Question

The half-life period of a first-order reaction is 10 minutes. The time required for the concentration of the reactant to change from \(0.08 \mathrm{M}\) to \(0.02 \mathrm{M}\) is (a) 10 minutes (b) 20 minutes (c) 30 minutes (d) 40 minutes

Short answer

The time required for the concentration of the reactant to change from 0.08 M to 0.02 M is 20 minutes (option b).

Step-by-step solution

Understand the half-life concept

The half-life of a reaction is the time required for the concentration of the reactant to reduce to half its initial value. In a first-order reaction, the rate of reaction is directly proportional to the concentration of the reactant.

Apply the concept to the given problem

The half-life period given in the problem is 10 minutes. This means that 0.08 M would reduce to 0.04 M in 10 minutes.

Calculate the time for the required concentration change

To go from 0.04 M to 0.02 M, yet another half-life period would pass, which is another 10 minutes. Thus, the time required for the concentration to change from 0.08 M to 0.02 M would be twice the half-life period, or 20 minutes.

Key concepts

Half-Life Period

The concept of half-life is an essential aspect of reaction kinetics, especially for first-order reactions. For any reaction, the half-life is defined as the time taken for the concentration of a reactant to decrease by half. It is a fundamental measure to understand how fast a reaction proceeds over time.

• In a first-order reaction, the half-life is independent of the initial concentration of the reactant. This means that no matter how much reactant you start with, the time it takes for half of it to react will always be the same.
• The formula for the half-life of a first-order reaction is given by: \[ t_{1/2} = \frac{0.693}{k} \] where \( t_{1/2} \) is the half-life and \( k \) is the rate constant of the reaction.
• This concept helps in determining the number of half-lives required for a certain concentration change.

Understanding half-life is crucial as it helps in determining how quickly a reactant will disappear and how this affects the product formation over time.

Rate of Reaction

The rate of a chemical reaction refers to how quickly or slowly the reactants turn into products. In first-order reactions, the rate is directly proportional to the concentration of the reactant. This key feature makes first-order reactions unique.

• For a first-order reaction, the rate can be expressed as: \[ ext{Rate} = k[A] \] where \( k \) is the rate constant and \([A]\) is the concentration of the reactant.
• The units of the rate constant \( k \) for a first-order reaction are \( s^{-1} \).
• This direct relationship implies that if you double the concentration of the reactant, the rate of reaction also doubles.

The understanding of reaction rate is essential for predicting how changes in conditions can speed up or slow down a reaction. By knowing the rate, chemists can design processes to either enhance or control the speed of chemical reactions effectively.

Concentration Change Calculations

In the context of a first-order reaction, concentration change calculations are pivotal for understanding how a reactant depletes over time. These calculations help in determining the time taken for a specific concentration transformation.

• In first-order kinetics, the concentration of reactants decreases exponentially, as described by the formula: \[ [A] = [A]_0 e^{-kt} \] where \([A]\) is the concentration after time \( t \), \([A]_0\) is the initial concentration, \( k \) is the rate constant, and \( e\) is the base of the natural logarithm.
• In our original exercise, understanding that going from 0.08 M to 0.02 M means two half-lives have elapsed, allows us to calculate the total time of 20 minutes, since one half-life is 10 minutes.
• Often, logarithms are used to rearrange the equation for problem-solving: \[ ext{ln}([A]) = ext{ln}([A]_0) - kt \]

Such calculations regularly inform how long it will take for a reaction to reach a particular stage, which is critical in industries ranging from pharmaceuticals to environmental science.