= Copolymer composition equation
{title2=$F_1=(r_1f_1^2+f_1f_2)/(r_1f_1^2+2f_1f_2+r_2f_2^2)$}
= Mayo-Lewis equation
{c}
{synonym}
The \b[<copolymer composition equation>], also called the \b[<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. Then
$$
F_1=\frac{r_1f_1^2+f_1f_2}{r_1f_1^2+2f_1f_2+r_2f_2^2},\qquad F_2=1-F_1,
$$
where $r_1,r_2$ are <monomer reactivity ratios>.<IUPAC Compendium of Chemical Terminology, fifth edition (Gold Book)>[Copolymerisation composition equation, entry 15391, definition and composition-drift notes]{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 $k_{12}E_1c_2=k_{21}E_2c_1$. The two positive consumption rates are
$$
v_1=k_{11}E_1c_1+k_{21}E_2c_1,\qquad
v_2=k_{12}E_1c_2+k_{22}E_2c_2.
$$
Substitution yields $v_1/v_2=c_1(r_1c_1+c_2)/[c_2(c_1+r_2c_2)]$. 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 $F_1=1/7\approx0.143$. 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.<Radical Copolymerization Reactivity Ratios by IUPAC>[Announcement, conversion/composition observations and integrated analysis]{ref} Useful experiments distinguish current feed, conversion and product composition instead of treating all three as the same label.
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