Arrhenius Equation Calculator

"Oh, you small molecules in my beaker, invisible to my eye, at what rate do you react?" pondered Svante Arrhenius in 1889 – probably (also probably in Swedish). But instead of doing all your calculations by hand, as he did, you, fortunately, have this Arrhenius equation calculator to help you do all the heavy lifting.

We're also here to help you answer the question, "What is the Arrhenius equation?", as you may have been idly daydreaming in class and now have some dreadful chemistry homework in front of you. To also assist you with that task, we provide an Arrhenius equation example and Arrhenius equation graph, and how to solve any problem by transforming the Arrhenius equation in ln. It won't be long until you're daydreaming peacefully.

The Arrhenius equation is a formula that describes how the rate of a reaction varied based on temperature, or the rate constant. If we look at the equation that this Arrhenius equation calculator uses, we can try to understand how it works:

where:

• $k$ — The rate constant, with units that depend on the order of reaction, $n$, in the general form $\text{M}^{1-n}/\text{s}$: we have a tool dedicated to this important concept, the rate constant calculator;
• $A$ — The pre-exponential factor, sometimes called the Arrhenius constant, in the same units as the rate constant;
• $\text{e} \approx2.72$ = Euler's number (exponent);
• $E_{\text{a}}$ — The the activation energy of the reaction, in $\text{J}/\text{mol}$;
• $R$ — The universal gas constant, equal to $8.314\ \text{J}/(\text{K}\!\cdot\!\text{mol})$; and
• $T$ — The temperature, in $\text{K}$.

The $n$ noted above is the order of the reaction being considered.

Now, how does the Arrhenius equation work to determine the rate constant? Well, we'll start with the $R \cdot T$. By multiplying these two values together, we get the energy of the molecules in a system in $\text{J}/\text{mol}$, at temperature $T$.

We can then divide $E_{\text{a}}$ by this number, which gives us a dimensionless number representing the number of collisions that occur with sufficient energy to overcome the activation energy requirements (if we don't take the orientation into account - see the section below). This number is inversely proportional to the number of successful collisions. So the lower it is, the more successful collisions there are. For example, for a given time $t$, a value of $E_{\text{a}}/(R \cdot T) = 0.5$ means that twice the number of successful collisions occur than if $E_{\text{a}}/(R \cdot T) = 1$, which, in turn, has twice the number of successful collisions than $E_{\text{a}}/(R \cdot T) = 2$.

As you may be aware, two easy ways of increasing a reaction's rate constant are to either increase the energy in the system, and therefore increase the number of successful collisions (by increasing temperature T), or to provide the molecules with a catalyst that provides an alternative reaction pathway that has a lower activation energy (lower $E_{\text{a}}$). To make it so this holds true for $E_{\text{a}}/(R \cdot T)$, and therefore remove the inversely proportional nature of it, we multiply it by $-1$, giving $-E_{\text{a}}/(R \cdot T)$.

Note that increasing the concentration only increases the rate, not the constant!

As a reaction's temperature increases, the number of successful collisions also increases exponentially, so we raise the exponential function, $\text{e}$, by $-E_{\text{a}}/RT$, giving $\text{e}^{-E_{\text{a}}/RT}$. This represents the probability that any given collision will result in a successful reaction.

Now, as we alluded to above, even if two molecules collide with sufficient energy, they still might not react; they may lack the correct orientation with respect to each other so that a constructive orbital overlap does not occur. Therefore a proportion of all collisions are unsuccessful, which is represented by $A$. We multiply this number by $\text{e}^{-E_{\text{a}}/RT}$, giving $A\cdot \text{e}^{-E_{\text{a}}/RT}$, the frequency that a collision will result in a successful reaction, or the rate constant, $k$.

You may have noticed that the above explanation of the Arrhenius equation deals with a substance on a per-mole basis, but what if you want to find one of the variables on a per-molecule basis? Well, in that case, the change is quite simple; you replace the universal gas constant, $R$, with the Boltzmann constant, $k_{\text{B}}$, and make the activation energy units $\text{J}/\text{molecule}$:

where,

• $k$ — The rate constant, with units that depend on the order of reaction, $n$, in the general form $\text{M}^{1-n}/\text{s}$;
• $A$ — The pre-exponential factor, sometimes called the Arrhenius constant, in the same units as the rate constant;
• $\text{e}\approx 2.72$ — The Euler's number (exponent);
• $E_{\text{a}}$ — The activation energy of the reaction, in $\text{J}/\text{molecule}$;
• $k_{\text{B}}$ — Boltzmann constant, equal to $1.380649\times10^{−23}\ \text{J}/\text{K}$; and
• $T$ — The temperature, in $\text{K}$.

This Arrhenius equation calculator also allows you to calculate using this form by selecting the per molecule option from the topmost field.

What's great about the Arrhenius equation is that, once you've solved it once, you can find the rate constant of reaction at any temperature. The difficulty is that an exponential function is not a very pleasant graphical form to work with: as you can learn with our exponential growth calculator; however, we have an ace in our sleeves. If you want an Arrhenius equation graph, you will most likely use the Arrhenius equation's ln form:

This bears a striking resemblance to the equation for a straight line, $y = mx + c$, with:

• $y = \ln(k)$;
• $m = -E_{\text{a}}/R$;
• $x = 1/T$; and
• $c = \ln(A)$.

This Arrhenius equation calculator also lets you create your own Arrhenius equation graph! All you need to do is select Yes next to the Arrhenius plot? field at the bottom of the tool once you have filled out the main part of the calculator. You can also change the range of $1/T$ and the steps between points by selecting the checkbox Change graph options. This functionality works both in the regular exponential mode and the Arrhenius equation ln mode and on a per molecule basis.

Hopefully, this Arrhenius equation calculator has cleared up some of your confusion about this rate constant equation. Still, we here at Omni often find that going through an example is the best way to check you've understood everything correctly. So, without further ado, here is an Arrhenius equation example.

At $320\ \degree \text{C}$, $\text{NO}_2$ decomposes at a rate constant of $0.5\ \text{M}/\text{s}$. It was found experimentally that the activation energy for this reaction was $115\ \text{kJ}/\text{mol}$. What is the pre-exponential factor?

First thing first, you need to convert the units so that you can use them in the Arrhenius equation.

• Temperature conversion: $\small 320\ \degree\text{C} + 273.15 = 593.15\ \text{K}$; and
• Energy conversion: $\small 115\ \text{kJ}/\text{mol}\cdot 1000 = 115000\ \text{J}/\text{mol}$.

Now that you've done that, you need to rearrange the Arrhenius equation to solve for $A$. Divide each side by the exponential:

Then you just need to plug everything in.

Which returns approximately: