Question

At \(2000^{\circ} \mathrm{C}\), the equilibrium constant for the reaction $$ 2 \mathrm{NO}(g) \rightleftharpoons \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) $$ is \(K_{c}=2.4 \times 10^{3}\). If the initial concentration of \(\mathrm{NO}\) is \(0.175 \mathrm{M}\), what are the equilibrium concentrations of \(\mathrm{NO}\), \(\mathrm{N}_{2}\), and \(\mathrm{O}_{2}\) ?

Short answer

The equilibrium concentrations of NO, N2, and O2 are approximately 0.053 M, 0.061 M, and 0.061 M, respectively.

Step-by-step solution

Write down the given information

We have the following information: Initial concentration of NO = 0.175 M Equilibrium constant, Kc = 2.4 x 10^3 The given reaction is: \(2 \mathrm{NO}(g) \rightleftharpoons \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g)\)

Set up the ICE table

To find the equilibrium concentrations, we can use the ICE table which stands for Initial, Change, and Equilibrium. The table will help us determine the amounts of NO, N2, and O2 at equilibrium. Reaction: \(2 \mathrm{NO} \rightleftharpoons \mathrm{N}_{2}+\mathrm{O}_{2}\) Initial: \(0.175\) \(0\) \(0\) Change: \(-2x\) \(+x\) \(+x\) Equilibrium: \(0.175-2x\) \(x\) \(x\)

Write the expression of the equilibrium constant and substitute the equilibrium concentrations

Next, we need to write the expression for the equilibrium constant (Kc) using the equilibrium concentrations. \(K_c = \dfrac{[\mathrm{N}_{2}][\mathrm{O}_{2}]}{[\mathrm{NO}]^2}\) Now substitute the equilibrium concentrations from the ICE table. \(2.4 \times 10^{3} = \dfrac{x(x)}{(0.175-2x)^2}\)

Solve the equation for x

To find the equilibrium concentrations, we need to solve the above equation for x. \(2.4 \times 10^{3}(0.175-2x)^2 = x^2\) It is more convenient to solve the equation by quadratic formula, considering that \(a = 4, b = -3.5, c = 10^{-3} + k \), where k is defined by: \(k = 2.4 \times 10^{3} \times 0.175^2 -1\) Then find the roots x1 and x2 such as: \(x_{1,2} = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) Pick the root that makes physical sense (the one that results in a positive concentration).

Calculate the equilibrium concentrations of NO, N2, and O2

Once we found the valid value for x, use the equilibrium expressions from the ICE table to find the concentrations of NO, N2, and O2. NO equilibrium concentration = \(0.175-2x\) N2 equilibrium concentration = \(x\) O2 equilibrium concentration = \(x\)

Key concepts

ICE Table

An ICE table is a useful tool when solving equilibrium problems in chemistry. It allows you to systematically track the changes in concentrations of each species throughout a chemical reaction. The acronym ICE stands for Initial, Change, and Equilibrium, representing the stages each substance goes through.

• Initial: You begin by noting the initial concentrations of the reactants and products. In many cases, the initial concentration of the products is zero if the reaction has not yet started.
• Change: This row shows the change in concentrations as the reaction proceeds to equilibrium. For reactants, you input a negative change whereas for products, you use a positive change. In the given exercise's reaction, NO decreases by \(-2x\) because it is being consumed, while both \(\mathrm{N}_2\) and \(\mathrm{O}_2\) increase by \(+x\).
• Equilibrium: The equilibrium row consists of the expressions that combine the initial concentrations with their respective changes. After setting up these expressions, plug them into the equilibrium equation to find the unknown values.

By organizing this information in a table, it becomes easier to visualize the entire process and make accurate calculations.

Quadratic Formula

The quadratic formula is a mathematical tool used to solve quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). This formula is particularly helpful when you need to find the values of variables that satisfy this type of equation. The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here's how it applies within the context of equilibrium problems in chemistry:

• Set up the equation you need to solve by substituting your expressions from the ICE table into the equilibrium expression.
• Identify the coefficients \(a\), \(b\), and \(c\) from your equation based on how it corresponds to the standard quadratic form.
• Plug these coefficients into the quadratic formula to solve for \(x\), where \(x\) represents the change in concentration.
• Determine which solution makes physical sense. Typically, you’ll disregard any negative solution for concentration since concentrations cannot be negative.

By understanding and using the quadratic formula, you can solve for the changes in concentration that occur to reach equilibrium.

Equilibrium Concentration

Equilibrium concentration refers to the concentrations of reactants and products present once a chemical reaction has reached equilibrium. At this point, the forward and reverse reactions occur at the same rate, resulting in constant concentrations.

• From the ICE table, you express the concentration changes in terms of \(x\), where \(x\) represents the amount of product formed and reactant consumed.
• After solving for \(x\) using the quadratic formula, substitute it back into the equilibrium expressions from your ICE table.
• Calculate the equilibrium concentration of each species. For example, in the original problem, the concentration of \(\mathrm{NO}\) becomes \(0.175 - 2x\), whereas \(\mathrm{N}_2\) and \(\mathrm{O}_2\) both have concentrations \(x\).
• This process allows you to quantitatively predict the concentrations of all substances in the reaction mixture at equilibrium, which is crucial for understanding the extent of a reaction.

By knowing equilibrium concentrations, scientists can better understand and manipulate chemical reactions, which is critical in fields ranging from industrial chemistry to environmental science.