# Arrhenius Equation

Speed is a fascinating concept that can be applied to many things, like how fast you can run or how quickly an object falls. The rate at which things happen is affected by multiple factors, and this is true for chemical reactions too. Chemists use a formula called the Arrhenius equation to connect all of these factors together.

The Arrhenius equation is a mathematical formula that links the rate constant of a reaction (k) with the activation energy and temperature of that reaction. This formula is used in physical chemistry, and it's important to understand its different parts. Arrhenius plots are a graphical way of showing the Arrhenius equation, which we'll explain. Lastly, we'll talk about how changing certain variables in the Arrhenius equation can affect the outcome of a reaction. So, let's dive in!

## 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:

The Arrhenius equation may seem intimidating at first, but let's break it down into smaller parts. The equation has several components, including k, which is the rate constant, and A, also known as the Arrhenius constant or pre-exponential factor. There's also e, which is Euler's number, and Ea, which represents the activation energy of the reaction. R is the gas constant, and T represents the temperature.

Overall, the equation shows us the proportion of molecules that have enough energy to react. It's normal if you're not familiar with some of these symbols, but don't worry! We'll explain each part in more detail so you can understand how to use the equation.

**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, which is related to the number of collisions between reacting molecules. It is also known as the pre-exponential factor, and its units can vary depending on the rate constant. Both the Arrhenius constant and the rate constant always have the same units.

Now let's talk about the letter e in the equation. It represents Euler's number, which is a special and important number in mathematics. It's approximately equal to 2.71828, but you don't need to memorize this value because your calculator has a button for it.

Euler's number is an irrational number, meaning it can't be expressed as a simple fraction. It has an infinite number of decimal places, but it has some cool properties. For example, the gradient of any point on the graph y=ex also equals ex. This allows us to use e in contexts of constant growth and decay, such as modeling income growth in economics or nuclear decay in physics. There are many more interesting things about e, but we won't be able to cover them all here. If you're interested, you can check out resources on natural logarithm and growth and decay.

**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.

**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. 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 . 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**

To recap, the original Arrhenius equation is shown below:

And here's a table comparing the different parts of the equation:

SymbolMeaningkRate constantAArrhenius constant (pre-exponential factor)eEuler's number (~2.71828)EaActivation energyRGas constantTTemperature

Rearranging the Arrhenius equation involves using the natural logarithm, ln. This is because the original equation uses the power of e. Logarithms are the inverse of powers, so taking the natural log of both sides of the equation allows us to rearrange it and solve for different variables. For example, we can rearrange the equation to solve for the activation energy, Ea.

Knowing how to rearrange the Arrhenius equation isn't necessary for exams, but it can be helpful in understanding the relationship between the different variables. If you want to practice calculations with the Arrhenius equation in various forms, you can check out 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 . 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:

Note the following:

ln(k) represents the y-coordinates of points on the line. You work out ln(k) using values determined experimentally. represents m, the gradient of the line. represents the x-coordinates of points on the line. You work out 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 . 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 , 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 , making k equal 0.368.

## Arrhenius equation importance

To summarize, the Arrhenius equation is important because it helps us understand how changes in temperature and activation energy affect the rate of reaction. Increasing the temperature or decreasing the activation energy increases the rate constant, k, which in turn increases the rate of reaction. The equation can be rearranged and plotted on an Arrhenius plot to visualize this relationship. The equation takes the form , where k is the rate constant, A is the Arrhenius constant, Ea is the activation energy, R is the gas constant, and T is the temperature. By understanding the Arrhenius equation, we can predict and control the rate of chemical reactions.

## Arrhenius Equation

**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.