528
NANCY CORBIK, MARY .4LEXANDER, AND GUST.4V EGLOFF
(5) MILLIGAN AND MERTES.J . Phys. Chem. 60,465 (1946). (6) PASSERINI: Gam. chim. ital. 60, 644 (1930). (7) SIMON, FISCHER, AND SCHMIDT: Z. anorg. allgem. Chem. 186, 1 O i (1930).
(8) WEIBER:Inorganic Colloid Chemistry. Vol. I I . The HUdrous Oxides and Hydroxides. JohnWileyandSons, Inc.,KewYork (19351. (9) WEISERAND MILLIGAN: J. Phys. Chem. 38,1175 (1934). (10) WEISERAND MILLIGAN: J. Phys. Chem. 40,1071 (1936). J.Phys. Chem. 44, 1081 (1940). (11) WEISERAND MILLIQAN: (12) WRETBLAD: Z. anorg. allgem. Chem. 189,329 (1930).
T H E ANTOINE VAPOR-PRESSURE EQUATION FOR MONONUCLEAR AROMATIC HYDROCARBONS KAKCY CORBI?;, hlARY ALEXANDER,
AND
GUSTAV EGLOFF
Uniewrsal Oil Products Company, Chicago, Illinois Recsived September 84,1946
The present study was undertaken to determine whether the variation of the constants of the Antoine equation logp = A
-
B
__
t+C
with the number of carbon atoms could be utilized to predict vapor pressures and boiling points of compounds for which experimental data are inadequate. The results indicate that these properties can be reasonably predicted for the normal akylbenzene series and the 2-methyl-2-phenylalkane series. For other types of akylbenzenes the variations are too great to permit satisfactory use. The vapor pressure and boiling point can, however, be estimated for phenyl-substituted normal alkanes if the boiling point a t 760 mm. is known. The Antoine equation is valid over a greater pressure range then is the widely used equation logp = A
- BT-
=
A
-
B
t
+ 273.16
where p = pressure in mm., T = absolute temperature, and t = temperature in "C. A plot of the reciprocal of the absolute temperature against the logarithm of the pressure usually exhibits a slight curvature, which increases at lower pressures. Although the linear equation represents the data quite well between about 200 and 800 mm., it cannot be extended much below 200 mm. A more convenient representation of vapor-pressure data is given by the Antoine equation when the constant C is chosen to give as nearly a linear function as possible in the pressure range under consideration. This equation makes possible the use of data between about 800 mm. and 10 mm.
ANTOINE EQUATIOX FOR MONOXUCLEAR AROMATIC HYDROCARBOXS
529
When precise vapor-pressure data are available, the value of the constant C may be accurately evaluated (3, 4). However, the equation is useful even though the data are insufficient to evaluate C accurately, because the value of 230 for C gives a good representation of vapor-pressure data for many compounds (3). The wider range of validity of equation 1 as compared with equation 2 markedly increases the number of mononuclear aromatic hydrocarbons for which pressure-temperature relationships may be determined. During a recent critical evaluation (2) of physical constants of these hydrocarbons, it was found that vapor-pressure data between 700 and 200 mm. were lacking for a large number of compounds, thus precluding the possibility of evaluating the constants of equation 2. For many of these compounds, however, data were available a t 10 to 100 mm. and a t or near 760 mm., permitting evaluation of constants of the ,intoine equation. .I recent st.udy of vapor pressures and boiling points of some paraffi, alicyclic, and alkylbenzene hydrocarbons (4) indicates that within a homologous series the values of the constants B and C of the Bntoine equation vary in a regular manner with tlie number of carbon atoms. In the present study the variation of constants il and B n i t h the number of carbon atoms was investigated for two homologous series of mononuclear aromatic hydrocarbons. MGTHODS .4ND RESULTS
Experimental data for all the compounds studied except the first three memhers of the n-alkylbenzene series were insufficient to permit evaluation of constant C. The values of C for these three compounds (4) indicate that C decreases with increasing number of carbon atoms, and is approximately 200 for n-alkylbenzenes of about eleven carbon atoms. In order t'o determine the variation of constants : Iand B with the number of carbon atoms, calculations were carried out by the method of least squares lvith C taken as 200, 210, 220, and 230. The vaporpressure data were taken from Willingham rt a / . (4) for the first three compounds, and from Egloff (2) for the rest of the compounds. A plot of R versus n (the niimhcr of carbon atoms) indicates t,hat>a linear relation in the form
B=a+bn
(3)
is adequate. One compound, la-pentylbenzene, deviated widely from n.hat would be expected on the basis of the other values (figure 1) and w-as therefore given a low weight in the calculations. The first three compounds of the series werp given high weights, because of the quality and completeness of the data. Constants a and b of equation 3 were evaluated by the method of least squares for each of the four values of C used. These constants were found to be simple functions of C, expressed hy the folloiving equations:
a = -4.280 - 3.24934C b
=
6G.945G
+ 0.O2368SsC2
+ 0.295086C
(4)
(5)
530
NANCS CORBIN, ,MART ALEXANDER, AKD GUSTAV EGI OFF
Values of a and b computed from the experimental data, and values calculated from equations 4 and 5 a t the four values of C used in this study are shown in table 1. Values of constant computed from the experimental data and values calculated in two different ways are shown in table 2. The columns headed “ B (calculated) (1)” contain values of B computed from equation 3 and the “observed” values of Q and b in table 1. The columns headed “R (calculated) (2)” contain values of B computed from equation 3 and the “calculated” values of a and b in table 1. Constant A was evaluated by the method of least squares from the experimental vapor-pressure data and the final calculated values of B. A plot of the resulting values showed A to be a simple function of n. This function has the form
A =k
+ m + p2
(6)
TABLE 1 Constants of the B versw n equation
c l
cExPexLm*;
I-
200 210 220 230
~
292.897 359.509 425.902 501.990
1
(C*IC&ED)
(EXPEPPYENm
~
~
1
293.392 358.022 427.389 501.494
125.9075 129.2433 131.3699 135.0348
‘
1i
b (CALCOLATED)
125.9626 128.9134 131.8643 134.8151
ANTOINE EQUATION FOIL hIONONUCLE.4R
AROMATIC HYDROCARBONS
.-_, ^ m&f .!%E
5: f
b:
v. $i
4
ro N t-
6
E
2 1 0
$I
4i $I
531
532
NAXCY CORBIX, MARY ALEXLXDER, .iKD GUSTAV EGLOFF
O = OBSERVED V A L U E S X = C A L C U L A T E D VALUES
7
1
26001
C=230
3
/ / ,:
1900t
1700~
1600-
1300
Id I
I
I
1
I
I
I
I
9 10,II 12 13 14 15 FIG.1. Values of constant H of the Antoine cqusrion as a function of the num1)w of carbon atoms for 1-phenylalkanes.
7
8
TABLE 3 Cotislorits of the A uersus n equotion i(EXPEPlkYENTAL) (C*LCULATED)'(EXPEPI*aNIAL) k ' r
/____#____' 200'
230
I
6.095773 6.269501 6.493183 6.704629
, I ~
6.095664 ~-0.071107 0.062557 6.289830 6.492855 0.053278 6.704739 0.042686
j
'' I
(CALCJLAIED) ((EXPEI&ENTAI)i
___ '
'
~
0.071078 0.062645 0.053190 0.042715
0.001363 0.001667 0.001977 0.002334
I
I
1 1
(CAldLATED)
0.001354 0.001664 0.001980 0.002333
evaluated a t C = 200 and C = 230, and their relation to n was determined. Both A and B were found t o be substantially linear functions of n, as expressed by the following equations:
A = 6.07272 B = 399.278
+
+ 103.9747~1
(10) (11)
533
ANTOINE EQUATIOS FOR bIOKOiiUCLEAH AROMATIC HYDROCARBONS
TABLE 4 Values 01 A for 1-phenvlalkanes C = 230 A (enperi.
-__ 7.. . . . . . . . . 8.. , . . . . . . . 9.. , . . . . .. . 10.. , . . . . . . 11.. . . . . . . . . 12. ..... . . . . 13. . . . . . . . . , 14.. , . . , . . . . ' 15.. . . , . . . .
. .
.
~
6.66395 6.74825 6.85329 6.94036 7.04502 7.14230 7.26756 7.35815 7.47197
mental1
- - -I
6.66103 6.75286 6.84747 6.94484 7.04499 7.14790 7.24357 7.36202 7.47323
6.81178 6.80988 6.89171 6.89749 6.99329 1 6.98838 7.07657 1 7.08268 7.17896 7.18027 7.27451 j 7.28119 7.39839 7.3S-543 7.48763 7.49300 7.60126 7.60391
I
i ~
~
I
6.96570 6.96221 i ,11654 7.04098 7.04510 7,19791 7.13894 7.13195 7.28252 7.21775 7.22276 7.35722 7.31793 7.31753 7.45539 7.41171 7.41626 7.54742 7.53407 7.51895 7.66842 7.62210 7.62560 7.75546 7.73532 I7.73621 7.86836
,
I
' ~
A
(calcu. latedl
7.11806 7.1957i 7.27815 7.36519 7 45690 7.55327 7.65431 7,76002 7,87039 I
O=OBSERVED V A L U E S X= C A L C U L A T E D VA L U E S
7.97.8-
7.77.67.574A
7.3 -
6.9
6.7 6.6
8
9
IO
II
k. .-
12
13
14
15
FIG.2. Values of constant A of the Antoine equation as a function of the number of carbon atoms for 1-phenyalkanes.
A
= 6.64523
+
(12) (13)
Experimental and calculated values of constants A ctnd B are shown in table 5 .
534
NANCY CORBIN, MARY ALEXANDER, AND GUSTAV EGLOFF APPLICATION OF THE RESllLTS
By means of the Antoine equation and values for its constants calculated from equations 3 through 13, the vapor pressure over a considerable temperature range, or the boiling point at any pressure between 10 and 800 mm., may be calculated for the compounds covered by this study, using any desired value of C between 200 and 230. By extrapolation vapor pressures and boiling points of higher memhrrs of the two series may be estimated. The reliability of TABLE 5 Values of A and B for the B-rnelhyl-8-phenylulkuiie s e n e 8 -
c
I
-_
= 200
C = 230
TABLE 6 Boilirig points ut lair- pressures OJ higher n-olkylbenzenes 1 !EXPPPIYF.NTAL)
I
I (CALCULATED)
"C. 181.7 179.i
172.8 186.8 195.2 22
24 . . .. . . . . . . .
..
..
..
"
'I
15 16
230.1 228.4
15
246. S
extrapolations cannot be definitely ascertained until accurate experimental data are available beyond the range covered in this study. However, some indication of the accuracy of extrapolated values may be obtained from data already a t hand. Boiling points a t low pressure have been recorded for several higher 12-alkylbenzenes (2) and a comparison of the values for observed and calculated boiling points given in table 6 shows agreement which is as good as could be expected on the basis of the inconsistency of the experimental values. Prediction of vapor pressures of compounds for which experimental data are lacking is limited to members of the two homologoils series covered in this study.
liNTOINE EQUATIOX FOR MONONUCLEAR dROMATIC HYDROCARBONS
535
In an effort to enlarge the scope of the applicability of the Antoine equation to other types of aromatic hydrocarbons, the relation of constant B to the normal boiling point was investigated. .4s a basis for comparison, the final calculated values of B a t C = 230 were plotted against the boiling points (computed from the calculated values of .1 and B ) for n-alkylbenzenes of from seven to twentptwo carbon atoms. I n the same way a plot of B versus boiling point was made for the four tertiary alkylbenzenes covered in this study. In both series, B was found to be a function of the boiling point. The relationships are expressed by the tolloi\ ing equations: For n-alkylbenzenes:
+ 8.53311 - 0 021871? + 0.045101P 13 = -141.297 + 12.33921 + 0.0397tit' + 0.0480GGt3
13 = 421.037
C = 200
(14)
C = 230
(15)
T 113r,t i .3-ormui alXiiues wltli a phenyl s d s l i l u e n t c
= 200
I
C
=
230
1
Isopropylbenzene , , . lCp.31 2:Phenylbutane. , . . , . . . 112.8 2-Phenylhexane. . . . . . . 3-Phenylpentane., . . . . . :.Phenylhexane. . , , , . . 210.5,
'"4:: ~
For Z-mefhyl-2-phe~~ylalX.anes:
+ 4.87681 + 0.0017721? B = 817.540 + 5.08331 + 0.001816P B = 559.197
C
= 200
(16)
C = 230
(17)
where t = the boiling point at 760 mm. These equations reproduce the final calculated values of B with standard deviations of j=5.888 at C = 200 and i. 11.929 a t C = 230 for the n-alkylbenzenes, and =k 5.842 at (7 = 200 and i.6.089 a t C' = 230 for the 2-methyl-2phe nylalksnes . The values for constant B for a number of alkglbenzenes of types not included in the series mere computed from the available vapor-pressure data (2), and were compared with the \ d u e s of R calculated from equations 14 through 17. These comparisons are shown in tables i to 13. From table 7 it is apparent that the position of substitution of the phenyl group in normal alkanes has little effect on the valiie of B as a function of t.he boiling point. Thus, the value of constant 13 for any phenyl-substituted normal alkane whose boiling point at 780 mm. is known may he calculated from eqria-
536
NANCY CORBIN, MARY ALEXANDER, AND OUBTAV EOLOFF
tion 14 or 15. Constant A may then be evaluated from B and the boiling point. Using these values, the vapor pressure at a variety of temperatures or the boiling point a t pressures between 800 and 10 mm. may be estimated. TABLE 8 Phenul-substituted branched-chain alkanes with no auatemarg carbon atoms c BOILING
COYPOUND
POW'
B (experimental)
-
C
200
(d'culated)
AB
(er;eri-
- _ _ _ _ _ _ _ _ _ _ - - mental)
-
230
k%
AB
'C.
l-Phenyl-2-methylpropane, , 170.11564.4581490.829 $73.629 1886.9191786.683f100.236 1-Phenyl-2-methylbutane. , . 194.11984,0591626.445+357.6142357.0131928.126$428.887 1-Phenyl-2-methylpentane. . 205.01959.6541690.764$268.8902308.688 1994.704$313.984 1-Phenyl-3-methylpentane. . 221.31895.0231791.918$103,105 2236.4172098.165+138.252 l-Phenyl-3,7-dimethyloctane... . . . . . . . . . . . . . . . . . 275.O 2334.4782174.720$159.758 2700.8862505.015$195.871 2-Methyl-4-phenylpentane. . 197.91754.8211650.033+104.788 2089.5671953.9341-kl35.633
TABLE 9 Phenyl-substituted alkanes with the phenyl group on a quaternary carbon atom COMPOUND
2,3-Dimethyl-2-phenylpentane.. . . . . . . . . . . . . . . . 223.41743.56J1737.110 +6.459 2050.2812045.278 +5.003 2,4-Dimethyl-2-phenylpentane,. . . . . . . . . . . . . . . . 21i.711780.4371704.857 +75.5802098.0972011.6621-86.435 2,3-Dimethyl-2-phenyl' hexane.. . . . . . . . . . . . . . . . . 236.71773.1631812.815 -39.6522080.8982124.183 -43.285 2,4-Dimethy1-2-phenyl1824.254 - 11.6922121.9502136.104 -14.154 hexane.. . . . . . . . . . . . . . . 238.7,1812.562 2,5-Dimethyl-2-phenylhexane,. . . . . . . . . . . . . . . . . 2-Methyl-2-phenyl -3-ethyl pentane., ............... 3-Methyl-3-phenylpentane. 3-Ethyl-3-phenylpentane. .
4-Methyl-4-phenylheptane 3-Ethyl-3-phenylhexane. . 2-Methyl-3-ethyl-3-phenylpentane.. . . . . . . . . . ,
Branching of the side chain in alkylbenzenes has a pronounced influence on constant B, as can be seen from table 8. The effect in every case is to increase the value of R beyond what mould be expected from the boiling point.
ANTOINE EQUATIOS FOR MOXONUCLEAR AROY.4TIC HYDROCARBOKS
537
Compounds with five carbon atoms in the side chain appear to be anomalous. In all cases the value of constant B is higher than nould be anticipated from other compounds of similar structure. Whether these compounds really are anomalous, or whether inaccurate data lead to a high result in every case, is a question which cannot be settled without further experimental study.
_ - _ _ _ ~
TABLE 10 Dialkylbenaenes
c = 200
I
C
= 230
COYPOUND
1,3-Dipropylbenzene . . . , 216.5~1799.909~1761.080~ f38.829i2119.088~2067.496 +51.592 1,4-Dipropylbenzene.,. . . 221.1~1820.37S/1790.014~ +30 364i2139.084209i.506~$41.578 1,3-Diisopropylbenzene. . . . . 204.0~166S.7~S~1684.7i5~ - 16:027ll98i,79411988.508~ -0,714
Table 9 shov b results of calculation6 for compounds containing a quaternary carbon atom. The deviations from expected values of R are variable in sign, but in general of lesser magnitude than the deviations of branched-chain alkylbenzenes with no quaternary carbon atoms. However, in most cases the experimental values of Bare outside the limits of error, and therefore reliable estimates of vapor pressure or boiling points cannot be made for compounds of this type.
TABLE 11 Trialkylbentenes
I
I
I
c * 200
c
-
230
1,2-Dimethyl-4-ethylbenzene... . . . . . . . . . . . . . !. 167.61697.6411589.001f108.6402025.5701889.249f136.321 1,4-Dimethyl-2-ethylben- 1 . . . . . . . . . . . . . . . .j 185.41576.2711576.463 -0.189 1891.9731876.245 +i5.728 l-Ethyl-2,4-dimethylbenzene... . . . . . . . . . . . . I 185.61573.6131577.600 .- 3.9871886.3261877.428 $8.898 l-Propyl-2,4-dimethylbenzene.. . . . . . . . . . . . . . . i. 208.11623.3381709.452 - 86.1141921.8542014.039 -92.185 1,3-Dirnethyl-5-propylbenzene.. . . . . . . . . . . . . . 208.01639.8061709.846 -69.040 1942.1032013.413 -71.310 1,2-Dimethy1-4-isopropylbenzene... . . . . . . . . . . . . . . 199.21707.1261656.283 $50.843 24327.5101959.026 +68.484 l-Isopropyl-2,4-dimethylbenzene.. . . . . . . . . . . . . . . . 194.51643.2941628.770 +14.524 1959.7051930.535 +29.170 1-Methyl-2-e thyl-4450propylbenzene.. . . . . . . . . 209$31858.6771716.7371141.9402192.9672021.578f171.389 . . . 217.51795.7651767.328 f28.437 2117.9282073.973 f43.955 1,2,4-Triethylbenzene.. 1,3,5-Triethylbenzene.. . . . 216.21757.3501759.209 -1.8592075.4732065.558 f9.915 l-Methyl-2-propyl-4-iso- j propylbenzene.. . . . . . . . . . . 225.21945.1211816.230 +128.8912285.9892124.744+161.245 1,2,4-Triisopropylbenzene.. 240.418862.92 1917.2801 -30.388 2208.4052230.405 -22 .ooO ~
~
TABLE 12 Tetra-, penta-, and hexaalkylbenzenes
c = 200
I
C
= 230
1,2,3,4-Tetrame thylbenzene... . . . . . . . . . . . . . . . 205.11664.9151691.364-26.4491963.095 1995.325 -32.230 1,2,3,5-Tetramethylbenzene.. . . . . . . . . . . . . . . . 197.11694.2541643.945f5O.3092016.391 1946.253 f70.138 1,3,5-Trimethyl-2-ethylbenzene.. . . . . . . . . . . . . . . . 210.51726.5881724.051 f2.537 2042.OO5 2029 .148 f12.857 l12,4-Trimethyl-5-ethylbenrene... . . . . . . . . . . . . . . . 209.O 1729.4341714.913f14.5212O48.333 2019.690 f28.643 Pentamethylbenzene.. . . . . . 231.91928.2101859.990+68.220 2262.259 2170.334 f91.925 1,2,3,4-Tetraethylbenzene..253.51955.6552009.862-54.24372276.144 2328.045 -51.901 1,2,4,5-Tetrsisopropyl benzene... . . . . . . . . . . . . . . . 258.41994.2702045.948-51.67823131049 2366.049 -53.OOO Hexapropylbenzene.. . . . . . . . 333.52836.8392726.739-90.100 3002.576' 3118.228-115.652
* Boiling point for C = 230 is 332.9"C. 538
ANTOINE EQUATION FOR MONONUCLEAR AROMATIC HYDROCARBONS
539
Tables 10, 11, and 12 show observed and calculated values of constant B for polysubstituted benzenes with no branched-chain substituents. The values of B for the xylenes are based on excellent data (4),and indicate that B is very nearly the same for dialkylbenzenes as for hypothetical n-alkylbenzenes of the same boiling point. Values of B for all other polyalkylbenzenes are based on scant data (2), often only two or three values. The differences between experimental and calculated values of B are in most cases large, but up to a boiling point of about 240'C. the deviations apparently have a random distribution. Above 240°C. all experimental values are lower than the calculated ones. Up to this temperature it is probable that predictions of vapor pressures or of boiling points a t reduced pressures for polyalkylbenzenes on the basis of the calculated values of constant B will be no less accurate than much of the existing data. Table 13 compares values of B computed from experimental data with those calculated from the boiling points and equations 16 and 17, for a few di- and trialkylbenzenes with one tertiary side chain. The deviations are too great to TABLE 13 Polyalkylbenzenes with one tertiary substituent
-
I COMPOUND
OILING POINT
_
c zoo B (experimental)
_
1
I
~
B (calculated)
_
AB
_
I I 1
C
B (erperimental)
B
230
kk$_
_
IB
_
'C.
1-Methyl-4-fert-butylbenzene. . . . . . . . . . . . . . . . . . . . 190.21650.0781550.868 +99.2101968.5561851.165 +117.391 l-tert-Butyl-2,4-dimethylbenzene, . . . . . . . . . . . . . . . . 212.8 1740.805 1677.383 +63.4222058.938 1983.026 +75.912 lt3-Dimethyld-tert-butylbenzene. . . . . . . . . . . . . . . . . 204.9 1769.857 1632.849 +137.008 2101.151 1936.611 +164.540 1-n-Butyl-4-tert-butylbenzene . . . . . . . . . . . . . . . . . . 249.0 1818.146 1883.386 -65.2402124.373 2197.735 -73.362
permit reliable estimation of vapor pressures or boiling points for this type of compound. The only classes of compounds for which reliable estimates of vapor pressures or of boiling points a t pressures between 10 and 800 mm. can be made in the absence of any experimental data are the normal alkylbenzenes and the 2methyl-2-phenylalkanes. Estimation of boiling points a t reduced pressures can be made for phenyl-substituted alkanes with the phenyl group not in the 1-position and for polyalkylbenzenes with no branched-chain or tertiary substituents if the boiling points a t 760 mm. are known. SUMMARY
Constants A and R of the Antoine equation logp = A
B -t + C
~
540
JOSEPH TIIIE
have been evaluated and related to the number of carbon atoms at several values of C for the normal alkylbenzene series and the 2-methyl-2-phenylalkane series. The relationship of A and B to C has been evaluated for the normal alkylbenzene series. Applications of the results for the prediction of vapor pressures a t different temperatures or boiling points a t pressures of 10 to 800 mm. are discussed. REFERENCES (1) ANTOINE,C . : Compt. rend. 107, 681 (1888). (2) EGLOFF, G.: Physical Constants of Hydrocarbons, Vol. 111. Reinhold Publishing Corporation, New York (1946). (3) THOUSOX, G. W.: Chern. Rev. 98, 1 (1946). (4) WILLINGHAM, C. B., TAYLOR, W . J., PIONOCCO, J. M., A N D Rossrwr, F. D . : J . Research Xatl. Bur. Standards 36, 219 (1945).
A MATHEMATICAL APPROACH TO REACTION MECHANISMS JOSEPH THIE
University of Dayton, Dayton, Ohio
Received J u l y 16, 1946 INTRODUCTION
Several articles (3, 6, 7 , 9, 10) have already been written in which chemical equations are treated in a purely mathematical manner. However, there is more to be said about this subject, especially regarding the occurrence of intermediates and practical applications. The purpose of this article is fourfold: A. To give a rule for determining whether or not a given group of intermediates can occur alone in the production of a given reaction. B. To give a method of obtaining all possible component equations containing a given group of intermediates which have been postulated for a certain reaction. C. To give a method of obtaining all possible mechanisms by which a given group of intermediates, postulated for a certain reaction, can occur. D. To apply these new principles to practical problems of the kinetics of chemical reactions. METHOD AVD EXAMPLES
Concerning those groups of intermediates which may occur alone (Le., those for which mechanisms can be written containing only those intermediates and terms found in the original equation) we have the following principle: All those groups of intermediates, and only those, may occur along which, when inserted into a given equation, yield a resulting equation having the following properties: (I) It has all the terms of the original equation and one or more additional
