Question

The equilibrium constant for the reactions are \(\mathrm{H}_{3} \mathrm{PO}_{4} \rightleftharpoons \mathrm{H}+\mathrm{H}_{2} \mathrm{PO}_{4}^{-} ; \mathrm{K}_{1}\) \(\mathrm{II}_{2} \mathrm{PO}_{4} \rightleftharpoons \mathrm{II}^{+}+\mathrm{HPO}_{4}^{2} ; \mathrm{K}_{2}\) \(\mathrm{IIPO}_{4}^{2} \rightleftharpoons \mathrm{II}^{+}+\mathrm{PO}_{4}^{3} ; \mathrm{K}_{3}\) The equilibrium constant for \(\mathrm{H}_{3} \mathrm{PO}_{4} \rightleftharpoons 3 \mathrm{H}^{\prime}+\mathrm{PO}_{4}^{3-}\) is (1) \(\mathrm{K}_{1} / \mathrm{K}_{2} \cdot \mathrm{K}_{3}\) (2) \(\mathrm{K}_{\mathrm{1}} \times \mathrm{K}_{2} \times \mathrm{K}_{3}\) (3) \(\mathrm{K}_{2} / \mathrm{K}_{1} \cdot \mathrm{K}_{3}\) (4) \(\mathrm{K}_{\mathrm{1}}+\mathrm{K}_{2}+\mathrm{K}_{3}\)

Short answer

(2) \(K_1 \times K_2 \times K_3\)

Step-by-step solution

Understand the given reactions

Identify the given reactions and their respective equilibrium constants. These are dissociation reactions of phosphoric acid:1. \(\text{H}_3\text{PO}_4 \rightleftharpoons \text{H}^+ + \text{H}_2\text{PO}_4^- ; K_1\)2. \(\text{H}_2\text{PO}_4^- \rightleftharpoons \text{H}^+ + \text{HPO}_4^{2-} ; K_2\)3. \(\text{HPO}_4^{2-} \rightleftharpoons \text{H}^+ + \text{PO}_4^{3-} ; K_3\)

Write the overall dissociation reaction

Combine the three steps to get the final dissociation of \(\text{H}_3\text{PO}_4\): \(\text{H}_3\text{PO}_4 \rightleftharpoons 3\text{H}^+ + \text{PO}_4^{3-}\).

Determine how equilibrium constants combine

Realize that when adding equilibrium reactions, the overall equilibrium constant is the product of the equilibrium constants for each step. Thus, for \(\text{H}_3\text{PO}_4 \rightleftharpoons 3\text{H}^+ + \text{PO}_4^{3-}\), it is \(K_1 \times K_2 \times K_3\).

Select the correct answer

Given the final combined equilibrium constant, the correct choice is (2) \(K_1 \times K_2 \times K_3\).

Key concepts

Equilibrium Constant

In chemistry, the equilibrium constant (K) helps us understand the balance point between reactants and products in a chemical reaction.
It is crucial for predicting the direction of the reaction and the concentrations of the involved species at equilibrium.
For a general reaction at equilibrium, \(\text{aA + bB} \rightleftharpoons \text{cC + dD}\), the expression for the equilibrium constant \(K_eq\) is: \[K_eq = \frac{[\text{C}]^c [\text{D}]^d}{[\text{A}]^a [\text{B}]^b}\text{.}\] Here, \([A], [B], [C], [D]\) are the concentrations of the reactants and products, and \(a, b, c, d\) are their respective coefficients in the balanced equation.
Higher values of \(K_eq\) indicate a greater amount of products at equilibrium, whereas lower values suggest more reactants.

Dissociation Reaction

A dissociation reaction involves breaking down a compound into its individual ions or simpler molecules.
For instance, when phosphoric acid (H₃PO₄) dissociates in water, it goes through several steps:

• The first dissociation: \[ \mathrm{H}_3\mathrm{PO}_4 \rightleftharpoons \mathrm{H}^+ + \mathrm{H}_2\mathrm{PO}_4^- \] with equilibrium constant \(K_1\).
• The second dissociation: \[ \mathrm{H}_2\mathrm{PO}_4^- \rightleftharpoons \mathrm{H}^+ + \mathrm{HPO}_4^{2-} \] with equilibrium constant \(K_2\).
• The third dissociation: \[ \mathrm{HPO}_4^{2-} \rightleftharpoons \mathrm{H}^+ + \mathrm{PO}_4^{3-} \] with equilibrium constant \(K_3\).

Combining these steps gives the overall dissociation of H₃PO₄:
\[\mathrm{H}_3\mathrm{PO}_4 \rightleftharpoons 3\mathrm{H}^+ + \mathrm{PO}_4^{3-}\text{.}\] The overall equilibrium constant for the reaction is the product of the intermediate equilibrium constants: \[K = K_1 \times K_2 \times K_3\text{.}\]

Phosphoric Acid

Phosphoric acid (H₃PO₄) is a weak acid commonly used in fertilizers, food flavorings, and cleaning products.
It can donate three protons (H⁺ ions) in a stepwise manner:

• The first dissociation: \[\mathrm{H}_3\mathrm{PO}_4 \rightleftharpoons \mathrm{H}^+ + \mathrm{H}_2\mathrm{PO}_4^- \]
• The second dissociation: \[\mathrm{H}_2\mathrm{PO}_4^- \rightleftharpoons \mathrm{H}^+ + \mathrm{HPO}_4^{2-} \]
• The third dissociation: \[\mathrm{HPO}_4^{2-} \rightleftharpoons \mathrm{H}^+ + \mathrm{PO}_4^{3-} \]

Each dissociation step has its own equilibrium constant: \(K_1, K_2,\) and \(K_3\).
The overall dissociation reaction can be expressed and has an equilibrium constant as \[\mathrm{H}_3\mathrm{PO}_4 \rightleftharpoons 3\mathrm{H}^+ + \mathrm{PO}_4^{3-} \quad \text{with} \quad K_{overall} = K_1 \times K_2 \times K_3\text{.}\]
Understanding these dissociation steps and their constants is essential in predicting the behavior of phosphoric acid in different chemical contexts.