# The Melt Enthalpy of Pu\({}_{6}\)Fe
## 3 Reanalysis
### Evaluating \(\Delta H_{\rm fus}^{\rm Pu}(T_{\rm eu})\)  
Our reanalysis requires the melt enthalpy of Pu at \(T_{\rm eu}\), which is far below the fusion temperature of Pu (\(T_{\rm eu}=412\) K, \(T_{\rm fus}^{\rm Pu}=913\) K) and therefore not physically observable. To estimate \(\Delta H_{\rm fus}^{\rm Pu}(T_{\rm eu})\), we begin with \(\Delta H_{\rm fus}^{\rm Pu}(T_{\rm fus})\) and subtract the (known) entropy increase in the solid due to heating from \(T_{\rm eu}\) to \(T_{\rm fus}\), do the same for the liquid based on the assumption that the specific heat does not change upon \"supercooling,\" then reevaluate the entropy difference at \(T_{\rm eu}\):  
\[\Delta H_{\rm fus}^{\rm Pu}(T_{\rm eu}) = T_{\rm eu}\Delta S(T_{\rm eu})=T_{\rm eu}\Big{\{}S^{l}(T_{\rm eu} )-S^{\delta}(T_{\rm eu})\Big{\}} \tag{4}\] \[= T_{\rm eu}\Bigg{[}S^{l}(T_{\rm fus})-\int_{T_{\rm eu}}^{T_{\rm fus }}\frac{C_{P}^{l}}{T}dT\Bigg{]}\] \[- T_{\rm eu}\Bigg{[}S^{\epsilon}(T_{\rm fus})-\int_{T_{\varrho^{ \prime}\to\epsilon}}^{T_{\rm fus}}\frac{C_{P}^{\epsilon}}{T}dT-\frac{\Delta H _{\varrho^{\prime}\to\epsilon}}{T_{\varrho^{\prime}\to\epsilon}}-\int_{T_{ \varrho\to\varrho^{\prime}}}^{T_{\varrho\to\epsilon}}\frac{C_{P}^{ \epsilon^{\prime}}}{T}dT-\frac{\Delta H_{\varrho\to\varrho^{\prime}}}{T_{ \varrho\to\varrho^{\prime}}}-\int_{T_{\rm eu}}^{T_{\varrho\to T_{\varrho^{ \prime}}}}\frac{C_{P}^{\epsilon}}{T}dT\Bigg{]}\] \[= T_{\rm eu}\Bigg{[}\frac{\Delta H_{\rm fus}}{T_{\rm fus}}+\frac{ \Delta H_{\delta\to\delta^{\prime}}}{T_{\delta\to\delta^{\prime}}}+\frac{ \Delta H_{\varrho^{\prime}\to\epsilon}}{T_{\varrho^{\prime}\to\epsilon}}\Bigg{]}\] \[+ T_{\rm eu}\Bigg{[}\int_{T_{\rm eu}}^{T_{\varrho\to T_{\varrho}}} \frac{C_{P}^{\delta}}{T}dT^{\prime}+\int_{T_{\varrho\to\varrho^{\prime}}}^{T_ {\varrho\to\epsilon^{\prime}}}\frac{C_{P}^{\delta^{\prime}}}{T}dT+\int_{T_{ \varrho^{\prime}\to\epsilon}}^{T_{\rm fus}}\frac{C_{P}^{\epsilon}}{T}dT-\int_ {T_{\rm eu}}^{T_{\rm fus}}\frac{C_{P}^{l}}{T}dT\Bigg{]}.\]  
\begin{table}
\begin{tabular}{l c c} \hline
**Quantity** & **J/mol/K** & **Source** \\ \hline \(C_{P}^{\epsilon}\) & \(17.619+45.552T\) & Reference [9] \\ \(C_{P}^{\delta}\) & \(27.416+13.060T\) & Reference [9] \\ \(C_{P}^{\gamma}\) & \(22.023+22.959T\) & Reference [9] \\ \(C_{P}^{\delta}\) & \(28.4781+0.010807T\) & Reference [9] \\ \(C_{P}^{\delta^{\prime}}\) & \(35.56\) & Reference [9] \\ \(C_{P}^{\epsilon}\) & \(33.72\) & Reference [9] \\ \(C_{P}^{\dagger}\) & \(42.80\) & Reference [9] \\ \(C_{P}^{\rm Fe}\) & \(46.024\) & Reference [15] \\ \(S^{\rm Fu}(298)\) & \(54.46\) & Reference [9] \\ \(S^{\rm Fe}(T_{\rm fus})\) & \(99.823\) & Reference [15] \\ \(S^{l}(T_{\rm fus})\) & \(8344\) & this work \\ \hline \end{tabular}
\end{table}
Table 2: Specific heats and entropies used in the reanalysis of Pu\({}_{6}\)Fe melt data.  
\begin{table}
\begin{tabular}{l c c} \hline
**Quantity** & **kJ/mol** & **Source** \\ \hline \(\Delta H_{\rm fus}^{\rm Pu}(T_{\rm fus}^{\rm Pu})\) & 2.766 & Reference [9] \\ \(\Delta H_{\rm fus}^{\rm Pu}\) & \(111.65\) (\(15.95\)) & Reference [16], this work \\ \(\Delta H_{\rm fus}^{\rm U}\) & \(8.470\) & Reference [9] \\ \(\Delta H_{\rm a\to\beta}\) & \(3.706\) & Reference [9] \\ \(\Delta H_{\beta\to\gamma}\) & \(0.478\) & Reference [9] \\ \(\Delta H_{\gamma\to\delta}\) & \(0.713\) & Reference [9] \\ \(\Delta H_{\delta\to\beta^{\prime}}\) & \(0.065\) & Reference [9] \\ \(\Delta H_{\varrho^{\prime}\to\epsilon}\) & \(1.711\) & Reference [9] \\ \hline \end{tabular}
\end{table}
Table 3: Transition enthalpies used in the reanalysis of Pu\({}_{6}\)Fe melt data. Quantity in parenthesis is in kJ/(mol atoms).