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. Submit your article to this journal Article views: 1 View related articles Citing articles: 3 View citing articles Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=unct20 Download by: [University of Florida] 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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