# 5.4: The Law of Mass Action for Related and Simultaneous Equilibria

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##### Learning Objectives
• To know the relationship between the equilibrium constant and the rate constants for the forward and reverse reactions.
• To write an equilibrium constant expression for any reaction.

## Relationship among Equilibrium Expressions

Chemists frequently need to know the equilibrium constant for a reaction that has not been previously studied. In such cases, the desired reaction can often be written as the sum of other reactions for which the equilibrium constants are known. The equilibrium constant for the unknown reaction can then be calculated from the tabulated values for the other reactions.

To illustrate this procedure, let’s consider the reaction of $$N_2$$ with $$O_2$$ to give $$NO_2$$. This reaction is an important source of the $$NO_2$$ that gives urban smog its typical brown color. The reaction normally occurs in two distinct steps. In the first reaction (1), $$N_2$$ reacts with $$O_2$$ at the high temperatures inside an internal combustion engine to give $$NO$$. The released $$NO$$ then reacts with additional $$O_2$$ to give $$NO_2$$ (2). The equilibrium constant for each reaction at 100°C is also given.

1. $$N_{2(g)}+O_{2(g)} \rightleftharpoons 2NO_{(g)}\;\; K_1=2.0 \times 10^{−25}$$
2. $$2NO_{(g)}+O_{2(g)} \rightleftharpoons 2NO_{2(g)}\;\;\;K_2=6.4 \times 10^9$$

Summing reactions (1) and (2) gives the overall reaction of $$N_2$$ with $$O_2$$:

1. $$N_{2(g)}+2O_{2(g)} \rightleftharpoons 2NO_{2(g)} \;\;\;K_3=?$$

The equilibrium constant expressions for the reactions are as follows:

$K_1=\dfrac{[NO]^2}{[N_2][O_2]}\;\;\; K_2=\dfrac{[NO_2]^2}{[NO]^2[O_2]}\;\;\; K_3=\dfrac{[NO_2]^2}{[N_2][O_2]^2}$

What is the relationship between $$K_1$$, $$K_2$$, and $$K_3$$, all at 100°C? The expression for $$K_1$$ has $$[NO]^2$$ in the numerator, the expression for $$K_2$$ has $$[NO]^2$$ in the denominator, and $$[NO]^2$$ does not appear in the expression for $$K_3$$. Multiplying $$K_1$$ by $$K_2$$ and canceling the $$[NO]^2$$ terms,

$K_1K_2=\dfrac{\cancel{[NO]^2}}{[N_2][O_2]} \times \dfrac{[NO_2]^2}{\cancel{[NO]^2}[O_2]}=\dfrac{[NO_2]^2}{[N_2][O_2]^2}=K_3$

Thus the product of the equilibrium constant expressions for $$K_1$$ and $$K_2$$ is the same as the equilibrium constant expression for $$K_3$$:

$K_3 = K_1K_2 = (2.0 \times 10^{−25})(6.4 \times 10^9) = 1.3 \times 10^{−15}$

The equilibrium constant for a reaction that is the sum of two or more reactions is equal to the product of the equilibrium constants for the individual reactions. In contrast, recall that according to Hess’s Law, $$ΔH$$ for the sum of two or more reactions is the sum of the ΔH values for the individual reactions.

##### Note

To determine $$K$$ for a reaction that is the sum of two or more reactions, add the reactions but multiply the equilibrium constants.

##### Example $$\PageIndex{6}$$

The following reactions occur at 1200°C:

1. $$CO_{(g)}+3H_{2(g)} \rightleftharpoons CH_{4(g)}+H_2O_{(g)} \;\;\;K_1=9.17 \times 10^{−2}$$
2. $$CH_{4(g)}+2H_2S_{(g)} \rightleftharpoons CS_{2(g)}+4H_{2(g})\;\;\; K_2=3.3 \times 10^4$$

Calculate the equilibrium constant for the following reaction at the same temperature.

1. $$CO_{(g)}+2H_2S_{(g)} \rightleftharpoons CS_{2(g)}+H_2O_{(g)}+H_{2(g)}\;\;\; K_3=?$$

Given: two balanced equilibrium equations, values of $$K$$, and an equilibrium equation for the overall reaction

Asked for: equilibrium constant for the overall reaction

Strategy:

Arrange the equations so that their sum produces the overall equation. If an equation had to be reversed, invert the value of $$K$$ for that equation. Calculate $$K$$ for the overall equation by multiplying the equilibrium constants for the individual equations.

Solution:

The key to solving this problem is to recognize that reaction 3 is the sum of reactions 1 and 2:

$CO_{(g)}+ \cancel{3H_{2(g)}} \rightleftharpoons \cancel{CH_{4(g)}} + H_2O_{(g)}$

$\cancel{CH_{4(g)}} +2H_2S_{(g)} \rightleftharpoons CS_{2(g)} + \cancel{3H_{2(g)}} + H_{2(g)}$

$CO_{(g)} + 3H_{2(g)} \rightleftharpoons CS_{2(g)}+H_2O_{(g)}+H_{2(g)}$

The values for $$K_1$$ and $$K_2$$ are given, so it is straightforward to calculate $$K_3$$:

$K_3 = K_1K_2 = (9.17 \times 10^{−2})(3.3 \times 10^4) = 3.03 \times 10^3$

##### Exercise $$\PageIndex{6}$$

In the first of two steps in the industrial synthesis of sulfuric acid, elemental sulfur reacts with oxygen to produce sulfur dioxide. In the second step, sulfur dioxide reacts with additional oxygen to form sulfur trioxide. The reaction for each step is shown, as is the value of the corresponding equilibrium constant at 25°C. Calculate the equilibrium constant for the overall reaction at this same temperature.

1. $$\frac{1}{8}S_{8(s)}+O_{2(g)} \rightleftharpoons SO_{2(g)}\;\;\; K_1=4.4 \times 10^{53}$$
2. $$SO_{2(g)}+\frac{1}{2}O_{2(g)} \rightleftharpoons SO_{3(g)}\;\;\; K_2=2.6 \times 10^{12}$$
3. $$\frac{1}{8}S_{8(s)}+\frac{3}{2}O_{2(g)} \rightleftharpoons SO_{3(g)}\;\;\; K_3=?$$
$$K_3 = 1.1 \times 10^{66}$$