Mayo–Lewis equation

The Mayo–Lewis equation or copolymer equation in polymer chemistry describes the distribution of monomers in a copolymer. It was proposed by Frank R. Mayo and Frederick M. Lewis. The equation considers a monomer mix of two components and and the four different reactions that can take place at the reactive chain end terminating in either monomer ( and ) with their reaction rate constants : The reactivity ratio for each propagating chain end is defined as the ratio of the rate constant for addition of a monomer of the species already at the chain end to the rate constant for addition of the other monomer. The copolymer equation is then: with the concentrations of the components in square brackets. The equation gives the relative instantaneous rates of incorporation of the two monomers. Monomer 1 is consumed with reaction rate: with the concentration of all the active chains terminating in monomer 1, summed over chain lengths. is defined similarly for monomer 2. Likewise the rate of disappearance for monomer 2 is: Division of both equations by followed by division of the first equation by the second yields: The ratio of active center concentrations can be found using the steady state approximation, meaning that the concentration of each type of active center remains constant. The rate of formation of active centers of monomer 1 () is equal to the rate of their destruction () so that or Substituting into the ratio of monomer consumption rates yields the Mayo–Lewis equation after rearrangement: It is often useful to alter the copolymer equation by expressing concentrations in terms of mole fractions. Mole fractions of monomers and in the feed are defined as and where Similarly, represents the mole fraction of each monomer in the copolymer: These equations can be combined with the Mayo–Lewis equation to give This equation gives the composition of copolymer formed at each instant. However the feed and copolymer compositions can change as polymerization proceeds. Reactivity ratios indicate preference for propagation.