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Corrected Values for Boiling Points and Enthalpies of Vaporization of Elements in Handbooks

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Abstract

The scientific community relies upon the veracity of the scientific data in handbooks and databases. In a previous work, the authors developed a systematic, intelligent, and potentially automatic method to detect errors in such resources based on artificial neural networks (ANNs). This method revealed variations from (10 to 900) % in tables of property data for elements in the periodic table and pointed out the ones that are most probably correct. In this paper, we focus on the details of employing this method for analyzing the data of boiling points and enthalpies of vaporization recorded in different handbooks. The method points out the values that are likely to be correct. To verify the method employed, a detailed discussion of the data with reference to the original literature sources is given as well as factors that may affect the accuracy of the prediction.
Corrected Values for Boiling Points and Enthalpies of Vaporization of Elements
in Handbooks
Yiming Zhang
Ningbo Institute of Material Technology & Engineering, Chinese Academy of Sciences, No. 519 Zhuangshi Road, Zhenhai
District, Ningbo, Zhejiang Province, P.R. China 315201, and Department of Materials, Queen Mary University of London, Mile
End Road, London E1 4NS, United Kingdom
Julian R. G. Evans
Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, United Kingdom
Shoufeng Yang*
School of Engineering Sciences, University of Southampton, Southampton, SO17 1BJ, United Kingdom
The scientific community relies upon the veracity of the scientific data in handbooks and databases. In a
previous work, the authors developed a systematic, intelligent, and potentially automatic method to detect
errors in such resources based on artificial neural networks (ANNs). This method revealed variations from
(10 to 900) % in tables of property data for elements in the periodic table and pointed out the ones that are
most probably correct. In this paper, we focus on the details of employing this method for analyzing the
data of boiling points and enthalpies of vaporization recorded in different handbooks. The method points
out the values that are likely to be correct. To verify the method employed, a detailed discussion of the data
with reference to the original literature sources is given as well as factors that may affect the accuracy of
the prediction.
Introduction
It is well-known that both the boiling point and the enthalpy
of vaporization are important thermodynamic properties that are
required in product design and processes involving liquid and
vapor phase transitions such as distillation, vaporization, and
drying
1-3
and that, as a result, the quality and veracity of these
data in handbooks are important for the academic and industrial
scientific community.
4
However, it is accepted that errors in
handbooks and databases are inescapable and are anticipated
at rates of (1 to 5) %.
5-7
This work challenges the inevitability of such high error rates.
The values of boiling point and enthalpy of vaporization of the
elements were recorded for testing, because these data are
expected to be very reliable; only the elements with short half-
lives were excluded. Tables 1 and 2 show the boiling point and
enthalpy of vaporization values of elements in the periodic table
taken from five different handbooks. Considerable variation in
the values can be noticed, so it can be concluded that
inconsistencies are extant within these handbooks, and these
inconsistencies have persisted undetected into the 21st century.
In the authors’ related work,
8
we developed a systematic and
intelligent method based on artificial neural networks (ANNs)
to detect wide levels of inconsistencies in handbooks and so
stop their transmission to future researches and documents. In
this paper, we are going to show the details of application of
this method in the case of correcting values of the boiling point
and enthalpy of vaporization for the elements.
Experimental Details
Data Collection and Neural Network Construction. The data
are recorded from five different handbooks, including the
Chemistry Data Book (CDB),
9
The Lange’s Handbook of
Chemistry (LAG),
10
The Elements (ELE),
11
Table of Physical
and Chemical Constants (TPC),
12
and CRC Handbook of
Chemistry and Physics (CRC).
13
The neural networks are
constructed, trained, and simulated by employing MATLAB
7.4.0.287 (R2007a) software. A two-hidden layer network with
tan-sigmoid transfer function in the first hidden layer and a linear
transfer function in the second hidden layer was established. A
loop program was applied to redistribute the database to make
the training set cover the problem domain.
14,15
The detailed
process of data collection and neural network construction has
been described in the authors’ related works.
8
The ANN finds
indirect relationships between data sets and then identifies values
which disobey the relationship.
Stages of Error Correction. This includes four different
stages.
Stage 1. During the exploration of the indirect relationship
between boiling point and enthalpy of vaporization using data
from CDB (shown in Table 3) by an ANN, best linear fit
equations with a regression coefficient of R)0.973 and 0.972
were found (as shown in Figure 1). This result prompts us to
raise the correlation hypothesis that the correlation applies to
all of the elements.
Stage 2. In this stage, a conformity criterion is selected. On
the basis of the inconsistency within different handbooks,
* Authors to whom correspondence should be addressed. E-mail: s.yang@
soton.ac.uk. Tel.: 0044-23-8059-8697. Fax: 0044-23 8059 3016.
J. Chem. Eng. Data 2011, 56, 328–337328
10.1021/je1011086 2011 American Chemical Society
Published on Web 01/11/2011
Table 1. List of Boiling Points Divided by Kelvin from Five Handbooks
a
element CDB LAG
b
ELE TPC CRC (max/min -1) ·100 %
Ag
c
2483 2483 2485 2433 2435 2.14 %
Al
c
2743 2743 2740 2793 2792 1.93 %
Ar 87.15 87.15 87.29 87.29 87.3 0.17 %
As (gray)
d
886.2 886.2 889 883.2 876.2 1.46 %
Au 3243 3243 3080 3123 3129 5.29 %
Ba
e
1913 1913 1910 2173 2170 13.80 %
Be
e
2750 2750 3243
f
2743 2744 18.20 %
B
c
4203 4203 3931 4273 4273 8.70 %
Bi
c
1833 1833 1883 1833 1837 2.73 %
Br 331.7 332 331.9 332.1 332 0.12 %
C (graphite)
e
5103 5103 5100
d
- 4098
d
24.50 %
Ca 1760 1760 1757 1757 1757 0.17 %
Cd 1038 1038 1038 1043 1040 0.48 %
Ce
c
3743 3743 3699 3693 3716 1.35 %
Cl 238.5 238.5 239.2 239.2 239.1 0.29 %
Co 3173 3173 3143 3203 3200 1.91 %
Cr 2755 2755 2945 2943 2944 6.90 %
Cs 963.2 963.2 951.6 943.2 944.2 2.12 %
Cu 2868 2868 2840 2833 2835 1.24 %
Dy 2873 2873 2835 2833 2840 1.41 %
Er 3173 3173 3136 3133 3141 1.28 %
Eu 1713 1713 1870 1873 1802 9.34 %
F 85.15 85.15 85.01 85.05 85.03 0.16 %
Fe 3273 3273 3023 3133 3134 8.27 %
Ga 2673 2673 2676 2473 2477 8.21 %
Gd 3273 3273 3539 3533 3546 8.34 %
Ge 3103 3103 3103 3103 3106 0.10 %
H 21.15 21.15 20.28 20.28 20.28 4.29 %
He 4.15 4.15 4.216 4.37 4.22 5.30 %
Hf
g
5673 5673 5470 4873 4876 16.40 %
Hg 630.2 630.2 629.7 629.8 629.9 0.08 %
Ho
c
2873 2873 2968 2973 2973 3.48 %
I(I
2)
c
457.2 457.2 457.5 457.2 457.6 0.09 %
In 2273 2273 2353 2343 2345 3.52 %
Ir
g
5573 5573 4403 4703 4701 26.60 %
K 1047 1047 1047 1033 1032 1.45 %
Kr
c
121.2 121.2 120.9 120 119.9 1.08 %
La 3743 3743 3730 3733 3737 0.35 %
Li 1603 1603 1620 1613 1615 1.06 %
Lu 3603 3603 3668 3663 3675 2.00 %
Mg 1383 1383 1363 1363 1363 1.47 %
Mn 2373 2373 2235 2333 2334 6.17 %
Mo
g
5833 5833 4885 4913 4912 19.40 %
N 77.15 77.15 77.4 77.35 77.36 0.32 %
Na
c
1163 1163 1156 1153 1156 0.87 %
Nb
e
3573 3573 5015 4973 5017 40.40 %
Nd 3303 3303 3341 3343 3347 1.33 %
Ne 27.15 27.15 27.1 27.07 27.07 0.30 %
Ni
b
3003 3003 3005 3263 3186 8.66 %
O 90.15 90.15 90.19 90.19 90.2 0.06 %
Os
c
5273 5273 5300 5273 5285 0.51 %
P (white)
c
553.2 553.2 553 550.2 553.7 0.64 %
Pb 2017 2017 2013 2023 2022 0.50 %
Pd
e
4253 4253 3413 3233 3236 31.60 %
Pr
e
3403 3403 3785 3783 3793 11.50 %
Pt
b
,
h
4803 4803 4100 4093 4098 17.40 %
Rb 961.2 961.2 961 963.2 961.2 0.23 %
Re
c
5903 5903 5900 5873 5869 0.58 %
Rh
e
4773 4773 4000 3973 3968 20.30 %
Ru
h
5173 5173 4173 4423 4423 24.00 %
S (monoclinic)
c
718.2 718.2 717.8 717.8 717.8 0.06 %
Sb
g
1653 1653 1908 1860 1860 15.40 %
Sc 3003 3003 3104 3103 3109 3.53 %
Se
c
958.2 958.2 958.1 958.2 958.2 0.01 %
Si
g
2633 2633 2628 3533 3538 34.60 %
Sm
c
2173 2173 2064 2063 2067 5.33 %
Sn
e
2543 2543 2543 2893 2875 13.80 %
Sr 1653 1653 1657 1653 1655 0.24 %
Ta 5693 5693 5698 5833 5731 2.46 %
Tb
g
3073 3073 3396 3493 3503 14.00 %
Te
c
1263 1263 1263 1263 1261 0.16 %
Ti 3533 3533 3560 3563 3560 0.85 %
Tl 1733 1733 1730 1743 1746 0.92 %
Tm
e
2003 2003 2220 2223 2223 11.00 %
V
e
3273 3273 3650 3673 3680 12.40 %
W 6203 6203 5930 5823 5828 6.53 %
Journal of Chemical & Engineering Data, Vol. 56, No. 2, 2011 329
elements were then classified into four different categories based
on a 10 % variation between minimum and maximum values:
I. Boiling points of the elements are consistent, but enthalpies
of vaporization are inconsistent (shown in Table 4).
II. Enthalpies of vaporization are consistent, but boiling points
are inconsistent (shown in Table 5).
III. Both boiling points and enthalpies of vaporization are
inconsistent (shown in Table 6).
IV. Both boiling points and enthalpies of vaporization are
consistent (shown in Table 7).
Stage 3. We then use the data belonging to category IV to
train a second set of ANNs: ANN1 and ANN2. ANN1 is used
to predict enthalpies of vaporization from boiling points, while
ANN2 is used to predict boiling points from enthalpies of
vaporization. Here, ANN1 and ANN2 discover and capture an
indirect correlation between just two properties from category
IV data set, but this method can be applied in any situation
when the correlations between several properties can be captured
by ANNs.
8
We now suppose these two ANNs to have been
trained on “consistent” data providing a robust correlation
against which the consistency of other data can be judged. The
details of construction for these two ANNs can be found in the
authors’ related work.
8
Stage 4. Consistent values are now used to identify incon-
sistent data. At first, the ANN1 was used to predict enthalpies
of vaporization of elements in category I using the consistent
boiling points; then the outputs are compared with the handbook
data to isolate suspect data for the enthalpy of vaporization.
For example, Al has consistent boiling point values from (2740
to 2793) K (1.9 % difference) in all of the handbooks consulted,
but enthalpies of vaporization vary from (29 to 294) kJ ·mol-1
(910 % difference). The ANN-predicted enthalpy of vaporization
is 281 kJ ·mol-1, which is close to one handbook value, 284
kJ ·mol-1(CDB). The difference between the predicted and the
closest literature value is now 1.2 %.
The second part of stage 4 is to use the consistent enthalpies
of vaporization as input values in ANN2 to predict boiling points
of elements in category II for isolating suspect boiling point
data.
In the third part of stage 4, for the data in category III, a
method to calculate a comprehensive minimum of difference
was constructed.
8
Results and Discussion
In the first part of stage 4, the ANN1 is able to point out the
correct enthalpies of vaporization from the inconsistent values
from different handbooks. Most of the differences (11 out of
16, including Na, Al, B, Os, Kr, Ag, Te, Bi, Re, Ce, and Sm)
between the predicted and the closest recorded values of
enthalpy of vaporization are now less than 10 %. The results
of this stage are listed in Table 8. The average error for enthalpy
of vaporization in category I decreased from 231 % (Table 4)
to 8.8 % (Table 8). Inconsistencies for another five elements
(I, P (white), Se, Ho, S) have decreased dramatically as well
but are still slightly greater than 10 %. That could be because
none of the data in the handbooks was correct but that the ANN
gave a correct estimate. Those five elements will be discussed
later.
In the second part of stage 4, in a similar way, ANN2 is able
to point out the correct boiling point from inconsistent values.
Predicted and closest recorded values of boiling points for most
of the samples now differ by less than 10 % (10 out of 12
elements, including Pd, Nb, Yb, Rh, Y, C (graphite), Be, Sn,
V, Pr), as shown in Table 9. The average error of the boiling
point in category II decreased from 18.9 % (Table 5) to 6.4 %
(Table 9). Deviations of another two elements (Ba, Tm) have
decreased as well but are still higher than 10 %, and these will
be discussed later.
In the last part of stage 4, both ANN1 and ANN2 are used to
point out the correct enthalpies of vaporization and boiling point
of those elements (Tb, Hf, Ir, Mo, Sb, Si, and Zr). The predicted
and closest recorded values for both boiling point and enthalpies
of vaporization for category III now differ by less than 10 %
(Table 10).
Finally, after the correction made for data in category I,
category II, and category III, two new ANNs were trained with
the “corrected” values inserted (shown in Table 11), and the
results are shown in Figure 2. For forward and backward
predictions, respectively, statistics are given in the third and
fourth rows of Table 12. To make the comparison, the statistical
analysis for Figure 1 is shown in the first and second row of
Table 12. After correction, the moduli have decreased dramatically.
In this method, all that is required is a sufficient correlation,
which can itself be established by the ANNs that are then
employed to discriminate between the majority of well-
correlated data points and an outlying minority. Further, more
than two properties can be correlated, and the details of the
procedures for doing so have been discussed.
8
It also needs to
be pointed out that the “consistent” data in all handbooks does
not necessarily mean they are “correct”. However, the ANN
could detect them if they do not follow the trend. It is arguable
that the term “outlier” should not be used to justify a predictive
replacement, for it is often the case that unusual results are
indicators of new knowledge.
16
It needs to be pointed out that the boiling point and enthalpy
ofvaporizationareindirectlyrelatedbasedontheClausius-Clapeyron
equation (dPvap/dT))(vapHm)/(TVvap))(vapHm)/((RT2/
Pvap)Zvap), which in its integrated form gives Pvap )i
Table 1. Continued
element CDB LAG
b
ELE TPC CRC (max/min -1)·100 %
Xe 165.2 165.2 166.1 165.1 165 0.67 %
Yb
e
1703 1703 1466 1473 1469 16.20 %
Y
e
3203 3203 3611 3613 3618 13.00 %
Zn 1180 1180 1180 1183 1180 0.25 %
Zr
g
3853 3853 4650 4673 4682 21.50 %
a
Acronyms as in text. This table does not have exclusions based on judgement. The elements are in alphabetical order.
b
The boiling point is different
in different tables of LAG. For example, nickel (Ni) is 2884 °C in page 1.43 (Table 1.3) and 2730 °C in page 1.125 (Table 1.19). We selected most of
the boiling points from the later table.
c
Category I: the data of the boiling point for these elements are consistent in different handbooks, but those of
the enthalpy of vaporization are not.
d
Sublimation.
e
Category II: the data of the enthalpy of vaporization for these elements are consistent in different
handbooks, but those of the boiling point are not.
f
Under pressure.
g
Category III: the data of neither the boiling point nor the enthalpy of vaporization
for these elements are consistent in different handbooks.
h
The variation of boiling points of Pt and Ru are greater than 10 %, but here due to shortage
of data in range of (3603.15 to 5693.15) K in category IV, they were classified into category IV by identifying the closest value from the literature to
reduce the uncertainty in that range.
30,31
330 Journal of Chemical & Engineering Data, Vol. 56, No. 2, 2011
Table 2. List of Enthalpies of Vaporization Divided by kJ ·mol-1from Five Handbooks
a
element CDB LAG ELE TPC CRC (max/min -1) ·100 %
Ag
b
254 258 255 25.5 - 912 %
Al
b
284 294 294 29.1 294 910 %
Ar 6.53 6.43 6.53 6.52 6.43 1.56 %
As (gray) 32.4 - 31.9 - - 1.57 %
Au 342 324 324 324 324 5.56 %
Ba
c
149 140 151 140 140 7.86 %
Be
c
309 297 309 298 - 4.04 %
B
b
540 480 539 508 480 12.50 %
Bi
b
179 151 179 187 151 23.80 %
Br 30 30 30 29.6 30 1.35 %
C (graphite)
c
715 - 711 - - 0.56 %
Ca 153 155 150 155 - 3.33 %
Cd 100 99.9 99.9 99.9 99.9 0.10 %
Ce
b
- 398 314 314 - 26.80 %
Cl 20.4 20.4 20.4 20.4 20.4 0.00 %
Co 390 377 382 373 - 4.56 %
Cr 347 340 349 340 - 2.65 %
Cs 66.1 63.9 65.9 67.8 - 6.10 %
Cu 305 300 305 301 - 1.67 %
Dy - 280 293 - - 4.64 %
Er - 280 293 293 - 4.64 %
Eu - 176 176 176 - 0.00 %
F 6.32 6.62 6.55 6.54 6.62 4.75 %
Fe 354 340 351 350 - 4.12 %
Ga 256 254 256 256 254 0.79 %
Gd - 301.3 311.7 311.7 - 3.45 %
Ge 330 334 334 334 334 1.21 %
H 0.9 - 0.92 0.9 0.9 2.22 %
He 0.084 0.083 0.082 0.08 0.08 5.00 %
Hf
d
648 571 661 661 - 15.80 %
Hg 58.2 59.1 59.2 59.1 59.1 1.72 %
Ho
b
- 71 251 251 - 254 %
I(I
2)
b
22 41.6 41.7 41.9 41.6 90.50 %
In 225 232 226 226 - 3.11 %
Ir
d
636 232 564 564 - 174 %
K 79.1 76.9 77.5 76.9 - 2.86 %
Kr
b
10 9.08 9.05 9.03 9.08 10.70 %
La 400 402 400 400 - 0.50 %
Li 136 147 135 147 - 8.89 %
Lu - 414 428 - - 3.38 %
Mg 132 128 129 128 - 3.13 %
Mn 225 221 220 220 - 2.27 %
Mo
d
536 617 594 590 - 15.10 %
N 5.58 5.57 5.58 5.59 5.57 0.36 %
Na
b
101 97.4 89 97.4 - 13.50 %
Nb
c
694 690 697 690 - 1.01 %
Nd - 289 284 284 - 1.76 %
Ne 1.8 1.71 1.74 1.77 1.71 5.26 %
Ni 379 378 372 378 - 1.88 %
O 6.82 6.82 6.82 6.82 6.82 0.00 %
Os
b
678 738 628 628 - 17.50 %
P (white)
b
12.4 12.4 51.9 - 12.4 319 %
Pb 177 180 179 178 180 1.69 %
Pd
c
380 362 393 393 - 8.56 %
Pr
c
- 331 333 333 - 0.60 %
Pt 510 469 511 511 - 8.96 %
Rb 69 75.8 69.2 69.2 - 9.86 %
Re
b
636 704 707 707 - 11.20 %
Rh
c
531 494 495 495 - 7.49 %
Ru 619 592 568 568 - 8.98 %
S (monoclinic)
b
10 45 9.62 - 45 368 %
Sb
d
195 193 67.9 67.9 - 187 %
Sc 310 333 305 305 - 9.18 %
Se
b
14 95.5 26.3 26.3 95.5 582 %
Si
d
300 359 383 359 - 27.70 %
Sm
b
- 165 192 192 - 16.40 %
Sn
c
290 296 290 290 - 2.07 %
Sr 141 137 139 137 - 2.92 %
Ta 753 733 753 737 - 2.73 %
Tb
d
- 293 391 - - 33.50 %
Te
b
49.8 114 50.6 50.6 114 129 %
Ti 427 425 429 425 - 0.94 %
Tl 162 165 162 162 - 1.85 %
Tm
c
- 247 247 - - 0.00 %
V
c
444 459 459 447 - 3.38 %
W 774 807 799 806 - 4.26 %
Journal of Chemical & Engineering Data, Vol. 56, No. 2, 2011 331
exp[(vapHm)/(RT)], and the boiling point is the temperature at
which Pvap reaches ambient (iis a constant of integration).
17
The pre-exponential coefficient is not directly related. The ANN
can find clusters of properties that are indirectly related in this
way to identify suspect data.
Verification of the Method by Referring to Original
Sources. To verify the method, original sources were consulted
where possible to (1) check whether the refined values were
sensible, (2) to reveal and analyze the origin of the differences
between handbook data, and (3) to find whether the neural
Table 2. Continued
element CDB LAG ELE TPC CRC (max/min -1) ·100 %
Xe 12.6 12.6 12.7 12.6 12.6 0.79 %
Yb
c
- 159 159 - - 0.00 %
Y
c
390 365 393 393 - 7.67 %
Zn 115 124 115 115 - 7.83 %
Zr
d
502 573 582 591 - 17.70 %
a
Acronyms as in text. This table does not have exclusions based on judgement. The elements are sorted in alphabetical order.
b
Category I: the data
for the boiling point for these elements are consistent in different handbooks, but those for the enthalpy of vaporization are not.
c
Category II: the data
for the enthalpy of vaporization for these elements are consistent in different handbooks, but those of the boiling point are not.
d
Category III: the data
for neither the boiling point nor the enthalpy of vaporization for these elements are consistent in different handbooks.
Table 3. Data Set First Used to Train the ANNs Shown in Figure 1
a
TbvapHmTbvapHmTbvapHm
element K kJ ·mol-1element K kJ·mol-1element K kJ ·mol-1
Ag 2483 254 H 21.15 0.90 Pr 3403 331
Al 2743 284 He 4.150 0.08 Pt 4803 510
Ar 87.20 6.53 Hf 5673 648 Rb 961.2 69.0
As (gray) 886.2 32.4 Hg 630.2 59.1 Re 5903 636
Au 3243 342 Ho 2873 71.0 Rh 4773 531
B 4203 540 I (I2) 457.2 22.0 Ru 5173 619
Ba 1913 149 In 2273 225 S (mono.) 718.2 10.0
Be 2750 309 Ir 5573 636 Sb 1653 195
Bi 1833 179 K 1047 79.1 Sc 3003 310
Br 331.7 30.0 Kr 121.2 9.04 Se 958.2 14.0
C (graphite) 5103 715 La 3743 400 Si 2633 300
Ca 1760 153 Li 1603 136 Sm 2173 165
Cd 1038 100 Lu 3603 414 Sn 2543 290
Ce 3743 398 Mg 1383 132 Sr 1653 141
Cl 238.5 20.4 Mn 2373 225 Ta 5693 753
Co 3173 390 Mo 5833 536 Tb 3073 293
Cr 2755 347 N 77.15 5.58 Te 1263 49.8
Cs 963.2 66.1 Na 1163 101 Ti 3533 427
Cu 2868 305 Nb 3573 694 Tl 1733 162
Dy 2873 280 Nd 3303 289 Tm 2003 247
Er 3173 280 Ne 27.15 1.80 V 3273 444
Eu 1713 176 Ni 3003 379 W 6203 774
F 85.15 6.32 O 90.15 6.82 Xe 165.2 12.6
Fe 3273 354 Os 5273 678 Y 3203 390
Ga 2673 256 P (white) 553.2 12.4 Yb 1703 159
Gd 3273 301 Pb 2017 177 Zn 1180 115
Ge 3103 330 Pd 4253 380 Zr 3853 502
a
The majority were taken from the Chemistry Data Book (CDB) without judgement. A few data unavailable in CDB were taken from LAG and ELE.
(Tb: boiling point; vapHm: enthalpy of vaporization).
Figure 1. Prediction of (a) enthalpy of vaporization (vapHm) from boiling point (Tb), R)0.973; (b) boiling point (Tb) from enthalpy of vaporization
(vapHm), R)0.972; using data from CDB, LAG, and ELE. O, training data points; /, test data points; s, best linear fit; ----, exp. )prediction.
332 Journal of Chemical & Engineering Data, Vol. 56, No. 2, 2011
network can give values closer to the original literature. Some
errors are quickly attributable to incorrect unit conversions or
to misplaced decimal points, but some arise from different
reference conditions, and representative examples are shown
below.
Table 8 indicates that predicted and closest recorded values
of enthalpy of vaporization for most elements differ by less than
10 % after removal of incorrect data. The exceptions are I (15.6
%), P (17.2 %), Se (17.9 %), Ho (19.1 %), and S (26.7 %).
Data for elements I, P, and S depend on the polyatomic nature
of these molecules; this accounts for the differences shown in
Table 2, but handbooks do not always state how the value is
normalized. Because I, P, and S exist in polyatomic molecular
forms, the values for polyatomic forms should be clarified. From
Table 8, it is seen that the neural networks have a capability to
locate values which are in the right magnitude, although the
difference is still greater than 10 %. However, the values for
Ho and Se need more detailed investigation.
Ho. The predicted enthalpy of vaporization of holmium by
ANN is 299 kJ ·mol-1, and the values in handbooks are 71
kJ ·mol-1(LAG), 251 kJ ·mol-1(ELE), and 251 kJ ·mol-1(TPC)
(no record in CDB and CRC for Ho). If we select 251 kJ ·mol-1
as a correct value, this gives a 19.1 % deviation between ANN
prediction and the handbook value. From the following analysis
and search of original papers, we believe the ANN prediction
is more accurate than the handbook.
There are two sources of discrepancy. The first is the unit of
the values. Original research papers offer the values for vapHm,st:
301 kJ ·mol-1(71910 cal ·mol-1),
18
291kJ·mol-1(69500
cal ·mol-1),
19
314 kJ ·mol-1(75040 cal ·mol-1),
20
339 kJ ·mol-1
(81150 cal ·mol-1),
18
and 295 kJ ·mol-1(70600 cal ·mol-1).
21
Clearly these results indicate that the enthalpy of vaporization
of holmium is around 297 kJ ·mol-1(which is very close to
299 kJ ·mol-1predicted by ANN), rather than the 71 kJ ·mol-1
(the number in LAG but the unit is different) recorded in LAG,
and the source of error is the transcription of units. Indeed, in
the earlier 12th edition of Lange’s Handbook Chemistry,
22
the
value was 251 kJ ·mol-1(60 kcal ·mol-1), placing it in the
correct range.
The second source of discrepancy is the reference tempera-
ture. The values from the research papers listed above are
corrected to standard temperature. However, the category IV
Table 4. List of Category I Elements: Boiling Points of the Elements Are Consistent, but Enthalpies of Vaporization Are Inconsistent
element
TbvapHm/kJ ·mol-1
K CDB LAG ELE TPC CRC (max/min -1)·100 %
Ag 2483 254 258 255 25.5 - 912 %
Al 2743 284 294 294 29.1 294 910 %
B 4203 540 480 539 508 480 12.5 %
Bi 1833 179 151 179 187 151 23.8 %
Ce 3743 - 398 314 314 - 26.8 %
Ho 2873 - 71 251 251 - 254 %
I(I
2) 457.2 22 41.6 41.7 41.9 41.6 90.5 %
Kr 121.2 10 9.08 9.05 9.03 9.08 10.7 %
Na 1163 101 97.4 89 97.4 - 13.5 %
Os 5273 678 738 628 628 - 17.5 %
P (white) 553.2 12.4 12.4 51.9 - 12.4 319 %
Re 5903 636 704 707 707 - 11.2 %
S 718.2 10 45 9.62 - 45 368 %
Se 958.2 14 95.5 26.3 26.3 95.5 582 %
Sm 2173 - 165 192 192 - 16.4 %
Te 1263 49.8 114 50.6 50.6 114 129 %
average 231 %
Table 5. List of Category II Elements: Enthalpies of Vaporization Are Consistent, but Boiling Points Are Inconsistent
element
vapHmTb/K
kJ ·mol-1CDB LAG ELE TPC CRC (max/min -1) ·100 %
Ba 149 1913 1913 1910 2173 2170 13.8 %
Be 309 2750 2750 3243 2743 2744 18.2 %
C (graphite) 715 5103 5103 5100 - 4098 24.5 %
Nb 694 3573 3573 5015 4973 5017 40.4 %
Pd 380 4253 4253 3413 3233 3236 31.6 %
Pr 331 3403 3403 3785 3783 3793 11.5 %
Rh 531 4773 4773 4000 3973 3968 20.3 %
Sn 290 2543 2543 2543 2893 2875 13.8 %
Tm 247 2003 2003 2220 2223 2223 11.0 %
V 444 3273 3273 3650 3673 3680 12.4 %
Y 390 3203 3203 3611 3613 3618 13.0 %
Yb 159 1703 1703 1466 1473 1469 16.2 %
average 18.9 %
Table 6. List of Category III Elements: Both Boiling Points and Enthalpies of Vaporization Are Inconsistent
element
Tb/K vapHm/kJ ·mol-1
CDB LAG ELE TPC CRC CDB LAG ELE TPC CRC
Tb 3073 3073 3396 3493 3503 - 293 391 - -
Hf 5673 5673 5470 4873 4876 648 571 661 661 -
Ir 5573 5573 4403 4703 4701 636 232 564 564 -
Mo 5833 5833 4885 4913 4912 536 617 594 590 -
Sb 1653 1653 1908 1860 1860 195 193 67.9 67.9 -
Si 2633 2633 2628 3533 3538 300 359 383 359 -
Zr 3853 3853 4650 4673 4682 502 573 582 591 -
Journal of Chemical & Engineering Data, Vol. 56, No. 2, 2011 333
values used for training ANNs 1 and 2 are all referenced to the
boiling point (with the exception of four elements). Values for
the enthalpy of vaporization of Ho at the boiling point are
available:
21,23
280 kJ ·mol-1(67 kcal ·mol-1) and 270 kJ ·mol-1
(64.7 kcal ·mol-1). These values are close to the value predicted
by the ANN (which is 299 kJ ·mol-1), and this demonstrates
the remarkable discernment of the ANN in detection of
inconsistency and identification of true values.
Se. The handbook values are (14, 26.3, and 95.5) kJ ·mol-1,
and the ANN predicts 78.4 kJ ·mol-1from the well-established
boiling point of 958 K. The source of 95.5 kJ ·mol-1can be
found
24
and is supported by Hultgren et al.
18
who gives a value
at 625 K where, of the eight chemical allotropes for gaseous
Se, Se6predominates.
25
The value of 13.8 kJ ·mol-1for atomic
Se corresponds to 82.8 kJ ·mol-1for Se6.
18
This may explain
the handbook value of 14 kJ ·mol-1if that value is for atomic
Se. The inconsistent handbook value of 26.3 kJ ·mol-1looks
like a transcription of units error of Se6by 4.18 J ·cal-1(i.e.,
should be 26.3 kcal ·mol-1or 82.8 kJ ·mol-1). The selected value
of 95.5 kJ ·mol-1still deviates by more than 10 % from the
ANN predicted value 78.4 kJ ·mol-1. A possible reason for this,
as discussed by Bagnall
25
and Reid et al.,
26
is associated with
constants in the Clapeyron equation and is described in the
following section.
Factors that Affect the Accuracy in the Prediction. In this
section, the factors that affect the accuracy of prediction of
boiling points and enthalpies of vaporization in this case can
be identified as follows.
1. Determination of the Constant after Clausius-Clapeyron
Integration. As mentioned in Reid et al.,
26
it is not easy to
trace the origin of many experimental enthalpies of vaporiza-
tion. A few were determined from calorimetric measurements,
but in a large number of cases the values were obtained
directly from Clausius-Clapeyron equation (d ln Pvap)/(d(1/
T)) )-(Hvap)/(RZvap), in which the Zvap were determined
separately, and (d lnPvap)/dTwas found by numerical dif-
ferentiation of experimental vapor pressure data or by
differentiating some Pvap-Tcorrelation analytically. The
constants in one equation may be optimized for correlating
vapor pressures, but it does not necessarily follow that these
same constants give the best fit for computing slopes. For
this reason, the uncertainty is present in using any analytical
vapor pressure-temperature equation to obtain accurate
values of slopes (d lnPvap)/dT.
Table 7. List of Category IV Elements: Both Boiling Points and Enthalpies of Vaporization Are Consistent
TbvapHmTbvapHm
element K kJ·mol-1element K kJ ·mol-1
Ar 87.15 6.53 K 1047 79.1
As 886.2 32.4 La 3743 400
Au 3243 342 Li 1603 136
Br 331.7 30.0 Lu 3603 414
Ca 1760 153 Mg 1383 132
Cd 1038 100 Mn 2373 225
Cl 238.5 20.4 N 77.15 5.58
Co 3173 390 Nd 3303 289
Cr 2755 347 Ne 27.15 1.90
Cs 963.2 66.1 Ni 3003 379
Cu 2868 305 O 90.15 6.82
Dy 2873 280 Pb 2017 177
Er 3173 280 Pt 4100 510
Eu 1713 176 Rb 961.2 69.0
F 85.15 6.32 Ru 4423 619
Fe 3273 354 Sc 3003 310
Ga 2673 256 Sr 1653 141
Gd 3273 301 Ta 5693 753
Ge 3103 330 Ti 3533 427
H 21.15 0.90 Tl 1733 162
He 4.15 0.0840 W 6203 774
Hg 630.2 58.2 Xe 165.2 12.6
In 2273 225 Zn 1180 115
Table 8. List of Category I Elements with Predicted and Selected
Correct Values of Enthalpy of Vaporization and Difference
Percentage
Tb
predicted enthalpy
of vaporization
selected enthalpy
of vaporization
element K kJ ·mol-1kJ ·mol-1
difference
percentage
Na 1163 97.6 97.4 0.21 %
Al 2743 281 284 1.06 %
B 4203 499 508 1.77 %
Os 5273 664 678 2.06 %
Kr 121.2 8.8 9.03 2.55 %
Ag 2483 246 254 3.15 %
Te 1263 107 114 6.14 %
Bi 1833 168 179 6.15 %
Re 5903 752 707 6.36 %
Ce 3743 427 398 7.29 %
Sm 2173 207 192 7.81 %
I2457.2 35.1 41.6 15.6 %
P (white) 553.2 43 51.9 17.2 %
Se 958.2 78.4 95.5 17.9 %
Ho 2873 299 251 19.1 %
S 718.2 57 45 26.7 %
average 8.81 %
Table 9. List of Category II Elements with Predicted and Selected
Correct Values of the Boiling Point and Difference Percentage
vapHm
predicted
boiling point
selected
boiling point
element kJ ·mol-1KK
difference
percentage
Pd 380 3221 3233 0.37 %
Nb 694 5078 5017 1.22 %
Yb 159 1671 1703 1.88 %
Rh 531 4137 4000 3.43 %
Y 390 3317 3203 3.56 %
C (graphite) 715 5326 5103 4.37 %
Be 309 3100 3243 4.41 %
Sn 290 3060 2893 5.77 %
V 444 3939 3680 7.04 %
Pr 331 3081 3403 9.46 %
Ba 149 1600 1910 16.2 %
Tm 247 2655 2223 19.4 %
average 6.43 %
334 Journal of Chemical & Engineering Data, Vol. 56, No. 2, 2011
2. Enthalpy of Vaporization Varies with Temperature.
vapHmis always treated as a weak function of temperature,
and an assumption is made that this value does not vary with
temperature
26,27
during the integration of the Clausius-Clapeyron
equation. However, it is not true for each element.
18
For some
elements, the variation is narrow, such as Ga (from 258
kJ ·mol-1at normal boiling point to 272 kJ ·mol-1at room
temperature, 5.2 % difference), Zr (from 58 kJ ·mol-1at normal
boiling point to 61 kJ ·mol-1at room temperature, 5.0 %
difference), W (from 823 kJ ·mol-1at normal boiling point to
Table 10. List of Category III Elements with Predicted and Selected Correct Values and the Difference Percentage (Only the Corrected Pairs
Are Shown)
predicted enthalpy
of vaporization
selected enthalpy
of vaporization
predicted
boiling point
selected
boiling point
element kJ ·mol-1kJ ·mol-1difference percentage K K difference percentage
Hf 604 648 6.79 % 4610 4876 5.46 %
Ir 530 564 6.03 % 4158 4403 5.56 %
Mo 605 617 1.94 % 4375 4885 10.4 %
Sb 176 193 8.81 % 1968 1908 3.14 %
Si 395 383 3.13 % 3250 3533 8.01 %
Tb 374 391 4.35 % 3328 3396 2.00 %
Zr 569 591 3.72 % 4236 4650 8.90 %
Table 11. List of “Corrected” Values (Including Data from Category IV and Corrected Category I, II, and III)
TbvapHmTbvapHmTbvapHm
element K kJ ·mol-1element K kJ ·mol-1element K kJ ·mol-1
Ag 2483 254 H 21.15 0.9 Pr 3403 331
Al 2743 284 He 4.15 0.08 Pt 4100 510
Ar 87.15 6.53 Hf 4876 648 Rb 961.2 69
As 886.2 32.4 Hg 630.2 58.2 Re 5903 707
Au 3243 342 Ho 2873 251 Rh 4000 531
B 4203 508 I2457.2 41.6 Ru 4423 619
Ba 1910 149 In 2273 225 S 718.2 45
Be 3243 309 Ir 4403 564 Sb 1908 193
Bi 1833 179 K 1047 79.1 Sc 3003 310
Br 331.7 30 Kr 121.2 9.03 Se 958.2 95.5
C (graphite) 5103 715 La 3743 400 Si 3533 383
Ca 1760 153 Li 1603 136 Sm 2173 192
Cd 1038 100 Lu 3603 414 Sn 2893 290
Ce 3743 398 Mg 1383 132 Sr 1653 141
Cl 238.5 20.4 Mn 2373 225 Ta 5693 753
Co 3173 390 Mo 4885 617 Tb 3396 391
Cr 2755 347 N 77.15 5.58 Te 1263 114
Cs 963.2 66.1 Na 1163 97.4 Ti 3533 427
Cu 2868 305 Nb 5017 694 Tl 1733 162
Dy 2873 280 Nd 3303 289 Tm 2223 247
Er 3173 280 Ne 27.15 1.9 V 3680 444
Eu 1713 176 Ni 3003 379 W 6203 774
F 85.15 6.32 O 90.15 6.82 Xe 165.2 12.6
Fe 3273 354 Os 5273 678 Y 3203 390
Ga 2673 256 P (white) 553.2 51.9 Yb 1703 159
Gd 3273 301 Pb 2017 177 Zn 1180 115
Ge 3103 330 Pd 3233 380 Zr 4650 591
Figure 2. Prediction of (a) enthalpy of vaporization (vapHm) from the boiling point (Tb); (b) the boiling point (Tb) from the enthalpy of vaporization
(vapHm); using “consistent” data. O, training data points; /, test data points; s, best linear fit; ----, exp. )prediction. The general correlation performance
has been increased, and the values of M(the slope of the linear regression line) and Rare greater being M)0.99, R)0.994 and M)0.993, R)0.995
(for forward and backward predictions, respectively).
Journal of Chemical & Engineering Data, Vol. 56, No. 2, 2011 335
849 kJ ·mol-1at room temperature, 3.1 % difference) but for
others, the variation is wide, and the value of vapHmdecreases
with rising temperature, such as Ba (from 141 kJ ·mol-1at the
normal boiling point to 182 kJ ·mol-1at room temperature, 28.7
%) and Tm (from 191 kJ ·mol-1at the normal boiling point to
232 kJ ·mol-1at room temperature, 21.7 %). The data for the
enthalpy of vaporization are recorded in many handbooks in a
mixed fashion without mention of the temperatures to which
the values apply. This introduces uncertainties in finding the
correlation and reduces the accuracy of predictions. After
comparing the values in Category IV with the values recorded
in Selected Values of the Thermodynamic Properties of the
Elements,
18
which tabulates enthalpies of vaporization over a
temperature range, it is found that all but four (Dy, Eu, Gd,
and Lu) are recorded at the normal boiling points. Thus, Pb
(Category IV), for example, has a value 175 kJ ·mol-1at the
boiling point and 192 kJ ·mol-1at room temperature,
18
and we
infer that 177 kJ ·mol-1is the enthalpy of vaporization at the
boiling point. Thus, neural networks (ANN1 and ANN2) have
found the correlation between normal boiling points and the
enthalpy of vaporization under normal boiling points.
3. Clausius-Clapeyron Equation Is Not the Only
Equation for Estimating Enthalpies of Vaporization. Other
methods such as Pitzer’s acentric factor correlation, Riedel’s
method, Chen’s method, and Vetere’s method are also used.
More accurate estimates may be obtained when specific cor-
relations are employed and demand recourse to original sources.
These factors mean that differences between predicted and
corrected values cannot be avoided. The first and third factors
account for small differences (<10 %) since these just affect
the accuracy of the enthalpies of vaporization. The large
deviations (>10 %) may be attributed to the second factor. For
Category I, the problems of the second factor do not attend the
records of boiling point so the prediction of enthalpy of
vaporization for these elements from the consistent boiling point
can be treated as reliable. However, the second factor affects
predictions for the elements in Category II; so enthalpies of
vaporization used for these elements were rechecked with the
values recorded in Selected Values of the Thermodynamic
Properties of the Elements,
18
and another set of predictions
based on the enthalpy of vaporization referenced to the normal
boiling point was made. The results are shown in Table 13.
From the comparison of the results in Table 9 with Table
13, it can be seen that most of the boiling points are the same,
except for Yb and Tm. As a result, it is interesting to analyze
which prediction is more sensible.
For Yb, 159 kJ ·mol-1is the value corresponding to room
temperature according to Selected Values of the Thermodynamic
Properties of the Elements on page 564,
18
which gives 129
kJ ·mol-1at the normal boiling point and 152 kJ ·mol-1at room
temperature. A higher value of enthalpy of vaporization at the
normal boiling point corresponds to a higher value of the normal
boiling point, so 159 kJ ·mol-1was used to predict the boiling
point giving the higher value of boiling point for Yb. Two
factors need to be considered: (1) For different elements, those
having higher boiling points always have higher enthalpies of
vaporization at the boiling point; (2) for a given element, the
enthalpy of vaporization varies inversely with temperature. For
Yb, the boiling point is higher than ambient, and so the enthalpy
of vaporization at the boiling point is lower than the value at
ambience. Previously, the enthalpy of vaporization at room
temperature was used to predict the boiling point, and the first
factor means the boiling point was overestimated. Now the
enthalpy of vaporization at the boiling point is employed, and
the boiling point prediction is correct. Using 129 kJ ·mol-1, the
corresponding boiling point is 1460 K, which is closer to 1466
K (within 0.40 %), and this value is confirmed by the work of
Habermann and Daane.
28
In their work, the vapor pressures of
the rare-earth metals were measured by the Knudsen effusion
technique using a quartz-fiber microbalance, and then a com-
bination of second and third law methods were used to calculate
the normal boiling point for each rare-earth metal, and for Yb
this value is 1466 ((5) K.
For Tm, 247 kJ ·mol-1is the value corresponding to ambient
temperature, according to Selected Values of the Thermodynamic
Properties of the Elements on page 533,
18
which gives 191
kJ ·mol-1at the normal boiling point and 232 kJ ·mol-1at room
temperature and is greater than the value recorded at the normal
boiling point, which is about 191 kJ ·mol-1. For similar reasons,
247 kJ ·mol-1was used to predict the boiling point producing
a higher value. Using the value of 191 kJ ·mol-1, it is found
that the boiling point is 1943 K, which is closer to 2003 K
(within 3.00 %) and is consistent with the value obtained by
Spedding et al. after purifying this element at the Ames
Laboratory of the U.S. Atomic Energy Commission,
29
which
was 2000 K.
From this analysis, the prediction of boiling point in these
cases is more justifiable than before. So it emerges that although
there are several factors, especially the second, that may mislead
Table 12. Statistical Analysis for ANN Performance in Figures 1 and 2
conditions
test set whole set
MR
mean of
error
modulus
SD of error
modulus
mean of
percentage
error
modulus/%
SD of
percentage
error
modulus/% MR
mean of
error modulus
SD of error
modulus
mean of
percentage
error
modulus/%
SD of
percentage
error
modulus/%
Figure 1a 0.991 0.943 38.1 kJ ·mol-144.1 kJ ·mol-135.4 57.2 0.96 0.973 32.4 kJ·mol-136.9 kJ ·mol-1114 720
Figure 1b 0.982 0.954 409 K 354 K 36.9 64.1 0.963 0.972 269 K 284 K 45.9 139
Figure 2a 0.986 0.995 18.1 kJ ·mol-114.8 kJ ·mol-110.1 6.38 0.99 0.994 16.7 kJ ·mol-115.7 kJ ·mol-148.5 308
Figure 2b 1 0.986 196 K 167 K 35.9 109 0.993 0.995 119 K 108 K 14.7 50.5
Table 13. List of Category II Elements with Predicted and Selected
Correct Values of the Boiling Point and Difference Percentage
(Record of Enthalpies of Vaporization under the Normal Boiling
Point)
vapHm
predicted
boiling point
selected
boiling point
element kJ ·mol-1KK
difference
percentage
Yb 129 1460 1466 0.41 %
Nb 682 4944 4973 0.58 %
Y 363 3111 3203 2.87 %
C (graphite) 709 5254 5103 2.96 %
Tm 191 1943 2003 3.00 %
Rh 493 4144 4000 3.60 %
Pd 358 3093 3233 4.33 %
Be 292 3068 3243 5.40 %
Sn 296 3081 2893 6.50 %
V 451 3995 3680 8.56 %
Pr 297 3084 3403 9.37 %
Ba 142 1551 1910 18.8 %
average 5.53 %
336 Journal of Chemical & Engineering Data, Vol. 56, No. 2, 2011
the neural network method, when care is taken and critical
surveys are employed, it turns out to be robust and reliable.
Conclusions
The data in handbooks were thought to be very reliable,
especially the data for elements in the periodic table. The ANN
is able to find out many inconsistencies by a systematic and
automatic method and the development of this technique. Its
wider application and, more interestingly, its integration into
databases as an immune system suggest that it may no longer
be necessary to accept error rates at 5 %. The boiling point and
enthalpy of vaporization of most elements (except radioactive
elements) have been corrected in this paper, and the errors have
brought down from a maximum of 900 % to less than 10 %.
Factors affecting the accuracy of the prediction have been
discussed.
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Received for review October 26, 2010. Accepted December 15, 2010.
The authors are grateful to the School of Engineering and Materials
Science of Queen Mary, University of London, and the Central Research
Fund from the University of London (ref.: AR/CRF/B), to support this
work by providing a research studentship and research funds to Y.M.Z.
JE1011086
Journal of Chemical & Engineering Data, Vol. 56, No. 2, 2011 337
... Thermal inertia is defined as the thermal capacity of a material to store heat and to delay its transmission [30]. Thus, for a fixed Fig. 12. Temperature dependence of the specific enthalpy of tungsten given in unit of MJ/kg [28,29]. total heat capacity, a reduced heat conductivity is favored to mitigate the heat impact on the heat sink over the prolonged heat front travel time. ...
... Specific isobaric heat capacity c p (measured in J/(mol K))c p,s (T) = 21.87 + 8.07⋅10 − 3 ⋅T − 3.76⋅10 − 6 ⋅T 2 + 1.08⋅10 − 9 ⋅T 3 + 1.41⋅10 4 ⋅T − 2 (25 ∘ C ≤ T ≤ 2807 ∘ C)c p,s (T) = 2.02 + 1.32⋅10 − 2 ⋅T(2807 ∘ C ≤ T ≤ 3422 ∘ C) m (molar enthalpy of melting at 3422 • C): 46-54 kJ/mol (recommended value: 52 kJ/mol) ΔH v (molar enthalpy of vaporization at 5930 • C): 774-807 kJ/mol[29] ...
Article
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... Following the breakdown, the arc discharge, which gets its energy from the capacitors, occurs. Caused by the high sublimation temperatures of 3003K [14] for nickel and the low current of about 600mA, the maximum diameter of the cathode spots is 10 to 30µm [15] [2]. With the diameter as a measure for the plasma channel of the arc discharge, current densities between 8.5*10 4 und 7.6*10 5 A/cm² can be estimated. ...
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Thesis
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