Correlating Densities of Liquids A NEW NOMOGRAPH DONALD F. OTHMER, SAMUEL JOSEFOWITZ, AND A. E. SCHMUTZLER Polytechnic Institute of Brooklyn, Brooklyn, N . Y . Densities of liquids may be plotted directly to give straight lines on logarithmic Paper against a temperature temperature and denfrom the sities of a reference liquid. The method of correlation follows the technique used for vapor pressures, viscosities, surface tensions, and other physical properties of liquids and gases and is then expanded into a nomograph which direct determination Of the density Of Organic liquids at any temperature when the density for that liquid at two different temperatures is 1~x&vn. Mathematical derivations indicate the theoretical basis of this plot and the resulting nomograph.
DEVELOPMENT OF A NOMOGRAPH
In"genera1, any straight line or family of straight lines 011 a coordinate plot may be reproduccd immediately on a nomograph lines, one of which is calibrated to consisting of two correspond to the axis and the other malibrated to correspond to the y axis. If the two calibrations run in opposite directions, a point between the two scales represents a pivot to define a line of positive slope on the coordinate plot. Different values on the two scales, atsextremitjes of straight lines through the pivot, define the mathematical function in question. Different pivot lines of the original plot. points will each define Thus, the relation of densities of the materials of Figure 1 to the density of benzene a t the same critical difference temperature could be indicated on a parallel scale nomograph with inrevbus papers of this series it was shown that, for many dividual points corresponding to each of the several materials. I P I b i P e r t i e s of matter, simple and accurate Correlations could The calibration of the density scale f?r the several compounds be obtained by Plotting the Property in question of one material would correspond to the vertical scale of Figure 1 and of the versus the same O r a related Property of a ~ f e r e n c material e on reference substance scale (benzene) to that of the lower horizontal logarithmic Paper. I n this manner the vapor Pressures, gas scale. If it was desired to calibrate the second scale directly in solubilities, adsorption pressures, equilibrium Constants, vapor terms of the critical difference temperature, this wrould then compositions, relative volatilities, activities, ViSCOSitieS of liquids, correspond to the upper horizontal scale of Figure 1. and viscosities of gases have been correlated by plotting these Such a nomograph would have the disadvantage as Properties directly versus the vapor Pressure Of a ~ f e r e n c esubFigure 1 in requiring the use of critical difference temperatures, stance a t the same temperature @-lo). In a surface tension on a special scale, which would not be amenable to recalibration correlation, it was shown (8)that a more meaningful correlation in normal temperatures (a subtractive function of the critical n ttemperature based onthe Was obtained by using a n m ~ u ~ e m eof difference temperat,ure and a constant, the critical temperature). critical temperature of each compound as a ~ ~ f e r e n cPoint e for If, however, this critical difference temperature scale could be that compound. This temperature value was called the critical made arithmetic, or uniformly calibrated, then the values indifference temperature and represents the distance in degrees dicated thereon could be ordinary temperatures directly. This centigrade or Fahrenheit that the temperature in qilestion, T , may be done by graphically adjusting the density scale to correi s from the critical temperature of the material, T,. The spond; and in that case, of course, the function is somewhat symbol T D was adopted for the critical difference temperature different than that expressed by the coordinate plot of Figure 1. which thus was defined as: The nomograph of Figure 2 was constructed to permit the direct determination of densities of organic liquids if densities TD = To - T (l) a t only two temperatures are known. The right or temperature scale was calibrated as an arithmetic scale in order to allow direct DENSITIES ON REFERENCE SUBSTANCE PLOT subtraction and addition to that scale, so that critical difference temperatures wodd not have to be used. The left O r density It was found, as shown in Figure 1, that if the densities of one scale Was calibrated graphically with ethane as a reference subliquid are plotted versus the densities of another liquid at the same stance for densities of 0.35 to 0.55, benzene for densities of 0.5 lines are obtained on critical difference temperature, logarithmic paper Over the whole liquid range of the c o m p o u n ~ ~ to 0.8, acetic acid for densities of 0.75 to 1.05, and carbon tetrachloride for densities of 1.0 to 1.1. It was necessary to use more The data for ~i~~~~ 1 were obtained from standard handbook than one reference substance for the construction of this nomoalld tables, and in all cases it was found that this correlation graph, since no one material was able to cover the whole range of yielded straight lines if no change in polymeric structure takes place throughout the temperature range. Sulfur trioxide and densities d e s i ~ d . The center grid serves to locate the reference or pivot point water show breaks in the continuous functions to give, in each for each compound. This is located by noting the point of intercase, two straight-line functions. section of two straight lines joining values of two different denThe slopes of the lines of most of the liquids thus plotted are, sities a t two different temperatures of the compound in question. in each case, very close to unity, as would be expected from the This pivot point serves for the densities of that compound a t all mathematical consideration of this plot, A fairly safe extrapolaother temperatures; thus by extending a line from any value of tion of densities over a limited range of temperatures can t,herethe temperature on the right scale through this pivot point, the fore be made if only one density measurement and the critical density at that temperature is found by the intersection with the temperaiture are known. For accurate interpolation and extrapolation, it is necessary, however, to have measurements taken left scale. at two different temperatures and to know the critical temperaI n Table I are given the values of the coordinates on the ture for that material. center grid work for locating the reference point of 54 compounds.
883
x
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
a84
Critical Difference Temperature
TABLEI. X Compound Acetic acid Acetone Acetonitrile Acetylene Ammonia Isoamyl alcohol Aniline Benzene n-Butyric acid Isobutane Isobutyric acid Carbon dioxide Chlorobenzene Cyclohexane n-Decane n-Dodecane Diethylamine n-Eicosane Etlianc Ethanethiol E t h y l acetate E t h y l alcohol Ethyl chloride Ethylene E t h y l ether E t h y l formate E t h y l propionate E t h y l propyl ether E t h y l sulfide Fluorobenzene n-Heptadecane ’ n-Heptane n-Hexadecane
Vol. 40, No. 5
Y VALUESFOR DENSITYNOXOGRAPH
AND
X
Y
Compound
40.6 93.5 26.1 47.8 21.8 44.9 10.1 20.8 22.4 24.6 20.5 52.0 33.5 92.5 63.0 32.7 31.3 78.7 13.7 16.5 31.5 75.9 45.4 78.6 4 1 . 7 108.0 19.6 44.0 16.0 38.2 14.3 41.4 17.8 33.5 14.8 47.5 4.4 10.8 55.5 32 .O 66.0 35.0 48.6 24.2 62.4 42.7 17.0 3.5 35.8 22.6 68.4 37 .G 63.9 32.1 37 .O 20.0 23.7 56.3 86.7 41.9 15.6 45.7 12.6 29.8 1 5 . 8 45.0
x
s
13.5 37.3 40.1 25.8 31.5 33.0 52.3 27.2 25.0 46.4 36.5 31 .9 16 2 16.2 12.7 15,s 12.6 14.9 13.5 35.7 28.0 14.2 35.0 27.5 20.1 33.0 23.8 33.8 15.8. 15.3 17.9 14.4
hlethanethiol Methyl acetate Methyl alcohol Methyl n-butyrate Methyl isobutyrate Methyl chloride Methyl ether bIethyl ethyl ether Methyl foimate Methyl propionate hlethyl sulfide n-honane n-OctadGane n-Octane n-Pentadecane n-Pentane n-Nonadecane Isopentane Phenol Phosphine Propane Propionic acid Piperidene Propionitrile ProDvi acetate Propi4 alcohol Propyl formate n-Tetradeoane n-Tridecane Triethylamine n-Undecane
Y 27.0 59.5 70.3 49.1 65.5 64.1 62.9 30.1 34.4 74.6 68.3 57.4 36.5 46.5 32.5 44.2 22.6 47.0 22.5 103.8 22.1 12.2 83.5 60.0 44.6 65.5
50.8 66.7 43.3 42.4 37.0 39.2
N heie y is the surface tension, u is the specific volume, M is the molecular weight, and T and T , are the temperature and critical temperature, respectively.
Taking logarithms of Equation 1 log y
+ 213 log M + 2 / 3 log v
=
+ log (r/’o
log k
- 2’
-
6) ( 2)
It has alieady been shown (9) that log 7
n log
2‘D $-
c’
(3)
where nand C‘ are constants and T Dis the critical difference t e n perature. Density of Benzene
Substituting Equation 3 in Equation 2 :
plcc.
Figure 1. Density of Various Compounds Versus Density of Benzene at Same Critical Difference Temperature
12
Carbon tetrachloride Bromuhenzene Oxygen Acetic acid Water Propyl formate Propyl alcohol Propiunitrile 9. Ethylene 10. Methane Plotted densities for the following line8 should be multiplied by 10: 11. Xenon 12. Sulfur trioxide 13. Stannic chloride 14. Oxygen 15. Carbon tetrachloride 16. Bromobenaene
6)
(4)
+ log v = 3/2 log (2’0
(5)
10g~-3/210gQ=3’2log(?’~-6) -3/2nlog7’~
(6)
- 6)
Similarly log d
+ 3/2 log Q
=
312 n log ’I’D - 3 / 2 log (Y’U
- 6)
(7)
where d is the density. For another liquid a t the same critical difference temperatuie, there may be written similarly
Ramsay and Shields (11, Zf?) showed the following relation to be true :
- 6)
-
+ 3 / 2 log cl,
3 / 2 r~ log 2’u
THEORETICAL BACKGROUND
-T
2’
or
Table I can be supplemented for any organic liquid for which two or more density measurements are available. The densities of every compound indicated were checked at three points as a minimum and could be correlated on this scale except for the two materials (water and sulfur trioxide). The breaks in their lines are probably due to irregularities in their phase relations.
12(2’,
+ C‘ + 2 / 3 log M + 213 l o g o = log k + log (TC-
Combining constants into a new constant log Q and dividing by 213
1. 2. 3. 4. 5. 6. 7. 8.
y(iM~)”3 =
log TD
( 1)
log d‘ 4- 3 / 2 log Q’ = 3 / 2
n’
log T D - 3/2 log ( T I , - 6 ) (8)
Dividing Equation 7 by Equation 8 log d log d‘
+ 312 log Q - 312 12 log T D - 3 / 2 log ( T D - 6) + 312 log Q’ - 3 j 2 n’ log 2 ’ -~ 3 / 2 log ( T o - 6)
( 9)
For liquids which are far from the critical point, the critical diffrwnce temperature, Y‘D, is much greater than 6 and the term 3 / 2 log ( ? ’ D - 6) may therefore be written as 3 / 2 log T Dwithout introducing a11apprrciable error. Equation 9 can therefore be writtell
log -t i log CI:
+ 3 / 2 log Q
+ 31’2
log Q’
- ( n - 1) log T u (n’ 1) log T D
-
(10)
May 1948
’
INDUSTRIAL AND ENGINEERING CHEMISTRY
T
885
reduces therefore to 1 when n = m’. At the same critical difference temperature Equation 9 can therefore be written for these compounds log d log d’
+ 312 log Q
+ 3 / 2 log Q7 =
or
+
log d = log d’ 3 / 2 log Q’
- 3 / 2 log Q
(15)
Combining constants into a new constant C ’
+C
log d = log d‘
(16)
which is the equation of a straight line on logarithmic paper with slope 1. On examining Figure 1 it will be seen that for most compounds the slope is very close to unity, Equation 16 can also be written
Similarly, Equation 8 can be written f(d’)
=
~ ( T D ’4)Q
(18)
Substituting forf(d’) in Equation 17
where TD’is the critical difference temperature of a reference substance and K is a constant combining C and Q. Equation 19 has two variables and two constants and therefore i t is possible to express it as a point coordinate chart for each compound if one scale is calibrated in temperature and the other in density.
or
ACKNOWLEDGMENT
log d log d‘
+ 3 / 2 log & + 3 / 2 log &’ -
n
-
1
Appreciation is expressed to Donald Q. Kern for suggestions. (11) LITERATURE CITED
or log d = n - 1 logid’ n‘ - 1 ~
+n-1
3 / 2 log &’
- 3 / 2 log &
(1) Ferguson, A., Trans. Faraday Soc., 19,408 (1923). ENG.CHEM., 32,891 (1940). (2) Othmer, D. F., IND. (3) Ibid., 34, 1072 (1942). (4) Ibid., 36, 869 (1944). (6) Othmer, D. F., and Conwell, J. W., Ibid., 37, 1112 (1945). (6) Othmer, D. F., and Gilmont, R., Ibid., 36, 858 (1944). (7) Othmer, D. F., and Josefowitz, S., Ibid., 38,111 (1946). (8) Othmer, D. F., Josefowitz,!S., and 8chmutaler, A. E., Ibid., 40,
(12)
Combining constants into two new constants, m and C, there is obtained: log d = m log d’
+C
886 (1948). (9) Othmer, D. F., and Sawyer, F. G., Ibid., 35, 1269 (1943). (10) Othmer, D. F., and White, R. E., Ibid., 34,952 (1942). 63,1089 (1893). (11) Ranisay, W., andshields, J., J.Chem. SOC., (12) Ramsay, W., and Shields, J., Phil. Trans. Roy. SOC.,184A, 647 (1893). (13) Sugden, S., J. Chem. SOC.,125,1177 (1924).
(13)
which is the equation of a straight line on logarithmic paper hav-
ing a slope m‘as presented in Figure 1.
For many liquids this relationship holds true even for temperatures close to the critical, because the constant n in Equation 9 is almost identicaI for many organic compounds (1,19). The term
-
Presented before the Divlsion of I‘ndustrial and the 110th Meeting of the AMERICAN CHEMICAL
n log T D - log (TD 6) log T n - log (TD 6)
12’
SOCIETY, Chicago, Ill.
.,
*
