This invention is directed to a new low molecular weight copolymer (The PIB/UAR copolymer) that may be prepared by reacting a low molecular weight polyisobutene (PIB) having less than about 32 carbon atoms with an unsaturated acidic reagent in the presence of a free radical initiator. No chain transfer agent is required to prepare the low molecular weight PIB/UAR copolymer. The PIB/UAR copolymer may be useful as is, or as an intermediate for (1) polysuccinimides, (2) detergents or dispersants for lube oil or fuels, (3) pour point depressants and (4) surface sizing agents for paper. The PIB/UAR copolymer may be used by itself or as the ester, amide, imide or metal salt derivative of the PIB/UAR copolymer. The PIB/UAR copolymer may be liquid at ambient temperature.