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# Arrhenius Equation

When we think about speed, whether that's in the context of how fast someone can run, or how quickly a falling object reaches the ground, we know that there are always at least a couple of factors that affect the rate at which these things happen. Chemical reactions are no different.

Chemists like to neatly link all these factors together using formulae such as the Arrhenius equation.

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.

• We'll start by looking at the different parts of the equation in more detail.
• We'll then explore Arrhenius plots, a graphical way of representing the Arrhenius equation.
• Finally, we'll consider the significance of changing certain variables in the 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.
• Ea 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.

e is an irrational number. This means that it can't be expressed as one integer divided by another, which would give you a nice round number. Instead, it has an infinite number of decimal places - if you were to try and write it out, it would go on forever!

e has a few special properties. Perhaps the most significant is that the gradient of any point on the graph y=ex also equals ex. This feature allows us to use e in contexts of constant growth and decay. For instance, we use it in economics to model income growth, or in physics to model nuclear decay. There are many more cool things about e, but we won't be to cover them all here, unfortunately. If you're still interested, you might want to check out Natural Logarithm and Growth and Decay for more.

### Activation energy, Ea

Ea 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.

You're probably used to expressing activation energy in kJ mol-1, not J mol-1. Make sure you look out for this in exams and convert between the two if needed. To convert from kJ mol-1 into J mol-1, you multiply by 1000. To convert from J mol-1 into kJ mol-1, you divide by 1000.

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.

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

You'll notice that the gas constant, R, is measured per unit Kelvin, shown as K-1. This means that we must also measure temperature in Kelvin, K, instead of degrees Celcius, °C.

### 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.

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.

 Symbol Meaning Units k Rate constant Variable units A Arrhenius constant Variable units ${e}^{}$ Euler's number No units Ea Activation energy J mol-1 R Gas constant J K-1 mol-1 T Temperature K

## 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(ex) = 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:

$\mathrm{k}=\mathrm{A}{e}^{\frac{-{\mathrm{E}}_{\mathrm{a}}}{\mathrm{RT}}}\phantom{\rule{0ex}{0ex}}\mathrm{ln}\left(\mathrm{k}\right)=\mathrm{ln}\left(\mathrm{A}{e}^{\frac{-{\mathrm{E}}_{\mathrm{a}}}{\mathrm{RT}}}\right)$

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(ex) = x:

$\mathrm{ln}\left(\mathrm{k}\right)=\mathrm{ln}\left(\mathrm{A}\right)+\mathrm{ln}\left({e}^{\frac{-{\mathrm{E}}_{\mathrm{a}}}{\mathrm{RT}}}\right)\phantom{\rule{0ex}{0ex}}\mathrm{ln}\left(\mathrm{k}\right)=\mathrm{ln}\left(\mathrm{A}\right)+\frac{-{\mathrm{E}}_{\mathrm{a}}}{\mathrm{RT}}$

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 Ea:

$\mathrm{ln}\left(\mathrm{k}\right)-\mathrm{ln}\left(\mathrm{A}\right)=\frac{-{\mathrm{E}}_{\mathrm{a}}}{\mathrm{RT}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{\mathrm{E}}_{\mathrm{a}}=-\mathrm{RT}\left(\mathrm{ln}\left(\mathrm{k}\right)-\mathrm{ln}\left(\mathrm{A}\right)=\mathrm{RT}\left(\mathrm{ln}\left(\mathrm{A}\right)-\mathrm{ln}\left(\mathrm{k}\right)\right)\phantom{\rule{0ex}{0ex}}$

The Arrhenius equation can be rearranged into the form $\mathrm{ln}\left(\mathrm{k}\right)=\mathrm{ln}\left(\mathrm{A}\right)+\frac{-{\mathrm{E}}_{\mathrm{a}}}{\mathrm{RT}}$. Knowing how to rearrange the equation into this form isn't necessary for your exams, but it can help your understanding.

If you want to try your hand at some calculations involving the Arrhenius equation in various different forms, head over to Arrhenius Equation Calculations.

## 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, Ea, 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 Ea, 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.

The point at which x=0 is the point at which the line crosses the y-axis, provided your axes start at zero. However, when working with Arrhenius plots, your axes often skip a few numbers and start at say, 3.00 x 10-3. In this case, you have to work out the value of c separately - you can't just read it off the graph.

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:

$\mathrm{m}=\frac{-{\mathrm{E}}_{\mathrm{a}}}{\mathrm{R}}\phantom{\rule{0ex}{0ex}}{\mathrm{E}}_{\mathrm{a}}=\mathrm{m}×-\mathrm{R}=-\mathrm{mR}$

### 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.

$\mathrm{ln}\left(\mathrm{A}\right)=\mathrm{c}\phantom{\rule{0ex}{0ex}}\mathrm{A}={\mathrm{e}}^{\mathrm{c}}$

## Example of Arrhenius equation

Here's an example of an Arrhenius equation graph (which you'll remember is also known as an Arrhenius plot).

Fig. 4 - Arrhenius graph example

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.

Want to know how we arrived at these values? Check out Arrhenius Equation Calculations, where you'll be able to practice working out a reaction's activation energy and Arrhenius constant from a graph of the Arrhenius equation.

## 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.

## Arrhenius Equation - Key takeaways

• The Arrhenius equation is a mathematical formula that relates the rate constant k of a reaction with the activation energy and temperature of that reaction. It takes the form $\mathrm{k}={\mathrm{Ae}}^{\frac{-{\mathrm{E}}_{\mathrm{a}}}{\mathrm{RT}}}$.
• k is the rate constant. It has variable units.
• A is the Arrhenius constant. It also has variable units.
• Ea is the activation energy, in J mol-1.
• R is the gas constant, in J K-1 mol-1.
• T is the temperature, in K.
• The Arrhenius equation can be rearranged into a logarithmic form, where $\mathrm{ln}\left(\mathrm{k}\right)=\frac{-\mathrm{Ea}}{\mathrm{RT}}+ \mathrm{ln}\left(\mathrm{A}\right)$. This relates to the general equation of a line, y = mx +c, and can be plotted on an Arrhenius plot.
• ln(k) represents the y-coordinates of points on the line.
• $\frac{-{\mathrm{E}}_{\mathrm{a}}}{\mathrm{R}}$represents m, the gradient of the line.
• $\frac{1}{\mathrm{T}}$ represents the x-coordinates of points on the line.
• ln(A) represents c, the y-coordinate of the point at which x = 0.
• The Arrhenius equation tells us that increasing the temperature of a reaction increases the rate of reaction.
• The Arrhenius equation tells us that decreasing the activation energy of a reaction increases the rate of reaction.

#### Flashcards in Arrhenius Equation 7

###### Learn with 7 Arrhenius Equation flashcards in the free StudySmarter app

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What is R in the Arrhenius equation?

The 'R' in the Arrhenius equation is the gas constant.

What is the Arrhenius equation?

The Arrhenius equation is an equation used in chemistry that links the rate, the activation energy, and the temperature of a reaction.

What is the importance of the Arrhenius equation?

The Arrhenius equation provides information about the rate of a chemical reaction under different conditions, and how changing conditions affect the rate.

How do you calculate the Arrhenius equation?

To calculate the Arrhenius equation you need to know the activation energy of a reaction, the temperature at which a reaction will happen, and the value for the Arrhenius constant. Once you know those values, all you need to do is input them into the equation to calculate an answer.

What is k in the Arrhenius equation?

'k' in the Arrhenius equation is the rate constant. It is unique for each reaction at a specific temperature.

## Test your knowledge with multiple choice flashcards

True or false? For reactions with an activation energy of around 50 kJ mol-1, increasing the temperature from 290K to 200K will triple the rate constant.

Which of these do not appear in the Arrhenius equation?

What are the units of Ea in the Arrhenius equation?

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