Question

Write the \(K_{\mathrm{P}}\) expression for the following gas-phase reaction: \(4 \mathrm{NO}_{2}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g}) \rightleftarrows 2 \mathrm{~N}_{2} \mathrm{O}_{5}(\mathrm{~g})\)

Short answer

\( K_{\mathrm{P}} = \frac{(P_{\text{N}_2O_5})^2}{(P_{\text{NO}_2})^4 \cdot (P_{\text{O}_2})} \)

Step-by-step solution

Understanding the Reaction

We start with the balanced chemical reaction given: \( 4 \text{NO}_2(g) + \text{O}_2(g) \rightleftarrows 2 \text{N}_2O_5(g) \). This depicts a reversible reaction involving gases.

Identifying the Gases

Identify and list all the gaseous substances involved in the reaction. The reactants are \(\text{NO}_2\) and \(\text{O}_2\), while the product is \(\text{N}_2O_5\).

Writing the General Formula for Kp

Recall that the equilibrium constant \( K_{\mathrm{P}} \) for a gas-phase reaction is given by the partial pressures of the products raised to their stoichiometric coefficients, divided by the partial pressures of the reactants raised to their stoichiometric coefficients.

Applying the Formula

For the reaction \(4\text{NO}_2(g) + \text{O}_2(g) \rightleftarrows 2\text{N}_2O_5(g)\), the expression for \( K_{\mathrm{P}} \) is derived as follows: the numerator consists of the partial pressure of \(\text{N}_2O_5\) raised to the power of 2 (stoichiometric coefficient), and the denominator consists of the partial pressures of \(\text{NO}_2\) and \(\text{O}_2\) raised to the powers of 4 and 1, respectively.

Writing the Expression

Substitute the stoichiometric coefficients into the expression to get the final \( K_{\mathrm{P}} \) expression: \[K_{\mathrm{P}} = \frac{(P_{\text{N}_2O_5})^2}{(P_{\text{NO}_2})^4 (P_{\text{O}_2})^1}\]

Key concepts

Gas-Phase Reaction

In chemistry, reactions often occur in the gas phase. This simply means that the substances involved are all in the gaseous state during the reaction.
Gas-phase reactions are important in various fields, from industrial applications to atmospheric chemistry.

• They involve reactants and products that are gases.
• The reaction can be represented by a chemical equation that shows the transformation of reactants into products.
• Equilibrium is a key concept for these reactions because they often reach a state where the formation of products and reactants occurs at the same rate.

For the reaction given in the exercise, all substances, like \(\text{NO}_2\) and \(\text{O}_2\) as reactants and \(\text{N}_2O_5\) as the product, are gases. This allows us to use partial pressures instead of concentrations in expressions like \(K_\mathrm{P}\).

Partial Pressure

The concept of partial pressure is vital when dealing with gas-phase reactions. It refers to the pressure exerted by a single type of gas in a mixture.
In a mixture of gases, each gas contributes to the total pressure based on its proportion in the mixture.

• Partial pressure is denoted by the letter \(P\) with a subscript indicating the gas, like \(P_{\text{NO}_2}\) for nitrogen dioxide.
• We use partial pressures when calculating equilibrium constants for gas-phase reactions, such as \(K_\mathrm{P}\).
• The total pressure of a mixture is the sum of the partial pressures of all gases present.

By using partial pressures in the equilibrium expression, we can determine how the reaction progresses over time and under different conditions.

Stoichiometric Coefficients

Stoichiometric coefficients are the numbers placed in front of compounds in a chemical equation to show the ratio in which they react or are produced.
These coefficients are crucial in determining the relationship between reactants and products in any reaction.

• In the equation \(4 \text{NO}_2(g) + \text{O}_2(g) \rightleftarrows 2 \text{N}_2O_5(g)\), the numbers 4, 1, and 2 are the stoichiometric coefficients.
• They indicate, for instance, that four molecules of \(\text{NO}_2\) react with one molecule of \(\text{O}_2\) to form two molecules of \(\text{N}_2O_5\).
• When writing equilibrium expressions like \(K_\mathrm{P}\), each partial pressure is raised to the power of its stoichiometric coefficient.

Understanding stoichiometric coefficients helps in predicting how much of a reactant is needed or how much product will form in a reaction.