Applications of Duhring’s Rule E. J. ROEHL
~
The International Nickel Company, Inc.. Bayonne, N. J.
T
HE considerable number of papers (1, 3, 4, 6,7 , 0 , l 4 , 15, 16, I S , 19, 20, 23, 24, 26, 27, 28) on the relation first suggested by Duhring (8)sixty ycars ago have clearly e+tablished its usefulness; this relation was discoverctl as an enipirical rule, suggested by the close similarity of vapor pressure curves. The purpose of the present paper is to poiiit out two further applications of tliis rule. There are a t least threc equations or variations of this general type. Diihring’s rule states tliat if tlie tenipcratures a t which two substances exert equal vepor pressures are plotted against one aiwtliPr, the points so obtained are foulid to approximate very closely a straight line: T J T B = T ’ A / T ’ B= constant (1) or T A = K T B b where T A ,T B = abwlute boiling points of two Fuhqtances A nnd B ut one pressure, and 2 ” ancl ~ T ‘ g ~at different pressure K = slope b = intercept
+
The second equation, due to Ramsay and Young (B),mas proposed on a thermodynamic basis and was later discussed by Jollnston (11) and Porter (81):
+
+
R’ =: R c (t’ - t ) C‘ (1’ - t ) 2 (2) where R = ratio of absolute temperatures or the two substmircs corresponding to any vapor prossure, the same for bnt 11
R’ = rat io at any other pressure, the same for both c, c’ = ccmtnnts t ’ , t = temperntures of one of the substances corresponding to the two vapor pressures
This form is applicable to cases where there is extreme chcniical dissimilarity between tlie two sulmtanccs. Wliere the dissimilarity is less extreme, the last term may be dropped; and for cliemically related substances, c becomes small, and tlie equation assumcs the simple form:
R’ = R or TA/Tn = T ’ ~ / T ’ = B constant
(3)
This simple form is thus the theoretical form of the Duhring rule. The third equation, due to Cox (6), 1/Td = constant X I / T B (4)
is another and lem convenient form of the rule. Kracck ( l a ) , in employing tlie Ramsay-Young rule for the correlation of vapor pressure, temperature, ancl cornposition, has shown that the method is theoretically sound from the standpoint of tlicrtiiotlynatnics, as cvitlenced by the clevelopment of an equation for the heat of vaporization of solutions and the numerical check of this equation through thc calculation of the heat of vaporization of water from the value Tor ammonia. In tlie past, Dtihring’s rule has been succcssfully applied to certain cases which cat1 be given some theoretical basis, and it has also been applied in an empirical nisiiner to many other cases. While an equation such ax this can rarely be employed for predicting data of a high order of prcridon except over narrow ranges, it will, however, give results which are of
A relation is indicated whereby a minimum of experimental points will allow the calculation of the vapor pressure of a salt solution over the whole of the concentration range and a wide temperature range (a further application of Duhring’s rule). A second relation, connecting the vapor pressure and the number of carbon atoms in the homologous series of the hydrocarbons, is also based on Diihring’s rule.
great value for engineering and for other moderately precise requirements.
Relation between Diihring Lines and the Composition of Aqueous Salt Solutions The applicability of Dlihring’R rule to aqueous solutions of inorganic salts \vas first s h o m by Baker and \Vaite (1). They worked with solutions of inorganic salts in water (where the solute exertcd no apprcciable vapor prcssure) ; they were able to show that a t pressures up to one atmosphere the boiling points of these solutions, whcn plottcd against the boiling points of water a t corresponding pressures, fcll on straight lines. Thcy concluded that Diiliring’s rule was applicable t o such solutions “without regard to whethcr tlie concentration is high or low, whether or not the solute ionizes, and whether or not the molecules or ions of the solute take up water of hydration or association.” Sever:iI otlicr investigators have sitice confirmed this application but Iinvc not shown any definite relation between tlie curves for differelit concentrations of the same salt. If two vapnr prcssure determinations are neccssary for each Diihring line of consttint composition, an rsccc(liiigly large numbcr would be required to cover the wholc of tlir qpncentration range. In tlie present work it is proposed that there is a simple relation connecting the const:itit-coinposition curves for any one salt, such that the dcterniiiiatioti of a miniT mum of four vapor preysure points would allow the calciilation of the vapor prcssure of the salt solution a t aiiy concentration and over a wide temperature range. Figure 1 illustratcs thc application of Diihring’s rule to the system sodium chlorate-watcr. On the o r h a t e are plotted absolute tenipcratures of the solutioris and on the abscissa, the corresponding tcmperntures at which the reference substance (in this case, water) shows vapor pressures equal to those of the solutions. A faniily of curves is obtained, not all of which have been indicated. Each curve is one of constant coniposition, which is indicated in each case. If the logririthrns of the intercepts of these curves are plottcd against the logarithms of the concentrations, a straight line is obtained and the equation expressing this curve is:
log b 0.916 log C - 0.960 (5) where b = the interrept C = the concentration, grams NaCIOs/lOO grams Ha0 1320
INDUSTRIAL AND ENGINEERING CHEMISTRY
NOVEMBER, 1938
If the slopes of these curves are determined and o. term, A , is calculated from each slope, where sloDe = or =
A
IC = ( A X 10-4) (K 1) x 104.
+1
(6) . .
TABLEI. VAPOR PRESSURES OF AQUEOUSSODIUM CHLORATE SOLUTIONS (IN MILLIMETERS) Solution. G./IOO
a. HyO
-'
then a plot of the logarithms of A against the logarithms of the concentrations gives a straight line whose equation may be represented as log A == 0.974 log C
+ 0.688
5 10 20 40
60
80 100 120 140 160 180 200
-0' Obsvd. 4.51 4 45 4 30 4.03 3.78 3.47
.. . . . .
.. . . . .
C.7 - 2 0 ' C . 7 -40' Calcd. Obsvd. Calcd. Obsvd. 54.6 4.53 1 7 . 3 17 3 53 7 17 0 1 7 . 0 4 45 1 6 . 5 52.0 16 5 4 30 48.7 15 4 1 5 . 5 4.01 45.7 14 5 1 4 . 5 3.74 3.48 13 G 1 3 . 6 42.9 12.8 1 2 . 7 40.4 38.2
.. . . . .
.. . . . .
log b = 1.411 log C
A = 1.313 log C
-1.614
(8)
+ 0.3025
(9)
where for both systems the Diihring equation is: T.011..
... ..
... . .
... . .
... . .
C.-
-6OOC.-
Calcd. Obsvd. 147.4 54.6 145.1 53.7 140.5 52.1 131.6 48.9 45.8 123.4 42.9 115.0 40.2 109.2 37.8 103.1 97.5
.. .. .. ..
. . . . . . . .
Calcd. 147.3 345.1 140.6 131.7 123.8 115.8 109.0 102.0 95.9
...... . . . . . . ......
-SOo
Obsvd. 350.5 345.1 334.2 313.0 293.5 275.7 259.7 245.1 232.0 220.0
C.Calcd. 350.3 345 0 334.2 314.0 295.1 276.0 259.0 244 0 229.7 216.6
. . . . . . . . . . . .
c--lOOo C.Obsvd. 749 4 737 9 714 4 669 1 627 4 580 4 555 1
Calcd. 740.2 738 2 717 0 674 7 633 4 592 2 558 5
0 3 0 6
404 7 4(i8 7 439 0 417.0
524 495 470 447 425
1
523 7
(7)
I n the same manner the system sodium bromide-water yields the equations, log
1321
K X T H ~-O b
(10)
T o test these equations, the data for sodium chlorate and sodium 1)roniitle (10) n w e examined, nnd tlie results are given in Tables I and 11. In some cases the error reaches several per cent, but in view of the ranges involved, the agreement throughout is excellent.
the two Duhring lines would be established; the slopes and intercepts would then be ca.lculated and the log-log curves plotted. Once these log-log curves were established (or their equations), the slope and intercept for a Duhring line a t any given concentration could be determined; and by Equation 10, the vapor pressure of the solution of given concentration could be calculated over the temperature range. Monrad (19) obtained the slopes of the Diihring lines for several salt solutions, and if these data are plotted in the manner indicated above, the points fall well on straight lines. Likewise, if the intercepts are calculated and then plotted in the indicated manner, they also form straight lines. Hornever, his data cover only rather narrow ranges and therefore are not very conclusive. Similar data of Baker and Waite (1) for calcium chloride solutions shorn breaks in tlie curves, which apparently correspond to phase changes. Thus, it is possible that the correlation which has been suggested in the present work may not hold through a change in phase, although i t apparently did in the case of the sodium bromide solutions; in the latter it would be necessary to establish equations for each phase. Even if this proves to Le true, the method will still possess considerable value.
Application to Homologous Series
TH~O
FIGURE1.
-
D U ~ R I NLINES G FOR THE SODIUM CHLOIUTE-WATEit
I
SYSTEM
Carr and Murphy (4) showed that Diihring's rule could be employed to represent the vapor pressure data for the hydrocarbons from methane to decane; they obtained a family of straight lines but did not establish any relationship between the curves. In the present work the Diiliring lines were plotted for the first nineteen hydrocarbons, using hexane as the reference substance. On plotting the logarithms of tlie slopes of these curves against the logarithms of the number of carbon atoms, a straight line results; and a similar plot of the logarithms of (100 b) gives a straight line. For these two curves the following equations apply:
+
Figure 2 shows, the solubility curvcs for sodium chlorate and bromitle (17): The data for sodium clilorate forms a smooth curve; that for sodium broniide sliows a break at log K = 0.4118 log N - 0.322 (11) 50.674" C . , corresponding to a phase change f r o m ~ a U r . 2 1 1 ~ 0t o NaBr. I n a p p l y i n g this method to a ncw case, a m i n i m u m of four points \voultl have to be establislic(l experiment a l l y . T h a t is, t h e vapor pressures would have t o be determined for solutions of two dif90 110 130 IS0 170 190 210 230 85 95 105 II5 ferent concentrations, G M S N ~ t l O j P E R 100 G M S . H 2 0 G M S NABR P E R 100 G M 5 H 2 0 each a t two t e m p e r s tures. From these points FIGUnE-2. SOLUBILITY CURVES FOR SODIUM CHLORATE AND BROMIDE
INDUSTRIAL AND ENGINEERING CHEMISTRY
1322
(log 100
+
+ b ) = 0.415
logN 1.678 (12) N = number of carbon atoms
TABLE111. VAPOR PRESSURES OF HYDROCARBONS (IN MIL,LIMETERS)
Propane
By the application of these two equations, the vapor pressure curve for hexane, and the Duhring equation,
Butane
Pentane 0a
0000
c 0 0 0 0 010 1 0 c
where
Thydro. = absolute boil-
ing point of a hydrocarbon at one
Heptane
pressure
Thexane = absolute boiling point of
Octane
hexane a t the same pressure Nonane
gd m0 0 ccc0 0 0 0 00 0 80 , ld w M di ~9a h m m c 0.1 i
i 333
VOL. 30, NO. 11
the vapor pressure of any hydrocarbon can be calculated a t any temperature. The data for a few points selected a t random are given in Table I11 (2, I S , dg), calculated from Equations 11, 12, and 13. The data do not agree as well here as they did in the case of the salt solutions, but there is the possibility that the experimental results are not as precise. This generalization is limited to the normal m e m b e r s of t h e series. A further limitation lies in the fact that two members of the series do not fall in line a t all (methane and ethane). Rossini (d5) found that the energy of dissociation of normal C,HZ, + 2 (gas) into gaseous carbon and hydrogen atoms is not a linear function of n below 6 but of n above 6. He also states that the e n e r g i e s of t h e atomic linkages in the molecules of the normal paraffin hyd r o c a r b o n s can be represented in such a manner that the dif-
Obsvd. 50 200 500 760 5,896 7,077 8,734 12,800
Calcd. 50 197 485 760 5,600 6,790 8,276 11,970
Obsvd. 50 100 760
Calod. 51 106 760
Dodecane
50 100 760
51 103 758
Tridecane
495 770
50 100 760
52 107 760
Tetradecane
114.3 420,2 2,119 16,540 21,190
115 440 2,214 16,834 21,575
30 50 760
31 50 760
Pentadecane
50 100 760
760
58.4 795.2 3,450 10,105 20,430
61 800 3,608 10,774 20,703
49.4 646.4 3,382 11,185 18,730
53 690 3,498 11,577 18,655
50
50 100 500 760
94
50
53 107 780
100
760 Decane
Undecane
50
Hexadecane
50
50
100
760
100
50 100 760
Heptadecane
50 100 760
100 760
Octadecane
30 50 760
31 50 745
Nonadecane
50 100 760
90 700
51
46
52 100 760
100
760
ference between successive members becomes constant when n is greater than 3. This application, of course, is of less value than the first; however, it is possible that it would be quite useful in the case of other series where data for some of the members are lacking.
Literature Cited (1) Baker and Waite, Chem. & Met. Eng., 25, 1137, 1174 (1921). (2) Burrell and Robertson, J . Am. Chem. Soc., 37, 1893 (1915). (3) Caldwell and Roehl, paper presented before Div. of Physical and Inorganic Chemistry a t 93rd Meeting of A. C. S., Chapel Hill, N. C., April 12 to 15, 1937. (4) Carr and Murphy, J . Am. Chem. Soc., 51, 116 (1929). (5) Carr, Townsend, and Badger, IND. ENG.CHEM.,17, 643 (1925). (6) Cox, Ibid., 15, 592 (1923). (7) Dodge, Ibid., 14, 569 (1922). (8) Diihring, "Neue Grundgesetae zur rationelle Physik und Chemie," Leipaig, 1878. (9) Harris, ISD.ENG. CHEM.,24, 455 (1932). (10) International Critical Tables, Vol. 111,pp. 370, 371, New York, McGraw-Hill Book Co. (11) Johnston, 2. physik. Chem., 62, 330 (1908). (12) Kracek, J . Phys. Chem., 34, 499 (1930). (13) Kraft, Berl. Ber., 15, 1687 (1882). (14) Leslie and Carr, IND.ESG. CHEM.,17, 810 (1925). (15) Liempt, van, 2. anorg. allgem. Chem., 189, 287 (1930). (16) Loren2 and Hers, Ibid., 186, 165 (1930). (17) Mellor, "Comprehensive Treatise on Inorganic and Theoretical Chemistry," Vol. 11, pp. 328, 582, London, Longmans, Green & Co., 1922. (18) Mollier, Forschungsarb. Geb. Ingenierwessen, 63, 85 (1909). (19) Monrad, IND.EKG.CHEM.,21, 139 (1929). (20) Perry and Smith, Ibid., 25, 195 (1933). (21) Porter, Phil. Mug., 13, 724 (1907). (22) Ramsay and Young, Ibid., [5] 20, 515 (1885), 21, 33 (1885), 22, 37 (1886); 2. phyzik. Chem., 1, 237 (1887). (23) Reohenberg, von, "Die einfache und fraktionierte Destillation in Theorie und Praxis," Leipsig, 1923. (24) Riley and Bailey, Proc. R o y . Irzsh Acad., B38, 450 (1929). (25) Rossini, J . Research Nail. B u r . Standards, 13, 21 (1934). (26) Schulte, IND. ENQ. CHEW.,21, 557 (1929). (27) White, Ibid., 22, 230 (1930). (28) Wilson, Univ. Ill. Eng. Expt. Sta., Bull. 146 (1925). (29) Young, Proc. R o y . Dublin Soc., 12, 389 (1909). RECEIVED February 18, 1938.
