Infinite Points a ATICAL FO M. R. DREISBACN l i x ~ K. S. SPENCER T h e Dow Chemical Company, Midland, M i c h .
-
+
From the A n t o i n e equation, loglo I' = A B/jt C) w h e r e P is pressure in mm., A and B are constants f o r each compound, C i s equal to 230, and t is the t e m p e r a t u r e i n C . , the equations O
gave values which followed the determined data more closely than any of the other simple equations relating vapor pressure and temperature. R e gave results to verify this conclusion. Two major facts have tended t u discredit the accuracy olr the Antoine equatioii and the plot or log vapor pressure versus 1___ on the Cos chart,. These two facts have been the
I .
(t
and
have been developed. t o3 a n d P are the temperature a n d pressure of the so-called infinite point-i. e., the point at w h i c h the vapor p r e s s u r e - t e m p e r a t u r e curves of a family meet w h e n p l o t t e d on a Cox chart. The e q u a t i o n d t j d P =
B
O
- hgio PI2 gives the dt/dP ( r a t e of chaiage of
_____._
2.3026 P ( A
+ 230)
small size of the Cox charts which haGe been used and the inaccuracy of the literature data. I n order to read the tempcrature accurately t o within even 1' C. in the neighborhood of 200 C., thc Cox chart should be about 5 X 3 feet in size. The ordinate, or pressure scale, is laid off logarithmically so that the eight cycles cover a length of 56 inches. The abscissa, or temperature scale, is laid off bv means of the following formula: 1
temperature w i t h pressure) at a n y pressure accurately to the boiling point and probably to several atmospheres. Since the dtldP values are known now, the l a t e n t h e a t a t a n y t e m p e r a t u r e may be accurately calculated b y use of the Clapeyron equation and Haggenmacher's correction factor. N o m o g r a p h s are produced f o r giving correction t o the barometric boiling p o i n t and to the boiling point at 760 mm. f o r the benzene family, and for reading pressure corresponding to a n y t e m p e r a t u r e and vice versa. C h a r t s are p r e s e n t e d showing error introduced b y u s i n g C = 230 for all compounds instead of a separate value for each cornpound. Tables are s h o w n w h i c h give A, B , and dt/dP
a
=
37.17 t 230.1 .- t
O
O
I
+
where t = temperature in O C., and y = distance in inches, positive or negative, from the 0' C. point. The values of t cover the range from -100" to 10,000" C., and the distance is 38.75 inches. The 0 " C. point is 17.25 inches from the left, or -100" C. point of the chart. The value of y is determined for each value of t. This rcpresunts the distance in inches from thr 0 C. point, positive or negative, for the value of 1. The valuea of t are determined every 10" C. from --100" to +200" C.; every 50' C. from 200" to 600" C.; every 100" C. iroiir 600 ' to 2000 C. ; every 500 C. from 2009 ' to 4000 C.; and every 1000' C. from 4000" to 10,000" C. The spaces between each 10' C. are divided into tenths representing 1O C., from - 100 O to 0 C. ; into fifths from 0 to 200 C. representing 2 " C. into tenths from 200" to 600" C. representing 10" C.; and into halves from 600" to 1000" C. representing 50" C. The divisions above 1000° C. are too sinall to be subdivided. The readings obtained from such a chart give values adequately accurate for any ordinary laboratory work. The inaccuracy of the literature data may be judged from the fact that Beilstcin, for instance, niay give from one to five different boiling points, which may be as much as 6 " C. apart at barometric pressure, for the same compound. How to judge which one, if any, is correct presents a problem. The work of plotting the vapor pressure-temperature curve of a compound on a Cox chart is complicated further by the fact that the ternperatures corresponding to pressures below atmospheric are a p t to be even more in error. TJ7hen an attempt is made to use the Antoine equation or make a plot on the Cox chart, the ralculated values will, of course, not check the determined values, and when one or more of the determined values are in error, the determined values will not lie on a straight line on the Cox chw t. As a result numerous workers in recent years have developed their own equations for each compound, when in most cams the Antoine equation would have given better correlation with the determined values. While the value of C increases in each famlly with an increasing boiling point, the value 230 still gives very good results.
values f o r every 10" C . These charts cover the benzene, naphthalene, and aliphatic h y d r o c a r b o n families.
N 1888 Antoine
330.1 2/ or ~37.17 y
( 1 ) developed a n equation which gave the
pressure corresponding to any temperature and vice versa for various compounds and elements. As a further development of the relationship between vapor pressure and temperature, Cox (3) plotted the logarithm of the vapor pressure against temperature, the result of which plotting has been called the Cox chart. This plot as set up by laying off a logarithmic scale as the ordinate and drawing a straight line a t an angle. This line was taken to represent water, and by using certain pressure values the corresponding determined temperature values were marked on the abscissa. Davis ( 4 ) and later Calingaert and Davis ( 2 ) further developed the Cox chart so that when the logarithm of the pressure was plotted 1 against --__- for any compound, the result was a straight line. (t 230) Further, when the straight lines of related compounds were extended, they met at a common intersection point, the so-called infinite point of the "Cox chart families." The Cox chart families are those related groups of conipounds which have a coninion intersection point when plotted on a Cox chart. Thomson in his paper (9) on this equation gives (t very complete discussion of the various equations used t o relate vapor pressure and temperature. As a result of his research on the subject, he arrived at the conclusion that thc Antoine equation
+
176
O
January 1949
INDUSTRIAL AND ENGINEERING CHEMISTRY
177
~~
as is shown below. I n some cases the value of C should be much different, and in such cases a Cox chart with the new value of C in place of 230 in the formula would have to be set up. However, it is not necessary to have a Cox chart, as the formulas shown later in this paper eliminate the need for such a chart. The selected point method for obtaining the value of C by the use of three accurately determined values of pressure and their corresponding temperatures is the simplest method. This method is explained by Thomson (9).
TABLE I.
CONSTANT VALUEOF C ~
AS
230.1
a t 1710' _ Pressure _ _in Mm. _0 ~ - -C.--1
L
0
Calod. Calcd. A' and B ' , from A , E, and C C' = 230.1 Compound 1 c. 500.66 601.2 Henaenec'. b.p. = 80.104° C 67.135 104.02 103,67 26.888 45.96 45.44 ... 10 26.264 26.78 ... 0 7.793 7.443 20 36,404 33,530 3b:412 288.5 501.3 500.6 96,559 500.65 Tolriencb, b.p. = 110.625° C. 103.66 104.18 103.636 52 802 12,423 12.895 .., 10 6.72 7.062 ... 0 1.700 1.631 - 20 29,200 31,300 aO:il6 320.8 500.52 500.37 Ethyl brnnrne". b . p . 136.189' C. 121.312 500.66 151.4 a4.599 149.428 149.40 4.324 3.751 ... 10 0.5080 0.4799 0 27,300 29,045 352 . O 29',i80 6 Cyclohexaned, h.p. = 80.740" 500.7 500.8 67.467 600.73 149.4 149.88 150.3 34.821 4 7 . 4 9 4 8.34 10 ... 28.57 27.84 0 , . 8 .583 8.170 20 29,300 28,290 2Qi472 280 n-Propylcyclohexsnee, b.p. ,500.68 500.65 500.70 140.93 69.28 67.16 156.714' C. 80 805 67.175 1.5184 1.894 10 .. 0.7218 0.9531 0 ,.. 0 ,1983 0 . 1 2 8 6 20 . . h = 6.89322 B = 1203 835 C = 219 924 A ' = 7 06018 and B' = 1296.0. b A = 6.95334: B = 1343:943: C = 219:377: A' = ?.11141', and B',= 1441.0 E B = 6.94995, B = 1419.315, C = 212.611, A' = 7.14990, and B = 1564.0 d A = 6.84495, B = 1203.626, C = 222.863, A' = 6.94174, and B' = 1262.3. e'.$ = 6.88284, B = 1457.64, C = 207.511, A' = 7.14125, and B' = 1648.0
Literature Value 500.66 103.635
-
-
c.
Recently the Bureau of Standards has carried out some very accurate work on the pressuretemperature relationship of many compounds in the aliphatic hydrocarbons and the mononuclear aromatic or benzene family (7). These are the most extensive, really accurate data ever published on this subject. These data have been utilized to show the very close agreement between the determined values and the results obtained by the use of the Antoine equation. They have also been used t o verify the accuracy of the new equations. The Antoine equation relating temperature and pressure is B log,oP = A _(1)
- t+C
where P = vapor pressure in mrn., t = temperature in C., and A , B , and C are constants. The bureau determined a different value of C for each compound. When a different value of C is used for each compound, a slightly curved line is obtained on the Cox chart, the amount of curvature depending on the deviation of the value of C from 230; but the various compounds of a family do not, meet in a point. In the references mentioned (6,4 ) the value of C was set a t 230 for all compounds. This value of C gives straight lines on the chart, for by definition all accurately determined values lie almost entirely on a line. Furthermore, compounds of a family mpet in a small circle, the so-called infinite point. The calculated results, when using C a t the constant value of 230, do not check the determined values as accurately as when C is determined for each compound. This fact is to be expected. Hom7ever, if C is taken as constant for all compounds and only one value of pressure and temperature is known for a compound as well as the infinite point of the family t o which the compound belongs, then various values of pressure can be determined for the corresponding temperatures or the temperature relating t o any pressure. The accuracy obtainable by this method is shown in Table I. A t the critical temperature, the pressures computed by using C equal to 230 are much nearer the literature values than when a different value of C is used for each compound. The calculated vapor pressures at the critical temperature obtained by means of C = 230 are within 0.02 to 3% of the literature values of critical pressure. If the boiling points a t two pressures, P and PO,are known, then B can be determined by
This is Equation 40 of the Thomson paper (10).
c AND
COMPARISON O F VALUES OBTA41NEDBY USING DIFFERENT VALUES O F
-
-
4-0.54 $0.35 +0.52 +0.52 f0.35
0.022 +0.7 f0.52 f0.45 $0.34 + O . 16 -0.15 $2.00 +0.57 +0.028
2.2
0.12
$0.1
+n
R
+o:a5 + O . 63 $0.41
0.6
+0.05 1-1.12 +0.37 $0.23 +O 07
A is obtained by solving for A in Equation 1 and using C
&s
230. (3)
If, as happens in many cases, only the boiling point a t one pressure is known, the above equation may still be used, if the values of P , and T , are known-Le., the pressure and temperature of the so-called infinite point where the vapor pressuretemperature lines of a family tend to converge. This infinite point can be obtained by means of an equation developed by Spencer. This can be used when the A and R values of two compounds of a family are accurately known.
This equation is very sensitive and even with the best values gives somewhat different values of t m and Pm. Hence, t o be able to obtain an accurate infinite point, the A and B values of at least three compounds of a family should be known. Thomson (IO) gave a very simple method for determining whether different compounds belonged to the same family. When the value of A is plotted against the value of B , the points must lie on a straight line if the compounds belong to the same family. In order for the above to hold true, the C value must, of course, be the same in each case. This relation arises from Equation 6,
which is simply Equation 3 with P and t replaced by P, and t,. Another formula for the B value is as follows:
where t~ = boiling point at pressure P. The value for dt/dP, the rate of change of boiling point with pressure, is another value that is even more diffioult to obtain
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Vol. 41, No. 1
-e10
-em
-190
-1-
-170
-140
IS0
-140
-130
---
-ID0
-110
--
-100
-00
-
--
80
Figure 1. Boiling Point Correction Nomograph for Alkyl Benzenes
accurately than the vapor pressure-temperature relationship. Actually, the difficulty of obtaining an accurate value of dt,ldP led to the above variations of the Antoine equation. The slopes of the boiling point curves in a Cox chart family change in a consistent manner, since the straight lines on the Cox chart radiate from a common point. Spencer developed the formula, Equation 8, giving this value accurately from the pressure and the A and B values.
The range of applicability of Equation 8 is the same as that of the Antoine equation. This is discussed belox. I n order to verify the accuracy of the d t / d P values obtained
OF CALCULATED dt/dP VALVESAT 760 TABLE 11. COMPARISON M M . WITH STALUES DETERMINED B Y BUREAU O F STANDARDS
dt/dP Calcd. Compound by Eq. 8, C. Benzene Toluene Ethylbenzene 1 2-Dimethylbenzene 1:3-Dirnethylbenzene 1,4-Dimethylbenzene lsoprepylbenaene
+
280 310.103 340.623 366.187 374.414 369.102 368.348 382.393
IB
Benzene Toluene Ethylbenzene 1 2-Dimethylbenzene 1:3-Dimethylbenaene 1 4-Dimethylbenzene dopropylbenzene
dt/dP Bureau Value, O C.
Difference, O C. -0.000054 +o. 0001 11 -0.00001 - 0 IO0018 -0.00027 -0.000042 -to. 00005 dt/dP X l o 3 Bureau of Standards Equation 9 42 . ? 42.7 46.3 46.2 49.0 49.0 49.7 49.9 49.0 49.3 49.2 49.2 60.7 50.8
Figure 2. Boiling Point Correction Nomograph for Alkyl Benzenes
by means of Equation 8, a comparison of the determined values and the experimental values obtained by the Bureau of Slandards (8) is shown in Table 11. The A and B values of three of the compounds were computed from the boiling points a t various pressures as determined and an average taken. Then these averages were used to compute the infinite point and also to determine the values of d t l d P from Equation 8. The infinite point as obtained is t = 2020' C. and P , = 2.87 X 106 mm. for the benzene family. By using these values and the boiling points of the compounds as given by the bureau a t 760 mm., the d t / d P values at 760 mm. were determined. Rossini ( 8 ) considers their error on the dt/dP values to be *0.00025" C. It will be seen from Table I1 that the greatest difference between the determined and calculated value is 0.00027" C. Thomson (9) also called attention to a variation of Equation 8: dt/dP where and
=
(t
+ C) ( t m
- t)/2.3026P ( t m
+ C)(v/m - y)
(9)
yP = loglaPm ?J =
log lop
+
so that dt/dP/(t C) is linear v-ith temperature for each family at a given pressure. The infinite point can be obtained from such a plot. Since the dt/dP value in Equation 9 is obtained from the boiling point and infinite point, i t should give a more stringent test of the calculated values of di/dP than Equation 8. Table I1 is a comparison of the dt/dP values from Equation 9 with those determined by the Bureau of Standards. Equation 8 will give the value of d t j d P accurately at any pressure, but this cannot be used to correct barometric readings to 760 mm., since the dt/dP value varies with pressure. Sponccr devoloped the following three formulas for obtaining this correction: A T = ( T -4- 230)
(sz)
IN DUS T R I A L A N D E N G IN EER IN G C HEM I STR Y
January 1949
179
culated value of the pressure for the same temperature from the Antoine equation using C = 230 P . Mm. and the A and B values obtained P , Mm., Calcd. d t i d P from % Literature dt % from the infinite point and the Temp., Literature Using % dP Error TR C. Values C = 230 Error E , A'1m.a Error Values P boiling point of benzene are 2,625 2,849 +0.85 2,838 +0.5 0,0144 0.01437 -2.1 0.716 130 found in column 3. The calcu.., 0.734 140 3,516 3,557 +1.1 3,536 4-0.5 150 4,331 4,390 +i.4 4,354 +0.5 0.01'045 o.oio35 -1.0 0.752 lated values using the A , B , and 160 5,277 5,360 +1.6 5,304 f0.5 ... ... ... 0.770 .., .0.787 C values as determined by the 170 6,368 6,476 +1.7 6,367 +0.3 7,621 7,758 +1.6 7,639 +0.2 0,00687 0.06668 -2.7 0.805 180 Bureau of Standards are located . . . . . . . . . 0.823 9,050 9,211 f1.7 9,047 -0.3 190 200 10,669 10,650 +1.7 10,627 -0.4 ... ... ... 0.840 in column 5 ; and the values of o,859 12,495 12,685 + 1 . 5 12,390 -0.8 210 dt/dP from Equation 8 and the 14,546 14,728 +1.2 13,347 -1.4 0,00460 0.00472 -'
