## 2 Theory  
In this work, both quantities under consideration are calculated numerically by using the following thermodynamic relations, respectively (hereafter in Section 2, all macroscopic quantities are given per atom):  
\[c_{p}=T\bigg{(}\frac{\partial S}{\partial T}\bigg{)}_{p}, \tag{1}\]  
\[\alpha_{p}=\rho\bigg{(}\frac{\partial\Omega}{\partial T}\bigg{)}_{p}, \tag{2}\]  
where \(T\) is the temperature; \(p\) is the pressure; \(S\) is the entropy; \(\rho\) is the density; \(\Omega(\equiv 1/\rho)\) is the mean atomic volume. The calculations are performed on the basis of the right-hand side of the Gibbs-Bogoliubov inequality [35; 36; 37] by means of the minimization of the Helmholtz free energy, \(F\), in the reciprocal space (see, for example, Refs. [29; 38]):  
\[F=\frac{3}{2}k_{\rm B}T+E_{e}+E_{1}+E_{2}+E_{\rm M}-T(S_{\rm HS}+S_{e}), \tag{3}\]  
where \(k_{\rm B}\) is the Boltzmann constant; \(E_{1}\) and \(E_{2}\) are the first- and second-order terms of the pseudopotential perturbation theory (\(E_{2}\) has the physical meaning of the band structure energy); \(E_{\rm M}\) is the Madelung energy arising from the direct pairwise Coulomb repulsions between ions; \(S_{\rm HS}\) is the entropy of the hard sphere (HS) reference system (we use the analytical expressions for the one-component HS fluid and for the binary HS mixture obtained in the work [38]); \(E_{e}\) and \(S_{e}\) are the electron-gas contributions to the internal energy and the entropy, respectively (hereafter in Section 2, all characteristics are given in atomic units (a.u. )):  
\[E_{e}=\frac{3}{10}zk_{\rm F}^{2}-\frac{z}{2}\bigg{(}\frac{\pi k_{\rm B}T}{k_{ \rm F}}\bigg{)}^{2}+E_{\rm EC}, \tag{4}\]  
\[S_{e}=zT\bigg{(}\frac{\pi k_{\rm B}}{k_{\rm F}}\bigg{)}^{2}, \tag{5}\]  
where \(z\) is the valence (for the binary alloy, \(z=\overline{z}=c_{1}z_{1}+c_{2}z_{2}\), where \(z_{i}\) is the valence of the \(i\)-th component; \(c_{i}\) is the concentration of the \(i\)-th component in the alloy); \(k_{\rm F}=\left(3z\rho\pi^{2}\right)^{1/3}\) is the Fermi wave vector; \(E_{\rm EC}\) is the exchange-correlation energy taken here in the Nozieres-Pines approach [39], which was successfully used earlier for liquid Na-K [38] and for other alkali metal liquids [12; 14; 15]. The contribution \(E_{1}\) is expressed for pure metal and for the binary metal alloy, respectively, as follows:  
\[E_{1}^{\rm pure}=z\lim_{q\to 0}\biggl{(}\omega(q)+\frac{4\pi\rho\,z}{q^{2}} \biggr{)}, \tag{6}\]  
\[E_{1}^{\rm alloy}=\overline{z}\sum\limits_{i=1}^{2}c_{i}\lim_{q\to 0}\biggl{(}\omega_{i}(q)+ \frac{4\pi\rho\,z_{i}}{q^{2}}\biggr{)}, \tag{7}\]  
where \(\omega(q)\) and \(\omega_{i}(q)\) are the form factors of the unscreened-ion pseudopotentials in the pure metal and in the binary alloy for the \(i\)-th-kind ion, respectively; \(q\) is the modulus of the wave vector, \(\stackrel{{\rightarrow}}{{q}}\). In the present work, the Krasco-Gurskii (KG) model pseudopotential [40] and the Animalu-Heine (AH) model pseudopotential [41] in the local approximation [42] with the following form factors are used:  
\[\omega_{\rm KG}(q)\,=\frac{4\pi\rho\,z\bigl{(}(2a-1)q^{2}r_{\rm c}^{2}-1\bigr{)} }{q^{2}(1+q^{2}r_{\rm c}^{2})^{2}}, \tag{8}\]  
\[\omega_{\rm AH}(q)\,=-\frac{4\pi\rho}{q^{2}}\biggl{[}(z-AR_{\rm M})\cos(qR_{ \rm M})+\frac{AR_{\rm M}\sin(qR_{\rm M})}{qR_{\rm M}}\biggr{]}\exp\biggl{[}-0. 03\biggl{(}\frac{q}{2k_{\rm F}}\biggr{)}^{4}\biggr{]}, \tag{9}\]  
\[\omega_{i\rm KG}(q)\,=\frac{4\pi\rho\,z_{i}\bigl{(}(2a_{i}-1)q^{2}r_{\rm ci}^{2 }-1\bigr{)}}{q^{2}\bigl{(}1+q^{2}r_{\rm ci}^{2}\bigr{)}^{2}}, \tag{10}\]  
\[\alpha_{p}=\rho\bigg{(}\frac{\partial\Omega}{\partial T}\bigg{)}_{p}, \tag{2}\]0  
where \(a\), \(r_{\rm c}\), \(A\), \(R_{\rm M}\), and \(a_{i}\), \(r_{\rm ci}\), \(A_{i}\), \(R_{\rm M}i\) are the parameters in the pure metal and in the alloy for the \(i\)-th-kind ion (put here the same as the ones for the corresponding pure metals), respectively.