Question

(a) For a second-order reaction, what quantity, when graphed versus time, will yield a straight line? (b) How do the half-lives of first-order and second- order reactions differ?

Short answer

(a) For a second-order reaction, graphing \(\frac{1}{[A]_t}\) versus time (\(t\)) will yield a straight line. (b) The half-life of a first-order reaction is constant and independent of the initial concentration, while the half-life of a second-order reaction depends on the initial concentration, increasing with decreasing reactant concentration.

Step-by-step solution

Understanding what a second-order reaction is

A second-order reaction is a type of chemical reaction where the reaction rate is proportional to the square of the concentrations of one or more reactants.

Write down the rate law for a second-order reaction

The rate law for a second-order reaction can be written as: \[rate = k[A]^2\] where \(rate\) is the rate of the reaction, \(k\) is the rate constant, and \([A]\) is the concentration of the reactant A.

Write down the integrated rate law for a second-order reaction

Integrating the rate law, we have the integrated rate law for a second-order reaction: \[\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}\] where \([A]_t\) is the concentration of A at time \(t\), and \([A]_0\) is the initial concentration of A.

Identify the quantity that yields a straight line

From the integrated rate law, we can see that when we graph \(\frac{1}{[A]_t}\) versus \(t\), we will get a straight line. The slope of this line is equal to the rate constant, \(k\). Question (b):

Write down the half-life expressions for first-order and second-order reactions

The half-life of a first-order reaction can be represented by: \[t_{1/2}^{(1)} = \frac{\ln 2}{k}\] The half-life of a second-order reaction can be represented by: \[t_{1/2}^{(2)} = \frac{1}{k[A]_0}\] where \(t_{1/2}^{(1)}\) and \(t_{1/2}^{(2)}\) are the half-lives for first-order and second-order reactions, respectively.

Compare the expressions for half-lives

In a first-order reaction, half-life is constant and independent of the initial concentration (\([A]_0\)). This means that as the reaction proceeds, the half-life remains the same. On the other hand, in a second-order reaction, half-life depends on the initial concentration (\([A]_0\)). As the reaction proceeds, the half-life will change. Specifically, as the concentration decreases, the half-life increases.

Summarize the differences between half-lives

To summarise, the main difference between the half-lives of a first-order and second-order reaction is that the half-life of a first-order reaction is constant and independent of the initial reactant concentrations, while the half-life of a second-order reaction changes as the reactant concentrations change- it increases with decreasing reactant concentration.