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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) \(\mathrm{K}_{\mathrm{1}}+\mathrm{K}_{2}+\mathrm{K}_{3}\)

Step by step solution

01

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. \(\text{HPO}_4^{2-} \rightleftharpoons \text{H}^+ + \text{PO}_4^{3-} ; K_3\)

02

Write the overall dissociation reaction

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

03

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 \(K_1 \times K_2 \times K_3\).

04

Select the correct answer

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 (H鈧働O鈧�) 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 H鈧働O鈧�:
\[\mathrm{H}_3\mathrm{PO}_4 \rightleftharpoons 3\mathrm{H}^+ + \mathrm{PO}_4^{3-}\text{.}\] 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 (H鈧働O鈧�) is a weak acid commonly used in fertilizers, food flavorings, and cleaning products.
It can donate three protons (H鈦� 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.