Now that we can write equilibrium reactions and expressions (K_eq), what happens if we change all the coefficients in the reactions? What about if two reactions are happening to create a third, unknown reaction, and we want to know if that third reaction is reactant or product favored? In this section, we will be looking at how to manipulate the equilibrium constant in these scenarios.

Equilibrium Constant Expressions and Stoichiometry

Consider the reaction:

Reaction 1:  2 NOCl(g)   ⇌   2 NO(g)  +  Cl₂(g)      K_eq,1 =[latex]\frac{[NO]^{2}[Cl_{2}]}{[NOCl]^{2}}[/latex]

This reaction and its resulting equilibrium constant will represent the ratio between the products and reactants at equilibrium. But what if we were more interested in studying one mole of NOCl reacting to form NO and Cl₂? If we divide the entire reaction by 2, the equilibrium reaction and equilibrium constant expression would be:

Reaction 2:  NOCl(g)   ⇌   NO(g)  +  ½ Cl₂(g)      K_eq,2 =[latex]\frac{[NO][Cl_{2}]^{\frac{1}{2}}}{[NOCl]}[/latex]

Notice the difference between the two equilibrium constant expressions. The products are still in the numerator and reactants in the denominator, but the exponents that each species is raised to differs because the coefficients are different. And in fact each coefficient in K_eq,2 is half that of K_eq,1, or each species is raised to the ½ power.

K_eq,1 = K_eq,2^½

Equilibrium Constant Expressions and Reversing Reactions

Sometimes we need to flip a reaction in order to properly represent the system we are trying to study. An example would be if we wanted to study:

Reaction 1:  2 NOCl(g)   ⇌   2 NO(g)  +  Cl₂(g)      K_eq,1 =[latex]\frac{[NO]^{2}[Cl_{2}]}{[NOCl]^{2}}[/latex]

But we only had information about the opposite reaction:

Reaction 2:  2 NO(g)  +  Cl₂(g)   ⇌   2 NOCl(g)      K_eq,2 =[latex]\frac{[NOCl]^{2}}{[NO]^{2}[Cl_{2}]}[/latex]

How do these two equilibrium constant expressions differ?

K_eq,1 = [latex]\frac{[NO]^{2}[Cl_{2}]}{[NOCl]^{2}}[/latex]       K_eq,2 =[latex]\frac{[NOCl]^{2}}{[NO]^{2}[Cl_{2}]}[/latex]

When we flip a reaction, the reactants and the products switch sides, so the numerator and the denominator in the equilibrium constant expression also switch and they are inverses of eachother.

K_eq,1 =[latex]\frac{1}{K_{eq,2}}[/latex]

Equilibrium Constant Expressions for Multiple Reactions

Suppose we wanted to study the reaction:

Reaction 1:  2 NOCl(g)  +  O₂(g)   ⇌   N₂O₄(g)  +  Cl₂(g)     K_eq,1 =[latex]\frac{[N_{2}O_{4}][Cl_{2}]}{[NOCl]^{2}[O_{2}]}[/latex]

Unfortunately for us, this is a completely unstudied reaction and we have no idea if this reaction is reactant- or product-favored, whether it will occur fast or slow, and whether it is safe for us to even perform the reaction! Luckily, we have two other similar reactions that we can use to give us insight into our desired reaction.

Reaction 2:  2 NOCl(g)   ⇌   2 NO(g)  +  Cl₂(g)     K_eq,2 =[latex]\frac{[NO]^{2}[Cl_{2}]}{[NOCl]^{2}}[/latex]

Reaction 3:  2 NO(g)  +  O₂(g)   ⇌   N₂O₄(g)     K_eq,3 =[latex]\frac{[N_{2}O_{4}]}{[NO]^{2}[O_{2}]}[/latex]

Notice that if we add Reaction 2 and Reaction 3 together, we are able to get our desired reaction, Reaction 1. But how do we know what the value of K_eq,1 is? Let’s compare K_eq,1, K_eq,2, and K_eq,3 a little more closely. What do we have to do with K_eq,2 and K_eq,3 in order to get K_eq,1?

K_eq,1 =[latex]\frac{[N_{2}O_{4}][Cl_{2}]}{[NOCl]^{2}[O_{2}]}[/latex] K_eq,2 =[latex]\frac{[NO]^{2}[Cl_{2}]}{[NOCl]^{2}}[/latex] K_eq,3 =[latex]\frac{[N_{2}O_{4}]}{[NO]^{2}[O_{2}]}[/latex]

Perhaps you noticed that if we multiply K_eq,2 and K_eq,3, it will be equal to K_eq,1.

K_eq,2 × K_eq,3 =[latex]\frac{[NO]^{2}[Cl_{2}]}{[NOCl]^{2}}\times\frac{[N_{2}O_{4}]}{[NO]^{2}[O_{2}]}=\frac{[N_{2}O_{4}][Cl_{2}]}{[NOCl]^{2}[O_{2}]}[/latex]= K_eq,1

Key Concepts and Summary

If K_eq is known for a specific reaction, K_eq can also be calculated if the chemical reaction is modified by changing the coefficients, reversing the reaction, or combining the given reaction with another reaction. Approaching this as you have approached Hess’ Law in the past is a great strategy, except remember that changing the coefficients requires you to raise K_eq to that change, reversing a reaction requires you to take the inverse of K_eq, and combining reactions requires you to multiply the K_eq for each reaction combined.