= Solution
In (i), any pair of compatible functional groups can react: <monomers>, oligomers and long chains all participate. For an equimolar, strictly bifunctional mixture without <cyclization in step-growth polymerization>, the <Carothers equation> gives
$$
\boxed{\overline X_n=\frac{1}{1-p}.}
$$
Here $p$ is functional-group conversion and $\overline X_n$ counts the original <monomer> <molecules>. Large <degree of polymerization> therefore requires nearly complete conversion.
In (ii), growth occurs only at the metal-bound chain end. If initiation is rapid and <chain transfer in chain polymerization> is negligible, the number of chains is fixed and
$$
\overline X_n\simeq \frac{[M]_0}{[I]_0}p.
$$
This is the linear conversion dependence of <living polymerization>. Actual zirconocene systems can undergo chain transfer, so linear growth is a limiting model, not a consequence of <coordination polymerization> alone.
In (iv), a newly initiated chain rapidly attains a large size and then undergoes <radical termination>. Further conversion mostly creates additional chains. Thus high <degree of polymerization> can occur at low conversion, without the systematic linear increase of an ideal living system. <temperature>, initiator depletion and <monomer> depletion can change that size during the run.
\Image[/past-exam-of-the-chemistry-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1-polymer-growth.png]
{title=Contrasting idealized degrees of polymerization during three growth mechanisms}
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