I suppose you have seen some chemical equations. A lot of them probably had a one sided arrow showing the direction of a reaction, telling you which side the reactants (that will eventually change into products) are on. Actually, most reactions aren’t like that. Most reactions go in both directions, forward and backward. That means that at the same time some reactants are changing into products but also some products are changing into reactants. That doesn’t necessarily have to be happening at the same rate. If the forward reaction is moving at a higher rate, more products are being formed.
The rate of a reaction changes over time, often together with the concentrations of the reactants. That means the reactions go slower and slower as they are proceeding. The rate of this reaction will decrease. The opposite is true for the backward reaction. We have more products being formed by the forward reaction which speeds up the backward reaction. The rates of these reactions will eventually become equal. When that happens, the forward and backward reactions are happening at the same rate and from the outside, it looks like nothing is happening. We call this state the equilibrium.
The equilibrium constant
Talking about equilibrium can be useful, so we have to figure out some way to do so. We usually use so-called equilibrium constants. Every chemical reaction with a given set of reaction conditions has its equilibrium constant, which isn’t affected by the initial concentrations of the substances and is often figured out empirically (experimentally). You can read about factors affecting equilibrium constant and position in later topics.
This equilibrium constant is denoted as K and for a general equation calculated this way:
\[\ce{aA + bB <=> cC + dD}\]
\[K = \frac{[C]^c[D]^d}{[A]^a[B]^b}\]
It is the ratio of equilibrium concentrations of products raised to their stoichiometric coefficients to the equilibrium concentrations of reactants raised to their stoichiometric coefficients. See the double sided arrow in the chemical reaction? We use it to denote that the reaction is reversible, going in both directions.
Let’s look at the formula. For reactions with very high equilibrium constant, the backward reaction is negligible, so we treat the reaction as essentially going to completion. That’s when we use the one sided arrow. On the other hand, with very low equilibrium constant, the reactants are dominating and we can say the reaction hardly proceeds.
Also, the constant is dimensionless. That isn’t trivial looking at the formula. Officially, we do that by dividing each concentration before plugging it in by standard concentration, say 1M (and this quantity is called the activity). But for our purposes, you can just lose the units when calculating and remember to use them in the end.
An important thing to remember is that, for aqueous phase reactions, the concentration of solids is just considered to be 1 (so basically the compound is ignored). Same thing goes for liquids and solids in the case of gas phase reactions, but also for water in the case of aqueous phase reactions.
For those interested to get a deeper understanding, this happens because:
In aqueous phase reactions, solid substances are always found in a very small, constant amount, inside the solution. So, just like we usually divide by the standard 1M concentration, in this case we divide by the small concentration.
For gas phase reactions, solids and liquids only have a very small amount of vapors, which is found in the gas phase and actually reacts. Just like before, we can divide by this very small amount instead of 1M, to get the activity
First, let’s try writing the expression of equilibrium constant for a simple equation. Consider the reaction \(\ce{H2 (g) + I2 (g) <=> 2HI (g)}\). The equilibrium constant will be \(K_{eq}=\frac{[HI]^2}{[H_2][I_2]}\). Hopefully that was easy.
When calculating something using the equilibrium constant we can use so-called ICE tables, sometimes called RICE tables. The name stands for Reaction, Initial, Change, Equilibrium. We write the reaction in the first row, initial concentrations into the second row, change in concentration (we usually don’t know it so we use x) and equilibrium concentration in the last row.
For the reaction above, let’s consider initial concentrations of hydrogen and iodine gas 1M. At a given temperature, the equilibrium constant is 49. Calculate the equilibrium concentrations of all gases.
The ICE table looks like this:
Reaction
\(\ce{H2}\) (g)
\(\ce{I2}\) (g)
\(\ce{2HI}\) (g)
Initial
1M
1M
0
Change
- X
- X
+ 2X
Equilibrium
1 - X
1 - X
2X
The change row is determined by the stoichiometry, we use the same amount of hydrogen and iodine to make twice as much of HI. We can now write the equilibrium constant expression.
We complete the equilibrium row and get the equilibrium concentrations of reactants 0.22M and product 1.56M.
This was a particularly pretty example; we could root the fraction. That is not always the case. We can sometimes use another trick, which is when the constant is very low. That means the change is negligible in the denominator, 1-X is almost the same as 1. We can neglect it and solve the equation that way. However, if the constant isn’t low, we cannot neglect the change. In that case, we have to use the quadratic equation to solve for the change.
Not in equilibrium yet?
Okay, we can talk about our systems in equilibrium pretty nicely now. But what if they aren’t in equilibrium yet? We can calculate so-called reaction quotient Q the same way as K just using the actual concentrations (not at equilibrium). We can then compare Q and K to figure out which way is the reaction proceeding. It makes sense that if Q = K, the reaction reached its equilibrium. Looking at the formulas, if Q < K there are too many reactants and the reaction will proceed in the forward direction. If Q > K there are too many products and the reverse/backward reaction will take place. You don’t have to memorize this, you can always figure it out just by looking at the expression.
Other equilibrium constants
Although until now we expressed the equilibrium constant in terms of concentrations, we can express it in terms of other quantities. We can denote the usual constant (expressed in terms of concentrations), by \(K_c\). Other constants are the one expressed in terms of partial pressures, \(K_p = \frac{p_C^cp_D^d}{p_A^ap_B^b}\), and the one expressed in terms of mole fractions, \(K_x = \frac{x_C^cx_D^d}{x_A^ax_B^b}\).
In the next lesson, we’ll see how to convert between the three.
Note: Like before, the partial pressures are divided by the standard 1atm pressure, in the formula of \(K_p\), so it is dimensionless.
Conclusion
Equilibrium constants can be useful in many different ways. You may have heard of solubility product constant \(K_sp\), basically telling us how soluble a salt is. We also have dissociation constants used with weak acids and bases \(K_a\) and \(K_b\) used to calculate the \(pK_a\) and \(pK_b\). We also have hydrolysis constants such as \(K_w\) and many more. They are all the same, just found to be so useful that they got their own names and subscripts.