Kraft lignin is an effective heavy metal adsorbent with potential industrial applications, but the lack of mechanistic understanding of the process is impeding the development of this area, which is provided in this study. It was found the Pb(II) adsorption on Kraft lignin follows an "S" dependency on the environmental pH suggesting the existence of multiple functional groups. Structural characterization revealed adsorption functional groups including phenolic hydroxyl, carboxyl groups and electriferous groups existed in the Kraft lignin sample. Multivalent binding of a bivalent heavy metal cation to more than one functional groups is also expected. By understanding the binding mechanisms, a novel adsorption theory, "1-n cooperative adsorption theory", was developed to explain this complex interaction. The new theory managed to interpret the experimental data by considering binding mechanisms when an adsorbate can bind with multiple active sites with adsorbent-adsorbent interactions (cooperativity). Kraft lignin exhibit a high adsorption affinity towards Pb(II) (49.8 mg/g-lignin at neutral pH) and the process can be easily reversed through pH adjustment, which makes it a promising industrial Pb(II) adsorbent with regeneration abilities. This study provides guidelines for Kraft lignin applications as a heavy metal adsorbent, as well as theoretical backgrounds for other complicated multi-valent surface interactions. (C) 2018 Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.