b$)$ When conducting a polyesterification using a diol and a diacid, a student decided to add the two monomers in a $1$:$1\mathrm{.}2$ molar ratio and heat the reaction in a flask under conditions that allows removal of water.

i$.$ Write out the Carothers equation and the modified Carothers equation that considers the impact of monomer stoichiometry, and define the terms in each equation.

ii$.$ Use the modified Carothers equation to calculate the maximum average chain length of the polymer that the student will theoretically synthesise.

iii. Using your choice of diol and diacid monomer structures, draw the predicted average product of the reaction including the end groups.

iv$.$ Explain two experimental changes that the student can make to future reactions to ensure that the polymerisation will form a much higher average chain length.