# 10.4: Calculations with Equilibrium Constants

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Iron is an important component of red blood cells. Patients who have low iron will usually be anemic and have a lower than normal number of red blood cells. One way to assess serum iron concentration is with the use of Ferrozine, a complex organic molecule. Ferrozine forms a product with $$\ce{Fe^{3+}}$$, producing a pink color. In order to determine factors affecting the reaction, we need to measure the equilibrium constant. If the equilibrium does not lie far in the direction of products, precautions need to be taken when using this material to measure iron in serum.

## Calculations with Equilibrium Constants

The general value of the equilibrium constant gives us information about whether the reactants or the products are favored at equilibrium. Since the product concentrations are in the numerator of the equilibrium expression, a $$K_\text{eq} > 1$$ means that the products are favored over the reactants. A $$K_\text{eq} < 1$$ means that the reactants are favored over the products.

Though it would often seem that the $$K_\text{eq}$$ value would have various units depending on the values of the exponents in the expression, the general rule is that any units are dropped. All $$K_\text{eq}$$ values will be reported as having no units.

Example $$\PageIndex{1}$$

Equilibrium occurs when nitrogen monoxide gas reacts with oxygen gas to form nitrogen dioxide gas:

$2 \ce{NO} \left( g \right) + \ce{O_2} \left( g \right) \rightleftharpoons 2 \ce{NO_2} \left( g \right)\nonumber$

At equilibrium at $$230^\text{o} \text{C}$$, the concentrations are measured to be $$\left[ \ce{NO} \right] = 0.0542 \: \text{M}$$, $$\left[ \ce{O_2} \right] = 0.127 \: \text{M}$$, and $$\left[ \ce{NO_2} \right] = 15.5 \: \text{M}$$. Calculate the equilibrium constant at this temperature.

Solution:

Step 1: List the known quantities and plan the problem.

Known

• $$\left[ \ce{NO} \right] = 0.0542 \: \text{M}$$
• $$\left[ \ce{O_2} \right] = 0.127 \: \text{M}$$
• $$\left[ \ce{NO_2} \right] = 15.5 \: \text{M}$$

Unknown

The equilibrium expression is first written according to the general form in the text. The equilibrium values are substituted into the expression and the value calculated.

Step 2: Solve.

$K_\text{eq} = \frac{\left[ \ce{NO_2} \right]^2}{\left[ \ce{NO} \right]^2 \left[ \ce{O_2} \right]}\nonumber$

Substituting in the concentrations at equilibrium:

$K_\text{eq} = \frac{\left( 15.5 \right)^2}{\left( 0.0542 \right)^2 \left( 0.127 \right)} = 6.44 \times 10^5\nonumber$

The equilibrium concentration of the product $$\ce{NO_2}$$ is significantly higher than the concentrations of the reactants $$\ce{NO}$$ and $$\ce{O_2}$$. As a result, the $$K_\text{eq}$$ value is much larger than 1, an indication that the product is favored at equilibrium.

The equilibrium expression only shows those substances whose concentrations are variable during the reaction. A pure solid or a pure liquid does not have a concentration that will vary during a reaction. Therefore, an equilibrium expression omits pure solids and liquids, and only shows the concentrations of gases and aqueous solutions. The decomposition of mercury (II) oxide can be shown by the following equation, followed by its equilibrium expression.

$2 \ce{HgO} \left( s \right) \rightleftharpoons 2 \ce{Hg} \left( l \right) + \ce{O_2} \left( g \right) \: \: \: \: \: K_\text{eq} = \left[ \ce{O_2} \right]\nonumber$

The stoichiometry of an equation can also be used in a calculation of an equilibrium constant. At $$40^\text{o} \text{C}$$, solid ammonium carbamate decomposes to ammonia and carbon dioxide gases.

$\ce{NH_4CO_2NH_2} \left( s \right) \rightleftharpoons 2 \ce{NH_3} \left( g \right) + \ce{CO_2} \left( g \right)\nonumber$

At equilibrium, the $$\left[ \ce{CO_2} \right]$$ is found to be $$4.71 \times 10^{-3} \: \text{M}$$. Can the $$K_\text{eq}$$ value be calculated from that information alone? Because the ammonium carbamate is a solid, it is not present in the equilibrium expression.

$K_\text{eq} = \left[ \ce{NH_3} \right]^2 \left[ \ce{CO_2} \right]\nonumber$

The stoichiometry of the chemical equation indicates that as the ammonium carbamate decomposes, $$2 \: \text{mol}$$ of ammonia gas is produced for every $$1 \: \text{mol}$$ of carbon dioxide. Therefore, at equilibrium, the concentration of the ammonia will be twice the concentration of carbon dioxide. So,

$$\left[ \ce{NH_3} \right] = 2 \times \left( 4.71 \times 10^{-3} \right) = 9.42 \times 10^{-3} \: \text{M}$$

Substituting these values into the $$K_\text{eq}$$ expression:

$K_\text{eq} = \left( 9.42 \times 10^{-3} \right)^2 \left( 4.71 \times 10^{-3} \right) = 4.18 \times 10^{-7}\nonumber$

## Using Equilibrium Constants

The equilibrium constants are known for a great many reactions. Hydrogen and bromine gases combine to form hydrogen bromide gas. At $$730^\text{o} \text{C}$$, the equation and $$K_\text{eq}$$ are given below.

$\ce{H_2} \left( g \right) + \ce{Br_2} \left( g \right) \rightleftharpoons 2 \ce{HBr} \left( g \right) \: \: \: \: \: K_\text{eq} = 2.18 \times 10^6\nonumber$

A certain reaction is begun with only $$\ce{HBr}$$. When the reaction mixture reaches equilibrium at $$730^\text{o} \text{C}$$, the concentration of bromine gas is measured to be $$0.00243 \: \text{M}$$. What is the concentration of the $$\ce{H_2}$$ and the $$\ce{HBr}$$ at equilibrium?

Since the reaction begins with only $$\ce{HBr}$$ and the mole ratio of $$\ce{H_2}$$ to $$\ce{Br_2}$$ is 1:1, the concentration of $$\ce{H_2}$$ at equilibrium is also $$0.00243 \: \text{M}$$. The equilibrium expression can be rearranged to solve for the concentration of $$\ce{HBr}$$ at equilibrium:

\begin{align*} K_\text{eq} &= \frac{\left[ \ce{HBr} \right]^2}{\left[ \ce{H_2} \right] \left[ \ce{Br_2} \right]} \\ \left[ \ce{HBr} \right] &= \sqrt{K_\text{eq} \left[ \ce{H_2} \right] \left[ \ce{Br_2} \right]} \\ &= \sqrt{2.18 \times 10^6 \left( 0.00243 \right) \left( 0.00243 \right)} = 3.59 \: \text{M} \end{align*}\nonumber

Since the value of the equilibrium constant is very high, the concentration of $$\ce{HBr}$$ is much greater than that of $$\ce{H_2}$$ and $$\ce{Br_2}$$ at equilibrium.

## Summary

10.4: Calculations with Equilibrium Constants is shared under a CC BY-NC license and was authored, remixed, and/or curated by LibreTexts.