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Nuclear Technology
ISSN: 0029-5450 (Print) 1943-7471 (Online) Journal homepage: http://www.tandfonline.com/loi/unct20
Smoothed Values of the Enthalpy and Heat
Capacity of UO2
Jerry F. Kerrisk & David G. Clifton
To cite this article: Jerry F. Kerrisk & David G. Clifton (1972) Smoothed Values of the Enthalpy
and Heat Capacity of UO2, Nuclear Technology, 16:3, 531-535, DOI: 10.13182/NT72-6
To link to this article: http://dx.doi.org/10.13182/NT72-6
Published online: 10 May 2017.
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Date: 25 October 2017, At: 17:44
SMOOTHED VALUES OF THE ENTHALPY
AND HEAT CAPACITY OF U02
MATERIALS
KEYWORDS: uranium clioxicle,
enthalpy, specific heat, calorimeters, high temperature,
temperature measurement, very
high temperature, temperature
c/epenc/ence, equations
JERRY F. KERRISK and DAVID G. CLIFTON
Los Alamos Scientific Laboratory,
Los Alamos, New Mexico 87544
Downloaded by [University of Florida] at 17:44 25 October 2017
Received May 11, 1972
Revised August 7, 1972
U02 enthalpy data measured by five different
investigators were fitted to a theoretically based
equation from room temperature to the melting
point. The equation can be used to calculate enthalpy, and differentiated to obtain heat capacity
in this temperature range. Two of the constants
in the equation are related to properties of the
uo2 lattice.
1. represent the enthalpy data within experimental error
I. INTRODUCTION
The enthalpy of U02 has been measured using
drop calorimetry by many different investigators
from room temperature to its melting point. In
some cases experimenters have reported the experimental enthalpy-temperature data, 1 - 6 while in
others only an equation fitted over the range of the
data has been reported. 7 - 9 Even when the observed data are reported, an experimenter generally fits his data to some function of temperature.
This function serves to smooth and interpolate the
data, or as is occasionally done to extrapolate the
data outside the temperature range of the measurements. The most common functions used are
polynomials in temperature and inverse temperature .
In applications, the equations representing the
enthalpy as a function of temperature are often
differentiated to obtain the heat capacity. This is
necessary because, for example, heat transfer
schemes used in computer codes usually require
the heat capacity to be supplied . When the heat capacity curves for U02 derived from the fits of enthalpy as a function of temperature are compared
NUCLEAR TECHNOLOGY
in a temperature range where they overlap, significant differences in the heat capacity ("'10%) and
its curvature are found. 6 • 8 If these heat capacity
functions are extrapolated outside the range where
they were fit, very large differences can be found.
This phenomenon is due to the different functional
forms used by the various experimenters to fit
their data, and the magnification of error in the
slope (heat capacity) of a fitted function (enthalpy) .10 • 11
The purpose of this work is to fit the published
U02 enthalpy data to one equation which can be
used from 298"K to the melting point ( "'3120°K).
The ideal equation would
VOL. 16
DECEMBER 1972
2. be easy to evaluate
3. when differentiated present a physically
meaningful form for the heat capacity over
the entire temperature range.
Initial work with polynomial and spline functions
was followed by the use of a more theoretically
based function, which ultimately was chosen as
best meeting these objectives.
Section II briefly describes the enthalpy data
used, and Sec. III discusses the equations examined and the constraints applied to them. Section IV presents the results with an estimate of
the uncertainty in the calculated values and a discussion of the theoretical significance of two of
the constants in the equation.
II. ENTHALPY DATA
The enthalpy data used for the least-squares
fits were obtained from five sources . Table I lists
the temperature range and number of data points
531
Kerrisk and Clifton
ENTHALPY AND HEAT CAPACITY
TABLE I
Downloaded by [University of Florida] at 17:44 25 October 2017
UO:! Enthalpy Data Sources
Reference
Temp
Range ("K)
Number
of Data
Points
Moore and Kelley 1
Ogard and Leary2 • 3
Hein, Sjodahl, and Szwarc 4
Liebowitz, Mishler, and Chasanov 5
Frederickson and Chasanov 6
483 to 1462
1338 to 2303
1174to3107
2557 to 3083
675 to 1434
14
13
31
12
24
from each investigator. All temperatures were
reported in degrees Kelvin, and all enthalpies
were reported as cal/mole relative to 298"K, i.e.,
H(T) - H(298), where H(T) is the enthalpy (cal/
mole) at T(°K).
Moore and Kelley1 measured the enthalpy of
one specimen of U02. They did not report enthalpy
measurements on standard materials in the same
apparatus, nor did they report any verification of
the stoichiometry of their specimen aside from a
measured uranium content.
Ogard and Leary2 •3 measured the enthalpy of
two specimens of U0 2 • The specimens were characterized by chemical analysis and a lattice
parameter measurement which indicated stoichiometric U02. Enthalpy measurements of a-Ah03
were also made in the same apparatus.
Hein, Sjodahl, and Szwarc 4 measured the enthalpy of two specimens of U02. The 0/U ratio
measured before and after the enthalpy measurements was 2.003 ± 0.003 by chemical analysis.
The enthalpies of sapphire and tungsten were determined concurrently with the U02 measurements
but these data were not reported.
Leibowitz, Mishler, and Chasanov 5 measured
the enthalpy of uo2 which was characterized by
chemical analysis (0/U = 2.015) and lattice parameter measurement. Specimens were heated to
look for lattice parameter changes and tungsten
(capsule material) solubility. No enthalpy measurements on standard materials were reported at
any temperature. (In the t e m p e r at u r e range
covered by this investigation no accurate standards exist.)
Frederickson and Chasanov6 measured the enthalpy of two specimens of U02 with an 0/U ratio
of 2.005. Enthalpy measurements were made on
sapphire before and after the U02 measurements.
Two questions that arise about these data are
(a) whether they should be combined, and (b) if
they are combined whether any of the sets should
be weighted relative to others. The only evident
difference among the specimens is a variation in
stoichiometry which was not deemed large enough
532
to preclude a combined treatment. An examination
of the data indicates some systematic differences
among the various investigators. For example,
the measurements of Moore and Kelley are 1 to
2% greater than those of Frederickson and Chasanov in the temperature range from 700 to 1000°K,
even though the internal precision of both sets of
data is better than 1%. Since Moore and Kelley did
not report Ah03 data, a further comparison is not
possible. At 1400°K, the low temperature data of
Ogard and Leary a~e 1 to 2% greater than the high
temperature data of Frederickson and Chasanov.
In this case, both investigators reported the enthalpy of Ab03, and their calculated values are
within 0.2% at 1300 and 1400°K. Based on the information available it does not seem possible to
determine whether these systematic differences
are mainly related to differences in the specimens
or whether they are due to the use of different
apparatus. For the least-squares fits it was decided not to preferentially weight any experimenters data more than the others. Thus each
measurement was given equal significance.
Ill. EQUATIONS
The form of the equation sought is
H(T)- H(298)
= t:.H(T) = F(T;P;)
(1)
where the P; are parameters whose values are
chosen by the least-squares procedure. The range
of T of interest is from 29SOK to the melting point
("'3120°K). Irrespective of the form ofF, certain
constraints are necessary or desirable. Since
t:.H(298) =0, we expect F(298;P;) = 0 for any choice
of the P;. A second constraint that is desirable,
but not necessary, is fixing Cp(T) =(dt:.H jdT) at
T = 298°K, It is possible to allow the leastsquares procedure to determine Cp(298) but this
ignores a body of independent measurements of
12 13 Also, the data
Cp(T) at low temperatures. '
fitted here are quite sparse below 700°K which
would result in a poor determination of Cp(298).
For these reasons Cp(298) was fixed, and the value
chosen by the IAEA in 1965 was used; Cp(298) =
15.2 cal/mole°K. 14
A least-squares fit where each data point is
equally weighted assumes the variance of the dependent variable is constant over the range of the
data. Some initial fits were done this way, but an
examination of the deviation (observed-calculated
enthalpy) plots indicated that the magnitude of the
deviations increased with increasing temperature
(or enthalpy). Further examination showed that
the percentage deviations were of the same magnitude over the entire range. Experimentally this
would relate to the experimental error being approximately a constant percentage of the enthalpy
NUCLEAR TECHNOLOGY
VOL. 16
DECEMBER 1972
Downloaded by [University of Florida] at 17:44 25 October 2017
Kerrisk and Clifton
over the range of data. Since this is a reasonable
experimental situation, and the data scatter seems
to substantiate it, further fits were performed
using weighting factors proportional to the reciprocal of the square of the enthalpy. In practice
this amounts to minimizing the sum of the squares
of the percentage deviations.
Initially polynomial and spline functions were
fit to the data since they are easy to evaluate.
Both types of functions provided an adequate fit of
the enthalpy-temperature data but when differentiated to obtain the heat capacity-temperature
function, the results were less pleasing. In particular the heat capacity curve exhibited a maximum near 2800 to 300°K, decreasing with
increasing temperatures beyond this point. Although many substances with heat capacity peaks
are known, the physical processes postulated for
15 16 Rather
U02 do not predict this phenomenon. •
than try to restrict the behavior of empirical
functions, a more theoretically based function was
sought.
The general shape of the heat capacity curve
was similar for all the polynomial and spline fits.
Below "'1400°K the curve looked like the Cp of a
traditional solid, but above this temperature Cp
increased very rapidly. Prior to the rapid rise,
the heat capacity should be represented by Cp =
Cv + CE, where Cv is the heat capacity at constant
volume which is generally given by the Debye
function, and CE = (Cp- Cv) = a 2 VT/1], where a=
1/V(oV /oT)p, 17 = -1/V(OV /oP)y, P is the pressure,
and Vis the molar volume. 17 Rather than use the
Debye function to represent Cv, it was decided to
use the Einstein function since the Einstein function is more easily evaluated and does represent
Cv data away from the low temperature region. To
approximate CE, the expansion contribution, we
assumed CE proportional to T.
The reason for the rapid rise in the heat capacity of uo2 at high temperatures has been discussed by a number of authors who have attributed
it to defect formation in the U02 lattice. 15 •16 The
excess heat capacity associated with defect formation, Cv, can be written as
A
Co (T) = T 2 exp(-E0 /RT) ,
(2)
where E 0 is the energy of formation of a defect
and A is a constant .16
The total heat capacity can be written as the
sum of Cv, CE, and C0 ,
+ K~f exp( -E0 /R T)
(3)
RT
NUCLEAR TECHNOLOGY
VOL. 16
DECEMBER 1972
ENTHALPY AND HEAT CAPACITY
where 8 is the Einstein temperature and K1, K2,
and K3 are constants. Since the data to be fitted
are enthalpy data, the heat capacity function must
be integrated. This results in
T
1
I).H(T) = 98 Cp(T) dT ""K18 {[exp(B/T) 1} + K2(T 2 - 298 2)
-[exp(e /298) -
h
1r
1r
+ K3 exp(-E0 /RT)
(4)
where the constants e, E 0 , K1, and K2, and K3 are
to be determined by least-squares fitting. It
should be pointed out that Eq. (4) is only an approximation to the integral since the term due to
the evaluation of the integral of Co at the lower
limit (298°K) was dropped as a simplification because its contribution to I).H(T) was expected to be
very small. The actual contribution of this term
is on the order of 10- 18 cal/mole with the final
values of K3 and E 0 , so that its neglect is justifiable.
Equation (4) as written constrains AH(298) to
be zero (to the order described above), but Cp(298)
is not fixed. From Eq. (3) we have (neglecting C0 )
C (298) =
P
K 182 exp(B/298)
+ 2 (298) ](.2
(298) 2 [exp(8/298) - 1]2
or
K = (298) 2 [Cp(298) - 2(298) K2 ][exp(8/298) - 1]2
1
2
8 exp(B/298)
(5)
as a relation between K1 and K2 necessary for the
calculated heat capacity at 298°K to be Cp(298).
Thus Eq. (4) with the constraint required by Eq.
(5) must be least-squares fitted to the enthalpytemperature data. There are five parameters in
Eq. (4) but the constraint of Eq. (5) relates two of
the parameters so that there are four free parameters to be determined by the least-squares fitting.
IV. RESULTS
Equation (4) was least-squares fitted to the
enthalpy-temperature data described in Sec. II.
Table II lists the values of the constants calculated by the least-squares program.18 Figure 1
shows a plot of the enthalpy data as a function of
temperature along with the calculated curve. Figure 2 shows a plot of the actual and percentage
deviations of the data from the calculated curve.
The systematic differences between the various
sets of data are evident from this plot. The maximum deviation of the data from the calculated
curve was 2. 7%.
Figure 3 shows a plot of the heat capacity, calculated from Eq. (3), as a function of temperature.
533
Kerrisk and Clifton
ENTHALPY AND HEAT CAPACITY
In addition, curves of measured heat capacity from
three different sources are shown for comparison12'13'16. None of these data were used in the
least-squares fit. The agreement between the heat
capacity calculated from this fit and the heat capacity measurements below 1000°K is very good.
The U02.oo curve of Affortit and Marcon is from
4% (at 1600°K) to 22% (at 3000°K) below that obtained here.
A useful result of this work would be the assignment of a statistically derived uncertainty or
confidence limits in the calculated enthalpy and
heat capacity values. One can carry out the calculation but in this case the result would be deceiving. One of the major assumptions necessary
Downloaded by [University of Florida] at 17:44 25 October 2017
Eo
K1
K2
K3
~0
0
0
<
cP
0
o
0
0
cog
0
X
10- 4
10 6
--X
X
e
oo
il
A
Vtf
A
a
A~
6
A A
vv
o" A
0
A
A
"
0
0
A
A
v<f.V
A
A
__. -800
~
"A
A
v
A
0
A
v
oV
0
t::;-1200
<
3
0
2
0
0
@
8
0
'b
9 0 Do
0
0
A
00
&0
0
"
B
0
0
Dc o 0
A
A
0 A
0
0
vv
v
fl
AA
A
6
A§
;;, o il ~
oo o
ll
-2
v
v
0
0
o..O
1.38
2.04
0
'0
0 00 0
0
golll f.P;;tt~o
" v <fv
AI?,
"" "
"
A
6 A
"
"
A
v
"
"
v
ov
0
0
-3
200
600
1000
8.86
2.39
X
0
Q:)
0
-400
Standard
Deviation
Value
"K
535.285
kcal/mole
37.6946
cal/mole "K
19.1450
7.84733
cal/mole "K2
cal/ mole
5.64373
6
z' 400
0
;:::
v
0
z;:::
Constants in Eqs. (3) and (4)
Units
v
0
800
w
t!>z 0
<o
w<
u- -I
o<>
ww
TABLE II
Constant
"
~ 1200
~
0
u
Fig. 2.
45
60
40
2600
3000
Deviations and percentage deviations of measured enthalpy from the calculated curve. Data
point legend shown on Fig. 1.
10- 4
10 6
MOORE AND KELLEY
OGARD AND LEARY
" HEIN,SJODAHL AND SZWARC
70
v LEIBOWITZ, MISHLER AND CHASANOV
o FREDRICKSON AND CHASANOV
1400
1800 2200
TEMPERATURE, °K
--THIS WORK
----- HUNTZICKER AND WESTRUM
------ GR0NVOLD et al.
----- AFFORTIT AND MARCON
o
o
if
0"
s8 35
~50
~
0u
>-'
>-' 40
5 30
~
.....
6
~
t:
Q..
__.
z
w
.....
~
30
25
J:
20
20
15
I
10
I
I
I
I
I
I
I
100
o~~~--~~~--~~~--L-~-L--L-~-L~
200
600
1000
1400 1800
TEMPERATURE, °K
2200
2600
3000
Fig. 3.
Fig. 1.
534
Enthalpy of UOa as· a function of temperature.
I
400
800
1200
1600 2000
TEMPERATURE, °K
2400
2800 3200
Heat capacity of U0 2 as a function of temperature.
NUCLEAR TECHNOLOGY
VOL. 16
DECEMBER 1972
Downloaded by [University of Florida] at 17:44 25 October 2017
Kerrisk and Clifton
to obtain a meaningful statistical uncertainty in a
fitted curve or its derivitive is that the deviations
of the data from the curve (or the percentage
deviation as used here) are normally distributed. 19
Due to the systematic differences between the
various sets of data (see Fig. 2), the actual distribution is bimodal rather than normal. Confidence
limits based on such data can only be considered
as order of magnitude estimates. Table III lists
9 5% confidence limits (± 2 standard deviations)
calculated for the smoothed values of enthalpy and
heat capacity. 18 The limits at 298"K are zero
since both ~H(298) and Cp(298) were fixed.
The main objective of this work was to generate an equation for the enthalpy and heat capacity,
not to estimate parameters of the U02 lattice. But
the equation used has some theoretical basis, in
particular the parameters e and Ev. The Einstein
temperature (535"K) obtained here compares very
well with the value of 542"K calculated for the
oxygen sublattice by the IAEA panel14 from neutron diffraction data. This is in line with the estimate that the major contribution to the lattice heat
capacity above 300"K comes from the oxygen sublattice. The form of the excess heat capacity due
to defect formation, Eq. (2), follows the work of
Affortit and Marcon, 16 and Szwarc .15 The value of
Ev obtained here (37. 7 kcaljmole) is in good
agreement with the values obtained by Affortit and
Marcon 16 from their heat capacity data; 38.4 kcalj
mole for uo2.00, and 32.8 kcal/mole for uo2.04•
Szwarc 15 estimated 2 Ev using a portion of the data
treated here and obtained 71.3 to 72.5 kcal/mole.
The implications of this estimate were discussed
by Szwarc.15
ENTHALPY AND HEAT CAPACITY
REFERENCES
1. G. E. MOORE and K. K. KELLEY, J. Am. Chem.
Soc., 69, 2105 (1947).
2. A. E. OGARD and J. A. LEARY, "Thermodynamics
of Nuclear Materials," Proc. IAEA Symp., September
4-8, 1967, Vienna, p. 651, International Atomic Energy
Agency, Vienna (1968).
3. A. E. OGARD and J. A. LEARY, Quarterly Status
Report on the Advanced Plutonium Fuels Program,
April 1-June 30, 1970, and Fourth Annual Report,
LA -4494-MS', p. 43, Los Alamos Scientific Laboratory
(1970).
4. R. A. HEIN, L. H. SJODAHL, and R. SZWARC,
J. Nucl. Mater., 25, 99 (1968).
5. L. LEIBOWITZ, L. W. MISHLER, and M. G.
CHASANOV, J. Nucl. Mater., 29, 356 (1969).
6. D.
R.
FREDERICKSON and M. G. CHASANOV,
J. Chem. Thermodynamics,2, 623 (1970).
7. R. A. HEIN, P. N. FLAGELLA, and J. B. CONWAY,
J. Am. Ceram. Soc., 51, 291 (1968).
8. T. K. ENGEL, J. Nucl. Mater., 31,211 (1969).
9. J. B. CONWAY and R. A. HEIN, J. Nucl. Mater.,
15, 149 (1965).
10. H. C. JOKSCH, SIAM Review, 8, 47 (1966).
11. H.
THOMANN and G. LINDSJO, Intern. J. Heat
Mass Trans., 9, 1455 (1966).
12. J.
J.
HUNTZICKER and E. F. WESTRUM, Jr.,
J. Chem. Thermodynamics,.3, 61 (1971).
13. F. GR<Z\NVOLD, N. J. KVESETH, A. SVEEN, and
J. TICHY, J. Chem. Thermodynamics, 2, 665 (1970).
TABLE III
Smoothed Enthalpy and Heat Capacity
Confidence Limits
95% Confidence Limitsa
Temp
("K)
Enthalpy
Heat Capacity
298
500
1000
1500
2000
2500
3000
3100
0
±0.5%
±0.5%
±0.5%
±0.5%
±0.5%
±0.7%
±1%
0
±2%
±2%
±2%
±2%
±3%
±6%
±10%
14. Thermodynamics and Transport Properties of Uranium Dioxide and Related Phases, Technical Report
Series No. 39, p. 23, International Atomic Energy
Agency, Vienna (1965).
15. R. SZWARC, J. Phys. Chem. Solids, 30, 705 (1969).
16. C. AFFORTIT and J. MARCON, Rev. Int. Hautes
Temper. et Refract., 7, 236 (1970).
17. A. H. WILSON, Thermodynamics and Statistical
Mechanics, p. 149, Cambridge University Press, Cambridge (1960).
18. R. H. MOORE and R. K. ZEIGLER, "The Solution
of the General Least Squares Problem with Special
Reference to High-Speed Computers," LA-2367, Los
Alamos Scientific Laboratory (1960).
a±2 standard deviations.
ACKNOWLEDGMENTS
This work was performed under the auspices of
the U.S. Atomic Energy Commission.
NUCLEAR TECHNOLOGY
VOL. 16
DECEMBER 1972
19. A. HALD, Statistical Theory with Engineering Applications, p. 526, John Wiley and Sons, New York
(1952).
535
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