Question

Define the half-life of a reaction. Explain on the molecular level why the half-life of a first-order reaction is constant.

Short answer

The half-life is the time taken for a reactant's concentration to halve. In first-order reactions, it is constant because the rate depends only on the concentration.

Step-by-step solution

Define Half-Life

The half-life of a reaction is the time required for the concentration of a reactant to decrease to half of its initial concentration.

Understand First-Order Reactions

For a first-order reaction, the rate of the reaction is directly proportional to the concentration of the reactant. This can be expressed with the rate law: \[ \text{Rate} = k[A] \] where \( k \) is the rate constant and \([A]\) is the concentration of the reactant.

Derive the Half-Life Formula for First-Order Reactions

The integrated rate law for a first-order reaction is: \[ [A_t] = [A_0] e^{-kt} \] where \([A_t]\) is the concentration at time \(t\), \([A_0]\) is the initial concentration, and \(k\) is the rate constant. To find the half-life \(t_{1/2}\), set \([A_{t_{1/2}}] = \frac{[A_0]}{2} \) and solve for \( t \):\[ \frac{[A_0]}{2} = [A_0] e^{-kt_{1/2}} \]Divide both sides by \([A_0]\):\[ \frac{1}{2} = e^{-kt_{1/2}} \]Take the natural logarithm of both sides:\[ \text{ln} \frac{1}{2} = -kt_{1/2} \]Since \(\text{ln} \frac{1}{2} = - \text{ln} 2\), we have:\[ - \text{ln} 2 = -kt_{1/2} \]Therefore, solving for \( t_{1/2}\):\[ t_{1/2} = \frac{\text{ln} 2}{k} \]

Explain the Constant Half-Life on the Molecular Level

In a first-order reaction, the fraction of reactants that react in a given time period is constant because the rate depends only on the concentration. As molecules react and the concentration decreases, the rate decreases proportionally so that it always takes the same amount of time for the concentration to halve.

Key concepts

first-order reaction kinetics

First-order reaction kinetics describe reactions where the reaction rate is directly proportional to the concentration of one reactant. This means as the reactant's concentration decreases, so does the rate of the reaction.

To visualize this, consider the reaction rate law for a first-order reaction: \[\text{Rate} = k[A]\],
where \[k\] is the rate constant and \[A\] is the concentration of the reactant.
Imagine you start with a certain concentration of reactant. As the reaction progresses and the quantity of reactant diminishes, the reaction rate falls proportionately. This linear relationship keeps the study of these reactions simple but powerful in understanding how time and concentration are interconnected.

rate constant

The rate constant, symbolized as \[k\], is a crucial factor in the kinetics of a reaction. It determines the speed of the reaction at any given concentration of reactant.

Think of \[k\] as a multiplier in the rate equation: \[\text{Rate} = k[A]\].
Its value depends on factors like temperature and the presence of a catalyst. Higher temperatures or the addition of a catalyst typically increase the rate constant, making reactions proceed faster.

For a first-order reaction, \[k\] is calculated using the formula for the half-life: \[t_{1/2} = \frac{\text{ln} 2}{k}\] going further, knowing \[k\] allows you to predict how long it will take for a reaction to reach a certain level of completion, which is invaluable in both laboratory and industrial settings.

integrated rate law

The integrated rate law for a first-order reaction helps to connect the concentration of reactants at a given time directly to the initial concentration and the time elapsed. This is particularly useful for calculating how much reactant remains after a certain period.

For a first-order reaction, the integrated rate law is given by: \[ [A_t] = [A_0] e^{-kt}\]. Here, \[ [A_t]\] is the concentration of the reactant at time \[t\], and \[ [A_0]\] is the initial concentration.

You can use this equation to see how the concentration changes over time. By inputting various time values into the equation, you can plot a decay curve, which showcases the exponential decrease in reactant concentration. This is essential for understanding how quickly a reaction proceeds and helps in predicting the duration required to reach specific concentrations.

reaction rate

The reaction rate is a measure of how quickly a reactant is consumed or a product is formed in a chemical reaction. For first-order reactions, the rate is directly tied to the concentration of one reactant.

This relationship is expressed as: \[ \text{Rate} = k[A] \], meaning that if the concentration of \[A\] changes, the rate changes proportionally.

The concept is intuitive: imagine you have a pile of chemical \[A\]; the larger the pile, the more 'events' (reactions) you can observe per unit of time. Hence, as \[A\] decreases, so does the observable rate of these reactions. Understanding this concept sheds light on why the half-life of first-order reactions remains constant, as a fixed fraction of the reactant is consumed uniformly over equal time intervals.