Thermodynamic functions of uranium carbides have been extensively reviewed by Holley etal.4 and, more recently, by Chevalier and Fisher.10 Numerical data are reported in Tables 5 and 6 and plotted in Figures 14 and 15.

A few authors measured the heat capacity of UC from low to high temperature. Holley et al4 assessed
the temperature coefficient g of the electronic heat capacity (18.9 ± 1mJ K-2 mol — ), the Debye temper­ature 0D = 328 K, and the high-temperature behavior for 298 K < T< 2780 K.

Most ofthe U and Pu carbides show steep increase in heat capacities at temperatures above 0.6 T11, attributed to the formation of defects.4

The 0 K randomization entropy is zero for stoi­chiometric UC, but an additional term S(0) = Rxln x should be added for nonstoichiometric UC1+x compositions. The formation enthalpy of stoichio­metric UC was also assessed by Holley et al4 Its value is composition-dependent and slightly decreas­ing in the hypostoichiometric carbide, as suggested by the uranium vaporization study by Storms128 and the carbon activity measurements of Tetenbaum and Hunt.129 The UC room-temperature Gibbs energy of formation was calculated from the enthalpy and the standard entropy, and the value AfG (UC, s, 298) = -98.89 kJ mol-1 was proposed by Holley et al. for the reaction U + C = UC. The error affecting this value was estimated to be around 2.1 kJ mol-1 from the uncertainty in the U and C activities, strongly

Table 5 The heat capacity Cp of uranium carbides at atmospheric pressure (in J K 1 mol 1)

Compound T < 10K 10K < T < 300K T > 300K

aNo satisfactory fit for these points, probably due to marked change in slope around 10 K.

Table 6 Thermodynamic functions of uranium carbides (in SI units). Compound A<H°(298) (kJ mol-1) DfG° (J mol-1) S°(298) (J K-1 mol-1) Transition DH° (J mol-1) Bulk modulus B = V-1 (d2E/dV2) (GPa) Critical parameters UC -97.953 -31465.6 — 499.228T + 64.7501 T ln(T) — 7984166/T — 0.0144T2 for 298 K < T < 2780 K 59.123 AmH°= 48900 180est Tc = 8990 K; Pc = 1580 bar; Pc = 1.3159gcm-3 [Gigli] U2C3 182.5 -732.422 — 806.686T + 107.049T ln(T) — 11285627/T- 0.03029T2 for 298 K < T< 2000 K 137.8 208 a-UC2 85.4 ± 4.2 for UC1 94 21591.6 — 930.689T + 123.806T ln(T) — 13384440/T — 0.03708T2 68.3 Aa! pH° = 10100est 216 — b-UC2 — — AmH°= 67000(R) — — (R)=Richard’s rule and est estimated.

T (K)

Figure 15 Comparison of Gibbs energies of formation of plutonium and uranium carbides. The values correspond to the reported chemical formulae.

dependent on composition and oxygen impurities. Sheth et al.130 proposed DmH°= 48.9k. JmoP1 for the enthalpy of fusion and the following data for liquid UC up to 4800 K:

Cp(UC, liquid) =49.887 + 7.794

x 10~3T (JK^moP1) [8]

H°(T) — H°(298)(UC, liquid) = 51362 + 49887T + 3.987 x 10~3T2(Jmol1) [9]

The recently assessed and optimized Gibbs energy data gave excellent fit with both thermodynamic properties and phase diagram data. Therefore, Gibbs energies of formation of binary compounds of both

U—C and Pu-C systems can be calculated using Gibbs energy functions given by Chevalier and Fischer10 and Fischer,131 respectively. To recalculate the Gibbs energy of formation of the compounds here, the free energy of the pure elements, in their stable reference state at a given temperature, is subtracted from that of the compounds. The following expression can be retained for UC from 298.15 K to the melting point:

Af G°(UC)( Jmol-1) =-31465.6 — 499.228T

+ 64.7501 Tln(T) — 7984166/ T — 0.0144T2 [10]

This temperature dependence of AfG (UC) is shown in Figure 15 and compared with the ones of other ura­nium and plutonium binary carbides.

The partial pressures of the actinide species play an important role in the redistribution of actinides and the restructuring of fuel elements during burnup (Figure 16).

In the case of U-C system, gaseous UCn molecules with n = 1-6 have been detected by mass spectrome­try.8 The partial pressure equations of UC2(g), C1(g), C2(g), and C3(g) are derived from the Gibbs energies
of formation and the activities of uranium and car­bon.4,132-134 In the composition range, C/U = 0.92­1.10, the partial pressure of U(g) is almost equal to the total pressure, the next predominant species being C1(g). The following equations4 can be used to calcu­late the U sublimation enthalpy in single-phase regions on the complete U-C system at 2100 K: 2.34 exp(29x) + 1 exp(- 10(x — 1)) + 1

[11]

exp(40x)+1 — 192.56x + 58.6exp(-100(x — 0.86)2) [12]

x = C/U — 1. The partial pressure of uranium decreases with increasing C/U, showing a steep change in the UC1+x phase field. Correspondingly, the U enthalpy of vaporization increases with C/U up to 711.62 kJ mol-1 at C/U ~ 1.08. The congruent vaporiz­ing composition was recommended as UC1n at 2300 K and UC184 at 2100 K.101 At the melting point,

the sublimation enthalpy is AsubH(UC, 2780) = 661 kJ mol, and the vaporization enthalpy AvapH (UC, 2780) = 611 kJ mol-1. The UC total pressure at the melting point is 2.8 x 10-5 bar.8 Sheth and Leibowitz133 and Joseph eta/.135 calculated the high-temperature vapor pressures of different species over liquid regions of UC1+x and MC + M2C3 systems. Sheth and Leibowitz calculated their values from fusion enthal­pies and Gibbs energies of formation for condensed and vapor species, whereas Joseph et a/. used a semi­empirical method based on the Principles of Corresponding States (PCS).135 They calculated the critical parameters of the compounds and the total vapor pressure over the liquid up to ~9000 K. Interest­ingly, the partial pressures of metal species dominate the vapor phase at low temperatures, but at very high temperatures, T> 4000 K, the partial pressures of carbon-bearing species prevail. Finn eta/.136 provided the following expression to estimate the total pressure (in bar) above liquid UC between 2800 and 10 000 K:

31704

log p = 6.110 — t + 0.197logT [13]

Ohse et a/.137 obtained the boiling point of UC by extrapolation of the vapor pressure curve to 4700 K. Gigli eta/.138 calculated the isotherms and isochores in the pressure-internal energy coordinates between 5200 and 13 600 K, proposing the following critical quantities: Tc = 8990K; pc = 1580bar; pc = 1.3159g cm-3. Thermal expansion of crystalline UC results from the lattice expansion and, at very high tempera­tures, the generation of Schottky defects.8 The follow­ing expression for the linear thermal expansion coefficient aT = /—1(d//dT) (/0 = sample length at 298 K) is a slight modification of an earlier one,139 which underestimated the values measured by Richards at 2773 K140:

aT = 10.08 x 10-6 + 5.802 x 10-9

(T — 273.16)(T inK) [14]

It can be useful to note that, as UC displays a higher thermal conductivity than does U2C3, there can be internal stresses for a two-phase mixture of UC + U2C3. This factor should be taken into account in multiphased samples.