Physical Unit 4.2.8. The Arrhenius Equation

​The Arrhenius Equation

​The Arrhenius equation shows how the rate constant k depends on temperature T and activation energy E_a ​ (the minimum energy needed for a reaction).

​The rate constant and activation energy

​The Arrhenius equation shows how the rate changes when temperature or activation energy changes.
If E_a ​ is high, the reaction is slow because fewer particles have enough energy to react, so fewer collisions are successful.

​(Frequency Factor)

​The frequency factor A is a constant in the Arrhenius equation that represents how often particles collide with the correct orientation to react.

​The rate constant and temperature

​The equation shows that k increases as temperature rises. This is because higher temperature means particles move faster, collide more often, and more collisions have enough energy to overcome the activation energy, increasing the reaction rate.

​You may be given four values in the Arrhenius equation and asked to find the fifth by rearranging the equation and substituting the known values.

​Using the Arrhenius equation

​Solving for Activation Energy (E_a)

​Solving for Frequency Factor (A)

​Solving for Temperature (T)

or

​Example : The decomposition of HI at 500 K has a rate constant of 3.20 × 10^⁻3 s⁻¹. The Arrhenius constant for this reaction is 2.50 × 10¹³ s⁻¹. Calculate the activation energy of this reaction. (R = 8.31 J K⁻¹ mol⁻¹)

​Example : The decomposition of NO₂ has a rate constant of 2.50×10⁻³ s⁻¹ . The Arrhenius constant for this reaction is 1.20×10¹³ s⁻¹ . The activation energy is 95.0 kJ mol⁻¹ . Calculate the temperature at which this reaction occurs. (R = 8.31 J K⁻¹ mol⁻¹)

​Arrhenius plots

​Writing the Arrhenius equation in logarithmic form usually makes it easier to work with. By plotting ln k against 1/T (an Arrhenius plot), we can use this linear relationship to determine key values. The gradient of the straight line is −Ea/R, while the y-intercept equals ln A.

Multiple Choice

What happens to the rate constant k when the temperature increases, according to the Arrhenius equation?

• It decreases because molecules move slower

• It remains constant because activation energy does not change

• It becomes zero at very high temperatures

• It increases because a greater fraction of molecules have energy ≥ activation energy

Multiple Choice

A graph of ln⁡k against 1/T​ is a straight line. What does the gradient represent?

• +Ea/R

• −Ea/R

• −RT/Ea

• +R/Ea

Multiple Choice

For the Arrhenius equation, which statement is correct about the effect of a catalyst?

• It increases the value of Ea​, making k smaller at all temperatures

• It decreases A, reducing the frequency of collisions

• It provides an alternative pathway with lower Ea​, increasing k

• It increases temperature, which directly reduces Ea​

Example : ​The rate constants for a reaction at a variety of temperatures are shown in the table below. Use the data to create an Arrhenius plot, and therefore work out the activation energy and the Arrhenius constant.

​If given T and k values for an Arrhenius plot, first calculate 1/T and ln k for each pair.
Next, plot 1/T against ln k on appropriate axes, then draw a line of best fit through the points.