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 ${\displaystyle M_{1}\,}$ and ${\displaystyle M_{2}\,}$ and the four different reactions that can take place at the reactive chain end terminating in either monomer (${\displaystyle M_{1}^{*}\,}$ and ${\displaystyle M_{2}^{*}\,}$) with their reaction rate constants ${\displaystyle k\,}$:

${\displaystyle M_{1}^{*}+M_{1}{\xrightarrow {k_{11}}}M_{1}M_{1}^{*}\,}$

${\displaystyle M_{1}^{*}+M_{2}{\xrightarrow {k_{12}}}M_{1}M_{2}^{*}\,}$

${\displaystyle M_{2}^{*}+M_{2}{\xrightarrow {k_{22}}}M_{2}M_{2}^{*}\,}$

${\displaystyle M_{2}^{*}+M_{1}{\xrightarrow {k_{21}}}M_{2}M_{1}^{*}\,}$

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.

${\displaystyle r_{1}={\frac {k_{11}}{k_{12}}}\,}$

${\displaystyle r_{2}={\frac {k_{22}}{k_{21}}}\,}$

The copolymer equation is then:

${\displaystyle {\frac {d\left[M_{1}\right]}{d\left[M_{2}\right]}}={\frac {\left[M_{1}\right]\left(r_{1}\left[M_{1}\right]+\left[M_{2}\right]\right)}{\left[M_{2}\right]\left(\left[M_{1}\right]+r_{2}\left[M_{2}\right]\right)}}}$

with the concentrations of the components in square brackets. The equation gives the relative instantaneous rates of incorporation of the two monomers.