The equilibrium constant

In this lesson, we will learn:

• To write an expression for the equilibrium constant K_eq.
• How to interpret the value of K_eq and describe the reaction using this value.
• How to use the equilibrium expression with equilibrium concentrations to solve for K_eq (and vice versa).
• To write an expression for the reaction quotient, Q, and learn the difference between Q and K_eq.

• We now know the definition of equilibrium; a chemical process where the forward reaction rate is equal to the reverse rate. Be careful – this tells us nothing about how much product or reactant is there! To find that, we need to use the equilibrium constant expression.


• Using measurements of reactant and product concentrations, it is possible to find what is called the equilibrium constant, K_eq, (sometimes K_c) of a given reaction at equilibrium. This is done using the expression:
  For the reaction at equilibrium:
  $aA + bB \rightleftharpoons cC + dD$
  K_eq is calculated by:
  $\large K_{eq} = \frac{[C]^c[D]^d}{[A]^a[B]^b}$
  
  Be clear with your language:
  • The whole equation is the equilibrium expression.
  • K_eq is the equilibrium constant.
  • K_eq is the general term – if the equilibrium is measuring concentration (in mol dm^-3) it might be called K_c. K_p would be used if it was partial pressures (for gases).
  
  
• The equilibrium constant is called a constant because it is not affected by changes in some conditions. Changes to concentration of reactants or products and changes in pressure do not affect the equilibrium constant!
• Remember Le Chatelier’s principle: the system will counteract any change made. If you add reactant, more product will be made to counteract the change. This keeps K_eq constant in the long run.
• Changing temperature WILL affect K_eq, depending on whether the reaction is endothermic or exothermic.
  Because of this, ALWAYS quote K_eq with a temperature.


• The equilibrium expression looks complicated, so breaking it down using simple math can help.
• It is a fraction. It has terms for amount of product and reactant.
• You can write fractions as ratios. So…
• It is a ratio of products to reactants in the reaction. Written as a decimal, the value of K_eq tells us something important:
• K_eq is smaller than 1: There is less product than reactant in the reaction mixture. The smaller the value of K_eq, the less product there is.
• K_eq is approximately 1: There is roughly the same amount of product as reactant in the reaction mixture.
• K_eq is larger than 1: There is more product than reactant in the reaction mixture. The larger K_eq is, the more product compared to reactant.


• When writing K_eq for heterogeneous systems, where the substances are not all in the same phase, ignore any substances in the solid state. Solid reagents do not affect the equilibrium constant; everything else in the K_eq expression is written as normal.
  For example, let’s look at the thermal decomposition of calcium carbonate:

CaCO_{3 (s)} $\, \rightleftharpoons \,$ CaO _(s) + CO_{2 (g)}

• If CO₂ escapes the reaction vessel because it’s open, equilibrium cannot and will not be established. CO₂ alone dictates whether the equilibrium happens, so that is all the K_eq expression contains.

K_eq = [ CO_{2 (g)}]

• Remember that K_eq is only used for reactions at equilibrium.
  The reaction quotient (symbol Q) is used for reactions not at equilibrium to reveal which direction a reaction favours. Practically, Q is a K_eq value for reactions that are not yet at equilibrium!
  It is calculated using largely the same information as K_eq would be.

For the reaction:

aA + bB $\,$→$\,$ cC + dD
We find Q by calculating:

$Q =$ $\large \frac{[C]^{c} [D]^{d}} {[A]^{a} [B]^{b} }$
This would be for reactants and products in the aqueous or gaseous phase, where concentration is measured by partial pressure (see K_p and partial pressure). Like with K_eq, pure solids and liquids are not included in this calculation.

For some reactions, Q isn’t very useful. Reactions that go to completion will have an infinitely large Q (all product, no reactant), and reactions that don’t proceed will have a Q value equal to zero. A reaction with Q = 1 is already at equilibrium.

But many aqueous and gaseous reactions can go to equilibrium. Recall Le Chatelier’s principle: Q is useful because comparing Q to K_eq lets us predict changes to concentration as the reaction tries to reach equilibrium.
You can think of Q then as a pendulum; K_eq is the point at rest and Q is where it currently is:
• If Q is smaller than K_eqfor a reaction, the products will be favoured. Q > K_eq suggests there are more reactants in the mixture than the ideal equilibrium concentrations, so according to Le Chatelier’s principle there will be a shift back toward the products.
• If Q is equal to K_eq, there is no favouring of products or reactants. Q = K_eq means the reaction is already at ideal equilibrium concentrations. In this situation, nothing would be expected to change.
• If Q is greater than K_eq, the reactants will be favoured. Q < K_eq suggests that there is more product in the vessel than the ideal equilibrium concentration, so the equilibrium will shift back toward the reactants to remove this product surplus.


This Q value is larger than the quoted K_eq. Which means there is currently more product than there is at ideal equilibrium conditions. Applying Le Chatelier’s principle, we expect that in the current conditions, the reaction will favour the reactants.