Conversion bookkeeping for binary copolymers (${\overline{F}}_1=[f_0-(1-X)f]/X$)

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 areTheir sum is $X$, so the cumulative type-1 fraction isThis 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.^[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 $0\le X<1$,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.

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.^[ref]