Question

The reaction \(\mathrm{NO}_{2}(g)+\mathrm{NO}(g) \rightleftharpoons \mathrm{N}_{2} \mathrm{O}(g)+\mathrm{O}_{2}(g)\) reached equilibrium at a certain high temperature. Originally, the reaction vessel contained the following initial concentrations: \(\left[\mathrm{N}_{2} \mathrm{O}\right]=0.184 \mathrm{M},\left[\mathrm{O}_{2}\right]=0.377 M\) \(\left[\mathrm{NO}_{2}\right]=0.0560 M,\) and \([\mathrm{NO}]=0.294 M .\) The concentration of the \(\mathrm{NO}_{2}\), the only colored gas in the mixture, was monitored by following the intensity of the color. At equilibrium, the \(\mathrm{NO}_{2}\) concentration had become \(0.118 \mathrm{M}\). What is the value of \(K_{c}\) for this reaction at this temperature?

Short answer

The equilibrium constant \(K_c\) for the reaction at this temperature is approximately 3.946.

Step-by-step solution

Write the equilibrium expression

For the reaction \( \mathrm{NO}_{2}(g) + \mathrm{NO}(g) \rightleftharpoons \mathrm{N}_{2} \mathrm{O}(g) + \mathrm{O}_{2}(g) \), the equilibrium constant expression \( K_c \) is defined as \( K_c = \frac{[\mathrm{N}_2\mathrm{O}][\mathrm{O}_2]}{[\mathrm{NO}_2][\mathrm{NO}]} \).

Determine the change in concentrations

The concentration of \( \mathrm{NO}_{2} \) at equilibrium is \( 0.118 \, \mathrm{M} \), up from \( 0.0560 \, \mathrm{M} \). This indicates that the concentration of \( \mathrm{NO}_{2} \) increased by \( 0.118 \, \mathrm{M} - 0.0560 \, \mathrm{M} = 0.062 \, \mathrm{M} \). Since the reaction is 1:1, the concentration of \( \mathrm{NO} \) will decrease by the same amount, resulting in a new equilibrium concentration of \( 0.294 \, \mathrm{M} - 0.062 \, \mathrm{M} = 0.232 \, \mathrm{M} \). The concentrations of \( \mathrm{N}_2\mathrm{O} \) and \( \mathrm{O}_2 \) will also increase by that amount, yielding new concentrations of \( 0.184 \, \mathrm{M} + 0.062 \, \mathrm{M} = 0.246 \, \mathrm{M} \) and \( 0.377 \, \mathrm{M} + 0.062 \, \mathrm{M} = 0.439 \, \mathrm{M} \), respectively.

Calculate the equilibrium constant \( K_c \)

Substitute the equilibrium concentrations into the equilibrium expression to find \( K_c \): \( K_c = \frac{[\mathrm{N}_2\mathrm{O}][\mathrm{O}_2]}{[\mathrm{NO}_2][\mathrm{NO}]} = \frac{(0.246 \, \mathrm{M}) (0.439 \, \mathrm{M})}{(0.118 \, \mathrm{M}) (0.232 \, \mathrm{M})} \). Calculating this gives: \( K_c = \frac{0.108014 \, \mathrm{M}^2}{0.027376 \, \mathrm{M}^2} \), which equals approximately 3.946.

Key concepts

Equilibrium Constant Expression

Understanding the equilibrium constant expression is crucial when analyzing chemical reactions at equilibrium. It quantifies the ratio of the concentrations of products to reactants, each raised to the power of their respective stoichiometric coefficients. In the reaction
( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( � (((( ( ( (Abstract Concepts:) ( � ( ( What looks like a simple ratio is in fact the pivot of understanding how reactions proceed depending on different conditions. For instance, a large value for the equilibrium constant indicates a reaction that favors product formation, as in the given example where the value of approximately 3.946 suggests the products are favored over the reactants at equilibrium. Conversely, a small value implies the reactants are more favored.

It's important to note that the equilibrium constant is temperature-dependent—if the temperature changes, so will the value of the equilibrium constant. In addition, the concentrations used in the expression are equilibrium concentrations—it says nothing about the rate of the reaction or how quickly equilibrium is achieved.

Reaction Quotient

The reaction quotient, denoted as Q, is a measure similar to the equilibrium constant, but for a system that has not necessarily reached equilibrium. Q is calculated using the same expression as the equilibrium constant, but with the current concentrations of the reactants and products regardless of whether the system is at equilibrium or not.

The reaction quotient is pivotal in predicting the direction of the reaction. If Q is less than the equilibrium constant, K, the reaction will proceed in the forward direction to form more products. This is because there are too many reactants present, and the reaction shifts to decrease them. If Q exceeds K, the reaction will proceed in the reverse direction to produce more reactants, as there are too many products. When Q equals K, the system is at equilibrium, and no net change in the concentration of reactants and products occurs.

Le Chatelier's Principle

Le Chatelier's principle is a foundational concept in chemistry that predicts how a change in conditions can affect a chemical equilibrium. Essentially, this principle states that if an external change is imposed on a system at equilibrium, the system adjusts to counteract the change and restore a new equilibrium.

For instance, changes in concentration, pressure, volume, or temperature can all shift equilibrium according to Le Chatelier's principle. If a reactant is added to the system, the equilibrium will shift to consume the added reactant and produce more products, and vice versa. A change in temperature, on the other hand, can result in a shift that either favors the endothermic or the exothermic direction of the reaction, depending on whether heat is added or removed. Pressure changes affect the equilibrium only in systems with gaseous components; increasing pressure by decreasing volume typically shifts the equilibrium towards the side with fewer gas moles.

Equilibrium Concentration Calculation

Calculating equilibrium concentrations requires a systematic approach, typically using an ICE (Initial, Change, Equilibrium) table to keep track of the concentrations of all species involved in the reaction. In the exercise provided, the increase in NO2 concentration at equilibrium allowed us to calculate the changes in concentration for all other species involved and ultimately determine the equilibrium constant, Kc.

The solution demonstrated how to adjust the initial concentrations by the amount they increased or decreased to find the final equilibrium concentrations. This type of stoichiometric calculation is essential in real-world applications, such as when chemists need to predict the concentrations of substances in a chemical process or in environmental reactions where changes in conditions might affect the direction and extent of the reactions.