CORRELATING VAPOR PRESSURES AND VAPOR VOLUMES Use of Reference Substance Equations DONALD F. OTHMER ERL-SHEEN YU
he simple, thermodynamic Clapeyron equation of T state is for a vapor in equilibrium with a liquid below its characterizing conditions-the critical temperature and p-e. I t relates in a differential form the pressure, P, volume, V, and absolute temperature, T, o f 1 mole of vapor with the energy, L, necessary to vaporize it.
dP
-=
dT
L T(V-
--- L D) TAV
AV is volume change in vaporizing; at low prawres AV approximates V . Many attempts have been made to simplify this P, V, T, and energy relation to a P,T relation in finite, integral form, based on usual temperaturea and presm. Clausius made the first simplification of neglecting liquid volume and approximating vapor volume by using the ideal gas laws and constant R. Then, if latent heat is assumed constant, integration can be approximate over a narrow range. Numerous empirical relations have attempted to correct for these approximations owing to the nonideality of saturated vapors and the changing latent heat; the best may be exempM& by that of Thek and Stiel (76). They accumulate as many as 12 to 15 terms involving power functions and a handful of constants-usually combined empirically to two or three to save the iterative calculations &n required-and can be used practically only by compute~~ for , which extensive programs must be written. Many of these empirical expressions have been carefully analyzed-e.g., by Miller (3) who considered experimental data in a high p-e range from the normal boiling point to the critical for 23 liquids (no organics except methane) and a low pressure range from 10 mm. to 1500 nun. Hg for 71 liquids. Equations often expressed well the data in only one of the two ranges. Temperatures and pressures of liquids are expressed best, but used with more difficulty, as “reduced” temperatures-ie., the ratios of ordinary values to those at the critical. T , = T / T . where T , is the reduced temperature, T is usual temperature, and T, is the ‘22
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
critical temperature. Similarly, P, = P/P.. Critical temperatures and pressures are at the extremes of the scales, and any experimental errors are then transferred to all other data. However, many expressions arc in the more theoretic reduced form; but Miller (3) notes that they usually will be superior at higher pressures, rather than below the atmospheric boiling point. Use of R.hnnc* Substcmcb Relalion
Values of physical properties for one liquid have been correlated against those taken at the same condition for another standard, well studied, reference liquid to give a relation more nearly exact than can be written for the one liquid alone. A property of one liquid under the same values of one ofthe conditions-e.g., t e m p e r a t u r e varies as does the property of the reference liquid because ofthe same physical laws; and a simpler and more exact thermodynamic expression may be written to relate this variation than can be given for the abaolute variation of the property in a single liquid. Vapor presshave thus been c m e l a t e d - x predicted (4-49, 72)-with a reference substance by an expression having only one or two constants and one or two tams. Besides the always necessary table of l o g a r i t h w because log P is involved-these simple equations require a table of vapor pressures of water. This is the most convenient reference substance; and vapor pressure values alone, or in the familiar steam tables, are in every handbook of chemistry, physics, and engineering which contains also the tables of logarithms. Results are usually more accurate from these equations than by multiterm equations which express absolute values of vapor prr*rwrar in terms of molecular concepts or in terms of empirical mathematical functions and constants.
A logarithmic reference relation has been shown ( 4 4 9 ) to be nearly linear: log P = rn,logP’
+C
(1)
Here, P is vapor prawre of one substance, and P’ is vapor pranrure of a reference substance, both taken at the same temperature. The straight line has a constant
This simple correlatm or predctm equation gives good results equivalent to those obtained3om more complicated equations. It suggests that the correlating techniques m y be almost within the range of accuracy af many e#pcrimental determinations
l ~ m kms l of Clapoym Equafionthrough D i v i h by Same E~p~afir~n for Reference .Subsfam W i n g Point and O n Other Vapor Preaum Point Reduced Farms Require 1. and Pt, as One Point)
I "mn
I
log
I
=
m p q I
- .-I E?
L.'?'Av' PW-P
-' mTn = LT'
P'
lo.
nT =
mp-
T
latent heak Internal enemies Compressibiliiies latent heak Critical pressures
LT. L'. PAV 1's L P'AV' = La\, ~
I
Vapor volumes Internal energies Lotent heok Vapor volumes Internal energies latent heats Critical temperature
log P'
log (log PI
-
-
log (log PI
7
F a C r i l i i l P-ex
log (log TI
log T'
+E
loa (loa TI
log T'
+I
. 1 and k m mpar PMW.
-
log Pr
ba P m ba Pm
Or,if To l,and m dbdlng poi* om bovnt log . P =
.P
Note This i s a reduced equation, but usuol (nonreduced) values oftemperatures are used because of double log form
powl know (hat TLand Ps at Tsh
log P*
F a Critical Tenperahmt
Note This i s a reduced equation, but uswl (nonreduced) values of pressures are used because of double log form
- ba R b a P k - loa Pa.
- nm l o . p 6
and Hro mpar p m m powl lrnam IR at 11and PJ01 Tds
V O L 6 0 NO. 1 J A N U A R Y 1 9 6 8
23
slope, mT. C is the constant of integration. Values for mT and C for over 500 compounds were computed (9) and have since been reprinted by Reid and Sherwood (74). A vapor pressure value for any of these 500 compounds may thus be calculated within a minute or two. A nomogram (9), also reprinted by Reid and Sherwood ( I d ) , allows graphical determinations within seconds, within the precision required for most engineering work and also within the accuracy of much published vapor pressure data. Temperatures in centigrade and fahrenheit and pressures in mm. Hg and atmospheres may be used directly or interchanged. The nomogram gives values of pressures for the 500 listed compounds, as well as for other compounds or their solutions for which the boiling point and one other vapor pressure value are known and for the concentrations and pressures of gases dissolved in liquids ( 7 7 ) or adsorbed on adsorbents (70). Later (5), a simpler, more exact equation gave a linear, logarithmic relation of reduced vapor pressures, always taken at the same reduced temperature: log P, = mTR log P,‘
where
handbooks have no experimental background; they have been calculated from two or more vapor pressures taken near the boiling point to obtain the differential of the vapor pressure curve; and the use of such values to correlate vapor pressures may add little to the experimental determination of the boiling point itself. Fourth, if values of vapor pressure are available, all of these data may be used to add their confirmation to the correlation -either geometrically by determining the best line through them, or algebraically by the method of least squares. Thus, all experimental vapor pressure points may be used immediately in determining the constants. This is of particular use in preparing tables of physical properties, such as vapor pressures, for a system wherein 10 or more experimental values are available. Empirical equations are usually used with their large number of terms set up for input of the temperature and the latent heat at the boiling point. Additional experimental data are hard to feed into the correlation, while, with the straight line function, any number of vapor pressure points are fed in to determine the best line through them or to be averaged by least squares in evaluating the constants. Ranges of Data and Their Correlations
The substantially invariant slope, mTR, of the straight line is the single and thermodynamically correct constant. I t also allows accurate determination of the values of latent heats (5, 74) at all temperatures. This precise equation is simpler and more readily used than other reduced expressions because it gives straight lines going through the point T , = 1 and P, = 1. Thus, the integration constant disappears to simplify the algebra and to give a sheaf of lines on a plot. The common point representing critical conditions is the point in the point-slope form of a linear equation, so only the boiling point or the latent heat is needed with the critical values. Input Data for Evaluating Equations
I n all equations for vapor pressure, it is desirable to use as input data two experimental points of vapor pressure data at some distance apart. One of these may be the normal boiling point; the other, particularly for reduced equations, may be the critical point. This gives the so-called two-point equation for a straight line, and it is preferable to the point-slope form obtained by taking as input data or considering one experimental point of vapor pressure (the boiling point) and the latent heat which determines the slope. The two-point form is preferable for four reasons. First, vapor pressure data are much more readily determined accurately than are latent heat data which require much more complicated experimental techniques with many sources of errors. Second, experimental values, carefully determined, are reported at two or more points for many more compounds than are experimental latent heat values. Third, most of the latent heat values in the 24
INDUSTRIAL A N D ENGINEERING CHEMISTRY
As noted above, correlations are often made by different equations in the ranges above the normal boiling point and below the normal boiling point, or sometimes by the same equation with different values of the constants. Obviously, different experimental techniques, equipment, and samples are used at low pressures as compared to those used at high pressures. Glass predominates in apparatus for determinations below atmospheric pressure and metal predominates for high pressure work. Different instruments for measuring the variables are usually used. In most cases, different laboratories and experimenters have determined vapor pressures in the two ranges using different techniques and, invariably, with different samples of what is supposed to be the same pure chemical compound. Because of these experimental differences, data reported for the low pressure range may not correlate with data reported for the high pressure range. Correlations usually break approximately at the normal boiling point. This is determined by experimenters in both ranges and is always a fixed or “anchor point” for both
Previous articles in this series have appeared in IND. ENG. CHEM. in 1940, 1942-46, 1948-51, 1953,1955, 1957, 1959, 1961, and 1965; in Ind. Eng. Chem. Process Design Develop. in 1962; and in Ind. Eng. Chem. Fundamentals in 1964. Other articles in the series were published in J. Chem. Eng. Dafo in 1956 and 1962; A.1.Ch.E. J. in 1960; Chem. Fng. in 1940; Chim. Ind. (Paris) in 1948; Euclides (Madrid) in 1948; Sugar in 1948; Petrol. Refiner in 1951-53; World Pefrol. Congr., 3rd Proc., The Hague, 1951; Intern. Congr. Pure Appl. Chem., I 7 fh Proc., London, 1947; KirkOthmer Encyclopedia of Chemical Technology, 1st ed., Vol. 4, 1955; 2nd ed., Vol. 6, 1965.
I
A single equation, or a single. line or curve, should represent the function throughout the entire range high and low lines, curves, equations, or sets of constants used in any subsequent correlation. Any deviation owing to expaimental techniques or to correlation of the experimental data may be expected to show a break point near the normal b o i i point. However, the molecuk mechanism developing vapor pressure does not anticipate a change in the wntrollig function or its mechanism at the atmospheric b o i i point; thus, either the experimental data or their mathematical expressions cannot be correct if there must be two equations or two sets of constants for one equation. A single equation, or a single line or curve, should represent the function throughout the entire range. But it is commonly accepted that an equation may be good in one or the other pressure range, but not in both. The dected data for the test compounds suggested by Miller (3) have been reduced to linear functions by Equations 1 or 2; and straight line plots and linear equations are obtained which often show a slight break or angle of 1' to 3' at the boiling point, slightly less than 180' on the upside. Complicated curve functions of other Cordations may not demonstrate visibly the slight breaks which are indicated when the straight lines of Equations 1 or 2 are used; these show even slightly different values of the slope. Because theac breaks arc most improbable physically, the reasons for this obvious error should be considered in either the data used from different observations above and below the b o i i g point or their correlation. I n some other casee, vapor pressure data available for same of these liquids--possibly raw experimental data, u n c d a t e d by other techniqu-ve a line with a slight curvature, also concave upward. This leads to the suspicion that most ofthe other data presently available in standard references have been correlated by empirical constants in two expressions-ane for use above the boiling point, and one for use below. The more logical expression, a consistent variation through the boiling point, might better be represented by a very slight curvature concave upward. Thiswill be discussed
below. An example of an empirical approach is Miller's equation RPME (30). This is said to be as good a p d i c t o r as other single reduced equations, and because it is subtantially simpler for calculations (than other empirical equations suggested), it has been m o m mended (&) for rapid estimation of vapor pressures over the whole liquid range. RPME combines 4 equations. -G
lo~P= p -[l Ti7
- Tp*+ k (3 + Tp) (1 - Tpz)]
However, G is a function of a, while k is a function of both G and a. In turn, (I is a function of temperatures and preapures. Thus,
G = 0.210
+ 0.200 a
and
Here TR8is the reduced temperature at the boiling point
To determine one point of reduced pressure for a compound when the boiliig point and the critical point are known, a is determined, then G, then k; and these values are substituted in RPME. Values so calculated in the high and low pressure ranges are evaluated in Table I, along with values calculated with reference substance equations having only one or two terms and no empirical constants to be determined separately. Use of Equation I (Nonmdued)
The simple nonreduced Equation 1 repxwents well, as a linear algebraic expression, graphical plot, or nomogram, vapor prdata over wide ranges of high and low pressures and for usual liquids, both organic and inorganic. The two constants have been evaluated for over 500 usual compounds (9). Othmer and Zudkevitch (72) showed that latent heats cannot be calculated exactly using the slope of the straight line as mT = L/L', because of the errors brought in by the Clausius modification. (Usually an approximate value of latent heat is obtained, within 2 to 6% at any pressure below one or two atmospheres.) However, the thermdynamic Clapeyron equation itself may be arranged and multiplied by P / P = 1 for one liquid, and by P'/P' = 1 for a reference liquid always at the same temperature:
dP PAY d T -._e-
P
L
T
and
dP' - . - =P'AV' P' L'
dT T
d T / T is eliminated for both the liquid in question and the reference liquid : dP _ = -P'AV' . _ _ L dP' P PAY L' P VOL 60
NO. 1 J A N U A R Y 1 9 6 8
25
I
Table 1.
I
Results of Equations for Vapor Pressures Comparisons with experimental data High and low High pressure range ( 1 ah.-critical) pressure range No. over Av. of 10% Av. max. Av. % No. Av. of high I O 4 X S (of 70error error aver av. max. and low Input 24 comp. 24) for 24 of 24 10% %error %error
low pressure range (10-1500 rnrn.) Av.
xs
Equation
103 7 1 comp.
Input
No. over 10%
(of 71)
% Av. % error for error 71 for71 max.
Reduced Equations (Require T,, Po) 4
8
Least squaree
5.47
0
2.15
0.59
Least squarea
40.6
0
2.58
0.95
0
2.37
0.77
To, IO mm,b
8.54
0
1 .93
0.75
Tb,
1,
114.2
0
3.16
1 .66
0
2.55
1.21
Least
3.3
0
1 .3
0.40
Least
25.5
0
1.58
0.58
0
1.44
0.49
59.6
0
2.15
0.94
0
1.70
0.72
squarea
T,,
squarea
0
5.25
10 mm.b
1 .24
0.50
Temperature at 1 0 - m m . pressure. Temperature a t 30-atm. presLeast-square average of eight to 15 vapor pressures b y linear equation. Equation developed as MRA b y Miller (3), results calculated b y Thek and Stiel (161, utilizing compounds in high pressure range and showing sure. somewhat better results comparatively than Miller's calculations (3). e Equation developed as RPME with evaluation flgures b y Miller (30). Equotion developed and calculated b y Thek and Stiel (16). @
*
here
L m
T
=
L
P'AV' '
'
M
Also, AZ = P A V / R T and AZ' = P ' A V ' / R T , where A Z and AZ' are the differences of the vapor and liquid compressibilities. The gas constant R cancels, as does T = T ' , because values for both are always at the same temperature : P'A V' AZ' =PAV AZ or
mT =
-L. - AZ'
L'
AZ
This is constant, so integration gives log P 26
= m,,log
P'
+- C
INDUSTRIAL A N D ENGINEERING CHEMISTRY
(1 1
Thus, Equation 1 may be used algebraically, as has been done since 1940, to express vapor pressure data linearly, or by the straight line plot or nomogram. This is substantially rigorous, because it comes from the Clapeyron equation alone using compressibilities. Othmer and Zudkevitch (72) used mT = AZ'L/AZL' and presented a nomogram and values of constants for 500 compounds to calculate accurate values of latent heats throughout the temperature range [reprinted in 1966 (74)1. There are various applications of the PAV and the P'AV' terms representing the external work done in vaporizing a compound. Thus, m,, the slope, which must be determined from two or more vapor pressure points, may be considered as the ratio of the latent heats times the inverse ratio of the external work terms. This is the same as compressibilities (12) except for cancellation of temperature, which is the same for both.
1
d
Use of Equation 2 (Reduced)
peratures may be used for any substance, from the fixed gases to metals. While theoretically, as with any reduced equation, Equation 2 is preferable because it expresses temperatures as a fraction of the critical temperature instead of in absolute values, practically it has the disadvantage of requiring the knowledge and use of critical temperatures. These have not been determined for many substances; and the experimental errors of their determination at the top of the pressure and temperature scales enter into and affect the probable accuracy of any reduced equation. Also, reduced properties have some inconveniencies in use which may be minimized with computer programming or by the nomograms previously developed (5). Equation 2 and its slope m T R = L T ,' / L T , come from the Clausius modifications of the Clapeyron equation. I t may be derived from the rigorous Clapeyron equation and compressibilities using T R = T R ' and AZ = PAV/RT. Equation 2 remains the same but the slope becomes mTR = L Tc'AZf/L'TcAZ and, as with Equation 1 above, the correction of the slope term is the ratio of compressibility differences A Z f / A Z . I n Equation 2, the properties are considered at the same reduced temperatures which means the properties are considered under conditions having the same relation to the respective critical conditions of the two substances. Hence, while L, AZ, and L/AZ vary with temperature ( 7 4 , the ratio of the properties for two compounds at the same temperature (as in Equation 1) varies much less because of the reference substance principle; the ratio at the same reduced temperatures (as in Equation 2) varies only infinitesimally because of the nature of the compressibility function. Thus, the term AZ'/AZ may be entirely neglected because of this combination of the effects of the reference substance relation, in this application of the principle of corresponding states, for the two materials at the same reduced temperatures. Determination of Critical Pressures. Equation 2 also calculates the critical pressure if critical temperature and two points of vapor pressure--e.g., the boiling point and another (6)-are known. The two reduced temperatures of the vapor pressure points are found and the corresponding reduced pressures for water at the same reduced temperatures are taken from the steam tables, using the known critical properties for water. A cut-and-try solution with a nomogram was demonstrated in 1942 (5). However, if the two vapor pressures of the compound at P I and Pz, and the corresponding reduced pressures for water at the same reduced temperatures are PIRl and P f R 2 respectively, , the point slope form gives:
Equation 2, log PR = mTR log PR', has the advantage of a single term and a single constant, mTR = L T,'/L 'T,. The constant of integration disappears; and the critical temperature and the boiling point determine the entire range of vapor pressures and, through the constancy of mTR as confirmed by Reid and Sherwood (74) and others, of latent heats. This considers vapor pressures at the same reduced temperature as that of a single reference substance, water, which in any range of tem-
The critical pressure Po can also be calculated from the normal boiling point and the latent heat. The slope mTR is readily evaluated from L T c ' / L f T , , as all terms are known. Thus, in atmospheres, and using normal boiling temperature (6))
This same correction of the value of the slope for Equation 3 discussed below was derived in 1940 (4). I t was also shown differently for Equation 1 by Palermo and Balch (73). The PAV terms cannot be obtained readily because AV is not usually known; and P comes on both sides of the equation and thus is not explicit. However, this does not change the constancy of mT, which is determinable from vapor pressure data. This shows that the constants mT and C defining the equation or line should not be used as a relation of latent heats, except for low pressures-Le., below one or two atmospheres-and should be derived from two or more points of vapor pressure data. However, PAV is the molar external energy of vaporization of the compound. Also, P'AV' is the same for water and is tabulated as B.t.u. per pound (kilocalories per kilogram) in the steam tables against values of temperature or pressure throughout the entire liquid range. Thus, mT, the slope of the vapor pressure relation determined by only two points may be combined with L/L' if the latent heat of the compound is known (or determined from the same vapor pressure data and Equation 2) ; and the latent heat for water is taken from the steam tables. Immediately there may be evaluated from mT = LP'AV'/L'PAV the external work term of vaporization, a quantity which, by itself, may often be of interest. Also, because the ratio of the internal energies equals the ratio of the compressibility differences, (PIAVI/PAV = A Z f / A Z ) ,at the same temperature, compressibilities may be readily calculated for any compound if L is determined. I n Equation 1 the reference substance is always considered at the same temperature; if water is the reference substance, it may be used only in the temperature range throughout which water is a liquid, about 0' to 374' C., the critical temperature of water, where the pressure is 217.72 atm. Vapor pressures of water have been determined more precisely and tabulated more conveniently than for any other substance; thus, water is the most useful reference substance. However, if Equation 1 is used with, say, cryogenic gases, another reference substance must be used-e.g., nitrogen. Or, in correlating the vapor pressure of metals or of the decomposition of solids (such as COz from calcium carbonate) or the relative heats of reaction or equilibrium constants of chemical reactions (8), Equation 1 requires the use of mercury or some other metal which is liquid in the temperature range of interest.
VOL 60
NO. 1
JANUARY 1968
27
Equation
3
gives a valuable tool for the correlation
and prediction of both vapor pressures and of sa tura ted vapor volumes log P, = -mrB log P f B T h e e equations using the critical temperature, the normal boiling temperature, and either another point of vapor pressure or the latent heat give P. within the usual experimental emom of its determinetion. ComlclHon a)Ih. Same P m i u m
Vapor Pressurea and*Vapor Volumes. The first article of this series (4)discussed a t length the correlation with temperatures taken a t the same pccmma as well as pressurea at the same temperatures. Reasonably correct equations orpressed temperature as reciprocals or, preferably, as logarithms as being simpler for either graphical or algebraic use. The relation for reduced reciprocal temperatures at the same reduced prearures may also be written &om the equation of the first article in exactly the same way as the reduced Equation 2 came from Equation 1, without improving the raults (15). The finalequation derived in the 1940 article (4)used the rigorous Clapcyron equation without the Clausius assumptions. I t gave, with values always taken a t the same prraaUnSi.e., P = P' or P/P' = 1:
L'AV log T' LAV'
logT=-
log T
OT
- - .L' t
P'AV'
+C
log T'
+C
(3)
Le.,
log T = mplog T ' + C where
L' PAV mp3---L P'AV'
L'AV LAV'
Usually, it is more convenient to consider, beaides the ratio of mqhr latent heats, the ratio AV/AV' in Equation 3, wherein the ratio o f pneacurar (P/P' 1) has bem eliminated, rather than the PAV/P'AV' ratio o f external energies found in Equation 1. The ratio of external energies is simpler than the wrmponding expression in the similar form of Equation 1, a h derived from the unmodified Clapeyron equation, because the operation is always at the same pccmma in Equation 3. Thus, as shown Mow, Equation 3 gives a valuable tool for the correlation and prediction of bothvapor pccmma and of saturated vapor volumes; also of external energies ofevaporation (work term). Range of Temperatures for Equation 3. Equation 1, with values taken at the same temperatures as those of water as a reference, is limited to those temperaturea
-
28
INDUSTRIAL A N D ENGINEERING CHEMISTRY
a t which liquid and vaporous water can be in cauilibrium -approximately 0' to 375' C. By comparison, Equation 3, taJcen at the same pressurea aa those of water as a reference may be used at any pressure at which liquid and vaporous water can be in equilibrium. However, the upper limit of pressure of the vapor-liquid water equilibrium is at 217.7 am.-the critical pragurc -which is higher than that for any other common material; the vapor pressure of supercooled water is tabulated in handbooks down to 0.25 m m of Hg. Thus,water may be uscd as the reference substance in the nonreduced Equation 3 throughout substantially the entire v a p prrssure range of any material from the cryogenic gases to the m a t refractory metals for which data are available. For pure materials, Equation 2 has somewhat Similar advantaged. HWeve~, Equation 3 has an advantage over the r c d u d Equation 2 in that critical properties arc not required. In many cam, they are unknown, indetcrminate-eg., water in caustic soda solutions-diicult to d&nc-e.g., water in alcohol dutionb-or unreIated-e.g., c o s out of CaCOa. Use of Equation 3 for Vapor P r e r u r u . Although Equation 3 is exprarsed in the form of temperaturea inatead of pressures, it is as readily usable as is Equation 1, and simpler in we, algebraically or geometrically, than any other vapor pressure equation of comparable aocuracyand range. With the b o i point and another point of vapor pragurc being known, three step readily give the two constants ofEquation 3, asis also the casefor Equation 1. Thus, TA is at PA; also TB is a t PB, preferably some distance apart One may be the normal boiling point. To obtain constants in Equation 3: - O b t a i n from the vapor pruaure tables for water, values corresponding to: TA'at PA', which equals PA; and o f TB'at PB', which equals PB. -Evaluate the slope from the geometric relation: mp
-
1%
log
TA- 1% TB TA'
- log TB'
-Substitute mp in Equation 3 at either TAand
TA'
Donold F. O t b is Dirringuirhed Projessor of Ckanicol Engincaing at fhe Poly&~hkInstitute oj &&p and Erl-Sheen Yu was a Research Asmckte in the same deebnent. M i . Yu is p e s d j with the h u s Co. The a u t h s gratefully acknowIedge the support of this m k by a grant from the &so Research and Eiginming Co. AUTHORS
or at T , and T,' (always at the same pressure) to determine C, the second constant.
Two simple steps determine at a temperature, T , the vapor pressure, P, for a liquid for which m p and C of Equation 3 are known: -The given value of T is inserted as log T along with mp and C in Equation 3, which is then solvsd for log T'. Hence, T' is obtained. -The vapor pressure table for water gives P' a t this temperature, T'; but P' equals P, the vapor pressure of the liquid-Le., the value sought.
If, however, a vapor pressure, P, of a liquid is given, and the corresponding temperature, T, is to be found, there are two converse steps:
h
-P = P' for water; and the value of T' corresponding to P' is found in the vapor pressure tables for water. Log T' is found in log tables. -Log T', mp, and C are inserted in Equation 3 and log T is calculated. The desired value, T, is found as the antilog. Relation at the Same Reduced Pressure. The expression for the reduced temperature at the same reduced pressure may be written. It compares to Equation 3 as 2 does to 1: log TR =
mpR
log T'R
(4)
where
L' L
mPR=-'-'-
L,
T , AV T,' AV'
Equation 4 loses its constant of integration, as does Equation 2. The single constant represents the slope. Equation 4 also represents a sheaf of lines through the point represented by the critical, and requires additionally only the boiling point or one other vapor pressure point. Calculation of Critical Temperatures from Critical Pressures. Critical temperatures are more often available than are critical pressures. However, for fewer compounds, the critical pressure is known, but not the critical temperature. Equation 4 allows its calculation from the critical pressure, the boiling point, and one or more other points of vapor pressure in the same way as Equation 2 allows the calculation of the critical pressure from the critical temperature and vapor pressure data. Thus, T I and T2 are the temperatures at which the compound has vapor pressures P1 and P2 and the corresponding reduced temperatures for water at the same reduced pressures are T'Rl and T'R2. The point slope form and the equation for log T , are analogous to that above for log P,:
Corrections for Changes of Slope, the m Term
The substantial constancy of the slopes-i.e., the linearity and straightness of lines in the corresponding plots-is remarkably good for Equations 1, 2, 3, and 4. Each equation contains one constant representing the slope defined as a ratio of energies; Equations 1 and 3 contain a second constant, that of integration. Nevertheless, over the entire liquid range of some compounds, there is a slight variation of the slope which may be noticeable in precise work. In general these equations instead of representing the two straight lines correlated by other equations, may actually represent curves of very slight upward concavity. This is difficult to establish because the discrepancies are at the border of accuracy of vapor pressure data and their presentationusually after various attempts at correlation or smoothing. For those data which have been smoothed by two sets of constants in empirical equations, two straight lines are found, breaking slightly at the boiling point; these two lines obviously present the slight curvature better than any one straight line, particularly if it must be drawn through the anchor point-the normal boiling point, somewhat near the center of the logarithmic range. The value for the slope is m p = L'AV/LAV' in Equation 3. It may be calculated at different temperatures for those few compounds for which have been tabulated values of latent heat, or for which values of latent heat may be calculated from some other equation. Listed values are available from the thermodynamic tables of some liquids-particularly those used as refrigerants. The slope m p of Equation 3, calculated for those liquids where latent heats and gas and liquid volumes were available, was calculated and found to vary linearly with the logarithm of the absolute temperature. The equation of this relation is: log T - F log T = E m p F or m p =
+
E where m p is the actual value of L'AV/LAV' at the same pressure. Here E and F are empirical constants. This value of m p may be substituted into the differential equation from which Equation 3 was derived-Le., d log T = m p d log T'. This gives: d log T 1 = - d log T' log T - F E Because F is a constant: d
(log T log T
- F)
-F
=:
1 d log
E
Integrate and combine constants:
- F ) = A log T' + B In all cases tried F/log T < 0.2 and this is substantially: log (log T
log (log T ) = A log T'
Similarly, if the normal boiling temperature, latent heat, vapor volume, and critical pressure are known, log T , can be evaluated by the following:
T'
+B
(7 ) This equation, for which values are taken a t the same pressure, is a more exact representation of a linear vapor pressure function because it empirically corrects for slight changes in the slope m p in Equation 3, already very good. Compared with all of the equations preVOL. 6 0
NO. 1 J A N U A R Y 1 9 6 8
29
viously tested (3, 6) by the reference technique, or with a large number of empirical terms and constants, it gives an excellent correlation. Similarly, a reduced form may be written empirically because straight lines are obtained in calculating and plotting mpR against log TR. In the differential form, because the critical temperature T , is a constant: d (log T R ) = d (log T - log T,)= d log
T
Thus, there results when T and T‘ are taken at the same reduced pressures: log (log T )
=
A log T’
+B
This gives a close representation of vapor pressure data for a reduced equation and is much easier to use than other reduced equations because temperatures rather than reduced temperatures are used. This eliminates one half the nuisance of using reduced values. Change of Slope in Equations at the Same Temperature
Two other equations have been considered, analogous to Equations 7 and 8, and they obviously follow within the same general accuracy because they are merely another mathematical grid structure for presenting the same data and relations. Thus, considering the slight bowing of the lines of Equation 1 and the corresponding variation of the slope mT,the more accurate but empirical relation develops, always at the same temperature : log (log P)
=
A log P’
+B
(5)
Similarly, there results from Equation 2 in correcting for the slight variance of mTR values always taken at the same reduced temperature : log (log P) = A log P’
+B
(6)
Equation 5 is obviously a better representation of vapor pressure data than is Equation 1, but, like Equation 1, it has the disadvantage that, with the use of water as a reference substance, its range is that of liquid water (approximately 0” to 375” C.). Equation G is good for any temperature within the pressure range of liquid water-i.e., up to its high critical value-and, although it is a reduced equation, conventional values of temperature are used, because T , drops out, to simplify its use. Vapor Pressure Results
The method of analysis used was that of Miller (3), with his systems of data for 71 compounds in the low and 24 compounds in the high vapor pressure regions. The average per cent error for the calculated values for the 71 compounds (over 1000 data points), as compared to the literature values, was determined, as were the average per cent maximum error shown by individual compounds and S, the summation of the squares of the standard errors of estimate. The high and low pressure ranges were considered together, without mathematical justification but as a general indication, by adding the number of errors over 10% for the two ranges; the average maximum per cent error and the average per cent error, respectively, for the two ranges were averaged. 30
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
Comparative data are shown in Table I, in somewhat the same form as before (3, 6 ) and with the results of Thek and Stiel (16) on their equation and on Miller’s equations MRA (3) and RPME ( 3 u ) . The least square method for averaging many points of vapor pressure was first used. I t determines the best line and the constants representing slope and intercept for utilizing all available data. However, this does not seem to be a less fair representation than the usual one of taking vapor pressure or latent heat data from tabulated values, which, by themselves, are not experimental data, but are already correlated or smoothed from several or more experimental points. Also, two vapor pressure data points are used as being more generally applicable and valid than one vapor pressure point and latent heat calculated therefrom or determined by much different and more complicated techniques. Reference substance equations may be evaluated without the latent heat term, which is implicit in most other equations. Thus, the following comparisons are not to be regarded as rigorous, because the input data are different; indeed, these equations always require also the well known and often tabulated properties of water. Use of Nonreduced Equation 3. When the slope is calculated by the least square method in the low pressure range, the square of the standard error S is less than the lowest values reported by Miller in either nonreduced or reduced equations. The average maximum per cent error shown by Equation 3 is comparable with the best of the values given by Miller. The largest error appears usually at the lowest point of vapor pressure, 10 mm. Hg. Equation 3 showed no compounds in greater error than lo%, while the equations in Miller’s tabulations showed from 6 to 66 compounds with errors over 10%. Miller’s semireduced equations were comparable to Equation 3. I n the high pressure range, the square of the standard error and the average maximum per cent error are much lower than any values among those given by Miller’s evaluation of nonreduced or semireduced equations, but somewhat less good than the best values of the reduced equations. When the slope is calculated by two points of vapor pressure in the low pressure range-i.e., boiling point and 10 mm. Hg-the value of S, the square of the standard error, and the average maximum per cent error are also much better than values given by Miller’s table for either the reduced or the nonreduced equation; again, no values are more than 10% in error. I n the high pressure range, if the two points which are used are the normal boiling point and the critical pressure, or if they are the normal boiling point and 20 atm., there is no important difference. Values of S and the average maximum per cent error are much lower than the values given by equations in Miller’s analysis in the nonreduced or semireduced equations-no compounds over 10% error us. 5 to 16 over 10% error. Equation 3 appears as an accurate as well as an easily used equation, and it may be used with water as single reference substance for any material from permanent
gases to high boiling metals if vapor pressures at two points sufficiently far apart are known. Use of Nonreduced Equation 7. When data were calculated by the least-square method in the low pressure range, the value of S and the average maximum per cent error are much lower than any analysis of Miller for either reduced or nonreduced equations. The maximum per cent error usually appears at the low vapor pressure of 10 mm. Hg. I n the high pressure range, the value of S and the average maximum per cent error are much lower than those given in Miller’s table for nonreduced or semireduced equations, and comparable with the best of the reduced equations. N o errors were over 10% in either range. When the slope was calculated by two points in the low pressure range, the two points were the normal boiling point and the lowest vapor pressure (usually 10 mm. Hg). The values of S and the average maximum per cent error are less than those given by Miller’s table for any equations, and again there are none with errors as high as 10%. (Propyne gives poorest results with a maximum 6.6% error at 400 mm. Hg.) Equation 7 appears to be the best nonreduced equation for calculating the vapor pressures in the low pressure range. I n the high pressure range, if the two points are the normal boiling point and the critical pressure, the value of S and the average maximum per cent error are very much less than any value given by Miller for nonreduced or semireduced equations, but are somewhat higher than those obtained by the best of the reduced equations. If T b and 20 atm. are used as the two input points, errors are slightly less. If critical properties are unknown, Equation 7 appears best for calculating vapor pressures in the high pressure range as well as in the low pressure range, because this nonreduced equation is more truly linear than any other. Table I shows 30 atm. instead of T , used as the second point of vapor pressure as input with Tb. Use of Reduced Equation 4. Using input of critical properties and temperatures at vapor pressures of 10 rnm. and 760 mm., or a least-square fit for 10 to 15 points for evaluating the low pressure range, maximum errors were comparable to the best analyses of Miller or of Thek and Stiel; and no compounds had errors greater than 10%. Errors were largest at lowest pressures; sulfur was worst, with -5,13Y0 error at 10 mm. Hg. The use of least-square input with Equation 4 shows much better correlation in the high pressure range than any of the reduced equations analyzed by Miller. When Equation 4 is analyzed with two-point input data, it is comparable to the best. With neither input does Equation 4 show up so well as the equation of Thek and Stiel. Hydrogen iodide shows the greatest error5.7970 at 11.4 atm. with two-point input and -4.93% a t 1.O atm. for least-square input. Use of Reduced Equation 8. When the Equation 8 is used with two points of vapor pressure in the low pressure range-Le., the normal boiling point and 10
mm. Hg-the average error for all points, the average maximum error, and the value of S are each much less than for equations evaluated by either the analysis of Miller or by Thek and Stiel. The equations analyzed by Miller had from nine to 62 out of 71 compounds showing errors over IO%, while Equation 8 has none. When input of least squares was used in Equation 8, even better results were obtained. I n the high pressure range, again with two points of vapor pressure as data input, no values were in error more than 10%. The values of S and the average maximum per cent error are not so low as those given by the analysis of Thek and Stiel or the best of the many analyzed by Miller. However, Equation 8 might be classed as a semireduced equation, because only one critical condition, P,,is used. ( T , is known more often.) I t shows no compound having an error more than 10% against six to 16 for Miller’s semireduced equations, which have a value of S varying from 17 to 67 times as great. Conclusions. The error analysis as tabulated is not exactly comparable to those of others (3, 76) because somewhat different compounds were evaluated in the high pressure range by the several investigators. Also, for reasons indicated above, two points of vapor pressure data were used in the two-point form of straight line as input. This is preferable to the less desirable values of boiling point and latent heat almost necessarily used instead in the involved algebraic expressions. Also, the least-square evaluation of many points was used as input to show its applicability when many data points are available. Table I shows that excellent results may be obtained using the boiling point and one other point of vapor pressures as input for these equations which are simple by comparison with those which are extremely involved in their use of somewhat different input data. For these reference substance equations, a high vapor pressure point with Tbfor the high range or Tband the critical constants suffice, while Tb and a low pressure point are the input for the low pressure range. For simplicity, nonreduced Equations 7 or 3 are recommended for vapor pressures. Only two points of vapor pressure data-e.g., one besides boiling pointare necessary for either-e.g., Tb and another point of vapor pressure which is some distance away. These simple equations, on average, give results at least as good as any others and may be used in any temperature range, always with water as the reference substance. Also, they may be applied immediately to adsorption of gases by solids (70) or absorption of gases by liquids (77). Equation 3 may also be used for correlating or predicting latent heats or vapor volumes as shown below. Saturated Molar Vapor Volumes
Equation 3 was shown in 1940 to relate vapor volumes (4) but was not tested :
log T = m p log T’ f C VOL
60
NO
1
JANUARY 1968
(3) 31
where mp = L'AV/LAV'
Error Type
For Equalion 3Evaluation of mp by Input Data
Any two vapor pressure points determine the line or the constants of this exact equation and allow calculation of m p , its slope, always operating at the same pressures. More than two points of vapor pressure can be used if available by a least-square averaging. Also, L may be known at one temperature, or it may be determined readily from Equation 2 (using the same vapor pressure data to determine the slope and, hence, L ) or from another suitable relation. Accurate values of the latent heat L' and of the specific volumes of both liquid and vaporous water, the reference substance, are listed a t every pressure and temperature in the steam tables. Thus, AV', the difference of the molar volumes of vapor and liquid, is readily determined. At low pressures, the liquid volume is insignificantly small and AV approximates V closely, Also, the density of the liquid a t one temperature is usually known, and it may be determined at a desired temperature by the Fishtine equation ( 7 ) or by the reference substance method of Othnier, Josefowitz, and Schmutzler (7). The value of the vapor volume, V, thus comes directly from m p of Equation 3. From the same data (or from one vapor pressure and latent heat) and critical temperatures and pressures, vapor volumes may be determined using Equation 4 at the same values of reduced pressures.
1
10 t o 15 vapor pressure points
where mpR =
L~IAV LRAV' ~
If L or V or A V are known at some one point of temperature and vapor pressure, the determination of m p or m P R is comparatively simple and the values of V throughout the entire range come more quickly. The vapor volume, V, was calculated by both Equations 3 and 4 for 27 compounds at each of seven to 10 temperature points at nearly equal intervals of temperature throughout the whole pressure range of interest. The compounds chosen were those for which thermodynamic properties, such as vapor pressure, latent heat, specific vapor, and liquid volumes, have been tabulated at these temperatures in the usual handbooks. The calculations were made with several different inputs-three each for each equation, as noted below. The total number of calculated points for the 27 compounds is 245. Differences were determined between the tabulated and calculated values and reported as errors, numbered from 1 to 6, for the respective input data. The vapor volumes, V , as calculated by each set of input data at each temperature were compared with listed values to obtain the percentage error. The average per cent error for all the points is summarized in Table 11. As a comparison, the specific vapor volume a t the tested temperature and presssure was also calculated by the ideal gas law. 32
INDUSTRIAL A N D ENGINEERING CHEMISTRY
11
(line slope by least squares) F;n;p;ressures
(two-
1
Input Type
A
and AV at boiling point (point slope form)
11
For Equation 4Evaluation of mpR by Input Data (Including Critical Constants)
10 t o 15 vapor pressure points !line slope by least squares) One vapor pressure (line goes through critical 1,l on logarithmic plot)
I
and A V a t boiling point !point shape f o r m )
I D
1:
Discussion of Results. Table I1 shows that the lowest average percentage error of all 27 compounds using the several possible sets of input data and Equation 4, is 1.?4y0, where the slope mPR is obtained from a number of vapor pressure points at various temperatures, giving a line evaluated by the least-square method, input D. The highest average error for the calculations of Equation 3 with different input for these 27 compounds is 2.72y0, where the slope mp is evaluated by the latent heat and a listed volume difference a t a temperature near the normal boiling point, input C. The highest average error for a single one of these 27 compounds is 5.47% for ethylene when the slope m p was evaluated from two vapor pressure points-i.e., input B. Mercury has lowest average error, 0.450j0, using many points by least squares (A), two points (B), or one point and latent heat (C) to evaluate the slope mp (the properties of mercury have usually been difficult to correlate). The average error obtained by evaluating the slope by the least-square method when 10 or more vapor pressure points are used is, of course, less than when the smaller amount of data input of the other methods is used. This is true in evaluating either Equation 3 or Equation 4. However, vapor pressures at many different temperatures are sometimes not available, and fewer data will have to be used to evaluate the slope. If critical properties are known, Equation 4 is recommended. The minimum input data necessary for evaluating the slope mPR, in addition to the critical temperature and the critical pressure, are one vapor pressure at any temperature-e.g., the normal boiling point-or the latent heat and the difference of volumes at any temperature. If the critical properties are not known but data for two vapor pressures or the latent heat and the volume difference at one temperature are available, Equation 3 is used. Critical constants are not known accurately for mercury; thus, Table I1 shows error 5 indicated as the
average of the other errors for mercury. As another example, the ideal gas law WGab0 uaed to calculate V and AV, and its percentage error when the pressure is lower than 1 am. is sometimes smaller than that calculated using the p e n t method. At higher pressurea, the error becomes large, often considerably over 100%. These calculations also showed that even the average percentage error of this group of 27 compounds when type E input data are used is slightly higher than that when Type F input data are used. However, in the high pressure range, type E input data showed less
percentage error than did type F. This is &ply be. cause, in using Type E input data, the slope m,, is evaluated by connecting the normal boiling point and the critical point on the log T p us. Log Tn' plot. Thus, near the critical point, the evaluated slope must be clw to the actual slope, and thus a better result is obtained. The slope of the line using type F input is evaluated from the latent heat and the volume ditTerence at a point at or near the normal boiling point. Therehe, type E may be expected to give a h i g k accuracy for the values of mpn near the normal b o i i g point and less accuracy far
Table II. CompdriSOn between ccllculafed and Li8l.d Saturated Vapor Volumes Error 1, mp evaluated by N vapor prwures, type A Ermr 2, mp evaluated by hvo vapor prarrures, type B Error 3, mp evaluated by 1 and AV near N.B.P., type C Error 4, mpR woluated by N vqpor pressures, type D Error 5, mpR evaluated by 1 vapor pressure, type E Error 6 mpp evaluated bv 1 and AV near N.B.P.. tme F
.
A". *nor 5,
Methane Ethane Propane
Butane
lsobutane Ethene 1,3-Butadiene Benzene Acetylene Methanol
1
344.2 550.1
664.0 767.4 732.7 509.8 765.6
1
45.8 48.3 42.0 37.5 36.0 50.5 42.7
1011.2 555.7 926.0 730.1 774.0 822.0 749.4
47.7 61.7 78.6 111.5 73.6 55.5
821 .o 930.0
53.3 54.1
043.0
43.2 40.6 48.7 59.2
co
694.0 665.0 877.2 239.3
COa
548.0
73.0
SO8
785.0 277.7 226.8 271.3 3282.0
77.7
NHa CHiNHz CIHINH~ CHIC1 CaHsCI CaHaClz CFCla CFtCla CHFpCl HCOOCHI
0 2
Na Ar Hg
65.8
34.5
49.7 33.5 48.0
-
200.0 A".
1
2.01 3.40 1.98 1.81 0.90 4.94 4.00
I
2.00 3.69 1.86 1.82 0.89 5.49 4.88
1
3.02 3.89 1.75 3.53 1.29 4.91 3.96
I
1.67 2.14 1.62 1.59 0.97 2.96 1.80
2.63 2.45 0.76 1.10 2.25 2.37 2.71
2.63 2.38 0.81 1.11 2.82 3.34 3.43
3.23 4.49 1.21 1.67 1.69 2.87 3.17
1.96 1.73 1.42 0.73 1.69 1.15 1.70
1.61 1.05 3.05 3.33 2.98 4.85 3.48
1.90 1.90 3.27 3.02 3.11 4.82 3.51
1.59 1.22 3.28 3.63 3.11 4.82 4.46
0.54 2.01 1.44 1.70 1.27 3.69 3.17
0.65 3.17 3.04 1.15 2.17 0.45
0.99 3.13 3.06 1.03 2.18 0.47
2.38 3.24 4.09 1.76 4.06 0.45
1.27 2.14 3.09 1.12 1.95 0.54
2.38
2.58
2.92
15 4
enor 6.
% 1.79 2.15
96
2.42 1.35 1 .a 0.97 3.49 2.59 1.78 1.69 1.43 1.23 3.98
2.39 1.10 2.55 1.01 2.93 1.79
2.05 3.56 2.12
4.90 - 673.00 0.270- 716.00 5.650- 57.5.00 30.000- 300.00
40.000- 400.00 0.018- 742,s 0,130- 220.50
2.21
1.29 1.38 1 .ei
-
1.423- 705.60 18.61 906.25 0.571-1155.00 0.882-1639.05 1.322- 110.61 0.740- 47.07 1.953- 263.90
1.17 3.55 2.26 2.05 2.18 3.20
-
0.53 2.66 1.58 1.92 1.32 3.63 3.37
-
2.200- 57.36 0.693- 17.15 0.739- 274.82 0.154- 502.54 0.278- 359.20 1.500- 46.15 2 . m - 507.58
1.26 4.05 3.15 1. IO 1.94 0.50
2.68 2.20 3.04 0.99 2.99 0.52
75.1oO-1069.40 0.294- 987.W 14.7730.59 14.7492.01 9.974- 705.69 0.402- 180.00
235
202
5.45
5.09
0.67
- -
-
V O L 6 0 NO. 1 J A N U A R Y 1 9 6 8
35
.
I.
.
-
The same.als0 cqqd when input data type B and type c Were used, ?hire the slope mp of type B input data is evaluated using hvo points of v a p o r p ~ u r e s and that oftype C input data is evaluated using only the latent heat and the volume difference at a poidt near or a t the normal boiliig point. Compariwn wit@Mehods of Fishtine a i d Lyderien. Fishtine (7) compared values of sIjecific vapor volumes calculated by his method and by that of Lydersen cf ul. (2). Fishtine u+ critical propaties and specific volumes of the liquid at Some basc temperature. To compare his calculated Specific vapor volume with the present work, the reduced Equation 4 was used. The same base temperature as used by Fishtine was taken for evaluating the values of mpR for input data type E. vapor volume were cilculated also at the same points oftemperature, and a total of 19 values was deterdned for five compounds. Average error was 1.2% by Equation 4, 2.0 and 3.6% by methods of L y d k n et al. and Fishtine, ,respectively. .. The comparison is listed in Table 111. B&t results were obtained with input type E where the slope mpB is evaluated oilly by T,,P., and vapor pressure at d e r ence state temperature. The specific vapor volumes are then calculated by insating the known latent heats
Table 111.
tine (7) defined one reference state temperature for each compound]. This gives yet another indication that the best input data are usually two or more points of vapor pressure. Again it is noted that, with input data of type E, better results are obtained at high pressure because all these tested points are at rather high pressure. The comparison shows that the present work can estimate saturated specific vapor volume accurately when the liquid density and the latent heat are known at that temperature and when a suitable method of calculating mpR is used. Equation 3 may be used immediately to express and predict the pressure and volume of a gas in a relation of adsorption on a solid (70) or of dissolution in a liquid (77). Vapor Volume Calculated from Correction of mp or mpB. A previous section shows that the slope of Equation 3, mp = L’AV/LAV‘, varies linearly with log T; also, the slope for Equation 4, mPR = LR‘AV/LRAV’,varies linearly with log TR:
log T = Emp
+F
and log TR = GmpR
+H
Comparison betwm Calculated and Listed Satumted Vapor Volumes
To calculate the vopor volume at one temperoture, T, the input data required are the vapor pressure, P, the latent heat, t, and the liquid volume, V bp.,. Canpound
NHr
Ref. dde, 7, F.
-
20
r,
P, p.ri. f.
80 120
-23
- 1M)
80 130 -50 0 40 M)
a0 70
,230 320 400 420
440 0
120 200 250
280 300
obwhne
153.0 286.4 98.8 194.9
93.8 219.7
385.0 494.2 630.7 69,O 251 .O 622.0 764.0 930.0 116.3 347.0 604.0 816.0 987.0
“.a*OpOI
re 0.74 0.79 -
0.78 0.85 0.75 0.M 0.91 0.95 0.98 0.75
0.84 0.93 0.95 0.97 0.75 0.85 0.91 0.95 0.98
-
-1.
cy. ftJh
1.9550 1.0470
0.4250
0.M80 1.3550
0.5754 0.3062 0.2164 0.1411 2.9700 0.8020 0 . M 0.2120 0.1510 0.7460 0.24M) 0.1290
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
-0.6 -2.4 - I .9 -1.1 -0.1 -0.5 1.1
1.7 3.3 5.4
0.0848 0.0599 Av. error of 19
34
-0.3 -1.1 0.2 1.3 0.2 0.5
In
-
1.2
I I
0.8 -1.1 -1.1 0.0 -4 6 -2.0
I
1
6.2 0.3 4.4 6.4 5.2 7.3
0.9 0.8 -2.8 -3.0 -1.8 -0.7 -0.7 -3.5 -3.4 -1.8
-3.6 -6.2 -4.0 -4.7 -2.0 0.7 2.2 2.3 2.5 3.0
2.0
3.6
C0"tto"h
mp
C
m-Xylene
1.2972 1.2742 1.2607 1.2469
o-Xylene
1.2501
D-Xvlene
1.2515
-0.8662 -0.7641 -0.6977 -0.6554 -0.6591 -0.6691
Benzene Toluene Ethyl benzene
1.0901 1.0506 1,0227
B
0.1929
-0.0978 -0.0782 -0.0663 -0.0594 -0.0587 -0.0617
0.1880
0.1853 0.1830 0.1830 0.1833
1.0088
1.0185 1.0124
Recent Vapor Pressure Determinations
During publication of this article, vapor pressures were carefully determined at the National Physical Laboratories. Data between the boiling and the critical temperatures were reported by D. Ambrose, B. E. Broderick, and R. Townsend [J.Chcm. Sac. (A), 633, 19671, for six aromatic compounds-benzene, toluene, ethyl benzene, o-xylene, m-xylene, and pxylene. Ten vapor pressure points were taken for each compound from the extensive data at temperature intervals as nearly equal as possible, and used to calculate the constants of the linear equations 3, 4,7, and 8 by the method of least squares. Vapor pressures then were calculated at these points by the equations using these constants. The percentage error at each point was calculated, as was the average percentage error. Water was used as the reference substance. The average percentage errors so determined for all compounds was, respectively, for Equation 8, 0.35%; 4, 0.42%; 7, 0.51%; and 3, 0.55%. The maximum error for Equation 7 was at the critical point, and for the other three equations was at the lowest pressure. Lower errors may result if one of the hydrocarbons, rather than water, is used as the reference substance. Equation 4 has only one constant; each of the others has two constants. T o express these data, Ambrose et al. used empirical, nonlinear equations, one with 4 constants, and one with 7 constants. Their error analysis is not immediately comparable to those used here; but they indicate that their equations give close fits. T h e constants, as calculated, for the Equations 3, 4,7, and 8,are listed above. Logarithms to the base 10 are used, the reference substance is water, absolute temperatures are in degrees Rankin, and pressures are in any desired value consistent with that used to express the vapor pressure of water.
EQUATION 8
A
m m
The constants E, F, G, and H can be determined when two vapor pressures and their corresponding latent heats and vapor volumes are known. mp or mPR is then calculated at the desired temperature. The improved value of mp or mPR gives a mnre exact value of vapor volume, but latent heat and liquid volume must be known at one point. I n most cases, the straight lines of these equations are more or less parallel. If this is assumed, the values of L and AV at the boiling point need be known, and a mean value of E or G may be assumed to calculate F o r H.
9
EQUATION 7
EQUATION. 4
EQUATION 3
A
B
0.1621 0.1550
0.1503 0.1481
0 . I491
0.I486
-0.0193 0.0058 0.0229 0,0297 0,0278 0.0279
-
NOMENCLATURE
A, B , E, F,G,H general constants C = constant of integration L = molar latent heat of vaporization m = slope of line, ratio of an energy function of the substance to that of the mference substance P = vapor pressure, absolute value
R
= gasconstant
S T V
= square
v
= molar liquid wlume
Z
= compressibility =
A
= change in property between its value in the liquid and its
of the standard ermr
= temperature, absolute value = molar vapor volume
PV/TR
value in the vapor state Superscript.
'
= value of reference substance under same condition
Subscripts
onc'set of conditions; PA at TA means vapor pressure at temperature value A
A
=
B
= another set of conditions;
c
Pa at Ts means vapor pressure at temperaturn value B = normal boiling; Ts means normal boiling point (1 atm.) = T o r P at critical
P
= at same pressure; m p is determined for temperature values
PR
= at same reduced pressure; mpR is determined for tempera-
b
taken at same pressure ~
ture valucs taken at same reduced pressure
R
= reduced
T o r P; TIT, is reduced temperature and PIP.
is reduced pressure
T
= at same temperature; mT is determined for prersum values
TR
= at same reduced temperature; mTR is determined for pressure values taken at same reduced temperature
taken at same temperature
REFERENCES S.H., END. EN^. C ~ e u F. u a o ~ w e 2~ (2). r ~ 149 ~ ~ (1963). (2) Lydclim A. L. Grccnkom, R. R., Hougcn. 0.A,, Univ. of Wlwonin, Enp. (1) Fiahtinc,
~ ~~ t ~~ . ,t ' i, ~.octobcr ~ ~ t . 195s. (3) Miller, D. G . . Iwo. Eno. Cxew. 56 (31,46 (1964). (3n) Miller, D. G., J.Phy. C h . 69, 3209 (1965). (4) Othmcr, D. F., INO. END. Cnnr. 32, 841 (1940). ( 5 ) Ibid., $4, 1072 (1942). (6) Othrncr, D.F., Huang, H.N., Ibid.,37 (lo), 43 (1965). (7) Othmer, D. F., loadowllz, S., Sehmufdrr, A. F., Ibid.> 40, 883 (1948). (8) Othmcr, D. F., Lulcy, A. H., Ibid.,58,408 (1946). ( 9 ) Othmer, D. F.,Msuer, P. W., Molinari, C. 1.. Kowalski, R. C.,Did., 49, 125
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(10) Othrncr,D.F.,Sawysr, F.G., Ibid., S5, 1269 (1943). (11) Othmu, D. F., White, R. E., Ibid., 34,952 (1942). (12) Orhrncr,D.F.,Zudkevirch,D., Ibid.,5t,791 (1959). (13) Palvmo J. A. Batch C. W. Sympoiurn: Thermodynamic8 of Fluids 59th Nntio&l M e e h g , AiChE, Oahaa, February 1966. (14) Reid, R., Shewood T. "Pro v t i e s of Gase and Liquids,'. McGraw-Hill. New York, l i t Ed., 19581 2dd ed., p966. (15) %line, L., INO.Eno. Cxeu. 38, 402 (1916). (16) Thck, R. E., Sfid, L. I., A.I.Ch.E. J . 12, 599 (1966).
VOL 60
NO. 1
JANUARY
1968
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