## 3 Results and Discussion
For both compounds, this function shows a steady growth; however, the difference between the values for NdNa\({}_{2}\)NiMnO\({}_{5}\) and NdK\({}_{2}\)NiMnO\({}_{5}\) is minimal. After thermodynamic investigations, electrophysical studies were conducted to clarify the nature of the identified second-order phase transitions. The studies of the electrophysical properties of NdNa\({}_{2}\)NiMnO\({}_{5}\) and NdK\({}_{2}\)NiMnO\({}_{5}\) in the temperature range of 293-483 K with 10 K increments, carried out using the LCR-781 device (Good Will Instrument Co., Ltd., New Taipei City, Taiwan) at a frequency of 1 kHz, demonstrate high measurement accuracy and rigorous methodology [31; 32; 33]. According to the specifications, the accuracy of measuring permittivity (\(\epsilon\)) and electrical resistance (R) is \(\pm\)0.05%, which ensures the reliability of the obtained data. A comparative analysis with the permittivity of the standard substance (BaTiO\({}_{3}\)) confirms the validity of the measurements, as the obtained value at 293 K (1296) aligns well with the recommended value of 1400 \(\pm\) 250 [34; 35; 36; 37]. Such studies help establish the relationship between changes in heat capacity and electrophysical characteristics, which is essential for determining the nature of second-order phase transitions. Measurements of dielectric permittivity and electrical resistance in this context can reveal structural anomalies, such as ferroelectric or paraelectric transitions, which are often accompanied by jumps in heat capacity. The results of the electrophysical measurements, presented in Table 8 and Figure 9, will likely provide deeper insights into how the properties of the material change within the studied temperature range and how these changes correlate with thermodynamic parameters. The activation energies of conductivity were calculated, representing the minimum energy required for a reaction to occur or a process to be activated. The calculation of the activation energy of electrical conductivity (\(\Delta E_{n}\)) was carried out as follows [38]. Based on the experimental data for \(T\) and \(R\), a scatter plot was constructed using a set of markers. By adding an \"exponential\" trendline, it was verified that the experimental graph could be described by a function of the form:  
\[R_{nn}=R_{0}e^{\frac{\Delta E_{n}}{2R^{2}}}. \tag{12}\]  
By plotting a scatter diagram of lg\(R_{nn}\) versus 10\({}^{4}\) T\({}^{-1}\) based on the experimental data in MS Excel, a linear trendline can be added, or a system of equations can be formed: lgy = a + bx, where y = lg\(R\) and x = 10\({}^{4}\)/T (Figures 10 and 11). \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**T, K** & **R, Om** & \(\epsilon\) & **lg\(\epsilon\)** & **lgR** & **R, Om** & \(\epsilon\) & **lg\(\epsilon\)** & **lgR** \\ \hline \multicolumn{8}{c}{**NdNa\({}_{2}\)NiMnO\({}_{5}\)**} \\ \hline
293 & 1,056,000 & 3489 & 3.54 & 6.02 & 2,005,000 & 298 & 2.47 & 6.30 \\
303 & 463,600 & 10,465 & 4.02 & 5.67  
Figure 9: Dependence of the dielectric permittivity (**a**), electrical resistivity (**b**) of NdNa\({}_{2}\)NiMnO\({}_{5}\) (**I**) and NdK\({}_{2}\)NiMnO\({}_{5}\) (**II**) on temperature. \(\cancel{\cancel{\cancel{\cancel{\cancel{\cancel{\cancel{\cancel{\cancel{\cancel{\cancel{\cancel{\cancel{\cancel{\cancel{\cancel{\cancel{\cancel{\cancel{\c{\c{\c{\c{\c{\c{\c{\c{\c{{\c{\}}}}}}}}}}}}}}}}\)–experimental data. For NdNa\({}_{2}\)NiMnO\({}_{5}\) in the temperature range of 293-353 K, the dependence \(\lg R\)-\(f(1/T)\) (Figure 10a) is described by the equation:  
\[y=-1.6581+\frac{2217}{T} \tag{13}\]  
and compared with the form:  
\[\lg R_{n}=\frac{\Delta E_{a}}{2K}\times T+lgR, \tag{14}\]  
then, with \(\Delta E/2\)k = 2217 K and \(\lg R_{0}=-1.6581\), we derive:  
\[\Delta E_{a}=\frac{2217\times 2\times 1.38\times 10^{-23}}{1.6\times 10^{-19}}=0. 382\;\mathrm{eV}, \tag{15}\]  
where \(k\) = 1.38 \(\times\) 10\({}^{-23}\) J\(\cdot\)K\({}^{-1}\) is the Boltzmann constant, and e = 1.6 \(\times\) 10\({}^{-19}\) C is the elementary charge.