  Designing new, technologically relevant superconductors has long been at the
forefront of solid-state physics and chemistry research. However, developing
efficient approaches for modeling the thermodynamics of superconducting alloys
while accurately evaluating their physical properties has proven to be a very
challenging task. To fill this gap, we propose an ab initio thermodynamic
statistical method, the Extended Generalized Quasichemical Approximation
(EGQCA), to describe off-stoichiometric superconductors. Within EGQCA, one can
predict any computationally accessible property of the alloy, such as the
critical temperature in superconductors and the electron-phonon coupling
parameter, as a function of composition and crystal growth conditions by
computing the cluster occurrence probabilities that minimize the overall mixing
Gibbs free energy. Importantly, EGQCA incorporates directly chemical ordering,
lattice distortions, and vibrational contributions. As a proof of concept, we
applied EGQCA to the well-known Al-doped MgB$_2$ and to niobium alloyed with
titanium and vanadium, showing a remarkable agreement with the experimental
data. Additionally, we model the near-room temperature sodalite-like
Y$_{1-x}$Ca$_x$H$_6$ superconducting solid solution, demonstrating that EGQCA
particularly possesses a promising potential for designing in silico
high-$T_{\text{c}}$ superhydride alloys. Our approach notably enables the
high-throughput screening of complex superconducting solid solutions,
intrinsically providing valuable insights into the interplay between synthesis,
thermodynamics, and physical properties.
