2.2.3

# The Arrhenius Equation (A2 Only)

Test yourself

## The Arrhenius Equation

The rate constant can be expressed in terms of the activation energy by using the Arrhenius equation.

### The Arrhenius equation

• The Arrhenius equation is: $k = Ae^{\frac{-E_a}{RT}}$
• In this equation, k is the rate constant.
• A is the Arrhenius constant.
• R is the gas constant.
• T is the temperature.
• Ea is the activation energy.

### Using the Arrhenius equation

• The Arrhenius equation lets us find the rate constant at different temperatures.
• More importantly, if we measure the rate constant at a bunch of different temperatures, we can calculate the activation energy.

### Key points about the equation

• As the activation energy rises, the rate constant gets smaller (so the rate is slower).
• This is what you expect - think back to the Maxwell Boltzmann distribution.
• The higher the activation energy, the fewer the number of molecules with enough energy to react.
• As temperature rises, the rate constant gets larger, so the rate is faster.
• Again, you can understand this by thinking about the Maxwell Boltzmann distribution.

## Arrhenius Plots

An Arrhenius plot is the name given to the graph used to find the activation energy.

### The activation energy

• You can rearrange the Arrhenius plot to find the activation energy.
• The Arrhenius equation is:
• $k =Ae^{\frac{-E_a}{RT}}$
• This can be rearranged to:
• $ln(k) = \frac{-E_a}{RT} + ln(A)$
• This means that if we plot the graph of ln(k) against 1/T, you have a graph with the gradient as ${\frac{-E_a}{R}}$.

### An example

• The gradient of the above graph is -6.014.
• This means that the activation energy is:
• 6.014 × R = 50 J

## Arrhenius Calculations

You need to be able to calculate the activation energy given a rate constant and vice versa.

### Calculating the activation energy

• The Arrhenius equation is:
• $k = Ae^{\frac{-E_a}{RT}}$
• If you want to calculate the activation energy, the easiest way is to rearrange the equation into:
• $ln(k) = \frac{-E_a}{RT} + ln(A)$
• You can then substitute in the values for k and A that you’re given to get the activation energy out.

### Calculating the rate constant

• To calculate the rate constant, you can just use the original Arrhenius equations and throw all of the values into your calculator.