Question

Determine the pH of a \(0.50-M\) solution of \(\mathrm{NH}_{4} \mathrm{OCl.}\) . See Exercise \(181 .\) )

Short answer

The pH of the 0.50 M solution of NH4OCl can be calculated by first finding the K_a and K_b values for the weak acid NH4+ and the weak base OCl-. Based on their respective parent acids and bases (NH4+ from NH3 with K_a = 5.6 × 10^-10 and OCl- from HOCl with K_b = 2.9 × 10^-7), we can solve for K_a and K_b. After solving the quadratic equations in the expressions for x (H3O+ concentration) and y (OH- concentration), we can find the pH by computing -log10(x), the negative base-10 logarithm of the H3O+ concentration.

Step-by-step solution

Write the dissociation reaction in water

NH4OCl is a salt that is formed from the weak acid NH4+ and the weak base OCl-. When it dissolves in water, it will dissociate into those two ions. Thus, the dissociation reaction can be written as follows: NH4OCl (aq) → NH4+ (aq) + OCl- (aq)

Write the equilibrium expressions for the weak acid and weak base

Since NH4+ is a weak acid and OCl- is a weak base, we need to write K_a and K_b equilibrium expressions. For NH4+ (from NH4+ + H2O (l) ⇌ NH3 (aq) + H3O+(aq)): \[K_a = \frac{[\mathrm{NH}_3][\mathrm{H}_3\mathrm{O}^+]}{[\mathrm{NH}_4^+]}\] And for OCl- (from OCl- (aq) + H2O (l) ⇌ HOCl (aq) + OH-(aq)): \[K_b = \frac{[\mathrm{HOCl}][\mathrm{OH}^-]}{[\mathrm{OCl}^-]}\] We are given information about the concentration of NH4OCl, which is 0.50 M, and hence both NH4+ and OCl- will have an initial concentration of 0.50 M.

Set up an equilibrium table and solve the expressions

To determine the concentration of H3O+ and OH- at equilibrium, we can set up an equilibrium table for the weak acid and weak base reactions. Let x be the moles of NH4+ that dissociate into NH3, then the equilibrium concentrations are: NH4+: (0.50 - x) M NH3: x M H3O+: x M In a similar way, let y be the moles of OCl- that dissociate into HOCl, then the equilibrium concentrations are: OCl-: (0.50 - y) M HOCl: y M OH-: y M Now we can substitute these concentrations into the K_a and K_b expressions: \[K_a = \frac{x^2}{0.50-x}\] \[K_b = \frac{y^2}{0.50-y}\] The K_w value, which is equal to K_a * K_b, can be used to simplify the equation. K_w is equal to 1.0 × 10^-14 at 25°C. K_w = K_a * K_b = (x^2 / (0.50 - x)) * (y^2 / (0.50 - y))

Solve for x and y, and compute the pH

First, we need to find the Ka and Kb values for the weak acid NH4+ and the weak base OCl- respectively from their respective parent acids and bases (NH4+ from NH3 and OCl- from HOCl with Ka given as 5.6 × 10^-10 and Kb given as 2.9 × 10^-7, respectively). We can solve for Ka and Kb from provided values, K_a = Kw / Kb and K_b = Kw / K_a. Then, we can use the values of Ka and Kb to solve for x and y, by solving the quadratic equations in the expressions. After we find x and y, we can find the pH by computing -log10(x), the negative base-10 logarithm of the H3O+ concentration. The PKa value of NH4+ and the pKb value of OCl- can also be calculated. If needed, you can also find that when x and y are very small, the concentration of NH4+ and OCl- remains 0.50 M approximately. Once you have these values, you can calculate the pH of the 0.50 M solution of NH4OCl by taking the -log10 of the concentration of H3O+.

Key concepts

Weak Acid

A weak acid is a type of acid that only partially ionizes in a solution, which means that not all of its molecules dissociate into ions. Because of this partial ionization, weak acids have higher pH levels compared to strong acids.
One typical example of a weak acid is \(NH_4^+\), the ammonium ion. When dissolved in water, weak acids like ammonium do not release all their hydrogen ions (\(H^+\)) into the solution, making it less acidic than if all ions were released.
This behavior can be expressed with the acid dissociation constant, \(K_a\), which indicates the strength of an acid in solution.

• Lower \(K_a\) means weaker acid.
• Only partially dissociates in water.
• Associated with specific equilibrium expressions.

Weak Base

Weak bases are compounds that partially accept hydrogen ions (protons) when dissolved in water. This results in an equilibrium between the base and its conjugate acid form.
One example of a weak base is the hypochlorite ion (\(OCl^-\)). When it reacts with water, only a fraction of its molecules will accept a proton to form hypochlorous acid (\(HOCl\)).
Like weak acids, weak bases are characterized by a dissociation constant, \(K_b\), which measures how well a base picks up hydrogen ions in a solution.

• Lower \(K_b\) indicates a weaker base.
• Partial ionization affects equilibrium and pH.
• Equilibrium favors undissociated form.

Equilibrium Expressions

Equilibrium expressions help us understand how acids and bases behave in solutions. They provide a way of writing down the concentrations of reactants and products at equilibrium.
For weak acids such as \(NH_4^+\), the equilibrium expression is based on the reaction with water, producing \(NH_3\) and \(H_3O^+\) ions, written as:
\K_a = \frac{[NH_3][H_3O^+]}{[NH_4^+]}\
For weak bases like \(OCl^-\), the equilibrium expression involves forming \(HOCl\) and \(OH^-\), given by:
\K_b = \frac{[HOCl][OH^-]}{[OCl^-]}\
These expressions show the relationship between concentrations and the equilibrium constants (\(K_a\) or \(K_b\)).

• Used to find pH and concentration changes.
• Expressions depend on initial concentrations and changes at equilibrium.
• Essential for understanding acid-base reactions.

Ammonium

Ammonium (\(NH_4^+\)) is a positively charged ion derived from ammonia (\(NH_3\)). It acts as a weak acid in aqueous solutions.
When ammonium is present in water, it donates protons to water molecules, forming hydronium ions (\(H_3O^+\)) and ammonia. This process is described by the following equilibrium reaction:
\NH_4^+ + H_2O \leftrightarrow NH_3 + H_3O^+\
The ability of ammonium to donate protons and form hydronium ions is why it is classified as a weak acid.

• Contributes to the acidity of a solution.
• Partnered with weak bases in salt formations.
• Important for calculating solution pH through \(K_a\) equations.

Hypochlorite

Hypochlorite (\(OCl^-\)) is an ion known for its role in bleach and disinfectants but also as a weak base in chemistry.
In solution, it can accept protons to form hypochlorous acid (\(HOCl\)), through the following equilibrium reaction:
\OCl^- + H_2O \leftrightarrow HOCl + OH^-\
Because hypochlorite is a weak base, it only partially reacts with water, and the equilibrium will favor the reactants. This means less hydroxide ion (\(OH^-\)) concentration in the solution, affecting pH calculations.

• Characterized by \(K_b\) value in reactions.
• Favors formation of hypochlorous acid rather than conversion to \(OH^-\).
• Integral part of salt solutions with weak acids.