Problem 29 - Chemistry The Molecular Nature of Matter

Question

Write the equilibrium law for the following reactions in aqueous solution. (a) \(\mathrm{Ag}^{+}(a q)+2 \mathrm{NH}_{3}(a q) \rightleftharpoons \mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}(a q)\) (b) \(\mathrm{Cd}^{2+}(a q)+4 \mathrm{SCN}^{-}(a q) \rightleftharpoons \mathrm{Cd}(\mathrm{SCN})_{4}^{2-}(a q)\)

Step-by-Step Solution

Answer

\( K_{eq} \) for (a) is \( \frac{[\mathrm{Ag}(\mathrm{NH}_3)_2^{+}]}{[\mathrm{Ag}^{+}][\mathrm{NH}_3]^2} \), and for (b) is \( \frac{[\mathrm{Cd}(\mathrm{SCN})_4^{2-}]}{[\mathrm{Cd}^{2+}][\mathrm{SCN}^{-}]^4} \).

Step 1: Writing the Equilibrium Law for Reaction (a)

The equilibrium law, also known as the law of mass action, states that at a given temperature, the ratio of product concentrations to reactant concentrations, each raised to the power of their coefficients in the balanced reaction, is constant. For reaction (a), the equilibrium expression is: \[ K_{eq} = \frac{[\mathrm{Ag}(\mathrm{NH}_3)_2^{+}]}{[\mathrm{Ag}^{+}][\mathrm{NH}_3]^2} \] where \( K_{eq} \) is the equilibrium constant.

Step 2: Writing the Equilibrium Law for Reaction (b)

Using the same principle for reaction (b), we write the equilibrium expression as: \[ K_{eq} = \frac{[\mathrm{Cd}(\mathrm{SCN})_4^{2-}]}{[\mathrm{Cd}^{2+}][\mathrm{SCN}^{-}]^4} \] Here again, \( K_{eq} \) represents the equilibrium constant for reaction (b).

Key Concepts

Law of Mass Action

The Law of Mass Action is a fundamental principle in chemistry that describes the relationship between the rates at which chemicals react and the concentrations of the reacting species. It forms the basis for understanding how chemical reactions reach a state of balance, known as equilibrium.

According to this law, for a reversible reaction at a constant temperature, the rate at which a reaction proceeds is directly proportional to the product of the active masses (concentrations) of the reactants, each raised to a power equal to their stoichiometric coefficients in the balanced equation. In simpler terms, the higher the concentration of the reactants, the faster they can react to form products, and vice versa.

For instance, consider the following generic reversible reaction in aqueous solution:

General Reaction Example

For reactants A and B, with stoichiometric coefficients a and b, reacting to form products C and D with coefficients c and d, respectively:
\[ aA + bB \rightleftharpoons cC + dD \]
The Law of Mass Action would express the reaction rate as:\[ Rate = k[A]^{a}[B]^{b} \]Where:

• \( k \) is the rate constant,
• \( [A] \) and \( [B] \) represent the molar concentrations of reactants A and B,
• Exponents a and b indicate the influence of reactant concentration on the rate.

Understanding this concept allows us to predict how changes in concentrations can affect the reaction's progress.

Equilibrium Constant

The equilibrium constant, symbolized as \( K_{eq} \), is the ratio of the concentration of the products to the concentration of the reactants at equilibrium, with each concentration raised to the power of its corresponding stoichiometric coefficient in the balanced chemical equation.

An important point to note is that the equilibrium constant is only valid at a particular temperature; changes in temperature can alter the value of \( K_{eq} \). The numerical value of the equilibrium constant gives us insight into the position of the equilibrium. A larger \( K_{eq} \) indicates a greater concentration of products at equilibrium, suggesting the reaction strongly favors the formation of products. Conversely, a smaller \( K_{eq} \) suggests the reaction favors the reactants.

Interpreting Equilibrium Constant Values

• \( K_{eq} > 1 \): The reaction favors the formation of products.
• \( K_{eq} < 1 \): The reaction favors the formation of reactants.
• \( K_{eq} = 1 \): The reactants and products are present in roughly equal amounts at equilibrium.

The equilibrium constant is central to the quantitative aspects of chemical equilibria and allows chemists to predict the extent to which a reaction will occur under certain conditions.

Chemical Equilibrium

Chemical equilibrium refers to the state of a reversible chemical reaction where the rate of the forward reaction equals the rate of the reverse reaction. Under these conditions, the concentrations of reactants and products remain constant over time, though they are not necessarily equal.

It's important to distinguish that equilibrium does not mean the reactants and products are in equal concentrations, but that their concentrations have stabilized in a fixed ratio as defined by the equilibrium constant \( K_{eq} \). At equilibrium, a dynamic process is occurring where reactants are continually being converted to products and products back to reactants at the same rate.

Reaching chemical equilibrium involves changes in reactant and product concentrations until they settle into this balance. Factors such as temperature, pressure (for reactions involving gases), and the presence of catalysts can affect the position of the equilibrium. Understanding how to manipulate these factors can be crucial in industrial processes where maximizing yield is essential.

Dynamic Nature of Chemical Equilibrium

While the system appears static at the macroscopic level, at the microscopic level there is continuous and dynamic interchange between reactants and products. This highlights the beauty and complexity of chemical equilibrium as a fundamental concept in chemical reactions.