*Interactive Case Study: *Arrhenius - Linearizing an exponential equation

Produced by Graham Currell, University of the West of England, Bristol, in association with:, and

● Royal Society of Chemistry, 'Discover Maths for Chemists' website

*●*

**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

gives the rate constant, *k*, of a chemical reaction as a function of temperature, *T*, where *E _{a}* 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:

where R is the gas constant with a value of 8.314 J K^{-1}mol^{-1},

calculate:

activation energy,E_{a}, 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: E_{a} = 87.9 kJ mol^{-1}, and A = 13.9 × 10^{3} 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

Eand_{a}A.

Please send any comments and suggestions to:

graham.currell@uwe.ac.uk

17/09/09