Question

Find the ratio of the t₀ and t for a first-order reaction.

Answer 1

The ratio of t₀ and t for a first-order reaction is given by the equation:

\[ \ln \left( \frac{[A_0]}{[A]} \right) = kt \]

Explanation:

This equation represents the relationship between the time taken for a reaction to occur at various stages in a first-order reaction. Let's break it down:

• t₀: This is the initial time of the reaction, often considered as the time when the reaction starts, or the time at which the concentration of reactants is at its maximum.
• t: This is the time at which the reaction is measured or observed, where the concentration of reactants has decreased due to the progress of the reaction.
• k: This is the rate constant for the first-order reaction, which dictates how fast the reaction proceeds. It is a measure of the reaction's speed, with higher values of k indicating faster reactions.

The equation \(\frac{t_0}{t} = \frac{2.303}{k}\) illustrates the inverse relationship between the reaction time and the rate constant. As the rate constant increases, the time t required for the reaction to progress decreases. This equation can be used to estimate the rate constant k if the times t₀ and t are known.

Important Concept: The factor 2.303 comes from the mathematical derivation of the integrated rate law for a first-order reaction, which involves logarithms. The general form of the rate law for a first-order reaction is:

\[ \ln \left( \frac{[A_0]}{[A]} \right) = kt \]

Where [A₀] is the initial concentration of reactant A, and [A] is the concentration of A at time t. In this case, the equation is simplified to give the relationship above when using logarithms to solve for reaction time.