Creator:Joseph S. Rosen Date Created:February 23, 1949 Place Created:Kansas City, Missouri Keywords:refractive indeices,dieletric constants Context:article reprinted from "The Journal of Chemical Physics" ************************************************** Reprinted from The Journal of Chemical Physics, Vol. 17, No. 12, 1192-1197, December, 1949 Printed in U. S. A. Refractive Indices and Dielectric Constants of Liquids and Gases under Pressure Joseph S. Rosen University of Kansas City, Kansas City, Missouri (Received February 23, 1949) Several interpolation formulas are given which reproduce refractive index and dielectric constant data for liquids and gases at high pressures. These formulas involve functions of the refractive indices and dielectric constants which appear in the formulas of Lorentz-Lorenz, Clausius-Mosotti, Gladstone and Dale, the empirical Eykman formula, etc. One formula shows the reciprocals of these functions to be linear in the specific volumes at various pressures. Another interpolation formula, involving the reciprocals of the same functions, ff contains a logarithmic term which is similar to that appearing in the Tait equation for compressibilities of liquids. The parameters involved in the Tait equations for compressibilities and in the analogous interpolation formula for refractive indices and dielectric constants are discussed. The article also shows that the Tait equation which has been so successfully adapted to compressibility data of liquids can also be applied to gases. INTRODUCTION TN a recent article1 it was shown that the compres-sions2 of alcohol, water and their mixtures calculated from the refraction formulas of Lorentz-Lorenz and of Gladstone and Dale are a linear function of the observed compressions of the solution. A logarithmic equation is given which reproduces with great precision the refractive indices of alcohol, water, and their mixtures under pressure; the logarithmic term is the same as that ur.tfe Tait equation for compressibilities, and the part of tb$ Equation containing the refractive indices involves the same functions of np which appear in the formulas of Gladstone and Dale, Lorentz-Lorenz, etc. It is the purpose of this paper to show that the results of the above article can be extended to existing data of both refractive indices and dielectric constants of liquids as well as to gases at high pressures. Of the important formulas for the refractive index or dielectric constant of a substance which give a specific refractive or specific polarization "constant," all have the form f(vp)vp= C, where f(vp) is a function of the refractive index nP or the dielectric constant tP, vP is the specific volume and C the "constant." Where the formula is applied to solutions on which the pressure is varied, the expression i-C/("o)//M]=V CD would equal the observed compression, kp, if C remained constant. Characteristic functions are the Lorentz-Lorenz expression («2—1)/(»2+2), Gladstone and Dale's (»—1), the Newton expression («2—1) and the empirical function of Eykman («2—1)/(«+.4). In all of these expressions np- may be replaced by tr to 1 J. S. Rosen, J. Opt. Soc. Am. 37, 932 (1947). 1 The symbols used in this article are the following: np, ep and vp are the refractive index, dielectric constant and specific volume, respectively, at pressure /'. The subscript 0 indicates atmospheric pressure (P = 0). The bulk compression, kp, is — (vp—ro)/t'o. Where the symbol rp occurs in the function /(vp) it will designate tp or np3, and }(rp) will refer to the same function of tp or nr. obtained the corresponding formula involving dielectric constants. It has generally been found that with none of these functions f(vP) can the observed compressions, kP, of the solutions be computed from Eq. (1) with any great accuracy;1,14 but Eq. (1) has frequently been used as a criterion in selecting the most suitable formula f(vp)vp=C applicable to a particular substance. It will, however, be shown here that the compressions, kp', computed from Eq. (1), may be taken as a linear function of the observed compressions kP. Accordingly, we may write kp'=m'kp+b', (2) where m! and b' are constants for a given substance. A more useful equation which is equivalent to (2) is obtained if we replace kp by (va—vp)/v0 and kP by the left member in Eq. (1). We obtain the expression 1 /f(yp) = mvp+b, (3) where m and b are constants independent of the pressure. Equation (3) will be shown to hold for existing data at high pressures both for the refractive indices and dielectric constants of liquids and gases. The forms of the function f{vP) in Eq. (3) may be those of theoretical significance already mentioned as well as others which have no theoretical foundations, such as for example, f(vp) = pp, etc. Though a number of these forms have been examined, only the Clausius-Mosotti and Lorentz-Lorenz functions, because of their theoretical interest, will be considered. The equation, proposed by Tait for water, dkP/dp=C/(B+P), has been shown in recent years,1-5 notably by Gibson, ' W. C. Rontgen and L. Zehnder, Ann. d. Physik 44, 49 (1891); G. Quincke, ibid. 44, 774 (1891). 4 R. E. Gibson and J. F. Kincaid, J. Am. Chem. Soc. 60, 511 (1938). 5 Harned and Owen, The Physical Chemistry of Electrolytic Solutions (Reinhold Publishing Corporation. New York, 1943),. p. 270; P. W. Bridgman, Rev. Mod. Phvs. 18, 17 (1946); H. Carl, Zeits. f. physik. Chemie 101, 238 (1922). 1193 REFRACTIVE INDICES to represent successfully the compressibilities of liquids. This equation involves two parameters6 c and b both independent of the pressure; b is a function of the temperature, concentration (in mixtures),1 and the properties of the solution; while C is approximately independent of the temperature. It will be shown in this article that Tait's equation can also be applied to gases. In the integrated form this equation is vP.-vp = Coc In | (/>+ b)/ (p'+ b) | (4) where the- initial pressure, usually one atmosphere, isP'. From Eq. (3), we have 1 1 -=m(vP' — rL f(vp') f(vP) If we substitute (4) into this equation, we obtain 1 1 /P+B\ -----= .4 log I----) f(vp.) /(„„) Kf+bJ (5 I Table I. The constants m and b of the equation l/fp—mvp+b, where l//j> = (ep+2) yep— 1), evaluated by the method of least squares from dielectric constant data for liquids at high pressures. In the last two columns are shown the average and maximum percent difference between the observed and computed values of the dielectric constants. The pressures and volumes are in the units of the original data. where a and b are constants for a given substance. If initially the pressure is the atmospheric pressure, and if the pressure range is large, it is most convenient to set p' equal to zero. A better adjustment of the data may be obtained by replacing (P'+ b) by a third parameter, but in general this is not justified either by the accuracy of the data or the additional computation involved. In this article we will justify Eqs. (.5) and (5) as interpolation formulas *or the considerable body of refractive index and dielectric constant measurements at high pressures for both liquids and gases. The Clausius-Mosotti and Lorentz-Lorenz functions will be adjusted by the method of least squares to these equations, but for Eq. (5) we will also study the form f(vp)=vp. THE INTERPOLATION FORMULAS The literature on the subject of refractive indices and dielectric constants at high pressures contains some precedence for Eq. (3) and (5). Danforth,7 in a much quoted paper on the dielectric constants at high pressure for some ten organic liquids, finds that the reciprocal of the Clausius-Mosotti expression pP/f(ePt when plotted against the density pP gives a straight line for most liquid.-. This is evidently similar to Eq. (3), with l/f(eP) = (tP+2)/(tP— 1). Owen and Brinkley,8 on theoretical grounds, derive the Eq. (5) with j{vP) = tp and P'= 1, and adapt the equation to data on dielectric constants at high pressure, and to the refractive indices of one non-polar liquid (benzene). Keyes and Kirk- 6 When the true compressibility dk/dP at atmospheric pressure is known only one parameter B is necessary. This is especially important when adjusting data to the equation by the method of least squares. See reference 1. 7 W. E. Danforth, Jr., Phys. Rev. 38, 1224 (1931). 8 B. B. Owen and S. R. Brinkley, Jr., Phys. Rev. 64, 32 (1943). Liquid i°C Pmax* m b (%>av (%W Data of Danforth** Ethyl ether 30 12000 1.17659 0.31125 1.21 2.74 Ethyl ether 75 12000 1.42582 0.06676 1.16 2.52 Pentane 30 12000 2.63595 0.36305 0.06 0.33 Ethyl alcohol 30 12000 0.12407 0.97643 0.38 1.08 Ethyl alcohol 0 12000 0.09448 0.99580 0.57 1.38 Hexyl alcohol 30 ' 4000 0.23520 0.96426 0.59 0.83 Hexyl alcohol 75 8000 0.38421 0.90341 0.58 0.92 Bromobenzene 30 4000 1.01806 1.01761 0.15 0.35 Bromobenzene 75 8000 1.16001 0.94717 0.08 0.23 Chlorobenzene 30 4000 0.91212 0.77397 0.27 0.43 Chlorobenzene 75 8000 0.96532 0.76365 0.17 0.33 Carbon bisulfide 30 12000 3.03747 0.38371 0.48 1.03 Carbon bisulfide 75 12000 3.15413 0.31914 0.21 0.38 Eugenol 30 3000 0.48120 0.87206 0.39 0.79 Glycerine 30 12000 0.16428 0.94074 0.41 0.93 Glycerine 0 8000 0.13400 0.95619 0.20 0.38 i-Butyl alcohol 0 12000 0.12782 0.99198 0.42 0.86 z'-Butyl alcohol 30 12000 0.17220 0.96906 0.36 0.67 Data of Chang" Carbon bisulfide 75 12000 3.31543 0.19076 0.23 0.37 . Carbon bisulfide 30 12000 3.27849 0.22453 0.41 1.77' z-Amyl alcohol 22.4 12000 0.26914 0.89999 1.85 2.19 Ethyl ether 30 12000 1.10326 0.35194 1.04 2.93 Ethyl ether 75 12000 1.34818 0.11343 0.62 2.93 Toluene 30 9500 2.43974 0.34378 0.13 0.31 Data of Kyropoulosb Carbon bisulfide 20 3000 3.35849 0.16093 0.08 0.16 Ethyl alcohoi 20 3000 0.10254 0.99178 0.25 0.37 Methyl alcohol 20 3000 0.08320 0.98660 0.14 0.21 Water 20 won 0.04890 0 98860 0.08 0.31 Acetone 20 3000 0 14357 0.96469 0.27 0.48 Ethyl ether 20 3000 1.29268 0.08743 0.53 0.90 * 1 hp pressures ot Cluing and KvropouUo are in kg » (%)«. Dielectric Constants of Gases* Ufe- («+2)/(«- 1) Carbon dioxide 0 200" 128.733 -0.03195 0.06 0.12 (liquid) Carbon dioxide 35 100 132.4435 -0.7808 0.54 3.02 Carbon dioxide 100 151 133.7713 -0.7962 0.04 0.24 Ammonia 100 55 23.9453 -0.2956 0.03 0.10 Ammonia 175 100 27.5442 -0.2266 0.06 0.15 Refractive Indices of Gases and Liquids Carbon dioxide1" Nitrogen0 Ethylene11 Benzene** Waterf Alcoholf 32 25 100 25 25 25 122" 3294.03 2053 5092.274 2269 2082.868 868 ' 2.68806 1500' 4.17917 1500' 3.30631 0.0363 -0.1734 -0.0413 0.33320 0.66864 0.31927 0.13 0.09 0.19 0.002 0.010 0.020 0.31 0.29 0.64 0.003 0.013 0.025 • See reference 9. » In all of Keyes and Kirlcwood's data the pressures are in atmospheres and the volumes are" in liters/mol. The lower pressure limit is in all cases above atmospheric pressure. " Michels and Hammers, Physica ♦, 1002 (1937). X -5876A. • Michels, Lebesque. and De Groot. Physica 13. 337 (1947). X-5876A. <• Michels. Botzen, and De Groot. Physica 13, 343 (1947). X -5876A. ** See reference 4. t See reference 1. • In all of Michels' data the pressures are in atmospheres and the densities are given in Amagat units; the volumes used in calculating the constants above are the reciprocals of these densities. The lower pressure limit is (except for Nitrogen) the atmospheric pressure. ' Pressure in bars. X =>589 mj*. « Pressure in atmospheres. X ««579 mn. term as a compression would be consistent with Tam-mann's hypothesis that the introduction of a dissolved substance has the same effect upon a solvent as compressing the pure liquid under external pressure. In Tables I and II the constants m and b of Eq. (3) are given for liquids and gases whose dielectric constants and indexes of refraction have been measured under pressure by various investigators. These constants were determined by the method of least squares by minimizing the sums of the squares of the difference of the observed and computed values of the function 1 ///> (the line of regression of 1 ///> on vp). As has already been noted, Danforth7 plotted the reciprocals of the Clausius-Mosotti expression against the densities, i.e., pp/f(tp) = b"pp+m" (6) sion of 1 /fp on vp is not the same as the line of regression of pp/fp on l/vp). It can be shown that if instead of assuming observations of equal weights in fitting the line of Eq. (3), we ascribe to the observations the weights 1 /vp2 we will obtain the line of Eq. (6). Conversely, with the weights vp- the least-square line of Eq. (6) will be the line of Eq. (3). In practice, the two lines of regression (3) and (6) give essentially the same results for a number of cases tried. Thus, for carbon bisulfide at 75°C (data of Danforth) and for ammonia at 100 °C the two line of Eqs. (3) and (6) give values for tp whose average and maximum percent deviation from the observed dielectric constants are practically the same. Danforth observed that the reciprocal of the Clausius-Mosotti function pp/fp is linear in the density in the case of all polar substances except ethyl ether and the two more simple non-polar substances carbon bisulfide and pentane. This conclusion is not altogether admissible from the results shown in Table I. Ethyl ether does consistently show the greatest deviations from linearity in the data of Chang and Kyropoulos as well as in the work of Danforth. However, the least-square line for pentane is associated with the smallest deviations shown in Table I, while at 75°C carbon bisulfide (Danforth and Chang) fits comparatively well, and at 30°C the results for carbon bisulfide are better than those for ethyl alcohol (0°C), which according to Danforth behaves normally. Evidently, Danforth drew his conclusions on the exceptional behavior of these substances by relying on graphical representation to estimate linearity. Table III. Two least square formulas l//p = mvp+b for the refractive indices of a gas* for which the range of the function l//p= (»pJ+2)/(«p!— 1) is extensive. The equation 1/fp 3294.03fp+0.0363 was obtained by minimizing the sum of the squares of the difference of the observed from the computed value of the function l//p. The better fitting equation l///> = 3330.76i>j> +0.1108 was obtained by minimizing the sum of the squares ot the percent difference between the observed and computed values of the refractive indices of the gas. The last two columns show the percent difference between the observed refractive indices and those computed from these equations. where pp is the reciprocal of the specific volume vp and l//>=(«-{-2)/(«— 1). We prefer the arrangement of Eq. (3) which can more readily be related to the logarithmic form of Eq. (5). Theoretically, Eqs. (3) and (6) are the same, but it must be noted that in practice, because of the change of variables, the least-square method of fitting the best line of Eq. (3) does not give the same line as in fitting Eq. (6). (The line of regres- Pressure. Percent deviation atmos. n&8-« vp** 1 /fp obs d. of n 21.35 1.0097 0.047015 154.9250 0.002 0.010 49.24 1.0278 0.016316 55.2198 -0.057 -0.039 56.25 1.0345 0.013101 43.2000 0.025 0.042 63.35 1.0438 0.010337 34.5121 -0.012 0.004 67.41 1.0512 0.008827 29.5660 -0.021 -0.009 72.35 1.0676 0.006691 22.4638 -0.018 -0.019 77.92 1.1482 0.003103 10.4232 0.208 0.030 86.45 1.1642 0.002802 9.4421 0.235 0.002 96.74 1.1735 0.002655 8.9554 0.263 -0.003 106.46 1.1794 0.00256$ 9.6730 0.278 -0.010 121.60 1.1864 0.002476 8.3612 0.312 -0.007 Average percent deviaion of n Maximum percent deviation of n. 0.312, 0.042. 0.130 0.016 * Carbon Dioxide at 32.07S°C. Data of Michels -.d Hamers, Physica 4 1002 (1937). ** The volumes shown are the reciprocals of the diviities originally given in Amagat units. 1195 REFRACTIVE. INDICES Table IV. The parameters A and B of the Tail equation .4 logio(l+P/£) fitted by the method of least squares to the functions l//o— 1//p*> l/«o— 1 .Up and the compressions ro— 'p. The three sets of parameters are given for each substance in the foregoing order. The last two columns give the average and maximum percent deviations between the observed and calculated values of ep. The pressures and volumes are in the units of the original data. Substance /°C co Pma* A B{ (% Imax Pentane** 30 1.82 * 12000" J0P 0.9113 0.09106 0.3449 359 625 354 0.096 0.154 0.113 0.285 0.334 0.511 Glycerine** 0 49.9 8000 0.02315 0.07472 0.1464 3807 3826 . 2845 0.051 0.063 -0.134 0.098 0.163 0.190 Carbon bisulfide** 30 2.61 12000 0.5054 0.08231 0.2005 610 848 .1129 0.186 0.281 0.074 0.544 0.671 0.211 Bromobenzene** 75 4.87 8$X) * 1. 0.1732 " , „ p.039§9,-0.1^20 • 936 1049 983 0.121 0.124 0.108 0.242 0.268 0.238 Acetoneb 20 21.50 3000b 0.03911 0.01196 0.3202 795 799 1085 0.085 0.087 0.077 0.176 0.177. 0.229 Carbon tetrachloride13 18 2.246 1000b 0.8403 0.1033 1144 1376 934c 0.009 0.010 0.020 0.023 Ammonia f 100 1.0940d 55 28.789 1.2085 — 14.12" _t -14.05" 0.163 0.675 0.314 1.167 Benzeneft -15 1.48518 11 S8h 0.699133 0.082786 0.253355 859 1030 829' 0.003 0.002 0.005 0.004 * l//Jp = i'cp+ 2)/(ep —1). l//o is the value of the function at either the atmospheric or the initial pressure. { B is in the same units as the pressure. ** See reference 7. * Atmospheres All of Danforth's pressures are in this unit. b The data of Kyropoulos. His pressures are in kg/cm2. c The value ot Gibson, t See reference 9 t+ See reference 4. ,J The value at 20 atmos the lower pressure limit. e The funrtio rmed "erf is that «u F t\- at P' -20 atmos. tin- imti.i' pressure * The fu " ti"ii ■ j t;- -ann«)t. :i. 1.tiis ca^-. l»e represented b\ a rait equation « The index 01 n"Ya« ti'-u \ ntn . ep is replaced b\ hp- » the equations. Ji Pressure in bars WEIGHTED FORMULAS FOR GASES In the last two columns of Tables I and II are shown the average and the maximum percent difference between the observed and computed value of the dielectric constant or refractive index (IOOAh »')." This is the criterion used for the closeness of tit of the interpolation formula of Eq. (3), though in fitting this line by the method of least squares, we minimized the sum of the squares of A(l,//-!, the difference of the observed and the computed values of the function 1 fp. Obviously, this is not the proper function to minimize to give us the most favorable deviations of np (or ep). Generally, this inconsistency is not serious- and it has the advantage of lessening the computational work involved in applying the method of least squares. The parameters m and b shown in Tables I and II were all found by fitting Eq. (3) in this way. However, where the 11 An (or At) will denote the absolute value of the difference between the observed and computed value of np (or fp). The A will have the same significance when used with the function 1 If p. range of the function 1 'fp is extensive (e.g., in the case of a gas, see Table III), the simplification of minimizing the sums of the squares of the residuals of 1/fp tends to pile up at the higher pressures large percent difference^ between the observed and computed values of iii- or f,-. This is apparent from the second to last column of Table III, where the data for the refractive indices of carbon dioxide are given as an illustrative example. The distribution of percentages in this column resulted from fitting Eq. (3) in the usual way. The last column, however, shows a better distribution of the percent difference between the observed and computed values of np \ in fact, an over-all improvement in fitting the observed values of nP to Eq. (3) has been affected. To obtain the distribution of percentages indicated in the last column of Table HI, we fit the observed data to Eq. (3) in another manner. Instead of minimizing £{A(1//)}2 we minimize ClAm/«)2, i.e., the Sum of the squares of the percent difference between the observed sq^wggf1 '1 ft .''WBisepipijjip^ JOSEPH S. ROSEN 1196 and computed values of nP. We assume below that the function 1 //> involves the indices of refraction np, but obviously the same procedure will apply where the dielectric constants are considered. If we write the approximate relation Aw= A(l//)/(1//)', where {\/J)'=d(\/f)/dn- then L(Ak/H)»«L{A(1//)/»(1//)T is the new function we must minimize. This is equivalent to ascribing to the function \/fp a weight wP, which is for each observation given by w={l/n(l/f)'\\ The published results of most of the investigators whose data we have used do not include enough information to permit a precise comparison with the deviations from our interpolation formula. Danforth, from whom the bulk of our data for very high pressures is taken, gives no estimate of the uncertainties in his measurements. However, with few exceptions, the Eq. (3) will represent his data within a few units in the last significant figure in «. A reasonable estimate of the experimental error of Gibson's measurements of the indices of refraction of benzene is perhaps 0.01 percent, and it may be seen that our interpolation formula reproduces his results well within this experimental error. Keyes and Kirkwood's experimental error for carbon dioxide is about 0.2 percent (and presumably, though they do not indiiate this, it may be assumed to be the same for ammonia Table 11 shows that we have fitted their data within liiis experimental error except in the case of carbon dioxide at 35 1 but Keyes and Kirk wood point out that 'hey • insider Kmagat's compressibility data, whii ". they used. unreliable above 60 atmospheres, arid here, too, is uher- our interpolation formula fails. Except in this case and that of the gases discussed below there are no visible trends in the residuals from the interpolation formula for the substances in Tables I and II. The situation, however, for the refractive index measurements of Michels et al. for carbon dioxide, nitrogen. and ethylene shown in Table II is somewhat different. The average deviations indicated for these gases in Table II are considerably in excess of the general accuracy claimed for their measurements. Furthermore, in the second last column of Table III, where the deviations are enumerated in detail for carbon dioxide, a systematic trend may be observed at higher pressures. However, the last column, which gives the deviations for the weighted formula, shows that this trend has been obliterated. The average percent deviation of n indicated in this column (0.016 percent) is in accord with the accuracy which Michels usually claims, but the maximum percent deviation in n (0.042 percent) may be too high. Considering the difficulties recorded by the investigators because the measurements pass through the region of the critical point, this abnormal deviation may reflect only this experimental situation. It may reasonably be claimed that the interpolation formula (3) represents the data for liquids, and that the same weighted formula reproduces the results for gases within the experimental error. SIGNIFICANCE OF THE PARAMETER B Owen and Brinkely8 conclude that the parameter b in the equation 1/«o~ 1/tp=A log(l+P/B) (7) has the same value as in the analogous Tait equation for the compressions of the liquid, i.e., the same value of b may be determined independently from either the dielectric constants or the densities. Their evidence for this conclusion is, aside from considerations of electrostatic theory and Tammann's hypothesis, that Eq. (71 accurately reproduces the variations of the dielectric constant with pressure for some liquids with values for b obtained by Gibson12 from compressibility data. Since, however, it has been shown in this article that it is possible to replace 1 «in Eq. (7) by other functions, and especially the function l//=(e+2)/(e—1), it seemed necessary to investigate more rigorously the nature of the parameter b when it is determined from compressibility as well as dielectric constant data over the same range of pressures and with 1 fP as both it/>-t-2).'(e/>—1) and 1 tp. Another factor prompts a reconsideration of the significance of the parameter b in 'hese equations. The value of b in the Tait equation is not critical, and as the equation has another disposable constant it is understandable that widely varying values of b in J) may-yet enable the equation to express the isothermal variations of the dielectric constant of liquids with pressure. It is, therefore, not surprising to see values of b, determined from compressibility data to pressures of approximately 1000 atmospheres,13 successfully used by Owen and Brinkely in Eq. (7) to represent dielectric constant data to as high as 12,000 atmospheres. This assumes what is only approximately true, that the parameter b in Tait's equation is independent of the external pressure on the solution. But b is not quite constant for a given liquid but depends on the pressure range of the data.14 For this reason all values of b u Gibson's values of B are summarized by Harned and Owen. See reference 5. " Gibson's values of B were determined from compressibility data in this range. These were applied by Owen and Brinkely to Danforth's data whose maximum pressure range is 12,000 atmos. and to Kyropoulos's data to 3000 kg/cm'. " Besides the relevance of this observation here it should also have some bearing in speculation (see reference 4) on the theoretical significance of the parameter B in Tait's equation. This should not only take into consideration the dependence of B on the pressure range but should also consider the manner in which the curve was fitted to the data. mi i n - HMHI ifmrn-wriiffiii^ 1197 REFRACTIVE INDICES presented here for comparison have been determined from the same pressure range. In Table IV are given the parameters A and B of A log10(l+P/2?) fitted by the method of least squares to the functions l//0— l/fp, l/e0—l/ep and to the compressions v0—vP. For each substance in the table three sets of the two parameters A and B are given corresponding to the functions given in the above order; the pressure range for these three functions is, for any particular substance, the same.15 Generally, it will be observed, the function l//0— l/fp fits the data somewhat better than l/e0—l/ep, but the values of the parameter B corresponding to these two functions sometimes differ considerably. When tp is large (e.g., glycerine and acetone), it may easily be shown by expansion in series that the function l/fp is approximately linear in l/ep so that l//0— l/fp is proportional to l/e0—l/ep, in which case the parameters B in the logarithmic form of these functions necessarily have nearly the same value (but this value of B is not necessarily the same value of B obtained from the compressibility data). On the other hand, where the values of tp are small (as in pentane) the B in the logarithmic representation of the function l/e0—l/ep does not, if pentane is typical of this class, have even nearly the value of B obtained by fitting the dielectric constant data to the function l//p=(«p+2)/(€p-l). For ammonia where the values of ep are only somewhat larger than one, and for other gases that were tried (where B is negative), the function l/e0—l/ep gives, when plotted against the pressure, a curve that is concave upward, so that it is not possible to represent the function l/e0—l/ep by a Tait equation. However, the function l//p= (ep+2)/(ep— 1), as may be seen from Table TV, when adjusted to the data gives a value of B which agrees well with the value of B obtained from the compressibility data. In conclusion it might be said that the results of Table IV lend some plausibility to the thesis that the parameters B of the logarithmic representation of the function l//o— l//p (and sometimes the function l/e0—l/ep) is the same as the B determined by the Tait "The compressibility data used are those given by the investigator. It must be pointed out that though Danforth's work was done in Bridgman's laboratory the densities he used do not always agree with Bridgman's published data. Thus. Bridgman's compressibility data for glycerine would give a much higher value for B than that shown in Table IV, but this value of B, too, would be of dubious value since the glycerine Bridgman used is admittedly contaminated with water. P. W. Bridgman, Proc. Am. Acad. Sci. 67,11 (1932). equation for v0—vp. However, it is evident that more reliable data would be required before this proposition could be established conclusively; for it must be kept in mind that the parameter B is extremely sensitive to very small variations in the data from which it is determined and that small deviations in the measurements affect the value of B enormously. THE TAIT EQUATION FOR GASES As we have shown in Table IV, it has been possible to adjust vp' — vp and l/fp' — l/fp for ammonia at 100°C to the Tait function Alog{(P+B)/(P'+B)}, where vP>= 1.385 and l//p< = 32.915, the values corresponding to the initial pressure P'= 20 atmos. It has already been pointed out that it is not possible to use the function l/t0—l/eP for gases; and that B, unlike that for liquids, is negative. It may be observed that though the pressure range is relatively small (20-55 atmos.), the compressibility data of ammonia are not reproduced by the Tait equation as well as it is for liquids over greater ranges of pressure. That this is not because the data are inherently less accurate may be indicated by the fact that the ammonia densities used are the measurements of Beattie and Lawrence,16 who also represented them by an equation of state (over the range 17-58 atmos.) but with somewhat better results. Their maximum percent deviation between the observed and calculated volumes was about 0.8 percent as compared to the 1.17 percent (and 0.68 percent average) shown in Table IV. It is, however, quite noteworthy that the Tait equation, which can so well be adapted to compressibility data for liquids (and some solids), can also be used with gases.17 The somewhat inferior results obtained for gases emphasize what has already been pointed out— that the parameter B is not independent of the external pressure, and this dependence is particularly greater for gases as is demonstrated by the results obtained for ammonia and by other gases that were examined. 16 J. A. Beattie and C. K. Lawrence, J. Am. Chem. Soc. 52, 6 (1930). 17 It might be of some interest to point out that v0— vp is only the simplest function of the volumes that can be represented by the Tait equation. As in the case of the dielectric constants and refractive indices, other functions of vP may be used, e.g., the Lorentz-Lorenz function \/f(kvp) = {\+2kvp)/(\— kvp) where k is a constant for which l — kvp>0. 4