Problem 44 - Chemistry Matter and Change

Question

Write equilibrium constant expressions for these homogeneous equilibria. $$ \begin{array}{l}{\text { a. } 2 \mathrm{N}_{2} \mathrm{H}_{4}(\mathrm{g})+2 \mathrm{NO}_{2}(\mathrm{g}) \rightleftharpoons 3 \mathrm{N}_{2}(\mathrm{g})+4 \mathrm{H}_{2} \mathrm{O}(\mathrm{g})} \\ {\text { b. } 2 \mathrm{NbCl}_{4}(\mathrm{g}) \rightleftharpoons \mathrm{NbCl}_{3}(\mathrm{g})+\mathrm{NbCl}_{5}(\mathrm{g})}\end{array} $$

Step-by-Step Solution

Answer

The equilibrium constant expressions for the given homogeneous equilibria are: \( K_{c1} = \frac{[\mathrm{N}_{2}]^3 \cdot [\mathrm{H}_{2}\mathrm{O}]^4}{[\mathrm{N}_{2}\mathrm{H}_{4}]^2 \cdot [\mathrm{NO}_{2}]^2} \) \( K_{c2} = \frac{[\mathrm{NbCl}_{3}] \cdot [\mathrm{NbCl}_{5}]}{[\mathrm{NbCl}_{4}]^2} \)

Step 1: a. Equilibrium constant expression for the first reaction:

For the reaction: $$2 \mathrm{N}_{2} \mathrm{H}_{4}(\mathrm{g})+2 \mathrm{NO}_{2}(\mathrm{g}) \rightleftharpoons 3 \mathrm{N}_{2}(\mathrm{g})+4 \mathrm{H}_{2} \mathrm{O}(\mathrm{g})$$ We can write the equilibrium constant expression as: $$K_{c1} = \frac{[\mathrm{N}_{2}]^3 \cdot [\mathrm{H}_{2}\mathrm{O}]^4}{[\mathrm{N}_{2}\mathrm{H}_{4}]^2 \cdot [\mathrm{NO}_{2}]^2}$$

Step 2: b. Equilibrium constant expression for the second reaction:

For the reaction: $$2 \mathrm{NbCl}_{4}(\mathrm{g}) \rightleftharpoons \mathrm{NbCl}_{3}(\mathrm{g})+\mathrm{NbCl}_{5}(\mathrm{g})$$ We can write the equilibrium constant expression as: $$K_{c2} = \frac{[\mathrm{NbCl}_{3}] \cdot [\mathrm{NbCl}_{5}]}{[\mathrm{NbCl}_{4}]^2}$$ Therefore, the equilibrium constant expressions for the given homogeneous equilibria are: $$ \begin{array}{l}{K_{c1} = \frac{[\mathrm{N}_{2}]^3 \cdot [\mathrm{H}_{2}\mathrm{O}]^4}{[\mathrm{N}_{2}\mathrm{H}_{4}]^2 \cdot [\mathrm{NO}_{2}]^2}}\\ {K_{c2} = \frac{[\mathrm{NbCl}_{3}] \cdot [\mathrm{NbCl}_{5}]}{[\mathrm{NbCl}_{4}]^2}}\end{array} $$

Key Concepts

Homogeneous Equilibria

The concept of homogeneous equilibria refers to chemical reactions where all reactants and products are in the same physical state. Frequently, this state is gaseous, as it is in the given exercise examples. Understanding homogeneous equilibria is crucial because it simplifies the way we approach calculating equilibrium concentrations. When writing equilibrium constant expressions for such reactions, it's important to remember that the concentrations (denoted by square brackets) of all gaseous reactants and products are raised to the power of their respective coefficients from the balanced chemical equation.

For instance, in the reaction \(2 \text{N}_2\text{H}_4(g) + 2 \text{NO}_2(g) \rightleftharpoons 3 \text{N}_2(g) + 4 \text{H}_2\text{O}(g)\), the equilibrium constant is a reflection of the reaction's kinetics and is expressed as the ratio of the products' concentrations to the reactants' concentrations, each raised to the power of their stoichiometric coefficients. This relationship helps chemists understand how changes in conditions such as pressure, temperature, or concentration will influence the system's behavior.

Chemical Equilibrium

The state of chemical equilibrium is reached when a reaction and its reverse occur at the same rate, leading to no net change in the amounts of substances involved. However, it's important to note that this does not mean that the reactants and products are present in equal amounts; rather, the ratios of their concentrations become constant over time. The equilibrium constant (\(K_c\)) provides a quantitative measure of the position of equilibrium. High values of \(K_c\) usually indicate that, at equilibrium, products are favored and concentrations of the products are higher compared to reactants. Conversely, a low \(K_c\) suggests that reactants are favored.

Focusing on the details, such as the fact that only gaseous and aqueous species appear in the \(K_c\) equation while pure solids and liquids do not, is essential for students to solve these problems correctly. Moreover, equilibrium constants are temperature dependent; changing the temperature will affect the value of \(K_c\), reflecting the exothermic or endothermic nature of the process.

Reaction Quotient

Alongside equilibrium constants, the reaction quotient (\(Q\)) plays a pivotal role in determining the direction a reaction will proceed to reach equilibrium. Defined similarly to the equilibrium constant, \(Q\) represents the ratio of the concentrations of products to reactants at any point during the reaction, not just at equilibrium. By comparing \(Q\) to \(K_c\), one can predict whether a reaction will proceed forward or reverse to reach equilibrium.

If \(Q < K_c\), this indicates that the concentration of products is too low, and the reaction will shift to the right or proceed forward to produce more products. Conversely, if \(Q > K_c\), the system contains too many products, and the reaction will shift to the left or reverse to form more reactants. When \(Q = K_c\), the system is at equilibrium. This is an invaluable tool for chemists to predict and control the outcomes of reactions, making the reaction quotient a key concept in the study of dynamic chemical processes.