The Arrhenius Equation

I’ve done a bit of talking about rates of reaction; after all, the subject constitutes a third of my and Sapphire’s first Gen Chem II test. In my previous post, I mentioned that there were several things that affect the rate of a reaction. Concentration of gaseous and aqueous reactants is one, but it isn’t by any means the only one.

Intuitively, we know that heating things up makes molecules move faster (even if we don’t know that that’s exactly what’s happening). If you add a drop of food dye to a glass of warm water, it will diffuse much quicker than it would in a glass of cold water. Ice melts into water when it is heated because the molecules are moving around more. The same applies to water evaporating.

When working with temperature in chemistry, it is easy to immediately translate the word “temperature” into “energy.” Increasing the temperature at which a reaction runs adds energy to the system. This is important when wanting to increase the rate of a chemical reaction because it makes it easier for reactants to overcome the barrier set forth by the activation energy of a reaction.

The activation energy of a reaction, put simply, is the minimum amount of energy it takes to get a reaction running. It’s specific for every reaction, and it’s one of the things a catalyst can change to make a reaction run faster. There are pretty graphs that illustrate this, often used to make freshman chemistry students painstakingly explain the difference between exothermic and endothermic reactions, even though they totally know the difference. (Don’t worry about it—I’ll get to that later.) Essentially, the more molecules that can pay the “energy cost” to climb over the hill, the more molecules of reactants that get turned into products. Paying the energy cost gets easier as more energy is added to the system by raising the temperature.

Knowing this, we can logically work through rate change as a function of temperature. Think about a reaction that has high activation energy. At room temperature, how many molecules of reactants are going to get turned into products? Not many. Add energy to the system by raising the temperature, and more molecules can be converted into products. The reaction, following this logic, is running at a faster rate at a higher temperature, because more molecules of reactants get converted into products quicker at higher temperatures.

That’s nice, but what does the titular equation (that sounds absolutely horrific, by the way) have to do with it?

I’m glad you asked, curious and brave reader. You see, the very simple example of reaction rate increasing as temperature increases can be needlessly complicated by stating that, following this, the rate constant of a reaction increases with temperature. I sort of hinted at this in the previous post about reaction rates by saying that the rate constant is dependent on temperature. Don’t worry if you’re cringing—all of the freshman chemistry students in my 117 class did, too.

What this means in a practical sense is that there is a way to quantitatively relate the rate constant of a reaction to its activation energy, the temperature at which it is run and an ugly little thing called the frequency factorThis is where our lovely little Arrhenius equation comes in. Consider the following:

k = Ae-Ea/RT

Where e is the base of the natural logarithm (ln—remember that from pre-calculus?), Ea is the activation energy of the reaction, A is the frequency factor, R is the gas law constant dealing with energy (8.314 J/mol*K) and T is the temperature in Kelvin.

Horrid, right? We thought so, too.

To take an apt explanation from my chemistry textbook, it really isn’t a terrible equation to have around. A, the frequency factor, depends on how often molecules collide and how often they are in the correct orientation to react. The other part, the e-Ea/RT part, basically tells us how many of the total molecules have the energy to overcome the activation energy barrier.

Now that we know how to calculate how different things, including activation energy, affect the rate constant, it’s time we actually understood what happens once that energy barrier is overcome. Those cute little equations we see when we’re given a net reaction are usually oversimplified and condensed. Reaction mechanisms tell us what’s actually going on.

Questions? Comments? Drop them below!

Corrections? Feel free to add them, nicely. Flames will be doused with cheap coffee and Diet Mountain Dew.


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