The Arrhenius Equation (HL) (HL IB Chemistry)

• The rate equation shows how each of the reactants in a reaction affects the rate of the reaction and it includes the rate constant, k
• However, k only remains constant if the concentration of the reactants is the only factor which is changed
  • If the temperature is changed or a catalyst is used or changed, then the rate constant, k, changes
• At higher temperatures, a greater proportion of molecules have energy greater than the activation energy
• Since the rate constant and rate of reaction are directly proportional to the fraction of molecules with energy equal or greater than the activation energy, then at higher temperatures:
  • The rate of reaction increases
  • The rate constant increases

What is the Arrhenius equation?

• The relationship between the rate constant, the temperature and also the activation energy is given by the Arrhenius equation:

• Where:
  • k = Rate constant
  • A = Arrhenius factor (also known as the frequency factor or pre-exponential factor) which is a constant that takes into account the frequency of collisions with proper orientations
  • E_a = Activation energy (J mol^-1)
  • R = Gas constant (8.31 J K^-1 mol^-1)
  • T = Temperature (Kelvin, K)
  • e = Mathematical constant (can be found on your calculator - it has the approximate value of 2.718)
• E_a and A are constants that are characteristic of a specific reaction
  • A does vary slightly with temperature but it can still be considered a constant
• R is a fundamental physical constant for all reactions
• k and T are the only variables in the Arrhenius equation
• The Arrhenius equation is used to describe reactions that involve gases, reactions occurring in solution or reactions that occur on the surface of a catalyst

Using the Arrhenius Equation

• The Arrhenius equation is easier to use if you take natural logarithms of each side of the equation, which results in the following equation:

• The Arrhenius equation can be used to show the effect that a change in temperature has on the rate constant, k, and thus on the overall rate of the reaction
  • An increase in temperature (higher value of T) gives a greater value of ln k and therefore a higher value of k
  • Since the rate of the reaction depends on the rate constant, k, an increase in k also means an increased rate of reaction
• The equation can also be used to show the effect of increasing the activation energy on the value of the rate constant, k
  • An increase in the activation energy, E_a, means that the proportion of molecules which possess at least the activation energy is less
  • This means that the rate of the reaction, and therefore the value of k, will decrease
• The values of k and T for a reaction can be determined experimentally
  • These values of k and T can then be used to calculate the activation energy for a reaction
  • This is the most common type of calculation you will be asked to do on this topic

Finding the activation energy and Arrhenius factor

• A graph of experimental data can be used to determine the activation energy and the Arrhenius factor

Arrhenius equation graph

• A graph of ln k against 1/T can be plotted, and then used to calculate E_a
  • This gives a line which follows the form y = mx + c
    
    

  Graph of ln k against 1/T

The graph of ln k against 1/T is a straight line with gradient -E_a/R

• From the graph, the equation is in the form of y = mx + c is as follows:

ln k	=			+	ln A
y	=	m	x	+	c

• Where:
  • y = ln k
  • x = 
  • m =  (the gradient)
  • c = ln A (the y-intercept)
• E_a can be calculated by:
  
  • E_a = - gradient x R
• As the slope has a negative gradient, the minus signs cancel giving E_a as a positive value
• The Arrhenius factor, A, can be calculated by:
  • A = e^y-intercept