Question

Given the expression \(K_{\mathrm{a}}=\frac{\left[\mathrm{AsO}_{4}^{3-}\right]\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}{[\mathrm{HCN}]},\) write the balanced equation for the corresponding reaction.

Short answer

The balanced equation for the corresponding reaction is: \(\mathrm{HCN(aq)} + \mathrm{H}_{2}\mathrm{O(l)} \rightleftharpoons \mathrm{AsO}_{4}^{3-}(\textrm{aq)} + \textrm{H}_{3}\textrm{O}^{+}(\textrm{aq)}\).

Step-by-step solution

Identify the components of the Kₐ expression

From the given expression, we can identify the following components: - AsO₄³⁻, which is the conjugate base of the acid - H₃O⁺, which is the hydronium ion - HCN, which is the acid

Write the balanced equation

Now we can write the balanced equation using the components identified in the previous step. We have the acid, HCN, reacting with water (H₂O) to produce the conjugate base, AsO₄³⁻, and the hydronium ions, H₃O⁺. The balanced equation for the corresponding reaction is as follows: $$\textrm{HCN(aq)} + \textrm{H}_2\textrm{O(l)} \rightleftharpoons \textrm{AsO}_4^{3-}(\textrm{aq)} + \textrm{H}_3\textrm{O}^{+}(\textrm{aq)}$$

Key concepts

Chemical Equilibrium

Understanding chemical equilibrium is critical when examining reactions like acid dissociation. It's the point in a reversible chemical reaction when the rate of the forward reaction equals the rate of the backward reaction. This establishes a balance between the concentrations of reactants and products, which remain constant over time.

Chemical equilibrium is an essential concept in understanding why all reactions do not proceed to completion and how environmental factors can shift the balance to favor formation of either reactants or products. For example, when evaluating the acid dissociation constant (\textbf{K\(_a\)}) in the given exercise, chemical equilibrium helps explain how HCN reaches a state where it dissociates into its conjugate base and hydronium ions at a constant ratio, not necessarily equally. Knowledge of equilibrium principles is also imperative when manipulating conditions to shift an equilibrium in industrial processes to optimize yield.

Conjugate Base

In acid-base chemistry, a conjugate base is what remains after an acid has donated a proton during a chemical reaction. This concept is an important part of understanding how acids interact with bases. The strength of an acid is related to the tendency of its conjugate base to reaccept the proton.

In the exercise, AsO\(_4\)\(^{3-}\) represents the conjugate base of the acid HCN. Recognizing conjugate acid-base pairs is essential when predicting the direction of an acid-base reaction and understanding buffers, which rely on the presence of both an acid and its conjugate base to maintain pH stability. Strong acids have weak conjugate bases, while weak acids have relatively stronger conjugate bases — this plays into how readily they participate in chemical reactions and establish equilibrium.

Hydronium Ion

The hydronium ion, represented as H\(_3\)O\(^+\), is a positively charged ion formed when an acid dissolves in water and transfers a proton (H\(^+\)) to a water molecule. This formation is crucial for measuring the acidity of a solution — the concentration of hydronium ions is directly related to a solution's pH.

The presence of hydronium ions is indicative of an acidic solution. In practical terms, as seen in the provided solution, the reaction of the acid HCN with water results in the production of hydronium ions. Understanding the hydronium ion is also foundational when learning about pH calculations, the acidity of solutions, and the workings of acid-base indicators that change color at certain pH levels due to the presence of hydronium ions.

Balanced Chemical Equation

A balanced chemical equation is a representation of a chemical reaction where the number of atoms for each element is the same on both the reactant and product sides of the equation, obeying the law of conservation of mass. Properly balancing a chemical equation is a fundamental skill required to solve many chemistry problems, including stoichiometry and thermodynamics.

In balancing the equation for the dissociation of HCN in water, we ensure that there are equal numbers of hydrogen, carbon, nitrogen, arsenic, and oxygen atoms before and after the reaction. This balance is crucial for accurately predicting the quantities of products formed and for calculating related values such as the equilibrium constant or \textbf{K\(_a\)}. In teaching contexts, the ability to balance equations helps students grasp the tangible aspect of reactions — that they are not merely abstract concepts but involve a strict accounting of all involved atoms.