Book Cover Chemistry 8th Edition / Chang
Student Study Guide

Chapter 13: Chemical Kinetics


Index | 13.1 | 13.2 | 13.3 | 13.4 | 13.5 | 13.6 |

THE RELATION BETWEEN REACTANT CONCENTRATION AND TIME (13.3)

STUDY OBJECTIVES
  1. Determine for first- and second-order reactions the concentration of a reactant at any time after the reaction has started, or the time required for a given fraction of the sample to react.
  2. Determine the half-life of a first-order reaction.
  3. Apply rate law equations and graphs to determine whether a reaction is first order or second order.

First-Order Reactions. One of the most widely encountered kinetic forms is the first-order rate equation. In this case the exponent of [A] in the rate law is 1.

A products

For a first-order reaction the unit of the rate constant is reciprocal time, 1/t. Convenient units are s–1, h–1, etc.

The equation that relates the concentration of A remaining to the time since the reaction started is

This is a very useful equation called the integrated first-order equation. Here [A]0 is the concentration of A at time = 0, and [A] is the concentration of A at time = t. The rate constant k is the first-order rate constant. The concentration [A] decreases as the time increases. This equation allows the calculation of the rate constant k when [A]0 is known, and [A] is measured at time t. Also, once k is known, [A] can be calculated for any future time.

To determine whether a reaction is first order, we rearranged the first-order equation into the form:

ln [A] = – kt + ln [A]0

corresponding to the linear equation

y = mx + b

Here m is the slope of the line and b is the intercept on the y axis. Comparing the last two equations, we can equate y and x to experimental quantities.

y = ln [A] and x = t

Therefore, the intercept b = ln [A]0, and the slope of the line m = –k. Thus a plot of ln [A] versus t for a first-order reaction gives a straight line with a slope of –k as shown in Figure 13.1 below. If a plot of ln [A] versus t yields a curved line, rather than a straight line, the reaction is not a first-order reaction. This graphical procedure is the method used by most chemists to determine whether or not a given reaction is first order.

Figure 13.1 A plot of [A] versus time for a first-order reaction gives a curved line. A plot of ln [A] versus time gives a straight line for a first-order reaction.

Half-life. The half-life of a reaction, t1/2, is a useful concept. For a first-order reaction, the half-life is given by:

t1/2 =
ln 2
k

where ln 2 (0.693) is a constant and k is the rate constant. Knowledge of the half-life allows the calculation of the rate constant k. The half-life of a reaction is the time required for the concentration of a reactant to decrease to half of the initial value. After one half-life, the ratio [A]/[A]0 is equal to 0.5.

If the reaction continues, then [A] will drop by 1/2 again during the second half-life period as shown in Figure 13.2. After two half-life periods the fraction of the original concentration of A remaining, [A]/[A]0, will be 1/2 of the concentration remaining after the first half-life, so [A]/[A]0 = 0.5 x 1/2 = 0.25.

Figure 13.2 A plot of [A] versus time for a first-order reaction gives a curved (exponential) line. Over each half-life period, [A] drops in half.

EXAMPLE First-Order Reaction

Methyl isocyanide undergoes a rearrangement to form methyl cyanide that follows first-order kinetics.

CH3NC(g) CH3CN(g)

The reaction was studied at 199°C. The initial concentration of CH3NC was 0.0258 mol/L, and after 11.4 min analysis showed the concentration of product was 1.30 x 10–3 mol/L.

a. What is the first-order rate constant?
k = x 10^ min–1

b. What is the half-life of methyl isocyanide?
t1/2 = min

c. How long will it take for 90 percent of the CH3NC to react?
t = min

Second-Order Reactions. In a second-order reaction, the rate is proportional either (1) to the square of the concentration of one reactant,

A product

or (2) to the product of the concentrations of two reactants, each raised to the first power:

A + B product

This reaction is first order in A and first order in B, and so it is second order overall. For a second-order reaction, the rate constant has units 1/(molarity x time) which is 1/M·s or M–1 s–1.

An important equation called the integrated second-order equation, which applies to a second-order reaction with one reactant, is

where [A]0, [A], and t have their usual meaning, and k is the second-order rate constant. This equation allows the calculation of the concentration of A at any time t after the reaction has begun. Alternatively, if [A]0, [A], and t are known, the rate constant can be calculated. Example 13.6 in the text shows an example calculation using this equation.

The half-life for a second-order reaction is given by

Here we see that the half-life is inversely proportional to the initial concentration, [A]0. This situation is different from that for a first-order reaction, where t1/2 is independent of [A]0.

EXAMPLE Second-Order Reaction

At 230°C the rate constant for methyl isocyanide isomerization is 9.25 x 10–4 s–1.

CH3NC CH3CN

a. What fraction of the original isocyanide will remain after 60.0 min?

b. What is the half-life of methyl isocyanide at this temperature?
s

Zero Order Reactions. A zero-order reaction is one in which the rate does not depend on the concentrations of reactants.

A + B products

A plot of [A] versus time is a straight line for a zero order reaction. The rate does not slow down as the reactant is used up.

Summary. The integrated rate equations and graphical methods allow us to distinguish between the various overall orders of reaction.

  1. A reaction is first order when a plot of ln [A] versus t is a straight line. A plot of [A] versus time will be curved.
  2. A reaction is zero order when a plot of [A] versus t is a straight line.
  3. A reaction is second order when a plot of 1/[A] versus t is a straight line, and ln[A] versus t is curved.
  4. The half-life equations also provide a way to distinguish between first- and second-order reactions. The half-life of a first-order reaction is independent of starting concentration, whereas the half-life of a second-order reaction is inversely proportional to the initial concentration.

OBJECTIVE CHECK

Complete the following questions to check your understanding of the material. Select the check button to see if you answered correctly.

  1. A certain first-order reaction A B is 40% complete (40% of the reactant is used up) in 75 s. What is (a) the rate constant and (b) the half-life of this reaction?
    rate constant =
    half-life =
  2. a. Write the integrated rate equation for a first-order reaction? Define the variables.
    1. How can the equation be plotted to give a straight line? Define x, y, and the slope.
    2. What information do you need in order to calculate the first-order rate constant?
  3. The half-life of a certain first order reaction is 112 min. What percent of the initial concentration of reactant will remain after 89 min?
  4. N2O5 is an unstable compound that decomposes according to the following equation.
    2N2O5 4NO2 + O2
    The following data was obtained at 50°C.
      [N2O5] (M) Time (s)
      1.00 0
      0.88 200
      0.78 400
      0.69 600
      0.61 800
      0.54 1000
      0.48 1200
      0.43 1400
    1. Use the data to calculate the first-order rate constant.
    2. How long will it take for the concentration of N2O5 to fall to 0.25 M?
    3. What is the half-life?
  5. What are the units of a first-order and a second-order rate constant?
    first-order =
    second-order =
  6. A decompositon reaction has a rate constant of 0.12 y–1 at a certain temperature.
    1. What is the half-life of the reaction at the same temperature?
    2. How long will it take for the concentration of the reactant to reach 20% of its initial value?

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