Question

For the reaction $$ \mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \rightleftharpoons 2 \mathrm{NH}_{3}(g) $$ \(K=1.3 \times 10^{-2}\) at a given temperature. If the system at equilibrium is analyzed and the concentrations of both \(\mathrm{N}_{2}\) and \(\mathrm{H}_{2}\) are found to be \(0.10 \mathrm{M},\) what is the concentration of \(\mathrm{NH}_{3}\) in the system?

Short answer

The equilibrium concentration of ammonia (\(\mathrm{NH}_{3}\)) in the system is \(1.14 \times 10^{-3} \mathrm{M}\).

Step-by-step solution

Write the expression for the equilibrium constant

The expression for the equilibrium constant involves the concentration of each species at equilibrium raised to their respective stoichiometric coefficients: $$ K=\frac{[\mathrm{NH}_{3}]^{2}}{[\mathrm{N}_{2}][\mathrm{H}_{2}]^{3}} $$

Insert the provided values for the equilibrium constant and the concentrations of N2 and H2

We are given the values for the equilibrium constant, \(K\), and the equilibrium concentrations of nitrogen and hydrogen. Let \([NH_3] = x\), so we can write the equation for K as: $$ 1.3 \times 10^{-2}=\frac{x^{2}}{(0.10)(0.10)^{3}} $$

Solve the equation for the ammonia concentration (\(x\))

We can solve the equation to determine the value of \(x\), which represents the equilibrium concentration of ammonia: $$ x^{2} = 1.3 \times 10^{-2} (0.10)(0.10)^{3} $$ Computing the values on the right-hand side, we get: $$ x^{2} = 1.3 \times 10^{-6} $$ Now, taking the square root of both sides to solve for x: $$ x = \sqrt{1.3 \times 10^{-6}} $$ Calculating the value of x, we obtain: $$ x = 1.14 \times 10^{-3} $$

Write the answer

The equilibrium concentration of ammonia (\(\mathrm{NH}_{3}\)) in the system is \(1.14 \times 10^{-3} \mathrm{M}\).

Key concepts

Chemical Equilibrium

Chemical equilibrium is a state in a reversible chemical reaction where the rates of the forward and reverse reactions are equal, leading to no net change in the concentration of reactants and products over time. This dynamic balance occurs when the reactions proceed at the same rate in both directions.

Consider a simple reaction like \(A + B \rightleftharpoons C + D\), where A and B are reactants converting to C and D products. Equilibrium does not mean the concentrations are equal; it merely means they are constant. In our original exercise, the balancing of nitrogen (\text{N}_2\) and hydrogen (\text{H}_2\) to form ammonia (\text{NH}_3\) is an example of a system that can reach chemical equilibrium at a given temperature.

Equilibrium Expressions

Equilibrium expressions are mathematical representations of a chemical equilibrium situation. They are used to describe the ratio of the concentrations of products to reactants, each raised to the power of their respective coefficients as presented in the balanced chemical equation.

For a generic balanced reaction \(aA + bB \rightleftharpoons cC + dD\), the equilibrium expression is written as: \[K = \frac{{[C]^c[D]^d}}{{[A]^a[B]^b}}\] where K is the equilibrium constant. It is important to note that pure solids and liquids are omitted from the equilibrium expression. The exercise provided exemplifies how equilibrium expressions are set up and calculated.

Stoichiometry

Stoichiometry is the aspect of chemistry that involves calculating the relative quantities of reactants and products in chemical reactions. It is based on the conservation of mass and the concept of the mole. The coefficients in a balanced chemical equation provide the mole ratios of reactants and products.

For our equation \(\text{N}_2(g) + 3\text{H}_2(g) \rightleftharpoons 2\text{NH}_3(g)\), the stoichiometry reflects a 1:3:2 ratio. When using stoichiometry to solve for equilibrium concentrations, as demonstrated in the step by step solution, it's critical to maintain these ratios to get the correct results. Through stoichiometry, we understand the quantitative relationships that allow us to calculate unknown quantities, such as the concentration of \text{NH}_3\, in our exercise.

Le Chatelier's Principle

Le Chatelier's principle provides insight into how a system at equilibrium responds to a change in conditions. If a dynamic equilibrium is disturbed by changing the conditions such as concentration, temperature, or pressure, the system adjusts in a way that opposes the change to re-establish equilibrium.

When a reactant is added to the system, the equilibrium will shift to produce more products. If a product is removed, it will shift to produce more of that product. These shifts are the system's attempt to minimize the disturbance. This principle helps predict the effects of changes and is very useful for industrial chemical processes, such as the synthesis of ammonia in the Haber process referenced in the original exercise.