Some reactions can be represented as a sequence of two or more relations. For example, let’s consider the following reaction between NOBr and Cl₂ gases:

2NOBr(g) + Cl₂(g) ⇆ 2NO(g) + 2 BrCl(g)

We can represent this net transformation as a sum of two reactions:

1) 2NOBr(g) ⇆ 2NO(g) + Br₂(g)

2) Br₂(g) + Cl₂(g) ⇆ 2BrCl(g)

To confirm that these two give the target equation, simply add them up and cancel out the identical components on both sides of the equation:

1) 2NOBr(g) ⇆ 2NO(g) + Br₂(g)
+
2) Br₂(g) + Cl₂(g) ⇆ 2BrCl(g)

2NOBr(g) + Br₂(g) + Cl₂(g) ⇆ 2NO(g) + Br₂(g) + 2 BrCl(g)

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3) 2NOBr(g) + Cl₂(g) ⇆ 2NO(g) + 2 BrCl(g)

So, Br₂ appeared on both sides of the equation and by canceling it, we obtained the final equation (3).

Now, how is this related to the equilibrium constant?

It turns out that we can calculate the equilibrium constant for a reaction if we know the equilibrium constants for other reactions that add up to this reaction. For this, you simply need to multiply the equilibrium constants of the two reactions.

For example, suppose the equilibrium constants of reactions (1) and (2) are 0.012 and 7.2 respectively:

1) 2NOBr(g) ⇆ 2NO(g) + Br₂(g)

2) Br₂(g) + Cl₂(g) ⇆ 2BrCl(g)

To obtain the equilibrium constant for reaction (3), we multiply the ones for reaction (1) and (2):

2NOBr(g) + Cl₂(g) ⇆ 2NO(g) + 2BrCl(g) (3)

Notice that we got an expression for K₃ that matches what we’d write according to the chemical equation (3).

Therefore, the value of K₃ is the product of K₁ and K₂:

K₃ = K₁ x K₂ = 0.012 x 7.3 = 0.088

If this is confusing, you can review the Hess’s law which uses a similar strategy to calculate the enthalpy of a reaction based on the enthalpy of other reactions that add up to it.