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Understanding the concept of a limiting reactant is crucial in chemical reactions, as it dictates the maximum amount of product that can be formed. This reactant is the one that will be completely used up first, and thus limits the progression of the reaction. In a chemical equation, the mole ratio of reactants is defined by the balanced equation, and the reactant present in the lesser amount, according to this ratio, is the limiting one.

Consider a simple reaction where ingredient A reacts with ingredient B to form product C. If you run out of A before B, then A is the limiting reactant. Similarly, in our given exercise, hydrogen bromide (HBr) was identified as the limiting reactant because the product stream contained unreacted HBr, suggesting all the ethylene (C₂H₄) could not react as there wasn't enough HBr available.

The fractional conversion calculation quantifies the proportion of a reactant that has been used up in the reaction; it is a measure of the reaction's progress. The calculation is straightforward: it's the initial amount of the reactant minus the amount that remains, divided by the initial amount. Expressed as a fraction (hence, 'fractional'), it can also be converted into a percentage.

In our example, to calculate the fractional conversion of hydrogen bromide, you subtract the final moles of HBr from the initial moles (which can be found using the mole fraction of HBr in the feed) and then divide by the initial moles. Mathematically, this is represented by the formula \[\text{Conversion} = \frac{\text{initial moles} - \text{final moles}}{\text{initial moles}}\] When successfully calculated, this gives you direct insight into how much of your limiting reactant was consumed during the chemical reaction.

While the limiting reactant is completely used up, the excess reactant is the one left over after the reaction goes to completion. Knowing the percentage of the excess reactant is helpful for understanding the reaction efficiency and planning for resource management in industrial processes.

The excess reactant percentage is found by the formula \[ \text{Excess} = \frac{\text{initial moles} - \text{converted moles}}{\text{initial moles}} \times 100\% \] In this formula, 'converted moles' is the number of moles of the excess reactant that actually participated in the reaction, which can be deduced from the stoichiometry of the chemical equation. In our reaction, the ethylene was in excess; thus, we could determine how much ethylene did not react and remained in the reactor.

Calculating the extent of reaction gives us another piece of important information, it represents the quantity in moles of a product formed or reactant consumed in a given reaction. The extent is calculated by determining the change in moles (\(\text{Δn}\)) of reactants or products, and dividing by the stoichiometric coefficient (\(u\)), which is the 'balanced equation coefficient' for each substance in the reaction.

For the reaction between ethylene and hydrogen bromide, the stoichiometric coefficients for both reactants are equal to 1 (as per the balanced equation), simplifying the calculation. The extent of reaction tells us how far the reaction has gone in terms of molar consumption and is directly related to the flow rate of the product stream in continuous reactors, like in the case of our exercise.