Question

Consider the reaction: \begin{equation}2 \mathrm{NO}(g)+\mathrm{Br}_{2}(g) \Longrightarrow 2 \operatorname{NOBr}(g) \atop K_{\mathrm{p}}=28.4 \mathrm{at} 298 \mathrm{K}\end{equation} In a reaction mixture at equilibrium, the partial pressure of NO is 108 torr and that of \(\mathrm{Br}_{2}\) is 126 torr. What is the partial pressure of NOBr in this mixture?

Short answer

The partial pressure of NOBr in the reaction mixture at equilibrium is 290.4 torr.

Step-by-step solution

Understand the Equilibrium Expression

The equilibrium expression for the given reaction is written based on the balanced chemical equation. The equilibrium constant expression for this reaction is \( K_p = \frac{{[NOBr]^2}}{{[NO]^2 \times [Br_2]}} \), where the square brackets represent partial pressures.

Set up the Equilibrium Expression with Given Values

Insert the given partial pressures into the equilibrium expression. For \( NO \), it is 108 torr, and for \( Br_2 \) it is 126 torr. Assume the partial pressure of \( NOBr \) to be \( x \) torr.\( K_p = 28.4 = \frac{{x^2}}{{(108)^2 \times 126}} \).

Solve for the Partial Pressure of NOBr

Rearrange the expression to solve for \( x^2 \). Multiply both sides by \( (108)^2 \times 126 \) to isolate \( x^2 \) on one side of the equation.

Calculate the Partial Pressure of NOBr

Take the square root of both sides of the equation to solve for \( x \), the partial pressure of \( NOBr \). Ensure to consider only the positive root as pressure cannot be negative.

Verify Unit Consistency

Confirm that the units used for the pressures in the equilibrium expression are consistent and ensure the final answer is also reported in those units.

Key concepts

Understanding Equilibrium Constant Expression

At the heart of chemical equilibrium lies the equilibrium constant expression, represented by the symbol 'K'. This mathematical ratio allows chemists to understand the extent of a chemical reaction at equilibrium. In gaseous reactions, we primarily consider partial pressures and refer to the constant as 'Kp'. For the reaction involving nitrogen monoxide (NO) and bromine (Br2) to form nitrosyl bromide (NOBr), the equilibrium constant expression is written as \[\begin{equation}K_p = \frac{{[NOBr]^2}}{{[NO]^2 \times [Br_2]}}\end{equation}\]where each bracketed term represents the partial pressure of that gas. The equilibrium constant is a fixed value at a given temperature, signifying the balance point of a reaction. This balance isn’t static but dynamic; reactants and products are continuously formed and consumed at equal rates.

The Role of Partial Pressures in Equilibrium

Partial pressures gauge the impact of individual gas species in a mixture. They are crucial when analyzing gaseous equilibria, as they directly factor into the equilibrium constant expression. When dealing with partial pressures, remember that each gas in a mixture exerts a fraction of the total pressure, proportional to its mole fraction. By measuring these pressures (such as 108 torr for NO and 126 torr for Br2 in our example), we can predict the position of equilibrium and the pressures of other gases not initially known, provided we have the equilibrium constant, Kp.

Applying Le Chatelier's Principle

Le Chatelier's principle states that a system at equilibrium will respond to a stress in a way that counteracts it. This 'stress' can be a change in concentration, pressure, or temperature. For instance, increasing the pressure on the reactants will shift the equilibrium toward products to reduce pressure, and vice versa. Insights from Le Chatelier's principle allow chemists to manipulate conditions to favor the production of desired substances. Understanding this principle is key to predicting how equilibrium will shift when external conditions are altered.

Balancing Chemical Equations for Equilibrium

Properly balancing chemical equations is essential when calculating equilibrium constants. Each substance's stoichiometry directly affects the equilibrium expression. A balanced equation is a representation of the conservation of matter—atoms are neither created nor destroyed. For our NO and Br2 reaction, we see a balanced stoichiometry that leads to a squared term for NOBr in the equilibrium constant expression, highlighting the importance of balancing, as it dictates the relationship of the partial pressures in the calculation.

Linking Thermodynamics and Kinetics to Equilibrium

Thermodynamics and kinetics are two key areas in chemistry that give us a fuller understanding of equilibrium. Thermodynamics relates to the energy changes and favorability of a reaction, often indicating the position of equilibrium through quantities such as Gibbs free energy. On the other hand, kinetics provides insight into the reaction rates. Although a reaction may be thermodynamically favorable (products more stable than reactants), if it has a high activation energy barrier, it may proceed slowly, affecting how quickly equilibrium is reached. The dynamically balanced state of equilibrium is achieved when the forward and reverse reaction rates are equal, embodying both thermodynamic stability and kinetic accessibility.