It was proposed by Frank R. Mayo and Frederick M.
[1] The equation considers a monomer mix of two components
and the four different reactions that can take place at the reactive chain end terminating in either monomer (
: 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.
[2] The copolymer equation is then:[3][4][2] with the concentrations of the components in square brackets.
The equation gives the relative instantaneous rates of incorporation of the two monomers.
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 (
Substituting into the ratio of monomer consumption rates yields the Mayo–Lewis equation after rearrangement:[4]
It is often useful to alter the copolymer equation by expressing concentrations in terms of mole fractions.
represents the mole fraction of each monomer in the copolymer:
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.
From the definition of reactivity ratios, several special cases can be derived: Calculation of reactivity ratios generally involves carrying out several polymerizations at varying monomer ratios.
The copolymer composition can be analysed with methods such as Proton nuclear magnetic resonance, Carbon-13 nuclear magnetic resonance, or Fourier transform infrared spectroscopy.
The polymerizations are also carried out at low conversions, so monomer concentrations can be assumed to be constant.
One of the simplest methods for finding reactivity ratios is plotting the copolymer equation and using nonlinear least squares analysis to find the
pair that gives the best fit curve.
This is preferred as methods such as Kelen-Tüdős or Fineman-Ross (see below) that involve linearization of the Mayo–Lewis equation will introduce bias to the results.
[10] The Mayo-Lewis method uses a form of the copolymer equation relating
For each different monomer composition, a line is generated using arbitrary
Fineman and Ross rearranged the copolymer equation into a linear form:[11]
yields a straight line with slope
The Fineman-Ross method can be biased towards points at low or high monomer concentration, so Kelen and Tüdős introduced an arbitrary constant,
[12] The data can be plotted in a linear form
This distributes the data more symmetrically and can yield better results.
A semi-empirical method for the prediction of reactivity ratios is called the Q-e scheme which was proposed by Alfrey and Price in 1947.
Where P is a proportionality constant, Q is the measure of reactivity of monomer via resonance stabilization, and e is the measure of polarity of monomer (molecule or radical) via the effect of functional groups on vinyl groups.
An advantage of this system is that reactivity ratios can be found using tabulated Q-e values of monomers regardless of what the monomer pair is in the system.