The Arrhenius Equation (HL) (DP IB Chemistry: HL): Revision Note

The Arrhenius equation

• Rate equations include the rate constant k

  • This reflects collision frequency and energy requirements, not just concentrations

• k is constant when the only variable is concentration

  • k changes if temperature or catalysts change

• Increasing the temperature increases the proportion of molecules with energy greater than the activation energy, E_a

• As temperature increases, more molecules have enough energy to overcome the activation energy

• This means that:

  • The rate of reaction increases

  • The rate constant increases

What is the Arrhenius equation?

• The Arrhenius equation is a mathematical expression that describes the temperature dependence of reaction rates

  • It links the rate constant, temperature and activation energy

• The Arrhenius equation describes reactions that:

  • Involve gases

  • Occur in solution

  • Occur on the surface of a catalyst

• The Arrhenius equation can be found in the IB Chemistry data booklet (Section 1):

What are the terms in the Arrhenius equation?

• k is the rate constant

  • It determines how quickly the reaction proceeds

• A is the Arrhenius factor

  • This is also known as the frequency factor or pre-exponential factor

  • It is a constant that takes into account the frequency of collisions with proper orientations

  • The value of A:

    • Changes slightly with temperature but is still considered constant

    • Is characteristic of a specific reaction

• E_a is the activation energy

  • Activation energy is the minimum energy per mole of reactant particles required to start a chemical reaction

  • The common units of activation energy are J mol⁻¹ and kJ mol⁻¹

  • The value of E_a is characteristic of a specific reaction

• R is the gas constant

  • It is a fundamental physical constant for all reactions

  • The value of R is 8.31 J K⁻¹ mol⁻¹

  • It can be found in the IB Chemistry data booklet (Section 2)

• T is temperature

  • The unit for temperature in this equation is Kelvin, K

• e is a mathematical constant

  • It can be found on your calculator

  • It has the approximate value of 2.718

• k and T are the only variables in the Arrhenius equation

Using the Arrhenius equation

• Taking the natural logarithm of the Arrhenius equation gives a straight-line

• This makes it easier to:

  • Determine E_a and A from experimental data

  • Solve problems when two sets of k and T are known

• Both forms of the Arrhenius equation are given in the IB Chemistry data booklet (Section 1)

  • The natural logarithmic form is:

$\ln k\mathit{=}\mathit{-}\frac{E_a}{RT}\mathit{+}lnA$

• This version of the equation shows how:

  • Temperature affects the rate constant, k:

    • Increasing temperature increases ln k

    • This gives a higher value of k, increasing the rate of reaction

  • Activation energy, E_a, affects the rate:

    • A higher E_a means fewer molecules have enough energy to react

    • This decreases the rate of reaction, giving a lower value of k

• The values of k and T can be determined experimentally:

  • These can be used to calculate E_a for a reaction

  • This is the most common Arrhenius calculation

Graphing the Arrhenius equation

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

• Plotting ln k against 1/T gives a straight line:

  • The slope of the line is $\frac{-E_{\mathrm{a}}}{R}$

  • The y-intercept of the line is ln A

• The equation of a straight line is:

y = mx + c

• This matches the natural logarithmic Arrhenius equation:

$\begin{array}{rcl}\ln k& =& -\frac{E_{\mathrm{a}}}{RT}+\ln A\\ & & \\ \ln k& =& -\frac{E_{\mathrm{a}}}{R}\times \frac{1}{T}+\ln A\end{array}$

• The variables are:

  • y = ln k

  • x = $\frac{1}{T}$

  • m = $\frac{-E_{\mathrm{a}}}{R}$ (the gradient)

  • c = ln A (the y-intercept)

Calculating the activation energy

• The activation energy can be calculated from the slope:

E_a = - gradient x R

• Since the slope is negative:

  • The minus signs cancel

  • E_a is a positive value

Calculating the Arrhenius factor

• The Arrhenius factor, A, can be calculated from the y-intercept:

A = e^y-intercept