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A new enthalpy-of-vaporization equation
RIJlDPHAS[ EQUILIBRIA ELSEVIER
Fluid Phase Equilibria 137 (1997) 53-62
A new enthalpy-of-vaporization equation H.W. Xiang * Department of Thermal Engineering, Tsinghua University, Beijing 100084, China Received 22 November 1996; accepted 15 April 1997
Abstract
A new, simple, accurate three-parameter enthalpy-of-vaporization equation is constructed consistent with the renormalization-group theory of critical phenomena. The proposed equation is valid over the entire range from the triple point to the critical point for various pure compounds and reproduces data within the experimental accuracy. Since the correct near-critical behavior is built in, the new equation is often better than existing enthalpy-of-vaporization equations in that it can be used to not only accurately correlate but also effectively extrapolate from the usual range in which data are available both to the critical point and to the triple point. Fitted parameters are given for 30 pure substances. The result is compared with experimental data and previous equations. © 1997 Elsevier Science B.V. Keywords: Critical parameters; Critical power laws; Enthalpy of vaporization; Model; Pure; Vapor-liquid equilibrium
1. I n t r o d u c t i o n
The enthalpy of vaporization A H, sometimes referred to as the latent heat of vaporization, is the difference between the enthalpy of the saturated vapor and that of the saturated liquid at the same temperature. The enthalpies of vaporization for pure substances have not been effectively correlated and predicted yet [1,2]. Correlation of the temperature dependence of the enthalpy of vaporization from the triple-point temperature Tt to the critical temperature Tc particularly in the limiting regions, near Tt and Tc requires that the specific properties of the enthalpies of vaporization be taken into consideration. For associated substances, the properties near the normal boiling point must be considered as well. In the vicinity of the critical point, the singular behavior may be described by the renormalization-group theory of critical phenomena [3,4]. Although nearly two dozen relationships, including those good equations [5-9] recommended by Svoboda and Basarova [1], have been put forward to describe the temperature dependence of enthalpy of vaporization, all of those cannot
* Corresponding author. 0378-3812/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S 0 3 7 8 - 3 8 1 2 ( 9 7 ) 0 0 1 0 0 - 3
54
H.W. Xiang / Fluid Phase Equilibria 137 (1997) 53-62
Table 1 The system-dependent parameters ( b 0, b 1, and of vaporization of 30 pure compounds Compound
T range
Reference(s)
na
bo
bI
b2
SD a (kJ/mol)
AD ~
Simple compounds: Argon 150.68 Xenon 289.74 Nitrogen 126.20 Oxygen 154.59
T~ -To Tt-Tc Tt - Tc Tt-T~
[15] [16] [ 17] [18]
18 22 23 27
4.814206 5.077615 5.101708 4.745252
1.671718 1.435663 1.613651 2.032891
1.439639 1.342613 1.372702 1.638452
0.0041 0.017 0.014 0.026
0.084 0.16 0.39 0.54
0.47 0.59 2.4 3.6
Alkanes: Methane Propane Pentane Hexane Heptane Octane Decane
190.564 369.83 469.7 507.6 540.2 568.7 617.7
T t - Tc 277-360 Tt - 4 2 7 Tt - 4 9 8 Tt-372 Tt-444 Tt-444
[19] [20] [20,21] [20-23] [20,21] [20,21] [20,21]
26 16 30 34 14 29 18
4.804140 6.655498 6.658643 7.282132 9.379409 9.220436 9.831543
1.888509 0.7896260 1.287706 1.058655 0.3106415 0.4527416 0.4176178
1.626068 0.7618771 1.010219 0.8684716 0.5780824 0.5713246 0.4675401
0.024 0.007 0.029 0.14 0.037 0.091 0.19
0.38 0.05 0.087 0.42 0.074 0.18 0.27
2,3 -0,07 -0,32 -5.9 0.21 -0.69 1,3
Alkenes: Ethylene Butene
282.35 419.57
Tt - Tc 202-377
[24] [20]
37 9
5.535985 6.772635
1.440953 0.8924645
1.220321 0.8450156
0.022 0.062
0.24 0.21
3.3 0.53
Cycloalkanes: Cyclo553.46 hexane
292-422
[20,22]
27
5.848123
1.576477
0.9705132
0.075
0.19
0.58
293-469 Tt - Tc
[20] [25]
34 25
6.777218 7.141740
0.9055878 1.009179
0.7459176 0.8981869
0.034 0.083
0.090 0.51
0.34 -3.9
Tt- L
[26]
30
7.438501
0.8676944
0.7639405
0.041
0.18
- 1.6
T t - T~
[27]
36
7.322273
1.109993
0.8607591
0.018
0.14
-2.2
Tt - Tc
[ 10]
53
6.847669
1.738293
1.391250
0.040
0.17
1.3
508.06
3 0 0 - Tc
[20,28]
13
6.852180
1.232239
0.9309510
0.057
0.25
-0.81
512.6 516.3 536.8 508.3
2 9 8 - Tc 298-Tc 298-499 298-477
[20,22,29] [20,22,30] [20,22] [20,22]
36 26 29 27
8.581301 8.062954 5.460086 6.178857
1.475778 1.984336 4.330661 3.449017
1.360956 1.230064 1.346481 1.092369
0.11 0.35 0.21 0.18
0.41 0.94 0.45 0.44
4.7 5.6 2.2 1.6
Aromatics: Benzene Toluene
T~ (ITS-90) (K)
b 2) and calculated deviations of the new equation, Eq. (4), for the enthalpy
562.05 593.91
Halogenated alkanes: 2,2-di456.83 chloro1,1,1-trifluoroethane 1,1,1,2tetrafluoroethane
374.18
Inorganic oxide: Water 647.096 Ketone: Acetone Alcohols: Methanol Ethanol
1-Propanol 2-Propanol
MD a
H.W. Xiang / Fluid Phase Equilibria 137 (1997) 53-62
55
Table 1 (continued) Compound
Tc (ITS-90) (K)
T range
Reference(s) n ~ b o
bI
1-Butanol Isobutanol t-butanol 1-Pentanol 3-Methyl-l-butanol 2-Methyl-2-butanol
563.0 547.8 506.2 588.1 577.2 543.7
298-500 298-498 298-440 298-499 303-496 298-444
[20] [20] [20] [20] [20] [20]
4.413808 1.025306 3.444654 0.8334125 3.541207 0.5059685 2.121914 0.5218778 3.468238 0.7831131 2.480965 0.1338460
17 17 15 14 9 8
5.036601 5.669265 5.219783 6.826003 5.589978 5.330664
b2
SD a (kJ/mol)
AD a MD a
0.19 0.31 0.15 0.38 0.26 0.25
0.36 0.60 0.29 0.66 0.47 0.46
-0.83 -1.5 -1.1 1.5 1.3 -0.89
~SD = [~]'(A Hcald- AHexpt)2/(n- m)] 1/2 and AD = 100(]]'1'[1- AHcald/AHexpt[/n) are the standard deviation and the average deviation, and MD = 100(1 - A Hcald/A Hexpt) is the maximum percentage deviation, where the subscripts expt and
cald represent the observed data and the correspondingcalculated values from the new Eq. (4). All the deviations are for all the data points, n is the number of data points and m the number of system-dependentparameters of the new equation.
describe the temperature dependence of enthalpy of vaporization for water [1] within the experimental uncertainties [ 10]. Only the relation due to Torquato and Stell [3] has a semitheoretical basis, while all the rest are empirical. However, the Torquato-Stell equation has as many as six system-dependent parameters and shows unfavorable behavior when extrapolated beyond the range of the measured values used when deriving the system-dependent parameters. In summary, some problems have been considered as enumerated below. (1) Does there exist a correlation over the entire temperature region of existence of the vapor-liquid equilibrium from the triple point to the critical point within experimental error? (2) Does an equation remain valid up to the critical point and conform to the exponents from the renormalization-group theory of critical phenomena? (3) Can it be accurately extrapolated from the usual range in which data are available to the triple point and to the critical point beyond the range of the measured values used when deriving the system-dependent parameters? It is the purpose of this paper to make an effort to address the problems mentioned above.
2.
Development
The expression for the enthalpy of vaporization for a fluid incorporates singular behavior near the critical point and regular behavior away from the critical point, as in Torquato and Stell [3]. RTc = a o r ~ + a l r ¢ + a a 2 r 1 - ~ + ~ +
ai r i
(1)
i=1
As Torquato and Stell [3] have stated, the first three terms account for the behavior in the vicinity of the critical point and the finite sum may describe A H away from the critical point [3]. Here r = 1--T~ with Tr = T / T ~ is the reduced temperature. The critical exponents c~ and /3, which describe the asymptotic behavior of the order parameter along the coexistence boundary, are taken to be 0.11 and 0.325 [3,4]; The exponent A = 0.51 is Wegner's first gap exponent that accounts for the nonanalytic behavior of the first correction to the asymptotic power-law behavior [3,4]. R = 8.31451 J
H.W. Xiang / Fluid Phase Equilibria 137 (1997) 53-62
56 Table 2 Comparison
between the calculated values of the new equation and previous equations and the experimental
data with the
associated tolerances for water [10] T (ITS-90) (K)
Tolerance a
PD b
273.160
+ 0.064
- 0.072
278.149
0.064
- 0.057
283.148
0.065
288.146
PD c
PD d
PD e
0.72
- 0.57
- 0.16
0.62
- 0.42
- 0.11
- 0.045
0.51
- 0.30
- 0.061
0.066
- 0.039
0.41
- 0.19
- 0.028
293.145
0.067
- 0.032
0.32
- 0.10
0.002
298.144
0.067
- 0.028
0.23
- 0.027
0.025
303.142
0.068
- 0.026
0.14
0.032
0.041
308.141
0.069
- 0.022
0.07
0.081
0.055
313.140
0.069
- 0.017
0.00
0.12
0.066
318.138
0.070
-0.014
-0.07
0.15
0.073
323.137
0.070
-0.012
-0.13
0.17
0.074
328.136
0.071
- 0.008
- 0.18
0.19
0.075
333.134
0.072
0.001
- 0.23
0.20
0.076
338.133
0.072
0.006
- 0.27
0.21
0.073
343.132
0.073
0.011
- 0.31
0.21
0.068
348.130
0.074
0.017
- 0.34
0.20
0.061
353.129
0.074
0.022
- 0.36
0.20
0.052
358.128
0.075
0.033
- 0.38
0.19
0.047
363.127
0.076
0.039
- 0.39
0.18
0.037
368.126
0.077
0.047
- 0.40
0.17
0.027
373.124
0.077
0.051
- 0.41
0.15
0.013
383.122
0.080
0.065
- 0.41
0.12
- 0.009
393.120
0.084
0.079
- 0.39
0.091
- 0.031
398.119
0.084
0.082
- 0.38
0.073
- 0.045
403.118
0.086
0.088
- 0.36
0.059
- 0.055
413.116
0.089
0.093
- 0.32
0.030
- 0.077
423.115
0.092
0.094
- 0.27
0.003
- 0.096
433.114
0.10
0.093
- 0.20
- 0.017
443.112
0.11
0.085
-0.13
-0.035
-0.12
448.112
0.11
0.082
-0.10
-0.038
-0.12
453.111
0.12
0.077
- 0.06
- 0.042
- 0.12
463.111
0.12
0.056
0.02
- 0.051
- 0.12
473.110
0.13
0.035
0.10
-0.049
-0.11
4 8 3 . ! 10
0.14
0.008
0.18
- 0.044
- 0.092
- 0.11
493.110
0.15
- 0.019
0.27
- 0.031
- 0.062
503.110
0.17
- 0.057
0.35
- 0.020
- 0.030
513.110
0.18
- 0.089
0.43
0.000
0.016
523.110
0.19
-0.13
0.50
0.017
0.066
533.110
0.22
- 0.17
0.56
0.024
0.11
543.111
0.24
-0.20
0.61
0.032
0.17
553.111
0.26
-0.24
0.64
0.024
0.22
563.111
0.28
- 0.26
0.65
0.006
0.26
573.111
0.31
- 0.27
0.63
- 0.040
0.29
583.111
0.35
- 0.25
0.59
- 0.11
0.29
593.110
0.40
- 0.21
0.49
- 0.23
0.25
603.110
0.45
- 0.11
0.33
- 0.40
0.16
H.W. Xiang/ FluidPhase Equilibria 137 (1997) 53-62
57
Table 2 (continued) T (ITS-90) (K)
Tolerance a
PD b
PD c
PD d
PD e
613.110 623.109 633.108 643.107 644.107 647.096
0.51 0.69 1.1 2.4 2.6 0
- 0.06 0.32 0.77 1.1 1.3 0
0.07 - 0.36 - 1.1 - 3.3 -3.7 0
-
- 0.027 - 0.36 -0.86 - 1.5 - 1.3 0
0.65 1.0 1.5 1.8 1.5 0
aExcept for the entries in the first line, the sign (+_) of the percentage tolerance is omitted. bpD = 100(1 -- A Hc~d / A Hexpt) is the percentage deviation, where the subscripts expt and cald represent and the corresponding calculated values from the new Eq. (4). cPD = 100(1 - A Hc~d / A Hexpt) is the percentage deviation, where the subscripts expt and calcl represent and the corresponding calculated values from the first three-terms of Eq. (1) by Torquato and Stell [3]. aPD = 100(1 - A Hcald/AHexpt) is the percentage deviation, where the subscripts expt and cald represent and the corresponding calculated values from the three-parameter Eq. (5) by Somayajulu [5]. epD = 100(1 - A Hcald/A Hexpt) is the percentage deviation, where the subscripts expt and cald represent and the corresponding calculated values from the four-parameter Eq. (6) Somayajulu [5].
the observed data the observed data the observed data the observed data
mo1-1 K - 1 is the gas c o n s t a n t [11]. a i are s y s t e m - d e p e n d e n t p a r a m e t e r s . T o s i m p l i f y the a b o v e e x p r e s s i o n , Eq. (1) can be e x p r e s s e d as the f o l l o w i n g series, AH
RTc - - a o ( r t~ + a l r t ~ + a ) ( 1 + a2 T l - a + . . . )
(2)
T h e series (1 + az'r 1 - a q- . . . ) o f Eq. (2) a p p r o a c h e s the limit (1 - a z ' r l - a ) -1 if [a27-l-a] g 1. It will be f o u n d b e l o w that this c o n v e r g e n c e condition is satisfied. N e g l e c t i n g the effect o f a and o m i t t i n g the n e g a t i v e sign o f a 2, Eq. (2) gives, AH (,~+bl r~+a) -=b0 RT~ ( 1 + b2r)
(3)
H o w e v e r , Eq. (3) c a n n o t r e p r o d u c e the e n t h a l p y - o f - v a p o r i z a t i o n data within the e x p e r i m e n t a l uncertainties. T h e v a r i a b l e t = T c / T - 1 a p p e a r s to be m o r e a p p r o p r i a t e for e x t e n d i n g the t h e o r y to include n o n a s y m p t o t i c critical b e h a v i o r [4], the l e a d i n g - o r d e r t e r m r t3 is t h e r e f o r e r e p l a c e d b y t t~ to i m p r o v e the a c c u r a c y in p r o d u c i n g e x p e r i m e n t a l data a w a y f r o m the critical point. C o n s e q u e n t l y , the final f o r m is,
An - -
RT¢
( t ~ + b , r ~+a) =b 0
(1 + b z r )
(4)
E x p a n s i o n o f Eq. (4) yields, AH
R T c = b o ( t ~ + blr ~+A + b2r ' + ~ + . . . ) w h i c h can be c o m p a r a b l e to Eq. ( l ) w h e n the t e m p e r a t u r e a p p r o a c h e s the critical point. T h e s e three t e r m s in Eq. (4) a c c o u n t for the singular b e h a v i o r in the vicinity o f the critical point and c o n f o r m to
58
14.W. Xiang / Fluid Phase Equilibria 137 (1997) 53-62
the critical exponents from the renormalization-group theory of critical phenomena [3,4]. This form of the enthalpy-of-vaporization equation also characterizes the regular behavior away from the critical point. Eq. (4) contains only three system-dependent parameters (b 0, b l, and b 2) that are determined from the experimental data. The leading-order amplitude b o is agreement with the theoretical value which can be seen below. The new equation possesses a simple form to describe the temperature dependence of the enthalpy of vaporization along the entire vapor-liquid phase coexistence curve. The new equation is better than the first three terms of Eq. (1) in reproducing the experimental data for water, as shown in table below along with other correlations for comparison.
3. Results In testing the performance of the new equation, Eq. (4), the three system-dependent parameters for each substance are determined by fitting experimental data. Reliable data for thirty substances covering a wide temperature range are used. All data have been converted, whenever necessary, to the ITS-90 scale and temperatures used throughout this paper are on this scale [12]. The critical temperatures of substances that do not have data over the entire range are taken from Ambrose and Tsonopoulos [13] and Gude and Teja [14]. The three system-dependent parameters b o, b 1, and b 2 for each of these substances determined by a nonweighted least-squares fit are given in Table 1 together with the critical temperatures used in the fit. The standard deviations, the average deviations, and the maximum deviations are also presented in Table 1. Within the experimental uncertainties, the new enthalpy-of-vaporization equation is accurate for pure compounds over the entire range from the triple point to the critical point. For example, the deviations for water with the associated tolerances, argon, and l,l,l,2-tetrafluoroethane (R134a) are given in Tables 2-4. It can be seen that, for water, for which highly accurate experimental data are available, the deviations of 0.07% at the triple point, 0.05% at the normal boiling temperature, and 1% near the critical point, are almost within the tolerances given in the 1985 IAPS [10]. For water, the leading-order amplitude b o = 6.85 is in good agreement with 6.89 (/3 = 1 / 3 ) obtained by Torquato and Stell [3]. All existing equations [5-9] do not have accuracies within the data uncertainties, as can
Table 3 Comparison between the calculated values from the new Eq. (4) and the recommended values of argon [15] T (ITS-90) (K)
PD a
T (ITS-90) (K)
PD a
83.81 84.01 88.01 92.01 96.01 lO0.O1 104.01 108.01 112.01
-0.066 -0.062 -0.010 0.027 0.053 0.072 0.078 0.070 0.048
116.01 120.01 124.01 128.01 132.01 136.01 140.01 144.01 148.01
0.011 -0.033 -0.080 -0.115 -0.126 - 0.097 - 0.021 0.077 0.47
aPD = 100(1 - A Hc~ld/A Hexpt) is the percentage deviation, where the subscripts expt and cald represent the observed data and the corresponding calculated values.
59
H.W. Xiang / Fluid Phase Equilibria 137 (1997) 53-62
Table 4 Comparison between the calculated values from the new Eq. (4) and the recommended values of 1,1,1,2-tetrafluoroethane (R134a) [27] T (ITS-90) (K)
PD a
T (ITS-90) (K)
PD a
169.85 176.00 182.00 188.00 194.00 200.00 206.00 212.00 218.00 224.00 230.00 236.00 242.00 246.78 252.00 258.00 264.00 270.00
0.180 0.099 0.039 - 0.010 -0.045 -0.073 - 0.089 -0.093 - 0.093 -0.086 - 0.076 -0.063 - 0.046 - 0.038 -0.014 0.000 0.019 0.034
276.00 282.00 288.00 294.00 300.00 306.00 312.00 318.00 324.00 330.00 336.00 342.00 348.00 354.00 360.00 366.00 372.00 374.00
0.046 0,060 0.072 0.079 0.087 0.085 0.086 0.080 0.075 0.061 0.039 0.015 - 0.028 -0.086 - 0.18 -0.30 -0.45 - 2.2
aPD = 100(1 - A Hcald/AHexpt) is the percentage deviation, where the subscripts expt and cald represent the observed data and the corresponding calculated values. be seen f r o m Ref. [1 ]. C o m p a r i s o n s of accuracies with empirical S o m a y a j u l u three- and f o u r - p a r a m e ter equations for water are also presented in Table 2. It appears that the new equation has the best description for water. The m a x i m u m deviation occurs near the critical point where the predicted enthalpy of vaporization differs f r o m the data b y several parts in 100. In light of the experimental difficulties one confronts in this region, this deviation is m o s t likely within the experimental error. It can be seen that the new equation can reproduce the experimental data of these substances within the experimental uncertainties. Consequently, the equation presented here is capable of representing the enthalpy of vaporization for nonpolar, polar, associated, and h y d r o b o n d i n g c o m p o u n d s . Both theoretically and practically, extrapolation to Tt and Tc is always expected. Since there are only e x p e r i m e n t a l data m e a s u r e d in the relatively easily accessible temperature interval a w a y f r o m both Tt and T~ available, special attention should be given to the prediction around Tt and To. S v o b o d a and S m o l o v a [2] evaluated the previous enthalpy of vaporization equations and r e c o m m e n d e d an empirical three-parameter equation b y S o m a y a j u l u [5] as one of m o s t suited for extrapolation towards Tt and T~ for water. The empirical S o m a y a j u l u three- and four-parameter equations are, AH
RL
3 1~ _COT8 + C l t + C 2 T 8
()3
AH __~" ~ + d I RT~ = do ~'t
( )19 ()27
+ d 2 - - z ~- -k- d 3 --'r ~~-t ~-t
(5)
(6)
H. W. Xiang / Fluid Phase Equilibria 137 (1997) 53-62
60
Table 5 Extrapolation of the new equation, Eq. (4), and the Somayajulu three-parameter equation, Eq. (5), upon different temperature intervals for water [10] Temperature interval Tt (273.16)-T b 293.15-373.12 313.14-373.12 333.13-373.12 348.13-373.12 293.15-313.14 293.15-348.13 Tb (373.12)- Tc 373.12-603.11 373.12-533.11 373.12-453.11 453.11-603.11 503.11-603.11 Overall
Somayajulu equation
New equation
SD a (kJ/mol)
AD ~
MD a
SD" (kJ/mol)
AD a
MD a
0.15 0.14 0.13 0.12 0.13 0.16 0.15 0.11 0.10 0.12 0.18 O.18 0.58 O.18
0.61 0.55 0.50 0.47 0.51 0.64 0.59 0.21 0.23 0.45 0.30 0.37 1.0 0.49
-4.7 -4.4 -4.1 - 3.8 -4.2 - 4.6 - 4.5 1.4 - 1.5 - 3.7 1.6 1.2 2.2 3.2
0.06 0.14 0.14 0.10 0.16 0.20 0.15 0.09 0.07 0.14 0.18 O.18 0.32 O.15
0.19 0.44 0.45 0.22 0.72 0.70 0.46 0.18 0.27 0.59 0.77 0.33 0.55 0.45
1.0 - 1.9 -2.0 - 0.92 6.4 - 3.5 - 2.0 - 0.49 2.0 5.5 7.1 - 0.90 - 1.5 2.7
aSD : [ ~ ( A Hcal d - - A Oexpt)2/(H - - m)] 1/2 is the standard deviation, AD = 100(Y~'[1- A O c a l d / A Hexpt[/n) is the average deviation, MD = 100(1-AOcald/AOexpt ) is the maximum percentage deviation, where the subscripts expt and cald represent the observed data and the corresponding calculated values from each equation. All the deviations are for all the data points, n is the number of data points and m the number of system-dependent parameters of each equation. w h e r e ~'t = 1 - T t / T ~. E x t r a p o l a t i o n o f the S o m a y a j u l u equation, Eq. (5), and the n e w equation, Eq. (4), to Tt and Tc for water using different t e m p e r a t u r e intervals is g i v e n in Table 5. T h e n e w e q u a t i o n m o r e a c c u r a t e l y predicts the e n t h a l p y o f v a p o r i z a t i o n for water in m o s t temperature intervals.
4. Conclusions A new, simple, accurate t h r e e - p a r a m e t e r e n t h a l p y - o f - v a p o r i z a t i o n equation, c o v e r i n g the entire r e g i o n f r o m the triple point to the critical point, is c o n s t r u c t e d to c o n f o r m to the r e n o r m a l i z a t i o n - g r o u p t h e o r y o f critical p h e n o m e n a . This correlation fits the available data within e x p e r i m e n t a l errors better than existing three- and f o u r - p a r a m e t e r correlations b e c a u s e its functional f o r m incorporates the correct near-critical b a h a v i o r and the e x p o n e n t s and the s y s t e m - d e p e n d e n t parameters reflect the p h y s i c a l properties o f the substances. T h e n e w e q u a t i o n m o r e accurately correlates the experimental data and m o r e e f f e c t i v e l y extrapolates to the triple point and to the critical point for pure c o m p o u n d s . T h e e n t h a l p y - o f - v a p o r i z a t i o n e q u a t i o n o b t a i n e d here m a y be applied to fluids w h i c h are in the s a m e universality class as the substances cited in this paper.
5. List of symbols ai
AD bi
s y s t e m - d e p e n d e n t p a r a m e t e r s o f the Eqs. (1) and (2) a v e r a g e deviation ( - 100(~]']1 - A H c a l o / A Hexpt[/n)) s y s t e m - d e p e n d e n t p a r a m e t e r s o f the n e w Eq. (4)
H. W. Xiang / Fluid Phase Equilibria 13 7 (1997) 53-62
Ci
di AH m
MD r/
PD R SD t T
Tc T~= T I T c
61
system-dependent parameters of the Eq. (5) system-dependent parameters of the Eq. (6) enthalpy of vaporization n u m b e r o f system-dependent parameters m a x i m u m percentage deviation ( = 100(1 - A Hcala/A Hexpt) data points percentage deviation ( = 100(1 - A Hcalo/A nexpt) universal gas constant ( = 8.31451 J / ( m o l K)) standard deviation ( = [E~'(A H c a l d - - A Hexpt)2/( n -- m)] 1/2) TiT1 temperature critical temperature reduced temperature triple-point temperature
G r e e k letter a = 0.11 critical exponent /3 = 0.325 critical exponent A = 0.51 W e g n e r ' s first gap exponent r
1 -
"rt
1 - TIT t
T/T~
Acknowledgements The author would like to thank Prof. V. Svoboda and editor-in-chief Prof. H. Renon for their constructive comments and suggestions on this manuscript.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
V. Svoboda, P. Basarova, Fluid Phase Equil. 93 (1994) 167-175. V. Svoboda, H. Smolova, Fluid Phase Equil. 97 (1994) 1-11. S. Torquato, G. Stell, J. Phys. Chem. 85 (1981) 3029-3030. J.V. Sengers, in: E. Kiran, J.M.H. Levelt Sengers (Eds.), Supercritical Fluids: Fundamentals for Application, Kluwer, Dordrecht, 1994, pp. 231-271. G.R. Somayajulu, Int. J. Thermophys. 9 (1988) 567-575. R. Goodwin, W. Haynes, Natl. Bur. Stand., Monograph No.170, U.S. Government Printing Office, Washington DC, 1982. M. Radosz, A. Lydersen, Chem.-Ing.-Tech. 52 (1980) 756-757. E. Aerebrot, The Norwegian Institute of Technology, Trondheim, 1980. D.J. Graue, V. Berry, B.H. Sage, Hydrocarbon Process 45 (1966) 191-194. H. Sato, M. Uematsu, K. Watanabe, A. Saul, W. Wagner, J. Phys. Chem. Ref. Data 17 (1988) 1439-1540. E.R. Cohen, B.N. Taylor, J. Phys. Chem. Ref. Data 17 (1988) 1795-1803. R.L. Rusby, J. Chem. Thermodyn. 23 (1991) 1153-1161.
62 [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
H.W. Xiang / Fluid Phase Equilibria 137 (1997) 53-62
D. Ambrose, C. Tsonopoulos, J. Chem. Eng. Data 40 (1995) 531-546. M. Gude, A.S. Teja, J. Chem. Eng. Data 40 (1995) 1025-1036. R.B. Stewart, R.T. Jacobsen, J. Phys. Chem. Ref. Data 18 (1989) 639-798. O. Sifner, J. Klomfar, J. Phys. Chem. Ref. Data 23 (1994) 63-152. R.T. Jacobsen, R.B. Stewart, M. Jahangiri, J. Phys. Chem. Ref. Data 15 (1986) 735-909. R.B. Stewart, R.T. Jacobsen, W. Wagner, J. Phys. Chem. Ref. Data 20 (1991) 917-1021. U. Setzmann, W. Wagner, J. Phys. Chem. Ref. Data 20 (1991) 1061-1155. V. Majer, V. Svoboda, Enthalpies of Vaporization of Organic Compounds, A Critical Review and Data Compilation, Blackwell, Oxford, 1985. K. Ruzicka, V. Majer, J. Phys. Chem. Ref. Data 23 (1988) 1-39. F. Vesely, L. Svab, R. Provaznik, V. Svoboda, J. Chem. Thermodyn. 20 (1988) 981-983. C.J. Wormald, T.K. Yerlett, J. Chem. Thermodyn. 17 (1985) 1171-1186. M. Jahangiri, R.T. Jacobsen, R.B. Stewart, R.D. McCarty, J. Phys. Chem. Ref. Data 15 (1986) 593-734. R.D. Goodwin, J. Phys. Chem. Ref. Data 18 (1989) 1565-1636. B.A. Younglove, M.O. McLinden, J. Phys. Chem. Ref. Data 23 (1994) 731-779. R. Tillner-Roth, H.D. Baehr, J. Phys. Chem. Ref. Data 23 (1994) 657-729. T.K. Yerlett, C.J. Wormald, J. Chem. Thermodyn. 18 (1986) 371-379. T.K. Yerlett, C.J. Wormald, J. Chem. Thermodyn. 18 (1986) 719-726. M.D. Vine, C.J. Wormald, J. Chem. Thermodyn. 21 (1989) 1151-1157.
Fluid Phase Equilibria 137 (1997) 53-62
A new enthalpy-of-vaporization equation H.W. Xiang * Department of Thermal Engineering, Tsinghua University, Beijing 100084, China Received 22 November 1996; accepted 15 April 1997
Abstract
A new, simple, accurate three-parameter enthalpy-of-vaporization equation is constructed consistent with the renormalization-group theory of critical phenomena. The proposed equation is valid over the entire range from the triple point to the critical point for various pure compounds and reproduces data within the experimental accuracy. Since the correct near-critical behavior is built in, the new equation is often better than existing enthalpy-of-vaporization equations in that it can be used to not only accurately correlate but also effectively extrapolate from the usual range in which data are available both to the critical point and to the triple point. Fitted parameters are given for 30 pure substances. The result is compared with experimental data and previous equations. © 1997 Elsevier Science B.V. Keywords: Critical parameters; Critical power laws; Enthalpy of vaporization; Model; Pure; Vapor-liquid equilibrium
1. I n t r o d u c t i o n
The enthalpy of vaporization A H, sometimes referred to as the latent heat of vaporization, is the difference between the enthalpy of the saturated vapor and that of the saturated liquid at the same temperature. The enthalpies of vaporization for pure substances have not been effectively correlated and predicted yet [1,2]. Correlation of the temperature dependence of the enthalpy of vaporization from the triple-point temperature Tt to the critical temperature Tc particularly in the limiting regions, near Tt and Tc requires that the specific properties of the enthalpies of vaporization be taken into consideration. For associated substances, the properties near the normal boiling point must be considered as well. In the vicinity of the critical point, the singular behavior may be described by the renormalization-group theory of critical phenomena [3,4]. Although nearly two dozen relationships, including those good equations [5-9] recommended by Svoboda and Basarova [1], have been put forward to describe the temperature dependence of enthalpy of vaporization, all of those cannot
* Corresponding author. 0378-3812/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S 0 3 7 8 - 3 8 1 2 ( 9 7 ) 0 0 1 0 0 - 3
54
H.W. Xiang / Fluid Phase Equilibria 137 (1997) 53-62
Table 1 The system-dependent parameters ( b 0, b 1, and of vaporization of 30 pure compounds Compound
T range
Reference(s)
na
bo
bI
b2
SD a (kJ/mol)
AD ~
Simple compounds: Argon 150.68 Xenon 289.74 Nitrogen 126.20 Oxygen 154.59
T~ -To Tt-Tc Tt - Tc Tt-T~
[15] [16] [ 17] [18]
18 22 23 27
4.814206 5.077615 5.101708 4.745252
1.671718 1.435663 1.613651 2.032891
1.439639 1.342613 1.372702 1.638452
0.0041 0.017 0.014 0.026
0.084 0.16 0.39 0.54
0.47 0.59 2.4 3.6
Alkanes: Methane Propane Pentane Hexane Heptane Octane Decane
190.564 369.83 469.7 507.6 540.2 568.7 617.7
T t - Tc 277-360 Tt - 4 2 7 Tt - 4 9 8 Tt-372 Tt-444 Tt-444
[19] [20] [20,21] [20-23] [20,21] [20,21] [20,21]
26 16 30 34 14 29 18
4.804140 6.655498 6.658643 7.282132 9.379409 9.220436 9.831543
1.888509 0.7896260 1.287706 1.058655 0.3106415 0.4527416 0.4176178
1.626068 0.7618771 1.010219 0.8684716 0.5780824 0.5713246 0.4675401
0.024 0.007 0.029 0.14 0.037 0.091 0.19
0.38 0.05 0.087 0.42 0.074 0.18 0.27
2,3 -0,07 -0,32 -5.9 0.21 -0.69 1,3
Alkenes: Ethylene Butene
282.35 419.57
Tt - Tc 202-377
[24] [20]
37 9
5.535985 6.772635
1.440953 0.8924645
1.220321 0.8450156
0.022 0.062
0.24 0.21
3.3 0.53
Cycloalkanes: Cyclo553.46 hexane
292-422
[20,22]
27
5.848123
1.576477
0.9705132
0.075
0.19
0.58
293-469 Tt - Tc
[20] [25]
34 25
6.777218 7.141740
0.9055878 1.009179
0.7459176 0.8981869
0.034 0.083
0.090 0.51
0.34 -3.9
Tt- L
[26]
30
7.438501
0.8676944
0.7639405
0.041
0.18
- 1.6
T t - T~
[27]
36
7.322273
1.109993
0.8607591
0.018
0.14
-2.2
Tt - Tc
[ 10]
53
6.847669
1.738293
1.391250
0.040
0.17
1.3
508.06
3 0 0 - Tc
[20,28]
13
6.852180
1.232239
0.9309510
0.057
0.25
-0.81
512.6 516.3 536.8 508.3
2 9 8 - Tc 298-Tc 298-499 298-477
[20,22,29] [20,22,30] [20,22] [20,22]
36 26 29 27
8.581301 8.062954 5.460086 6.178857
1.475778 1.984336 4.330661 3.449017
1.360956 1.230064 1.346481 1.092369
0.11 0.35 0.21 0.18
0.41 0.94 0.45 0.44
4.7 5.6 2.2 1.6
Aromatics: Benzene Toluene
T~ (ITS-90) (K)
b 2) and calculated deviations of the new equation, Eq. (4), for the enthalpy
562.05 593.91
Halogenated alkanes: 2,2-di456.83 chloro1,1,1-trifluoroethane 1,1,1,2tetrafluoroethane
374.18
Inorganic oxide: Water 647.096 Ketone: Acetone Alcohols: Methanol Ethanol
1-Propanol 2-Propanol
MD a
H.W. Xiang / Fluid Phase Equilibria 137 (1997) 53-62
55
Table 1 (continued) Compound
Tc (ITS-90) (K)
T range
Reference(s) n ~ b o
bI
1-Butanol Isobutanol t-butanol 1-Pentanol 3-Methyl-l-butanol 2-Methyl-2-butanol
563.0 547.8 506.2 588.1 577.2 543.7
298-500 298-498 298-440 298-499 303-496 298-444
[20] [20] [20] [20] [20] [20]
4.413808 1.025306 3.444654 0.8334125 3.541207 0.5059685 2.121914 0.5218778 3.468238 0.7831131 2.480965 0.1338460
17 17 15 14 9 8
5.036601 5.669265 5.219783 6.826003 5.589978 5.330664
b2
SD a (kJ/mol)
AD a MD a
0.19 0.31 0.15 0.38 0.26 0.25
0.36 0.60 0.29 0.66 0.47 0.46
-0.83 -1.5 -1.1 1.5 1.3 -0.89
~SD = [~]'(A Hcald- AHexpt)2/(n- m)] 1/2 and AD = 100(]]'1'[1- AHcald/AHexpt[/n) are the standard deviation and the average deviation, and MD = 100(1 - A Hcald/A Hexpt) is the maximum percentage deviation, where the subscripts expt and
cald represent the observed data and the correspondingcalculated values from the new Eq. (4). All the deviations are for all the data points, n is the number of data points and m the number of system-dependentparameters of the new equation.
describe the temperature dependence of enthalpy of vaporization for water [1] within the experimental uncertainties [ 10]. Only the relation due to Torquato and Stell [3] has a semitheoretical basis, while all the rest are empirical. However, the Torquato-Stell equation has as many as six system-dependent parameters and shows unfavorable behavior when extrapolated beyond the range of the measured values used when deriving the system-dependent parameters. In summary, some problems have been considered as enumerated below. (1) Does there exist a correlation over the entire temperature region of existence of the vapor-liquid equilibrium from the triple point to the critical point within experimental error? (2) Does an equation remain valid up to the critical point and conform to the exponents from the renormalization-group theory of critical phenomena? (3) Can it be accurately extrapolated from the usual range in which data are available to the triple point and to the critical point beyond the range of the measured values used when deriving the system-dependent parameters? It is the purpose of this paper to make an effort to address the problems mentioned above.
2.
Development
The expression for the enthalpy of vaporization for a fluid incorporates singular behavior near the critical point and regular behavior away from the critical point, as in Torquato and Stell [3]. RTc = a o r ~ + a l r ¢ + a a 2 r 1 - ~ + ~ +
ai r i
(1)
i=1
As Torquato and Stell [3] have stated, the first three terms account for the behavior in the vicinity of the critical point and the finite sum may describe A H away from the critical point [3]. Here r = 1--T~ with Tr = T / T ~ is the reduced temperature. The critical exponents c~ and /3, which describe the asymptotic behavior of the order parameter along the coexistence boundary, are taken to be 0.11 and 0.325 [3,4]; The exponent A = 0.51 is Wegner's first gap exponent that accounts for the nonanalytic behavior of the first correction to the asymptotic power-law behavior [3,4]. R = 8.31451 J
H.W. Xiang / Fluid Phase Equilibria 137 (1997) 53-62
56 Table 2 Comparison
between the calculated values of the new equation and previous equations and the experimental
data with the
associated tolerances for water [10] T (ITS-90) (K)
Tolerance a
PD b
273.160
+ 0.064
- 0.072
278.149
0.064
- 0.057
283.148
0.065
288.146
PD c
PD d
PD e
0.72
- 0.57
- 0.16
0.62
- 0.42
- 0.11
- 0.045
0.51
- 0.30
- 0.061
0.066
- 0.039
0.41
- 0.19
- 0.028
293.145
0.067
- 0.032
0.32
- 0.10
0.002
298.144
0.067
- 0.028
0.23
- 0.027
0.025
303.142
0.068
- 0.026
0.14
0.032
0.041
308.141
0.069
- 0.022
0.07
0.081
0.055
313.140
0.069
- 0.017
0.00
0.12
0.066
318.138
0.070
-0.014
-0.07
0.15
0.073
323.137
0.070
-0.012
-0.13
0.17
0.074
328.136
0.071
- 0.008
- 0.18
0.19
0.075
333.134
0.072
0.001
- 0.23
0.20
0.076
338.133
0.072
0.006
- 0.27
0.21
0.073
343.132
0.073
0.011
- 0.31
0.21
0.068
348.130
0.074
0.017
- 0.34
0.20
0.061
353.129
0.074
0.022
- 0.36
0.20
0.052
358.128
0.075
0.033
- 0.38
0.19
0.047
363.127
0.076
0.039
- 0.39
0.18
0.037
368.126
0.077
0.047
- 0.40
0.17
0.027
373.124
0.077
0.051
- 0.41
0.15
0.013
383.122
0.080
0.065
- 0.41
0.12
- 0.009
393.120
0.084
0.079
- 0.39
0.091
- 0.031
398.119
0.084
0.082
- 0.38
0.073
- 0.045
403.118
0.086
0.088
- 0.36
0.059
- 0.055
413.116
0.089
0.093
- 0.32
0.030
- 0.077
423.115
0.092
0.094
- 0.27
0.003
- 0.096
433.114
0.10
0.093
- 0.20
- 0.017
443.112
0.11
0.085
-0.13
-0.035
-0.12
448.112
0.11
0.082
-0.10
-0.038
-0.12
453.111
0.12
0.077
- 0.06
- 0.042
- 0.12
463.111
0.12
0.056
0.02
- 0.051
- 0.12
473.110
0.13
0.035
0.10
-0.049
-0.11
4 8 3 . ! 10
0.14
0.008
0.18
- 0.044
- 0.092
- 0.11
493.110
0.15
- 0.019
0.27
- 0.031
- 0.062
503.110
0.17
- 0.057
0.35
- 0.020
- 0.030
513.110
0.18
- 0.089
0.43
0.000
0.016
523.110
0.19
-0.13
0.50
0.017
0.066
533.110
0.22
- 0.17
0.56
0.024
0.11
543.111
0.24
-0.20
0.61
0.032
0.17
553.111
0.26
-0.24
0.64
0.024
0.22
563.111
0.28
- 0.26
0.65
0.006
0.26
573.111
0.31
- 0.27
0.63
- 0.040
0.29
583.111
0.35
- 0.25
0.59
- 0.11
0.29
593.110
0.40
- 0.21
0.49
- 0.23
0.25
603.110
0.45
- 0.11
0.33
- 0.40
0.16
H.W. Xiang/ FluidPhase Equilibria 137 (1997) 53-62
57
Table 2 (continued) T (ITS-90) (K)
Tolerance a
PD b
PD c
PD d
PD e
613.110 623.109 633.108 643.107 644.107 647.096
0.51 0.69 1.1 2.4 2.6 0
- 0.06 0.32 0.77 1.1 1.3 0
0.07 - 0.36 - 1.1 - 3.3 -3.7 0
-
- 0.027 - 0.36 -0.86 - 1.5 - 1.3 0
0.65 1.0 1.5 1.8 1.5 0
aExcept for the entries in the first line, the sign (+_) of the percentage tolerance is omitted. bpD = 100(1 -- A Hc~d / A Hexpt) is the percentage deviation, where the subscripts expt and cald represent and the corresponding calculated values from the new Eq. (4). cPD = 100(1 - A Hc~d / A Hexpt) is the percentage deviation, where the subscripts expt and calcl represent and the corresponding calculated values from the first three-terms of Eq. (1) by Torquato and Stell [3]. aPD = 100(1 - A Hcald/AHexpt) is the percentage deviation, where the subscripts expt and cald represent and the corresponding calculated values from the three-parameter Eq. (5) by Somayajulu [5]. epD = 100(1 - A Hcald/A Hexpt) is the percentage deviation, where the subscripts expt and cald represent and the corresponding calculated values from the four-parameter Eq. (6) Somayajulu [5].
the observed data the observed data the observed data the observed data
mo1-1 K - 1 is the gas c o n s t a n t [11]. a i are s y s t e m - d e p e n d e n t p a r a m e t e r s . T o s i m p l i f y the a b o v e e x p r e s s i o n , Eq. (1) can be e x p r e s s e d as the f o l l o w i n g series, AH
RTc - - a o ( r t~ + a l r t ~ + a ) ( 1 + a2 T l - a + . . . )
(2)
T h e series (1 + az'r 1 - a q- . . . ) o f Eq. (2) a p p r o a c h e s the limit (1 - a z ' r l - a ) -1 if [a27-l-a] g 1. It will be f o u n d b e l o w that this c o n v e r g e n c e condition is satisfied. N e g l e c t i n g the effect o f a and o m i t t i n g the n e g a t i v e sign o f a 2, Eq. (2) gives, AH (,~+bl r~+a) -=b0 RT~ ( 1 + b2r)
(3)
H o w e v e r , Eq. (3) c a n n o t r e p r o d u c e the e n t h a l p y - o f - v a p o r i z a t i o n data within the e x p e r i m e n t a l uncertainties. T h e v a r i a b l e t = T c / T - 1 a p p e a r s to be m o r e a p p r o p r i a t e for e x t e n d i n g the t h e o r y to include n o n a s y m p t o t i c critical b e h a v i o r [4], the l e a d i n g - o r d e r t e r m r t3 is t h e r e f o r e r e p l a c e d b y t t~ to i m p r o v e the a c c u r a c y in p r o d u c i n g e x p e r i m e n t a l data a w a y f r o m the critical point. C o n s e q u e n t l y , the final f o r m is,
An - -
RT¢
( t ~ + b , r ~+a) =b 0
(1 + b z r )
(4)
E x p a n s i o n o f Eq. (4) yields, AH
R T c = b o ( t ~ + blr ~+A + b2r ' + ~ + . . . ) w h i c h can be c o m p a r a b l e to Eq. ( l ) w h e n the t e m p e r a t u r e a p p r o a c h e s the critical point. T h e s e three t e r m s in Eq. (4) a c c o u n t for the singular b e h a v i o r in the vicinity o f the critical point and c o n f o r m to
58
14.W. Xiang / Fluid Phase Equilibria 137 (1997) 53-62
the critical exponents from the renormalization-group theory of critical phenomena [3,4]. This form of the enthalpy-of-vaporization equation also characterizes the regular behavior away from the critical point. Eq. (4) contains only three system-dependent parameters (b 0, b l, and b 2) that are determined from the experimental data. The leading-order amplitude b o is agreement with the theoretical value which can be seen below. The new equation possesses a simple form to describe the temperature dependence of the enthalpy of vaporization along the entire vapor-liquid phase coexistence curve. The new equation is better than the first three terms of Eq. (1) in reproducing the experimental data for water, as shown in table below along with other correlations for comparison.
3. Results In testing the performance of the new equation, Eq. (4), the three system-dependent parameters for each substance are determined by fitting experimental data. Reliable data for thirty substances covering a wide temperature range are used. All data have been converted, whenever necessary, to the ITS-90 scale and temperatures used throughout this paper are on this scale [12]. The critical temperatures of substances that do not have data over the entire range are taken from Ambrose and Tsonopoulos [13] and Gude and Teja [14]. The three system-dependent parameters b o, b 1, and b 2 for each of these substances determined by a nonweighted least-squares fit are given in Table 1 together with the critical temperatures used in the fit. The standard deviations, the average deviations, and the maximum deviations are also presented in Table 1. Within the experimental uncertainties, the new enthalpy-of-vaporization equation is accurate for pure compounds over the entire range from the triple point to the critical point. For example, the deviations for water with the associated tolerances, argon, and l,l,l,2-tetrafluoroethane (R134a) are given in Tables 2-4. It can be seen that, for water, for which highly accurate experimental data are available, the deviations of 0.07% at the triple point, 0.05% at the normal boiling temperature, and 1% near the critical point, are almost within the tolerances given in the 1985 IAPS [10]. For water, the leading-order amplitude b o = 6.85 is in good agreement with 6.89 (/3 = 1 / 3 ) obtained by Torquato and Stell [3]. All existing equations [5-9] do not have accuracies within the data uncertainties, as can
Table 3 Comparison between the calculated values from the new Eq. (4) and the recommended values of argon [15] T (ITS-90) (K)
PD a
T (ITS-90) (K)
PD a
83.81 84.01 88.01 92.01 96.01 lO0.O1 104.01 108.01 112.01
-0.066 -0.062 -0.010 0.027 0.053 0.072 0.078 0.070 0.048
116.01 120.01 124.01 128.01 132.01 136.01 140.01 144.01 148.01
0.011 -0.033 -0.080 -0.115 -0.126 - 0.097 - 0.021 0.077 0.47
aPD = 100(1 - A Hc~ld/A Hexpt) is the percentage deviation, where the subscripts expt and cald represent the observed data and the corresponding calculated values.
59
H.W. Xiang / Fluid Phase Equilibria 137 (1997) 53-62
Table 4 Comparison between the calculated values from the new Eq. (4) and the recommended values of 1,1,1,2-tetrafluoroethane (R134a) [27] T (ITS-90) (K)
PD a
T (ITS-90) (K)
PD a
169.85 176.00 182.00 188.00 194.00 200.00 206.00 212.00 218.00 224.00 230.00 236.00 242.00 246.78 252.00 258.00 264.00 270.00
0.180 0.099 0.039 - 0.010 -0.045 -0.073 - 0.089 -0.093 - 0.093 -0.086 - 0.076 -0.063 - 0.046 - 0.038 -0.014 0.000 0.019 0.034
276.00 282.00 288.00 294.00 300.00 306.00 312.00 318.00 324.00 330.00 336.00 342.00 348.00 354.00 360.00 366.00 372.00 374.00
0.046 0,060 0.072 0.079 0.087 0.085 0.086 0.080 0.075 0.061 0.039 0.015 - 0.028 -0.086 - 0.18 -0.30 -0.45 - 2.2
aPD = 100(1 - A Hcald/AHexpt) is the percentage deviation, where the subscripts expt and cald represent the observed data and the corresponding calculated values. be seen f r o m Ref. [1 ]. C o m p a r i s o n s of accuracies with empirical S o m a y a j u l u three- and f o u r - p a r a m e ter equations for water are also presented in Table 2. It appears that the new equation has the best description for water. The m a x i m u m deviation occurs near the critical point where the predicted enthalpy of vaporization differs f r o m the data b y several parts in 100. In light of the experimental difficulties one confronts in this region, this deviation is m o s t likely within the experimental error. It can be seen that the new equation can reproduce the experimental data of these substances within the experimental uncertainties. Consequently, the equation presented here is capable of representing the enthalpy of vaporization for nonpolar, polar, associated, and h y d r o b o n d i n g c o m p o u n d s . Both theoretically and practically, extrapolation to Tt and Tc is always expected. Since there are only e x p e r i m e n t a l data m e a s u r e d in the relatively easily accessible temperature interval a w a y f r o m both Tt and T~ available, special attention should be given to the prediction around Tt and To. S v o b o d a and S m o l o v a [2] evaluated the previous enthalpy of vaporization equations and r e c o m m e n d e d an empirical three-parameter equation b y S o m a y a j u l u [5] as one of m o s t suited for extrapolation towards Tt and T~ for water. The empirical S o m a y a j u l u three- and four-parameter equations are, AH
RL
3 1~ _COT8 + C l t + C 2 T 8
()3
AH __~" ~ + d I RT~ = do ~'t
( )19 ()27
+ d 2 - - z ~- -k- d 3 --'r ~~-t ~-t
(5)
(6)
H. W. Xiang / Fluid Phase Equilibria 137 (1997) 53-62
60
Table 5 Extrapolation of the new equation, Eq. (4), and the Somayajulu three-parameter equation, Eq. (5), upon different temperature intervals for water [10] Temperature interval Tt (273.16)-T b 293.15-373.12 313.14-373.12 333.13-373.12 348.13-373.12 293.15-313.14 293.15-348.13 Tb (373.12)- Tc 373.12-603.11 373.12-533.11 373.12-453.11 453.11-603.11 503.11-603.11 Overall
Somayajulu equation
New equation
SD a (kJ/mol)
AD ~
MD a
SD" (kJ/mol)
AD a
MD a
0.15 0.14 0.13 0.12 0.13 0.16 0.15 0.11 0.10 0.12 0.18 O.18 0.58 O.18
0.61 0.55 0.50 0.47 0.51 0.64 0.59 0.21 0.23 0.45 0.30 0.37 1.0 0.49
-4.7 -4.4 -4.1 - 3.8 -4.2 - 4.6 - 4.5 1.4 - 1.5 - 3.7 1.6 1.2 2.2 3.2
0.06 0.14 0.14 0.10 0.16 0.20 0.15 0.09 0.07 0.14 0.18 O.18 0.32 O.15
0.19 0.44 0.45 0.22 0.72 0.70 0.46 0.18 0.27 0.59 0.77 0.33 0.55 0.45
1.0 - 1.9 -2.0 - 0.92 6.4 - 3.5 - 2.0 - 0.49 2.0 5.5 7.1 - 0.90 - 1.5 2.7
aSD : [ ~ ( A Hcal d - - A Oexpt)2/(H - - m)] 1/2 is the standard deviation, AD = 100(Y~'[1- A O c a l d / A Hexpt[/n) is the average deviation, MD = 100(1-AOcald/AOexpt ) is the maximum percentage deviation, where the subscripts expt and cald represent the observed data and the corresponding calculated values from each equation. All the deviations are for all the data points, n is the number of data points and m the number of system-dependent parameters of each equation. w h e r e ~'t = 1 - T t / T ~. E x t r a p o l a t i o n o f the S o m a y a j u l u equation, Eq. (5), and the n e w equation, Eq. (4), to Tt and Tc for water using different t e m p e r a t u r e intervals is g i v e n in Table 5. T h e n e w e q u a t i o n m o r e a c c u r a t e l y predicts the e n t h a l p y o f v a p o r i z a t i o n for water in m o s t temperature intervals.
4. Conclusions A new, simple, accurate t h r e e - p a r a m e t e r e n t h a l p y - o f - v a p o r i z a t i o n equation, c o v e r i n g the entire r e g i o n f r o m the triple point to the critical point, is c o n s t r u c t e d to c o n f o r m to the r e n o r m a l i z a t i o n - g r o u p t h e o r y o f critical p h e n o m e n a . This correlation fits the available data within e x p e r i m e n t a l errors better than existing three- and f o u r - p a r a m e t e r correlations b e c a u s e its functional f o r m incorporates the correct near-critical b a h a v i o r and the e x p o n e n t s and the s y s t e m - d e p e n d e n t parameters reflect the p h y s i c a l properties o f the substances. T h e n e w e q u a t i o n m o r e accurately correlates the experimental data and m o r e e f f e c t i v e l y extrapolates to the triple point and to the critical point for pure c o m p o u n d s . T h e e n t h a l p y - o f - v a p o r i z a t i o n e q u a t i o n o b t a i n e d here m a y be applied to fluids w h i c h are in the s a m e universality class as the substances cited in this paper.
5. List of symbols ai
AD bi
s y s t e m - d e p e n d e n t p a r a m e t e r s o f the Eqs. (1) and (2) a v e r a g e deviation ( - 100(~]']1 - A H c a l o / A Hexpt[/n)) s y s t e m - d e p e n d e n t p a r a m e t e r s o f the n e w Eq. (4)
H. W. Xiang / Fluid Phase Equilibria 13 7 (1997) 53-62
Ci
di AH m
MD r/
PD R SD t T
Tc T~= T I T c
61
system-dependent parameters of the Eq. (5) system-dependent parameters of the Eq. (6) enthalpy of vaporization n u m b e r o f system-dependent parameters m a x i m u m percentage deviation ( = 100(1 - A Hcala/A Hexpt) data points percentage deviation ( = 100(1 - A Hcalo/A nexpt) universal gas constant ( = 8.31451 J / ( m o l K)) standard deviation ( = [E~'(A H c a l d - - A Hexpt)2/( n -- m)] 1/2) TiT1 temperature critical temperature reduced temperature triple-point temperature
G r e e k letter a = 0.11 critical exponent /3 = 0.325 critical exponent A = 0.51 W e g n e r ' s first gap exponent r
1 -
"rt
1 - TIT t
T/T~
Acknowledgements The author would like to thank Prof. V. Svoboda and editor-in-chief Prof. H. Renon for their constructive comments and suggestions on this manuscript.
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