$Zn_{0.5}Cd_{0.5}Al_{2}Se_{4}$ and $Zn_{0.5}Cd_{0.5}Al_{2}Se_{4}$:$Co^{2+}$ single crystals were grown by CTR method. The grown single crystals have defect chalcopyrite structure with lattice constant a=5.5966$AA$, c=10.8042$AA$ for the pure, a=5.6543$AA$, c=10.8205$AA$ for the Co-doped single crystal, respectively. The optical energy band gap was given as indirect band gap. The optical energy band gap was decreased according to add of Co-impurity Temperature dependence of optical energy band gap was fitted well to the Varshni equation. From this relation, we can deduced the entropy, enthalpy and heat capacity. Also, we can observed the Co-impurity optical absorption peaks assigned to the $Co^{2+}$ ion sited at the $T_{d}$ symmetry lattice and we consider that they were attributed to the electron transitions between energy levels of ions.