= Terminal model of copolymerization
{title2=$p_{ij}=k_{ij}c_j/\sum_\ell k_{i\ell}c_\ell$}
The \b[<terminal model of copolymerization>] represents a growing <copolymer> chain's addition kinetics using only the identity of its final <monomeric unit>. For two incoming <monomers>, there are two end types and four <chain-growth propagation> rate constants $k_{ij}$, where $i$ identifies the existing end and $j$ the incoming monomer. Earlier sequence history is omitted from this kinetic state description.<Radical Copolymerization Reactivity Ratios by IUPAC>[Published technical Introduction, terminal-model assumption]{ref}
At a fixed reacting-mixture composition, addition rates give a two-state <Markov chain>. With $f_1+f_2=1$ the monomer <mole fractions> and $r_1=k_{11}/k_{12}$, $r_2=k_{22}/k_{21}$ the <monomer reactivity ratios>, its <transition matrix> is
$$
P=\begin{pmatrix}
\dfrac{r_1f_1}{r_1f_1+f_2}&\dfrac{f_2}{r_1f_1+f_2}\\[6pt]
\dfrac{f_1}{f_1+r_2f_2}&\dfrac{r_2f_2}{f_1+r_2f_2}
\end{pmatrix}.
$$
Rows label the current terminal unit; columns label the newly added unit. Each row sums to one. This interprets a sequence at fixed feed and neglects initiation, termination and feed drift over the sampled run.
For an original example, take $f_1=f_2=1/2$ and $r_1=r_2=2$. Then $P_{11}=P_{22}=2/3$ and both cross-addition probabilities are $1/3$. The <stationary distribution> is $(1/2,1/2)$, so the long-chain mean composition is equimolar, while equal consecutive units are favored. A run of type 1 has a <geometric distribution> with stopping probability $1/3$ and mean length 3. Independent equimolar placement would instead give mean run length 2. Mean composition alone does not determine sequence correlations.
The <copolymer composition equation> concerns the instantaneous incorporated composition implied by this model. Applying it over a finite conversion interval requires following changes in the reacting mixture. The IUPAC recommendation explicitly bases ratio estimation on conversion and composition observations at different starting feeds.<Radical Copolymerization Reactivity Ratios by IUPAC>[Official recommendation announcement, measurement design]{ref}
Historically, <Frank R. Mayo> and <Frederick M. Lewis> published a 1944 theoretical comparison of monomers with a study of styrene and methyl methacrylate.<Copolymerization. I. A Basis for Comparing the Behavior of Monomers in Copolymerization; The Copolymerization of Styrene and Methyl Methacrylate by Frank R. Mayo and Frederick M. Lewis>[Publisher title, byline and publication record]{ref} A successful terminal-model fit is evidence about a chosen kinetic representation, not proof that every chemical influence vanishes beyond the last unit.
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