Question

For the reaction \(\mathrm{N}_{2} \mathrm{O}_{4}(g) \rightleftarrows 2 \mathrm{NO}_{2}(g),\) the equilibrium concentrations at \(298 \mathrm{~K}\) are \(\left[\mathrm{NO}_{2}\right]=0.0325 \mathrm{~mol} / \mathrm{L}\) and \(\left[\mathrm{N}_{2} \mathrm{O}_{4}\right]=0.147 \mathrm{~mol} / \mathrm{L}\) (a) What is the value of \(K\) at \(298 \mathrm{~K} ?\) Are reactants or products favored?

Short answer

The equilibrium constant \(K_c\) is 0.0072, indicating reactants are favored.

Step-by-step solution

Write the Expression for the Equilibrium Constant

The equilibrium constant \(K_c\) for the reaction \(\mathrm{N}_{2} \mathrm{O}_{4}(g) \rightleftharpoons 2 \mathrm{NO}_{2}(g)\) is given by the equation:\[K_c = \frac{[\mathrm{NO}_{2}]^2}{[\mathrm{N}_{2} \mathrm{O}_{4}]}\]Substitute the given concentrations into the expression.

Substitute Concentrations into K Expression

Substitute \(\left[\mathrm{NO}_{2}\right] = 0.0325\, \mathrm{mol/L}\) and \(\left[\mathrm{N}_{2} \mathrm{O}_{4}\right] = 0.147\, \mathrm{mol/L}\) into the expression:\[K_c = \frac{(0.0325)^2}{0.147}\]Calculate \(K_c\) using these values.

Calculate the Equilibrium Constant Value

Calculate \(K_c\):\[K_c = \frac{(0.0325)^2}{0.147} = \frac{0.00105625}{0.147} \approx 0.007186\]The value of \(K_c\) is approximately 0.0072.

Determine Reaction Favorability

Since \(K_c = 0.0072\) is much less than 1, the reaction favors the reactants. This means there are more reactants than products at equilibrium.

Key concepts

Equilibrium Constant

The equilibrium constant, denoted as \(K_c\), is a crucial factor in understanding chemical reactions at equilibrium.
It represents the ratio of the product concentrations to the reactant concentrations at equilibrium, raised to the power of their stoichiometric coefficients.
For the reaction \(\mathrm{N}_{2} \mathrm{O}_{4}(g) \rightleftarrows 2\mathrm{NO}_{2}(g)\), the equilibrium constant expression is given by:

• \(K_c = \frac{[\mathrm{NO}_{2}]^2}{[\mathrm{N}_{2} \mathrm{O}_{4}]}\)

Here, \([\mathrm{NO}_{2}]\) and \([\mathrm{N}_{2} \mathrm{O}_{4}]\) are the equilibrium concentrations of \(\mathrm{NO}_{2}\) and \(\mathrm{N}_{2} \mathrm{O}_{4}\), respectively.
This formula allows us to calculate \(K_c\), which provides insight into the extent of a reaction and how it shifts with changes in conditions.

Reaction Favorability

The concept of reaction favorability relates to whether a chemical reaction tends to favor the formation of products or the retention of reactants at equilibrium.
This is determined by comparing the value of the equilibrium constant \(K_c\) to 1.

When:

• \(K_c > 1\), the reaction is product-favored, meaning more products are present than reactants at equilibrium.
• \(K_c < 1\), the reaction is reactant-favored, indicating that reactants are more prevalent than products at equilibrium.

In our specific reaction, \(K_c = 0.0072\), which is significantly less than 1.
This means that the reaction favors the reactants, resulting in a higher concentration of \(\mathrm{N}_{2} \mathrm{O}_{4}\) compared to \(\mathrm{NO}_{2}\) at equilibrium.

Equilibrium Concentrations

Equilibrium concentrations refer to the concentrations of reactants and products when a reaction has reached a state of balance, meaning the rates of the forward and reverse reactions are equal.
In the given reaction, the equilibrium concentrations are:

• \([\mathrm{NO}_{2}] = 0.0325\, \mathrm{mol/L}\)
• \([\mathrm{N}_{2} \mathrm{O}_{4}] = 0.147\, \mathrm{mol/L}\)

These values are crucial for calculating the equilibrium constant \(K_c\) and understanding the direction of the reaction's favorability.
At equilibrium, these concentrations remain constant unless the system is disturbed by changes in concentration, pressure, or temperature.

Equilibrium Expressions

Equilibrium expressions are equations that define the relation between the concentrations of the reactants and products at equilibrium.
They are developed from the balanced chemical equation, using the concentrations raised to the power of their coefficients.
For the decomposition of \(\mathrm{N}_{2} \mathrm{O}_{4}\) to \(\mathrm{NO}_{2}\), the equilibrium expression is:

• \(K_c = \frac{[\mathrm{NO}_{2}]^2}{[\mathrm{N}_{2} \mathrm{O}_{4}]}\)

This expression not only helps in determining \(K_c\), but it also allows the prediction of how changes in conditions will affect the position of equilibrium.
Understanding this expression is essential for manipulating and predicting the behavior of chemical systems under various conditions, such as changes in concentration, temperature, or pressure.