Question

The half-life of a certain first-order reaction is \(15 \mathrm{~min}-\) utes. What fraction of the original reactant concentration will remain after 2.0 hours?

Short answer

The fraction of the original reactant that will remain after 2.0 hours is \( (1/2)^8 = 1/256 \).

Step-by-step solution

Understanding Half-Life

The half-life in a first-order reaction is the time required for the concentration of a reactant to decrease to half its initial value. Use the formula for the first-order reaction half-life, which is constant across the reaction.

Convert Time Units

The given time needs to be converted into the same units as the half-life. Here, convert 2.0 hours to minutes since the half-life is given in minutes. There are 60 minutes in an hour, so the time for the reaction is therefore 2.0 hours * 60 minutes/hour = 120 minutes.

Calculate the Number of Half-Lives

Divide the total time that has passed by the half-life to determine the number of half-lives that have occurred. Number of half-lives = Total time / Half-life.

Calculate the Fraction of Original Reactant

Plug the number of half-lives into the formula to find the fraction of the original reactant remaining. The initial concentration cancels out since we're looking for the fraction, which is a ratio of the final concentration to the initial concentration.

Key concepts

First-Order Reaction Kinetics

Understanding the kinetics of a first-order reaction is pivotal for grasping how reactants convert to products over time. First-order reactions are chemical processes where the rate is directly proportional to the concentration of one reactant. It means that if you were to double the concentration of that reactant, the rate of the reaction would also double.

In the realm of first-order reactions, the rate constant, denoted as 'k', is a crucial factor as it dictates how quickly a reaction proceeds. The unique aspect of first-order reactions is that their half-life, the time it takes for half of the reactant to be used up, is constant and does not depend on the initial concentration of the reactant. This constancy provides a straightforward way to predict how long it will take for a certain percentage of the reactant to react, regardless of how much reactant you start with.

For example, no matter the quantity of reactant present, if the half-life is 15 minutes, after 15 minutes, you will always have half the original amount. Utilize this property to predict the concentration of reactants at various times and to plan reactions in a laboratory or industrial setting.

Reactant Concentration Decay

The concept of reactant concentration decay is integral to understanding how substances react over time. For first-order reactions, this decay follows an exponential trend. In simpler terms, rather than decreasing by a fixed amount each half-life, the concentration decreases by half. As a result, plotting the concentration over time yields a curve—a hallmark signal of exponential decay.

In practical scenarios, this means that the concentration doesn't dwindle down to zero immediately. It gradually approaches, halving itself every half-life period. If you started with a 100% concentration, after one half-life, you'd have 50%, then 25%, and so on, cutting in half each time. This is why, even after several half-lives, a small amount of the original reactant can still be present.

Exponential decay is crucial in fields such as pharmacology, where it's important to understand how long a drug remains active in the bloodstream, or radioactivity, where it helps in determining the amount of time until a radioactive isotope decays to a safe level.

Calculating Reaction Half-Lives

Calculating reaction half-lives is a fundamental step when working with first-order kinetics. The half-life equation for a first-order reaction is a simple expression that involves only the rate constant, k, but in many practical situations, we might only have the half-life and the duration of the reaction.

For calculations, we apply the relationship that the half-life for a first-order reaction is defined as \( t_{1/2} = \frac{\ln(2)}{k} \), where \( \ln(2) \) is the natural logarithm of 2. Knowing the half-life allows us to determine how much reactant remains after any given time. As in the textbook problem, if the half-life is 15 minutes, we can discern that in 2 hours— which is 120 minutes, or 8 half-lives— the concentration would decrease by half 8 times.

To visualize: after 1 half-life (15 min), there's 50% left; after 2 half-lives (30 min), 25%; and this trend follows until 8 half-lives, where we calculate the fraction remaining by \( (\frac{1}{2})^8 \), which results in a very small fraction of the initial concentration. Such calculations are imperative in chemical manufacturing, to ensure proper reaction times, and in medicine, to calculate dosing schedules.