butyl chloride emf before capillary emf after %} °0 lict# dens, surf, tens measurement rise in cm measurement gms/co dynes/cm (volts) (volts) ,000944 1.625 .000944 24.1 .882 22,5 944 1.615 944 944 1.625 944 944 1.620 944 000173 1.825 .000180 181 1.830 180 - 4.8 .913 26.1 180 1.830 171 000723 1.920 .000739 745 1.920 750 -20.0 .930 28.0 752 1.920 756 001154 2,000 .001166 1172 1.980 1180 (1182 1.995 1185 -32.1 .943 29,5 (1187 1.995 1189 (1189 1.995 1188 000298 1.730 .000301 303 1.730 305 7.9 .899 24,5 306 1.730 308 000271 1.820 .000271 270 1.815 265 - 7,1 ,916 26,2 261 1.825 255 46 Table V (Cont'd) emf before capillary emf after t , °0 liq. dens. surf, tens measurement rise in om measurement gms/oo dynes/om (volts) (volts) ,001754 a. ioo .001760 176a a. 090 1765 -49,a ,96a 31.4 1766 a.095 1763 000875 1.415 .000875 aa.4 .838 18,6 875 1.410 875 875 1.405 875 000460 1.475 .000455 453 1.490 453 11.9 453 1.480 451 11.9 .849 19.8 000066 1.585 .000075 77 1.575 75' - a.i .864 , ai.3 75 1.575 • 69 000404 1.635 .000408 409 1.630 409 408 1.635 405 GO • o H 1 .878 aa.4 000818 i.7ao .000831 835 1,785 839 84S 1.710 843 -aa.4 .885 a3.8 842 1.700 840 00039a 1.480 .000394 (395 1*475 397 (397 1.485 399 lo.a .851 19.9 (400. 1.485 40E 47 Table V (Cont'd) emf "before capillary emf after t, °G liq. dena. surf, tens measurement rise in cm measurement gms/co dynes/om (volts) (volts) 000867 1.7000 .000877 882 1.710 844 884 1.705 884 -23.6 .885 23.7 883 1.710 878 Measurements of the Vapor Pressure of t-Butyl Chloride In order to apply equation (IS) to the calculation of the surface tension of t-butyl chloride, vapor pressure data is required. Since this data could not "be found in the literature, the vapor pressure of t-butyl chloride was de termined experimentally using essentially the simple apparatus described by Livingston.3* She data obtained is tabulated in table VII. This data was used to fit a log P - -A/T + b curve by calculating A and b using the method of least squares. The equation thus obtained la ( l > 3 Pic) = ' /.f5-3fc*lt>3*J_ t- fe.3T75- From the slope of this curve the head of vaporization was calculated, giving a H - 6650 calories/mole. VOjf, SI Livingston, Physico-Chemical Experiments, p. 65 (Macmillan) Hew York, 1939 48 Table VII - Vapor Pressure Data of t-Butyl Chloride vap, press in cm. log vap • t ,°Q press. (corr. for emerg, stem) T, °K 74.3 1.8710 49.1 322.2 74.0 1.8692 49.0 322.1 73.5 1.8663 48.8 321.9 55.8 1.7466 40.3 313.4 55.5 1.7443 40.2 313.3 55. 2 1.7419 40,1 313.2 55.0 1.7404 40.0 313.1 39.5 1.5966 30.7 303.8 39.4 (3)* 1.5955 (3) 30,7 (3) 303.8 (3) 25*0 1.3979 18.8 291.9 25.0 1.3979 18.9 292.0 25.1 • 1.3997 19.0 292,1 25.2 1.4014 19.1 292.2 17.9 1.2529 • 10.5 283.6 17.9 1.2529 10.7 283.8 18.0 1.2553 10.8 283.9 18.0 1.2553 10.9 284.0 13.9 1.1430 4.8 277.9 14.0 1.1461 4.9 278.0 14.0 1.1461 5.1 278.2 14.1 1.1492 5.2 278,3 49 Table Til (Cont'd) vap* press in cm* log vap. press. t} °0 (oorr. for emerg. stem) T°K 65.2 1.8142 45.3 318.4 64,8 1.8116 45.2 318.3 64.7 1,8109 45.1 318.2 46,9 (3) 1.6712 (3) 35.7 (3) 308.8 (3) 24.4 1.3874 18.4 291.5 24.4 (2) 1.3874 (2) 18.5 (2) 291.6 (2) 24*6 1.3909 18.5 291.6 11.4 (2) 1.0569 (3) 0,20 (2) 273.3 (2) 11.4 (2) 1.0569 (2) 0*30 (2) 273.4 (2) * lumbers in parenthesis indicate number of identical sets of figures. CHAPTER 7 Conclusion 50 In this paper a method for the calculation of the surface tensions of liquids from their vapor pressure and density has been developed. The resulting equation contains no empirical constants and gives fair agreement with experiment. This agreement confirms the validity of the theory from which the equation was developed. The main merit of the present treatment is its general ity permitting application to the case of solids. Solid free surfaoe energies are important in wetting phenomena and re lated to tensile strength and therefore are of value from both a practical and theoretical point of view. With one exception*^ there are no known methods by means of which solid free surfaoe energies can be calculated theoretically and there exists no accepted experimental method for the determination of this quantity. On the experimental side the surface tensions of iso- pentane n-butyl ohloride and t-butyl chloride were determined over a wide range of temperature. Experimental data on the surfaoe tensions of these substances has been heretofore available only at a few temperatures or not at all. The vapor pressure-temperature dependence of t-butyl chloride has also been determined and the heat of vaporization of t-butyl chloride has been calculated. X 2 3 4 5 6 7 8 9 XO XX 12 13 X4 X5 X6 X7 X8) 5X CHAPTER VI Bibliography Born, M., and Courant, C.f Physik. Z., 14, 731 (X9X3) Dorsey,, Bur. Standards Soi. Paper, £1, 563 (X936) Einstein, Ann. d. Physik, 34, X65 (19IX) EBtvos, Wied. Ann., 27, 456 (1886) Fowler, R.H., Proo. Roy. Soo. A 159. 229 (1937) Haul, R.B., Z. physik. Ghem., 53, 331 (1943) I.C.T., Vol. 3., 220 (1928) Kassel 1,S. and Muskat, M., Phys. Rev., 40, 627 (1932) Katayama, U., Tohoku Imp. Univ. So. Reports (1), (4), 373 (1916) landolt BSrnstein, Vol. I, 377 (1923) Landolt B5rnstein, 1st Supplement, 198 (1923) Landolt BSrnstein, Vol. I, 366 (1923) Lennard-Jones, J.E., and Corner, J., Trans. Farad. Soo., 36, 1156 (1940) Lennard-Jones, J.E., and Devonshire, A.F., Proo, Roy. Soo., A163, 53 (1937) Lennard-Jones, J.E., Roy. Soo., A 165, 1 (1938) Lennard-Jones, J.E., Proo* Roy* Soo., 169, 317 (1939) Lennard-Jones, J.E., and Dent, Proo. Roy. Soo., A 121, 247 (1928) Livingston, Physioo Chemical Experiments, p. (Maomillan) Hew York, 1939. 5a Bibliography 19) Madelung, E., Physik. Z., 14, 729 (1913) 20} Margenau, Phys, Rev., 538, 365 (1931) 21) Maas, 0., and Wright, C.H., <T. Am. Chem* Soo., 43^ 1107 (1921) 22) Macleod, Trans. Farad. Soc., 19» 38 (1923) 23) Ramsay and Shields, Phil. Trans., A 184, 647 (1893) 24) Lord Rayleigh, Proo. Roy. Soc., A 92, 184 (1915) 25) Richards, T.W., and Coombs, L.B., J. Am, Chem. Soc., 37, 1659 (1915) 26) Richards, T.W., J . Am. Chem. Soo., 3£, 1657 (1915) 27) Scott, R.B., Temp. Measurement and Control Symposium. Rational Bureau of Standards. 28) Shereshefsky, J.L., J. Phys. Chem., 35, 1712 (1931) 29) Stranski, I.H., Z. physik. Chem., B 38, 457 (1937) 30) Stranski, I.SI., and Kaischew, R., Z. physik. Chem., B 35, 427 (1937) 31) Timmermans, »T., <T. chim. phys., 27, 401 (1930)

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Creator Strisower, E. H (author)

Core Title A new method for the calculation of free surface energies of liquids and solids

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