Question

Consider the following reaction: $$ \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{NO}(g) $$ At a certain temperature, the equilibrium constant for the reaction is \(0.0639 .\) What are the partial pressures of all gases at equilibrium if the initial partial pressure of the gases (both products and reactants) is \(0.400 \mathrm{~atm} ?\)

Short answer

Answer: The approximate equilibrium partial pressures are 0.204 atm for N2 and O2, and 0.792 atm for NO.

Step-by-step solution

Write down the balanced chemical equation

The balanced chemical equation for the given reaction is $$ \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{NO}(g) $$

Set up the ICE table

The ICE table will help us keep track of the initial partial pressures, the changes in partial pressures during the reaction, and the equilibrium partial pressures of all the gases. The table is given below: | | N2(g) | O2(g) | 2NO(g) | |------|-------|-------|--------| | I | 0.4 | 0.4 | 0.4 | | C | -x | -x | +2x | | E | 0.4-x | 0.4-x | 0.4+2x |

Write the expression for the equilibrium constant, Kp

For this reaction, the equilibrium constant Kp is given by the expression: $$ K_p = \frac{(P_{NO})^2}{P_{N2}*P_{O2}} $$

Substitute equilibrium partial pressures into the Kp expression

Replace the partial pressures with their equilibrium partial pressures from the ICE table: $$ 0.0639 = \frac{(0.4 + 2x)^2}{(0.4 - x)(0.4 - x)} $$

Solve for x using a quadratic equation solver

To solve this equation, you can use a quadratic equation solver or a calculator with a numerical solver function. After solving, we find that x ≈ 0.196.

Calculate the equilibrium partial pressures

Now use the value of x to find the equilibrium partial pressures of all gases: $$ P_{N2} = 0.4 - x \approx 0.4 - 0.196 = 0.204 \,\text{atm} \\ P_{O2} = 0.4 - x \approx 0.4 - 0.196 = 0.204 \,\text{atm} \\ P_{NO} = 0.4 + 2x \approx 0.4 + 2(0.196) = 0.792 \,\text{atm} $$ The equilibrium partial pressures are approximately 0.204 atm for N2 and O2, and 0.792 atm for NO.

Key concepts

Chemical Equilibrium

The concept of chemical equilibrium lays the foundation for understanding how reactions proceed under certain conditions. It represents a state where the rate of the forward reaction is equal to the rate of the reverse reaction, meaning there is no net change in the concentrations of reactants and products. This doesn't mean that the reactions have stopped, but rather that they are occurring at the same rate.

During an equilibrium state, the reactions are said to be dynamic. This means that even though there's no observable change at the macroscopic level, molecules are continually reacting and forming products, while products are breaking down to form reactants. The equilibrium constant, denoted as K or Kp when involving gases, gives us quantitative information about the state of equilibrium. It is a ratio that describes the concentration (or pressure in the case of gases) of the products to the reactants raised to their stoichiometric coefficients.

Understanding the equilibrium constant is pivotal, as it can predict the direction of the reaction. A K value greater than 1 implies that the products are favored at equilibrium, while a value less than 1 implies that reactants are favored. In the context of the given problem, the Kp value of 0.0639 tells us that the reaction mixture will favor the reactants slightly more than the products at the given temperature.

Partial Pressures

Partial pressures are a measure of how much pressure a single gas exerts within a mixture of gases and are vital in quantifying the behavior of gases, especially in reactions held at constant temperature and volume. In relation to chemical equilibrium, the concept of partial pressures is used to determine the position of equilibrium in reactions involving gases.

Each gas in a mixture contributes to the total pressure proportionately to its mole fraction, as stated by Dalton's Law of Partial Pressures. For instance, if a gas mixture contains nitrogen and oxygen gases only, then the total pressure is the sum of the partial pressure of nitrogen and the partial pressure of oxygen. Understanding how to calculate and manipulate partial pressures can be crucial to solving problems in chemistry, especially when working with gas equilibria.

Importance in Equilibrium Calculations

When calculating the equilibrium constant for reactions involving gases (Kp), we use the partial pressures of the gas components. This distinguishes Kp from Kc, which is used for concentrations in molar units. As seen in the given problem, the equilibrium constant is expressed in terms of the partial pressures of nitrogen, oxygen, and nitrogen monoxide. Adjustments to these pressures, due to changes in reaction conditions or amounts, will directly impact the value of Kp, and subsequently, the position of equilibrium.

ICE Table

The ICE table, an acronym for Initial, Change, and Equilibrium, is a powerful tool used in chemistry to keep track of the changes in concentrations or partial pressures of reactants and products as a chemical system reaches equilibrium. It enables chemists to methodically approach problems involving equilibrium constants and to find unknown values based on stoichiometric relationships and initial conditions.

An ICE table starts with the initial concentrations or pressures of reactants and products, before any reaction has occurred. It then outlines the Change in these values as the system proceeds towards equilibrium. Finally, it provides the Equilibrium concentrations or pressures once the system has reached a state of balance. By writing down these values systematically, one can identify the relationships needed to solve for unknowns in equilibrium problems.

Building an ICE Table

To construct an ICE table, first write down the balanced chemical equation. Then, below it, organize three rows for the Initial, Change, and Equilibrium states, and make columns for each species in the reaction. Changes in the system are denoted by 'x', which correlates with the stoichiometry of the reaction. In our problem, for example, the loss of nitrogen (N2) or oxygen (O2) is 'x', whereas the gain of nitrogen monoxide (NO) is '2x', reflecting the 1:1:2 ratio in the balanced equation. The ICE table simplifies the process of solving equilibrium problems and lends a visual hand to understanding dynamic chemical systems.