## 3 Results and Discussion  
The discussion highlights that the \(\text{C}^{0}_{\text{p}}\)-f(T) graph for NdNa\({}_{2}\)NiMnO\({}_{5}\) and NdK\({}_{2}\)NiMnO\({}_{5}\) shows an anomalous jump in heat capacity at a temperature of 423 K. This jump indicates potential second-order phase transitions, which may be associated with Curie points, Neel points, or other structural changes that significantly influence the magnetic, electrical, and dielectric properties of the materials. To ensure the accuracy of the analysis, both experimental and calculated heat capacity values are presented on the graphs, confirming their reliability. Based on the temperatures of second-order phase transitions, equations describing the temperature dependence of heat capacity for the compounds, as detailed in Table 6, were derived. These equations enable more precise modeling of the heat capacity behavior of the material within the studied temperature range, taking into account the phase transitions and their impact on thermodynamic properties. The experimental data obtained were processed using the least squares method. The graphs presented in Figure 8, illustrating the \(\text{C}^{0}_{\text{p}}\)-f(T) dependence, demonstrate a good correlation between the experimental and calculated heat capacity values, confirming the adequacy of the derived equations. The equations describing the \(\text{C}^{0}_{\text{p}}\)-f(T) dependencies do not all merge smoothly; some intersect sharply, forming distinct peaks. For NdNa\({}_{2}\)NiMnO\({}_{5}\), the equations \(\text{C}^{0}_{\text{p(1)}}\)-f(T\({}_{1}\) \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Compound**} & \multicolumn{2}{c}{**Coefficients of Equation \(\text{C}^{0}_{\text{p}}\) = a + b\(\cdot\) T + c\(\cdot\) T\({}^{-2}\), J/(mol\(\cdot\)K)**} & \multirow{2}{*}{**AT, K**} \\ \cline{2-2} \cline{4-5}  & **a** & **b\(\cdot\)10\({}^{-3}\)** & & **c\(\cdot\)10\({}^{5}\)** & \\ \hline \multirow{4}{*}{NdNa\({}_{2}\)NiMnO\({}_{5}\)} & 602 \(\pm\) 24 & \(-\)(1298.0 \(\pm\) 50.9) & - & 298.15–348 \\  & (4305 \(\pm\) 169) & \(-\)(6458.6 \(\pm\) 253.2) & \(-\)(2309.8 \(\pm\) 90.5) & 348–423 \\  & 997 \(\pm\) 39 & \(-\)(1690.2 \(\pm\) 66.3) & - & 423–473 \\  & 538 \(\pm\) 21 & \(-\)(2.7 \(\pm\) 0.1) & \(-\)(759.3 \(\pm\) 29.8) & 473–673 \\ \hline \multirow{4}{*}{NdK\({}_{2}\)NiMnO\({}_{5}\)} & 128 \(\pm\) 6 & \(-\)(30.6 \(\pm\) 1.5) & (90.2 \(\pm\) 4.4) & 298.15–398 \\  & \(-\)(5013 \(\pm\) 243) & 13,029.5 \(\pm\) 630.6 & - & 398–423 \\ \cline{1-1}  & 2964 \(\pm\) 143 & \(-\)(5830.0 \(\pm\) 282.2) & - & 423–473 \\ \cline{1-1}  & (650 \(\pm\) 31) & \(-\)(54.8 \(\pm\) 2.7) & \(-\)(933.5 \(\pm\) 45.2) & 473–673 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Coefficients of equations of temperature dependence of heat capacity. and \(\rm C^{0}_{p(2)}\)\(\sim\)f(T\({}_{2}\)) intersect at 348 K, as this temperature corresponds to the values of both Equations (9) and (10), where \(\rm C^{0}_{p(1)}\) and \(\rm C^{0}_{p(2)}\) are equal. At 423 K, \(\rm C^{0}_{p(2)}\)\(\sim\)f(T\({}_{2}\)) and \(\rm C^{0}_{p(3)}\)\(\sim\)f(T\({}_{3}\)) intersect, with \(\rm C^{0}_{p(2)}\) and \(\rm C^{0}_{p(3)}\) having identical values, and so on. These graphs were generated using the \"COMPAS-3D LT V12\" software, which enabled the integration of experimental and calculated data for more accurate visualization of the results. To estimate the standard entropies of the compounds, the method of ionic entropy increments [29] was employed, yielding values of \(239\pm 7\) J/(mol\(\cdot\)K) for NdNa\({}_{2}\)NiMnO\({}_{5}\) and 264 \(\pm\) 8 J/(mol\(\cdot\)K) for NdK\({}_{2}\)NiMnO\({}_{5}\). The following equations [30] were used to calculate the heat capacity dependence at constant pressure (\(\rm C^{0}_{p}(T)\)) and thermodynamic functions (\(\rm S^{0}(T)\), \(\rm H^{0}(T)-H^{0}(298.15)\), and \(\rm\Phi^{xx}(T)\)).