Arrhenius Equation

Arrhenius equation definition

We've learnt what the Arrhenius equation is: a mathematical formula that relates the rate constant of a reaction, k, with the activation energy and temperature of that reaction. It's important because it allows us to see how a change in temperature affects rate of reaction.

The equation looks like this:

Fig. 1 - The Arrhenius equation

Let's break that down.

• k is the rate constant.
• A is the Arrhenius constant, also known as the pre-exponential factor.
• e is Euler's number.
• E_a is the activation energy of the reaction.
• R is the gas constant.
• T is the temperature.
• Overall, ${\mathrm{e}}^{\frac{-{\mathrm{E}}_{\mathrm{a}}}{\mathrm{RT}}}$is the proportion of molecules that have enough energy to react.

As you can see, there are a lot of 'moving parts' involved in this equation, and maybe a few symbols you aren't familiar with. Don't worry if you don't quite understand what's going on with the equation yet - we'll go through each part so you know how to use it.

The rate constant, k

If you've read the article Rate Equations, you'll know that rate of reaction is dependent on the concentration of certain products. We write this as rate = k [A]^m [B]ⁿ, where k is a rate constant that varies depending on the reaction. That same rate constant k appears here in the Arrhenius equation. It changes for different reactions at different temperatures. Its units vary too, depending on the reaction.

The Arrhenius constant, A

The letter A in the Arrhenius equation represents the Arrhenius constant. It is related to how many collisions there are between reacting molecules. It is also known as the pre-exponential factor and its units vary, depending on the rate constant. Both the Arrhenius constant and the rate constant always take the same units.

Euler's number, e

The next part we'll look at from the equation is that letter e, which is Euler's number. It's named after the mathematician who first discovered its significant properties. It equals approximately 2.71828, but you don't need to actually know that, because there is a button for e on your calculator that stores the value for you.

Activation energy, E_a

E_a is the activation energy of the particular reaction you are looking at. Like k, activation energy depends on the reaction. However, unlike k, it has fixed units: J mol^-1.

Gas constant, R

The letter R in the Arrhenius equation represents the gas constant. You might have come across this before in Ideal Gas Law. It is a constant that relates the pressure, volume, and temperature to the number of moles of a gas. It has a value of 8.31 and takes the units J K^-1 mol^-1.

Temperature, T

The final individual component in the equation is temperature, T. This is measured in Kelvin, K.

R, T, and Ea

In the Arrhenius equation, you'll see that e is raised to the power of the negative value of the activation energy, divided by the gas constant multiplied by the temperature. That's a mouthful - it is more simply written as $e^{\frac{-{\mathrm{E}}_{\mathrm{a}}}{\mathrm{RT}}}$. Overall, it represents the number of molecules with enough energy to react at a certain temperature. In other words, this means the number of molecules that meet or exceed the reaction's activation energy.

Summary table

Forgotten what the original Arrhenius equation looked like? Here it is again.

Fig. 2. - The Arrhenius equation

We've also made a table comparing the different parts of the equation.

Rearranging the Arrhenius equation

It's important to know how to rearrange the Arrhenius equation. You may come across questions where you'll need to change the subject of the equation to get to the answer. Because the original equation uses the power of e, rearranging the Arrhenius equation involves the natural logarithm, ln. Remember that logarithms (often shortened to logs) are the opposite of powers. If e⁵ = x, then ln(x) = 5. We can use various other rules of logs to derive another handy result: ln(e^x) = x. Taking natural logs of all parts of the equation is a way of getting rid of e, and makes it possible to find out the values of other variables in the equation.

Using the principles outlined above, we can find, for example, the particular activation energy of a reaction. We start by taking the natural log (ln) of both sides of the equation:

$$\begin{gathered} \mathrm{k}=\mathrm{A}{e}^{\frac{-{\mathrm{E}}_{\mathrm{a}}}{\mathrm{RT}}} \\ \ln \left(\mathrm{k}\right)=\ln \left(\mathrm{A}{e}^{\frac{-{\mathrm{E}}_{\mathrm{a}}}{\mathrm{RT}}}\right) \end{gathered}$$

Using another rule of logs, we can separate the right-hand side into two separate logs. We then simplify it, using the fact that ln(e^x) = x:

$$\begin{gathered} \ln \left(\mathrm{k}\right)=\ln \left(\mathrm{A}\right)+\ln \left(e^{\frac{-{\mathrm{E}}_{\mathrm{a}}}{\mathrm{RT}}}\right) \\ \ln \left(\mathrm{k}\right)=\ln \left(\mathrm{A}\right)+\frac{-{\mathrm{E}}_{\mathrm{a}}}{\mathrm{RT}} \end{gathered}$$

Having the Arrhenius equation in this form will become useful when we look at graphing the Arrhenius equation, but we could also rearrange it to find E_a:

_{$$\begin{gathered} \ln \left(\mathrm{k}\right)-\ln \left(\mathrm{A}\right)=\frac{-{\mathrm{E}}_{\mathrm{a}}}{\mathrm{RT}} \\ {\mathrm{E}}_{\mathrm{a}}=-\mathrm{RT}\left(\ln \right(\mathrm{k})-\ln (\mathrm{A})=\mathrm{RT}(\ln \left(\mathrm{A}\right)-\ln \left(\mathrm{k}\right)) \end{gathered}$$}

Arrhenius equation graphs

Let's move on to graphing the Arrhenius equation. An Arrhenius graph, or Arrhenius plot, is a graphical way of visualising the Arrhenius equation. It is a way of finding out the activation energy of a reaction, E_a, and the Arrhenius constant, A, using experimental data. By rearranging the Arrhenius equation into a form that relates to the general equation of a line, we can plot a line graph of ln (k) against $\frac{1}{\mathrm{T}}$. We can then not only use the gradient of the graph to work out E_a, but also use the point at which x= 0 to find A.

The general equation of a line

The equation of a line is y = mx + c, where:

• y is the y value of a point on the line.
• m is the gradient of the line.
• x is the x value of a point on the line.
• c is the y-coordinate of the point at which x = 0.

How does the general equation of a line relate to the Arrhenius equation? By shuffling the Arrhenius equation about, we can make it so each term in the Arrhenius equation maps onto a certain term from the general equation of a line. Here's how:

Fig. 3 - Arrhenius equation and the general equation of a line

Note the following:

• ln(k) represents the y-coordinates of points on the line. You work out ln(k) using values determined experimentally.
• $\frac{-{\mathrm{E}}_{\mathrm{a}}}{\mathrm{R}}$represents m, the gradient of the line.
• $\frac{1}{\mathrm{T}}$ represents the x-coordinates of points on the line. You work out $\frac{1}{\mathrm{T}}$ using values determined experimentally.
• ln(A) represents c, the y-coordinate of the point at which x = 0.

Activation energy

Looking back at how the rearranged Arrhenius equation is mapped onto the general equation of a line, we can see that the gradient of the line, m, is equal to $\frac{-\mathrm{Ea}}{\mathrm{R}}$. Because R is a constant, if we know the gradient of our line then we can easily rearrange to find what the activation energy is:

The Arrhenius constant

Using the same principle we used to determine activation energy, we can find the Arrhenius constant by looking at which part of the equation of a line it relates to. In the general equation of the line, we know that c is the y-coordinate of the point at which x = 0; in our Arrhenius graph, c is represented by ln(A). Once we know the point where x = 0, we can rearrange to solve for A.

Here, the gradient of the line, which is $\frac{-{\mathrm{E}}_{\mathrm{a}}}{\mathrm{R}}$, equals approximately -12300, so the activation energy equals 102 kJ mol^-1. We can also see that the y-coordinate of the point at which x = 0, which is ln(k), equals -1.0. This makes k equal 0.368.

Arrhenius equation importance

We know what the Arrhenius equation is, and how to work with it in its different forms. But why is it important? Well, it tells us how changing certain variables like temperature and activation energy affects the rate constant, k. This goes on to affect the rate of reaction. The rate constant is proportional to the rate of a reaction, so anything in the Arrhenius equation that increases the value of k increases the rate of reaction.

• Increasing the temperature of a reaction increases the value of $\frac{-{\mathrm{E}}_{\mathrm{a}}}{\mathrm{RT}}$. According to the Arrhenius equation, this increases the value of k and therefore increases the rate of reaction.
• Lowering the activation energy of a reaction increases the value of $\frac{-{\mathrm{E}}_{\mathrm{a}}}{\mathrm{RT}}$. According to the Arrhenius equation, this increases the value of k and therefore increases the rate of reaction.