 Copolymer composition equation

The copolymer composition equation, also called the Mayo-Lewis equation, relates the instantaneous monomer-consumption composition to the current reacting mixture under the terminal model of copolymerization. For binary mixture mole fractions $f_1+f_2=1$, let $F_1$ be the fraction of type-1 units among the units being incorporated at that instant. Thenwhere $r_1,r_2$ are monomer reactivity ratios.^[ref] The equation concerns molar incorporation, not the mass fraction or cumulative composition of a finished specimen.

A short derivation identifies the steady-chain-end assumption. Let $c_i$ be monomer concentrations and $E_i$ concentrations of active ends. Cross-additions interconvert the end types; balancing them gives $k_{12}{E}_1{c}_2=k_{21}{E}_2{c}_1$. The two positive consumption rates areSubstitution yields $v_1/v_2=c_1(r_1{c}_1+c_2)/[c_2(c_1+r_2{c}_2)]$. Normalizing $F_1=v_1/(v_1+v_2)$ gives the stated equation. This neglects additional sources of end-type changes and uses a terminal kinetic description.

For an original example with $r_1=r_2=2$ and $f_1=0.20$, the numerator is 0.24 and the denominator is 1.68, so $F_1=1/7\approx 0.143$. The instantaneous polymer is poorer in type 1 than the available mixture. At equimolar feed the same ratios give $F_1=1/2$; this composition equality does not imply independent sequence placement.

For a finite closed batch, conversion bookkeeping for binary copolymers relates the changing monomer pool to cumulative incorporated composition. Treating an initial $f_1$ as constant after substantial selective consumption can therefore misinterpret the final sample. The IUPAC recommendation addresses conversion and integrated composition data explicitly.^[ref] Useful experiments distinguish current feed, conversion and product composition instead of treating all three as the same label.