Question

(a) For a generic second-order reaction \(\mathrm{A} \longrightarrow \mathrm{B}\), what quantity, when graphed versus time, will yield a straight line? (b) What is the slope of the straight line from part (a)? (c) How do the half-lives of first-order and second-order reactions differ?

Short answer

For a generic second-order reaction, plotting \(\frac{1}{[\mathrm{A}]}\) versus time will yield a straight line. The integrated rate equation for a second-order reaction is \[\frac{1}{[\mathrm{A}]}-\frac{1}{[\mathrm{A}]_0}=kt,\] so the slope of the straight line is the rate constant, \(k\). The half-life (\(t_{1/2}\)) of first-order reactions is constant and can be expressed as \[t_{1/2}=\frac{\ln(2)}{k},\] whereas the half-life for second-order reactions depends on the initial reactant concentration, given by \[t_{1/2} = \frac{1}{k[\mathrm{A}]_0}.\]

Step-by-step solution

Understanding Second-Order Reactions

For a second-order reaction, the rate law can be represented as: \[rate = k[\mathrm{A}]^2\] where \(k\) is the rate constant and [\(\mathrm{A}\)] is the concentration of the reactant A.

Integrated Rate Equation

To obtain the integrated rate equation for the second-order reaction, we need to rewrite the rate law as a differential equation, and then integrate it. The result is: \[\frac{1}{[\mathrm{A}]}-\frac{1}{[\mathrm{A}]_0}=kt\] where \([\mathrm{A}]_0\) is the initial concentration of A at time \(t = 0\).

(a) Quantity to plot

For a second-order reaction, plotting \(\frac{1}{[\mathrm{A}]}\) versus time will give us a straight line. This is because the integrated rate equation shows that the relationship between \(\frac{1}{[\mathrm{A}]}\) and \(t\) is linear: \[\frac{1}{[\mathrm{A}]} = kt + \frac{1}{[\mathrm{A}]_0}\]

(b) Slope of the straight line

In the integrated rate equation, the slope of the straight line is given by the rate constant, \(k\). Thus: \[slope=k\]

(c) Half-lives of first-order and second-order reactions

The half-life (\(t_{1/2}\)) of a reaction is the time it takes for the reactant concentration to reduce by half. For a first-order reaction, the half-life is constant and can be expressed as: \[t_{1/2}=\frac{\ln(2)}{k}\] For a second-order reaction, the half-life is not constant and depends on the initial concentration of the reactant as follows: \[t_{1/2} = \frac{1}{k[\mathrm{A}]_0}\] As we can see, the half-lives of first-order reactions are independent of the initial concentration, while the half-lives of second-order reactions depend on the initial concentration of the reactant.

Key concepts

Integrated Rate Equation

The integrated rate equation is a key tool for understanding second-order reactions. In a second-order reaction, the rate at which reactants turn into products depends on the square of the concentration of one reactant. The rate law can be written as:

• \[ \text{rate} = k[\mathrm{A}]^2 \]

where

• \( k \) is the rate constant
• \([\mathrm{A}]\) is the concentration of the reactant.

By integrating this rate law, we arrive at the integrated rate equation, which reveals how the concentration of the reactant changes over time:

• \[ \frac{1}{[\mathrm{A}]} - \frac{1}{[\mathrm{A}]_0} = kt \]

In this equation:

• \( [\mathrm{A}]_0 \) represents the initial concentration of the reactant at the start \( t = 0 \).
• \( t \) is the time elapsed.

Plotting \( \frac{1}{[\mathrm{A}]} \) against time \( t \) yields a straight line, confirming a second-order reaction. The usefulness of this linear relationship is that it simplifies the analysis of the reaction's kinetics, and it makes it intuitive to determine various variables such as the concentration of reactants at any point in time.

Rate Constant

The rate constant, symbolized as \( k \), is an essential parameter in chemical kinetics. It varies depending on the reaction order:

• For a second-order reaction, the units of \( k \) are \( \text{M}^{-1}\text{s}^{-1} \), reflecting the change in concentration over time.

In the integrated rate equation for a second-order reaction:

• \[ \frac{1}{[\mathrm{A}]} = kt + \frac{1}{[\mathrm{A}]_0} \]

The rate constant \( k \) is the slope of the line when \( \frac{1}{[\mathrm{A}]} \) is plotted against time. This slope is a direct indicator of how rapidly a reaction occurs.
Factors affecting the rate constant include temperature and the presence of catalysts. Typically, increasing the temperature or adding a catalyst will result in a higher \( k \) value, which means a faster reaction.
Understanding the rate constant allows chemists to compare different reactions and predict how changes in conditions will affect the speed of a reaction. Moreover, by determining \( k \) experimentally, one can gain insights into the reaction mechanism and dynamics.

Half-Life

Half-life, denoted as \( t_{1/2} \), is the time it takes for half of the reactant to be consumed in a chemical reaction. The concept of half-life varies between reaction orders:

• First-Order Reactions

  In these reactions, the half-life is constant and independent of the initial concentration of the reactant. It can be calculated using the formula: \[ t_{1/2} = \frac{\ln(2)}{k} \]

• Second-Order Reactions

  Here, the half-life depends on the initial concentration of the reactant, and is calculated by the equation:\[ t_{1/2} = \frac{1}{k[\mathrm{A}]_0} \]

This dependency on initial concentration for second-order reactions means that as a reaction progresses and the concentration of the reactant decreases, the half-life increases. As a result, these reactions can initially proceed quite quickly but slow down over time as the reactants are depleted.
Understanding half-life is crucial for predicting reaction outcomes, particularly in processes where timing is critical, such as in pharmaceutical drug design and environmental chemistry. It helps chemists design reactions and processes that remain efficient throughout their duration.