A solution is presented to verify numerical computer codes of reactive transport with both equilibrium and kinetic reactions. A synthetic model of A <-> B <-> C -> chain reactions is proposed to describe operator-splitting numerical schemes used in numerical computer codes. A reaction matrix is derived for both the equilibrium and the first-order kinetic reactions and further decoupled as a diagonal matrix. Therefore, the partial differential equations (PDEs) coupled by the reaction matrix can be transformed into independent PDEs, for which closed-form solutions exist or can be derived. The solution derived in this study is compared with numerical results.