Question

In a first-order reaction, the conc. of reactant decreases from \(1.0 \mathrm{M}\) to \(0.25 \mathrm{M}\) in 20 minutes. The value of \(k\) is (a) \(0.6932\) (b) \(6.932\) (c) \(0.06932\) (d) none of these

Short answer

(a) 0.6932

Step-by-step solution

Identify Variables

From the problem, identify the initial reactant concentration (\[A_0\]) as 1.0 M, the final reactant concentration (\[A\]) as 0.25 M, and time (t) as 20 minutes or 1/3 hours.

Convert to natural logarithm

Convert the reactant concentrations to natural logarithm, which results in ln\[A_0\] = ln(1) and ln\[A\] = ln(0.25) = -1.386.

Substitute values into the rate equation

Substitute the values into the rate equation: -(-1.386) = k * 1/3 + 0. Solving for the value of k gives k = 0.6932 per hour.

Matching the answer

Math the calculated value of k with the options given in the problem. The answer is (a) 0.6932.

Key concepts

Reaction Kinetics

Understanding how reactions proceed over time is the essence of reaction kinetics. Simply put, it's about studying the speed or rate at which chemical processes occur. By analyzing reaction kinetics, scientists and engineers can predict how long it will take for a reaction to complete, which is critical for everything from pharmaceutical development to materials manufacturing.

For first-order reactions, the rate at which reactants transform into products depends directly on the concentration of one reactant. The unique feature of this order is that the half-life (the time required for half of the reactant to be consumed) remains constant and does not depend on the initial concentration. This property immensely simplifies the analysis and prediction of the reaction’s progress over time.

Rate Constant (k)

The rate constant, symbolized as 'k', is a critical component in the kinetics of a reaction, serving as a quantitative measure of how quickly a reaction occurs. For a first-order reaction, the rate law can be expressed as rate = k[A], where [A] is the concentration of the reactant. The rate constant has unique units for each reaction order; for first-order reactions, it's typically per second (s^-1) or per hour (hr^-1).

In our example problem, we calculated 'k' using the natural logarithm of the concentration changes over a given time. This approach is rooted in the integrated rate law for first-order reactions, which is ln([A₀]/[A]) = kt. Here, [A₀] is the initial concentration and [A] is the concentration at time 't'. A higher value of 'k' corresponds to a faster reaction.

Concentration of Reactants

The concentration of reactants is a cornerstone concept in reaction kinetics, especially for first-order reactions. The rate of a first-order reaction is directly proportional to the concentration of the singular reactant. As concentration diminishes, so does the reaction rate — it's a linear relationship.

In our textbook exercise, we examined the change in concentration of a reactant over time and used this information to determine the rate constant 'k'. The exercise demonstrated a fundamental characteristic of first-order reactions; a decrease in reactant concentration by a factor of four, as we saw when the concentration dropped from 1.0 M to 0.25 M. Understanding how to translate these concentration changes into kinetic data is crucial for mastering reaction kinetics.