Question

Consider the zeroth-order reaction: \(\mathrm{A} \longrightarrow\) product. (a) Write the rate law for the reaction. (b) What are the units for the rate constant? (c) Plot the rate of the reaction versus [A].

Short answer

(a) Rate = k; (b) Units: \(\text{mol L}^{-1} \text{s}^{-1}\); (c) Plot: Horizontal line.

Step-by-step solution

Identify the Order of the Reaction

The problem states that this is a zeroth-order reaction. In a zeroth-order reaction, the rate is independent of the concentration of the reactant.

Write the Rate Law

For a zeroth-order reaction, the rate is constant and given by the rate law: \[\text{Rate} = k\]where \(k\) is the rate constant.

Determine the Units for the Rate Constant

The rate of a reaction is typically expressed in units of concentration per unit time, such as \(\text{mol L}^{-1} \text{s}^{-1}\). For a zeroth-order reaction, the rate constant \(k\) must have the same units as the rate to ensure dimensional consistency. Therefore, the units for \(k\) are \(\text{mol L}^{-1} \text{s}^{-1}\).

Plot the Rate of the Reaction versus [A]

In a zeroth-order reaction, the rate is constant and does not depend on the concentration of \([A]\). Thus, when plotting the rate against the concentration \([A]\), the graph is a horizontal line, indicating that the rate remains constant as \([A]\) varies.

Key concepts

Rate Law for Zeroth-Order Reactions

In chemical kinetics, the rate law is an equation that describes the rate of a reaction in terms of the concentration of reactants. For a zeroth-order reaction like \(\mathrm{A} \longrightarrow \text{product}\), the rate is independent of the concentration of the reactant. This means the rate doesn't change even if the concentration of \([\text{A}]\) increases or decreases.

The rate law for a zeroth-order reaction is:

• \(\text{Rate} = k\)

Here, \(k\) is the rate constant. In zeroth-order reactions, since the rate doesn't change with \([\text{A}]\), it remains constant throughout the reaction.

Understanding the Rate Constant

The rate constant, symbolized by \(k\), is a crucial concept in understanding reaction rates. For a zeroth-order reaction, the units for \(k\) are the same as that of the rate itself because the rate is constant. Typically, the units are expressed as concentration over time.

For example, the units for \(k\) in a zeroth-order reaction are:

• \(\text{mol L}^{-1} \text{s}^{-1}\)

These units indicate that the reaction rate reflects how the concentration of the reactant changes per second or other time unit. It's essential that \(k\)'s units match the rate's units to maintain dimensional consistency across the equation.

Concentration vs. Time Plot

Plotting a concentration vs. time graph helps visualize the progress of a reaction. For a zeroth-order reaction, the concentration of \([\text{A}]\) decreases linearly with time. This is because the rate is constant regardless of the reactant's concentration.

If you were to plot the rate of the reaction against \([\text{A}]\), you would see a horizontal line. Here's why:

• The rate is constant, shown by a straight line parallel to the x-axis.
• The slope of the line is zero since changes in concentration do not affect the rate.

This graph clearly shows that the reaction progresses at a uniform speed, supporting the concept of a zeroth-order reaction where the reaction rate doesn't change with varying concentrations of \([\text{A}]\).