An Alternate Form of the Integrated First-Order Rate Equation

Radioactive processes and many chemical processes follow first order kinetics. The usual equations

found in general chemistry textbooks are:

a. ln Ao / A = kt , Where, Ao is the original amount of the sample, A is the amount at time t and k is the rate constant.

b. Changing the rate constant to half-life, ln2 = kt_1/2 , where t_1/2 is the half-life.

c. Solving equation 2 for k and substituting into equation 1 the result is lnAo / A = (ln2)t/t_1/2 .

d. Rearranging equation 3 gives / (ln2)t /t_1/2 o A = A e as indicated in [1].

e. Since eln2 = 2 , substitution into equation 4 yields / 2t /t1/2 o A = A the Alternate Form of the

Integrated first-order rate equation

Our students have found equation 5 to be relatively easier to use than equations 1 and 2. In equation 5, by dividing the time by the half-life, they get a number. On their calculators, they enter the number 2, y^x, the number and press=The result is divided into A_o, giving the value for A. For radioactive processes, the values of Ao and A may be in mass, such as grams, or activity in Becquerel’s (counts/second). For chemical processes, units for A_o and A may be written as rates, such as molarity/second.

1. Kenneth AC (1991) Chemical kinetics, the study of reaction rates in solution. VCH Publishers, USA, p. 496.