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• Quaternary adsorption of Cu2+, Co2+, Ni2+ and Ag+ ions was investigated. • Development of a new quaternary adsorption model. • Adsorption of Cu2+, Co2+ and Ni2+ ions was endothermic, while the Ag+ adsorption was exothermic. • Ag+ adsorption was reduced from single-compound to quaternary system. Quaternary adsorption of Cu2+, Co2+, Ni2+ and Ag+ ions on an adsorbent obtained from the alkaline fusion of wastewater treatment sludge was investigated. Adsorption tests in both single-compound and quaternary systems were carried out and the analysis of adsorption mechanism was done by the application of a new quaternary model. This model was developed from a general methodology based on the application of grand canonical ensemble in statistical physics. This adsorbent was characterized by different analytical techniques. Experimental data showed that the adsorption of Cu2+, Co2+and Ni2+ ions was endothermic, while the Ag+ adsorption was exothermic. Ag+ adsorption was significantly reduced from single-compound to quaternary system due to the competitive effect caused by the other divalent ions, while the adsorption capacities of Cu2+, Co2+ and Ni2+ were not significantly affected in the multicomponent solution. A significant competitive effect of Cu2+, Co2+ and Ni2+ ions on the adsorption of Ag+ ions occurred. Modelling analysis showed that the adsorption of Cu2+, Co2+ and Ni2+ ions was a multi-ionic process at all tested temperatures, while the adsorption of Ag+ was a multi-ionic process only at low temperatures (298 and 308 K). Finally, the statistical physics model confirmed that the adsorption energies corresponded to physical interactions forces and the water molecule adsorption played a relevant role during the removal of these trace metals. [ABSTRACT FROM AUTHOR]