Crimson Publishers Publish With Us Reprints e-Books Video articles

Full Text

Significances of Bioengineering & Biosciences

An Alternate Form of the Integrated First-Order Rate Equation

Frank E Stary*

Department of Chemistry, USA

*Corresponding author:Frank E Stary, Department of Chemistry, USA

Submission: June 11, 2019; Published: June 25, 2019

DOI: 10.31031/SBB.2019.03.000564

ISSN 2637-8078
Volume3 Issue3


Derivation of a first-order equation suitable for use in beginning energy science and chemistry courses is shown to be

A = Ao/ 2t /t1/2


Ao is the original amount of the sample

A is the amount

T is time t and

t1/2 is the half-life

Ao is larger than A

Derivation of the Alternate Form

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 = kt1/2 , where t1/2 is the half-life.

c. Solving equation 2 for k and substituting into equation 1 the result is lnAo / A = (ln2)t/t1/2 .

d. Rearranging equation 3 gives / (ln2)t /t1/2 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, yx, the number and press=The result is divided into Ao, 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 Ao 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.

© 2019 Frank E Stary. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and build upon your work non-commercially.

About Crimson

We at Crimson Publishing are a group of people with a combined passion for science and research, who wants to bring to the world a unified platform where all scientific know-how is available read more...

Leave a comment

Contact Info

  • Crimson Publishers, LLC
  • 555 Madison Avenue, 5th floor
  •     New York, NY 10022, USA
  • +1 (929) 600-8049
  • +1 (929) 447-1137