Interactive Case Study:    Arrhenius - Linearizing an exponential equation

book cover Produced by Graham Currell, University of the West of England, Bristol, in association with:
Royal Society of Chemistry, 'Discover Maths for Chemists' website
, and

Essential Mathematics and Statistics for Science, 2nd Edition
   Graham Currell and Antony Dowman, Wiley-Blackwell, 2009

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Measurement of the Activation Energy of a Reaction (Arrhenius equation)
download the following question as a Word document for educational use)

The Arrhenius equation 

Arrhenius equation

gives the rate constant, k, of a chemical reaction as a function of temperature, T, where Ea is the activation energy of the reaction and R is the gas constant. 

The following exercise gives a worked example with video answers to illustrate the mathematics of this problem.
You can then follow study links to other resources to help you with the mathematics.

The rate constant, k, of a chemical reaction is measured at several temperatures, T, as given in the table below:

T 600 700 800 900 1000 ºK
k 0.00034 0.0031 0.029 0.10 0.38 s-1

where T is in degrees Kelvin and k is measured in units of s-1.

The above data can be downloaded in the Excel (2003) file ArrheniusData
 

Answer the following questions:

Question 1        

Is this a first order or a second order reaction?

Question 2        

Assuming the data follows the Arrhenius equation:
Arrhenius equation
where R is the gas constant with a value of 8.314 J K-1 mol-1,
calculate:

  • activation energy, Ea, of the reaction, and
  • the value of the constant, A, in the equation.

Study guide:

The numerical and video answers are given in the sections below.
The following techniques are relevant to the calculations in this problem:

Standard techniques of rearranging equations (study guide)
Key properties of exponentials and logarithms (study guide)
Linear regression for the slope and intercept of a straight line (study guide)
Excel for data analysis (web): X-Y graphs using Excel (video), Linear regression using Excel (video)

Numerical Answers:

Question 1 It must be a first order reaction because the units of the rate constant, k, are given as: s-1.
  The units of k for a second order reaction would be given as: s-1 (mol L-1)-1 or L mol-1 s-1 .

Question 2 The correct answers are: Ea = 87.9 kJ mol-1, and A = 13.9 × 103 s-1

Video Answers: including specific links to revision/help in the relevant mathematical skills:

Linearising the Arrhenius equation  

Using Excel to plot the data and perform a linear regression to calculate Ea and A.

Please send any comments and suggestions to:
graham.currell@uwe.ac.uk
17/09/09