5 Steps to a 5 AP Chemistry

is a constant, independent of reactant concentration, and can be shown to have the follow-

ing mathematical relationship:

For second-order reactions, the half-life does depend on the reactant concentration and

can be calculated using the following formula:

This means that as a second-order reaction proceeds, the half-life increases.

Radioactive decay is a first-order process, and the half-lives of the radioisotopes are well

documented (see the chapter on Nuclear Chemistry for a discussion of half-lives with

respect to nuclear reactions).

Consider the following problem: The rate constant for the radioactive decay of

thorium-232 is 5.0 × 10 −^11 /year. Determine the half-life of thorium-232.

Answer: 1.4 × 1010 yr.

This is a radioactive decay process. Radioactive decay follows first-order kinetics. The

solution to the problem simply requires the substitution of the k-value into the appropri-

ate equation:

tl/2=0.693/k=0.693/5.0 × 10 −^11 yr−^1 =1.386 × 1010 yr

which rounds (correct significant figures) to the answer reported.

Consider another case: Hydrogen iodide, HI, decomposes through a second-order

process to the elements. The rate constant is 2.40 × 10 −^21 /M s at 25°C. What is the half-

life for this decomposition for a 0.200 M of HI at 25°C?

Answer: 2.08 × 1021 s.

The problem specifies that this is a second-order process. Thus, you must simply enter

the appropriate values into the second-order half-life equation:

t1/2=1/k[A] 0 =1/(2.40 × 10 −^21 /M s)(0.200 M) =2.08333 × 1021 seconds

which rounds to the answer reported.

If you are unsure about your work in either of these problems, just follow your units.

You are asked for time, so your answer must have time units only and no other units.

Activation Energy

A change in the temperature at which a reaction is taking place affects the rate constant k.

As the temperature increases, the value of the rate constant increases and the reaction is

faster. The Swedish scientist Arrhenius derived a relationship in 1889 that related the rate

constant and temperature. The Arrhenius equation has the form: k=Ae−Ea/RTwhere kis the

rate constant, Ais a term called the frequency factor that accounts for molecular orienta-

tion, eis the natural logarithm base, Ris the universal gas constant 8.314 J mol K−^1 , Tis

the Kelvin temperature, and Eais the activation energy, the minimum amount of energy

that is needed to initiate or start a chemical reaction.

The Arrhenius equation is most commonly used to calculate the activation energy of a

reaction. One way this can be done is to plot the ln kversus 1/T. This gives a straight line

whose slope is −Ea/R. Knowing the value of Rallows the calculation of the value of Ea.

Normally, high activation energies are associated with slow reactions. Anything that can

be done to lower the activation energy of a reaction will tend to speed up the reaction.

t

1/2 kA[] 0

=

1

t

1/2 kk

ln 2

==

0 693.

202  Step 4. Review the Knowledge You Need to Score High