= Conversion bookkeeping for binary copolymers
{title2=$\overline F_1=[f_0-(1-X)f]/X$}
\b[<Conversion bookkeeping for binary copolymers>] uses conserved component amounts to distinguish initial feed, residual feed and cumulative incorporated composition. It is an accounting constraint on a closed batch, separate from the kinetic prediction supplied by the <copolymer composition equation>.
Normalize initial total monomer amount to one mole. Let $f_0$ be its type-1 <mole fraction>, $X$ total molar conversion and $f$ the type-1 fraction in the remaining pool. Assume no additional feed, side losses or consumption outside the material being counted. The incorporated amounts are
$$
n_{1,p}=f_0-(1-X)f,\qquad
n_{2,p}=(1-f_0)-(1-X)(1-f).
$$
Their sum is $X$, so the cumulative type-1 fraction is
$$
\overline F_1=\frac{f_0-(1-X)f}{X},\qquad X>0.
$$
This is a molar-unit balance; conversion by mass requires the corresponding component <molar masses>. The IUPAC terminology distinguishes instantaneous composition from integrated conversion/composition relationships.<IUPAC Compendium of Chemical Terminology, fifth edition (Gold Book)>[Copolymerisation composition equation, entry 15391, integrated forms and composition drift]{ref}
For an original example, choose $f_0=0.20$, $X=0.50$ and residual $f=0.30$. The incorporated amounts are 0.05 and 0.45 mol, giving $\overline F_1=0.10$. Individual monomer conversions are $0.05/0.20=25\%$ and $0.45/0.80=56.25\%$. The residual mixture becomes richer in the less-consumed component even though the polymer contains less of it than the initial feed.
Nonnegative incorporated and remaining amounts impose, for $0\le X<1$,
$$
\max\left(0,\frac{f_0-X}{1-X}\right)\le f\le
\min\left(1,\frac{f_0}{1-X}\right).
$$
For $f_0=0.20$ and $X=0.10$, a reported residual fraction of 0.50 is impossible under the assumptions: it would leave $0.90(0.50)=0.45$ mol of type 1 when only 0.20 mol was present initially.
\Image[/copolymer-conversion-feasibility.png]
{title=Conserved monomer amounts bound residual-feed composition at each conversion}
{description=Original closed-batch molar bookkeeping for initial type-1 fraction 0.20. The feasible region requires both consumed amounts to be nonnegative; no reaction-rate constants are fitted. Residual composition is undefined at complete conversion.}
For a differentiable conversion trajectory, the instantaneous fraction is $F_1=d(X\overline F_1)/dX=f-(1-X)df/dX$. Equivalently, $(1-X)df/dX=f-F_1$. This is the bridge between conservation and kinetic composition drift. At complete conversion there is no residual mixture to assign an $f$ value; the total incorporated composition nevertheless equals $f_0$ under the closed-batch assumptions. The IUPAC experimental recommendation measures conversion and composition separately rather than replacing both by an initial feed label.<Radical Copolymerization Reactivity Ratios by IUPAC>[Official recommendation announcement, conversion and composition observations]{ref}
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