Final answer:
To find out how long it takes for 89.2% of a compound to decompose in a first-order reaction, we calculate the rate constant from the given decomposition percentage and time, and then apply the integrated rate law for first-order reactions with the new percentage.
Step-by-step explanation:
The question pertains to determining the time it takes for 89.2% of a compound to decompose in a first-order decomposition reaction if 48.4% decomposes in 7.7 minutes.
Since the half-life in a first-order reaction is constant, we can use it to calculate the time required for any other percentage of decomposition by applying the integrated rate law for first-order reactions.
The rate constant is not given, but it can be derived from the given information using the formula t = (ln(1/(1 - fraction decomposed)))/k, where t is the time, k is the rate constant, and the fraction decomposed is the percentage of compound that has decomposed, expressed as a decimal. Once the rate constant is known, the same formula can be used to find out how long it takes for 89.2% of the compound to decompose.