Question

For a first-order reaction, how long will it take for the concentration of reactant to fall to one-eighth its original value? Express your answer in terms of the half-life \(\left(t_{\frac{1}{2}}\right)\) and in terms of the rate constant \(k\)

Short answer

It will take three half-lives (\(3 \cdot t_{\frac{1}{2}}\)) or approximately 2.08 times the inverse of the rate constant (\(2.079 \cdot \frac{1}{k}\)) for the concentration of a reactant to drop to one-eighth its original value in a first-order reaction.

Step-by-step solution

- Understanding a First Order Reaction

For first order reactions, the rate of reaction decreases exponentially over time. We can express this as a mathematical relationship, i.e., \( N = N_0 \cdot (0.5) ^ \frac{t}{t_{\frac{1}{2}}}\) where \(N\) is the final concentration, \(N_0\) is the initial concentration, and \(t_{\frac{1}{2}}\) is the half-life.

Step 2- Applying the Relationship

We want the concentration to fall to one-eighth its original value, thus \( N = \frac{1}{8} N_0 \). Now we substitute \(N\) into the equation from step 1 and simplify which results in \( \frac{1}{8} = (0.5) ^ \frac{t}{t_{\frac{1}{2}}}\).

- Solving the Equation

We need to solve the equation from step 2 for \(t\). This involves utilising properties of logarithms. Solving the equation gives us \( t = 3 \cdot t_{\frac{1}{2}}\). This means that for a first order reaction, it will take three half-lives for the concentration of reactant to fall to one-eighth its original value.

- Express the Solution in Terms of the Rate Constant

For a first-order reaction, the half-life \(t_{\frac{1}{2}}\) is related to the rate constant \(k\) by the equation \(t_{1/2} = \frac{0.693}{k}\). To express our solution in terms of the rate constant, substitute this equation into our answer from step 3. We then get \(t = 3 \cdot \frac{0.693}{k} = 2.079 \cdot \frac{1}{k} \). This means that it will take approximately 2.08 times the inverse of the rate constant for the concentration of the reactant to fall to one-eighth its original value.

Key concepts

rate constant

In first-order reactions, the rate constant, denoted as \(k\), is a crucial parameter in determining how quickly a reaction proceeds. The rate constant is a fixed value that describes the rate of the reaction at a molecular level, independent of the concentration of the reactant. It essentially tells us how likely an individual molecule is to react in a given period of time.

Understanding the rate constant is important because it connects directly to the half-life and the speed of exponential decay in these types of reactions. The relationship between the rate constant \(k\) and the half-life \(t_{1/2}\) for a first-order reaction is given by the formula:

• \(t_{1/2} = \frac{0.693}{k}\)

This equation highlights that as \(k\) increases, \(t_{1/2}\) decreases, meaning the reaction completes faster. This direct inverse relationship helps in predicting reaction behavior using \(k\).

Knowing \(k\) allows chemists to make quantitative predictions about how the concentration of a reactant will change over time, which is invaluable in both laboratory and industrial settings.

half-life

Half-life, denoted as \(t_{1/2}\), is a core concept when studying first-order reactions. It is defined as the time required for half of the initial amount of reactant to be consumed. In the context of a first-order reaction, the half-life remains constant regardless of the initial concentration, making it a unique and useful measure for these reactions.

To visualize, if you start with a certain amount of substance, after one half-life, only half of it will remain unreacted. After another half-life, a quarter remains, and so forth. This regularity helps in predicting how long it will take for the concentration to reach a specific fraction of the initial amount. For example, in the exercise, it takes three half-lives for the concentration to decrease to one-eighth of the original.

This predictability stems from the mathematical relationship between concentration and time in first-order reactions. It's important to remember that the concept of half-life in this context fully relies on its definition relating to the rate constant \(k\), as given by \(t_{1/2} = \frac{0.693}{k}\). This ensures that half-life calculations are straightforward once \(k\) is known.

exponential decay

Exponential decay is a key aspect of first-order reactions. This term describes the process through which the concentration of a reactant decreases at a rate proportional to its current value, leading to a rapid reduction over time.

The mathematical form used to express exponential decay in first-order reactions is given by:

• \(N = N_0 \cdot e^{-kt}\)

Where \(N\) is the concentration at time \(t\), \(N_0\) is the initial concentration, and \(k\) is the rate constant.

Exponential decay characterizes how reactant concentrations fall off quickly at first, but then slow down as the reactant is depleted. It's a smooth, continuous process reflecting the nature of these reactions. The concept is particularly useful as it not only applies to concentrations but also describes other phenomena like radioactive decay.

In practical terms, understanding exponential decay allows scientists and engineers to predict how long a given amount of reactant will last, making it easier to plan chemical processes and manage resource use effectively. This concept serves as a foundation for much of reaction kinetics in chemistry.