A new version of the molecular thermodynamic model has been developed that takes into account the effect of ion specificity on the free energy of aggregation. The specificity of salt is reflected by differences in the bare ionic sizes and polarizabilities leading to the difference in the dispersion interaction of ions with the aggregate. The model also contains parameters that characterize the compactness of ionic pairs formed between a mobile ion and surfactant's headgroup. The values of these parameters show that more chaotropic heads form tighter pairs with chaotropic ions whereas more cosmotropic heads form more compact pairs with cosmotropic ions. The formation of compact pairs in the micelle corona diminishes the preferable curvature of the aggregates and promotes their growth. The model has been applied to aqueous solutions of cationic (alkyltrimethylammonium, alkyldimethylammonium, and alkylpyridinium) and anionic (alkylsulfate and alkylcarboxylate) surfactants in the presence of simple 1:1 salts. With a single set of parameter values, the model reproduces the critical micelle concentration-salinity curves and the sphere-to-rod transitions or the absence of thereof and describes the aggregate growth for different simple salts, in good agreement with experiment.