Point of Equilibrium

It is a well-known fact that a chemical reaction will not procede if the Gibbs Free Energy is positive. But as with so many well-known facts, there's more to the story than that.

Last week I wrote out the chemical reaction

CH4 + CO2 ===> 2CO + 2 H2

and noted that even at 300 degrees C, the Free Energy was positive. The obvious conclusion is that the hydrocarbons leaking into our leak detection system should not decompose, and if they exist they should be measurable at the hydrocarbon detector.

Anyone who has taken first year chemistry will be able to follow this straightforward logic. But it is an oversimplification of the truth. Now I'm going to explain why.

The much-repeated claim that the Gibbs Free Energy describes the spontaneity of a reaction is strictly true only when the components of the reaction are present in their stoichiometric proportions: which is to say, the proportions as given when the chemical reation is written out in its balanced form, as it is here. So if you mix one mole of methate, one mole of carbon dioxide, two moles of carbon monoxide and two moles of hydrogen, it is true that there will be no further production of the lighter species; in fact, given the right stimulus (e.g. a spark) the tendency would be for the reaction to go the other way, namely recombination into methane and CO2.

But in our leak detection system, the situation is very different. We are far from having the gasses present in their stoichiometric proportions. In fact we begin with pure CO2, and then introduce a tiny amount of methane, measured in the parts per million. What happens then?

It's a question of equilibrium. There comes a point where the rate of decomposition on the part of the heavier species equals the rate of recombination of the lighter species. The proportions of the mixture then stabilize. And not everyone remembers this, but in fact it is also part of the first-year chemistry curriculum (because otherwise I would have had no way of knowing it myself, not having any other education in the subject): you can use the Gibbs Free Energy to determine where exactly that point of equilibrium lies!

At this point I'm not going to explain why it works but I'm just going to write out the formula for equilibrium: it should look something like this:

k = [CO]^2 * [H2]^2 / [CH4] * [CO2]

The quantities in brackets are just the quantities of chemicals expressed in mole fractions. Reactants on the bottom, products on top. Because there are two moles of carbon monoxide in the balanced formula, you have to take the square of the concentration. Etcetera. Remember, I'm not explaining why this works, I'm just saying it's how the formula goes.

There's one more formula we need to make this work. The quantity k in the expression above is called the "equilibrium constant". To get the equilibrium constant you use the Gibbs Free Energy. It's an exponential formula: you take the ratio of the Gibbs Free Energy to the Ideal Gas Law constant for the temperature in question, and that ratio becomes the argument of the exponential function. If the ratio is positive, then the constant is greater than one; if negative, it is less than one.

If the Gibbs free energy is zero, then the equilibrium constant is equal to one. What does this mean? Just plug in the numbers. It means the numerator and the denominator of the fraction have to be equal. There are many ways you can do this. You can have one mole of each component. Or you can have, for example, two moles of methane, two moles of CO2, two moles of CO and one mole of H2...then your fraction comes to four on top and four on the bottom. Or whatever.

Even as a write these words, I can see that something is wrong. When the Gibbs Free Energy is zero, with k=1, the reaction ought to be balanced exactly as it is written. But if I try to plug those numbers into the equilibrium formula, I get 2^2 * 2^2 on top (because there are two moles each of hydrogen and carbon monoxide) and 1*1 on the bottom, so my equilibrium constant is way off. I'm going to take a pause here while I figure out what's wrong.