Final answer:
To find the time for 50% completion of a first-order reaction where 75% is done in 32 minutes, we calculate a single half-life by dividing the given time by 1.5 to get approximately 21.33 minutes, as the half-life is constant.
Step-by-step explanation:
The question involves determining the time it takes for 50% of a first-order reaction to complete, given that 75% completion takes 32 minutes. For a first-order reaction, the time required to reach a certain percentage of completion is not directly proportional to that percentage because of the exponential nature of the decay process.
In a first-order reaction, the half-life (time taken for half the reactant to react) is constant. This means it does not change no matter how much reactant has been consumed. Since 75% completion occurs in 32 minutes, we know that during this period, the reaction has gone through more than one half-life cycle (from 100% to 50%, then from 50% to 25%).
Thus, to determine when the reaction is 50% complete, we consider the duration of a single half-life. It takes one half-life to go from 100% to 50%, and another half-life to go from 50% to 25%. Since 75% completion is 32 minutes, that is the time for one and a half half-lives. Therefore, one half-life is 32 minutes divided by 1.5, which is approximately 21.33 minutes.