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Chapter 20: Problem 86

Calculate the \(\mathrm{pH}\) of a \(0.200 \mathrm{M} \mathrm{GaCl}_{3}(a q)\) solution at \(25^{\circ} \mathrm{C}\).

Short Answer

Expert verified
The pH of 0.200 M GaCl3 is slightly less than 7, due to weakly acidic Ga3+ ions.

Step by step solution

01

Understanding GaCl3 Dissociation

Gallium chloride, \( \text{GaCl}_3 \), dissociates in water to form gallium ions \( \text{Ga}^{3+} \) and chloride ions \( \text{Cl}^- \). The reaction is \( \text{GaCl}_3 (aq) \rightarrow \text{Ga}^{3+} (aq) + 3\text{Cl}^- (aq) \). Gallium ions act as a weak acid by hydrolysis in water.
02

Writing the Hydrolysis Equation

\( \text{Ga}^{3+} \) ions undergo hydrolysis in water following the equation: \( \text{Ga}^{3+} (aq) + 3\text{H}_2\text{O} (l) \leftrightarrow \text{Ga(OH)}_3 (s) + 3\text{H}^+ (aq) \). This reaction increases the \( \text{H}^+ \) concentration, influencing the \( \text{pH} \).
03

Determine the Concentration of \( \text{H}^+ \)

The initial concentration of \( \text{Ga}^{3+} \) is 0.200 M. If \( x \) is the concentration of \( \text{H}^+ \) produced, then the hydrolysis equilibrium can be approximated if \( K_{\text{hyd}} \) is known, but it's typically a very small value due to the weak acidic nature of \( \text{Ga}^{3+} \). For simplicity, assume \( x \approx \text{small} \) leading to \( x \approx 3\sqrt[3]{K_{\text{hyd}} \cdot 0.200} \), given the weakly acidic nature.
04

Calculating \( \text{pH} \)

Given \( x \) as the \( \text{H}^+ \) ion concentration is a small quantity, \( \text{pH} \approx -\log([\text{H}^+]) \). Typically for weak hydrolyzing species like \( \text{Ga}^{3+} \), the contribution is minimal, possibly affecting \( \text{pH} \) negligibly.

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Key Concepts

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

Acid-Base Equilibrium
Acid-base equilibrium is a principle that describes the balance between acids and bases in a solution. Whenever a compound like \( ext{GaCl}_3\) is dissolved in water, it dissociates into ions. In this case, it produces \( ext{Ga}^{3+}\) ions. These ions participate in the acid-base equilibrium by reacting with water. This reaction often aims to reach a state of equilibrium where the rate of the forward reaction (formation of ions) equals the rate of the reverse reaction (recombination of ions).
  • The equilibrium, once achieved, maintains the pH of the solution relatively stable.
  • Understanding this principle helps us determine how substances like \( ext{GaCl}_3\) can alter the pH in aqueous solutions.
Knowing this core principle is key to making sense of why the pH might change when \( ext{Ga}^{3+}\) ions are present as weak acids in a solution.
Hydrolysis
Hydrolysis is the chemical breakdown of a compound due to reaction with water. For gallium ions \( ext{Ga}^{3+}\), hydrolysis takes place when these ions interact with water molecules to form \( ext{Ga(OH)}_3\) and release \( ext{H}^+\) ions. This process raises the concentration of \( ext{H}^+\) ions in the solution, thus affecting the pH.Consider the equation:\[ \text{Ga}^{3+}(aq) + 3\text{H}_2\text{O}(l) \leftrightarrow \text{Ga(OH)}_3(s) + 3\text{H}^+(aq) \]
  • As the reaction proceeds, the formation of \( ext{H}^+\) ions pushes the equilibrium towards a more acidic environment.
  • This tendency to yield more \( ext{H}^+\) ions illustrates how hydrolysis directly influences the pH of the solution.
Understanding hydrolysis helps clarify why certain ions in a solution might cause an increase in acidity despite initially seeming neutral or weak as acids.
Weak Acids
Weak acids are substances that do not completely dissociate in water. In the context of \( ext{Ga}^{3+}\) ions, they act as weak acids because the hydrolysis reaction only partially proceeds to release \( ext{H}^+\) ions.Because the dissociation is incomplete:
  • A calculation for \( ext{H}^+\) concentration often involves assumptions regarding the equilibrium constant \(K_{\text{hyd}}\). This is usually a small value.
  • The result is that the \( ext{H}^+\) ions influence the pH minimally compared to stronger acids.
Navigating weak acids involves understanding their limited ability to lower pH significantly, as they only weakly increase the concentration of \( ext{H}^+\) in solution. Thus, even a compound like \( ext{Ga}^{3+}\), when appearing in solution as a weak acid, results in minor shifts in pH.
Gallium Ions
Gallium ions \( ext{Ga}^{3+}\) are the central players in the discussed reactions. When \( ext{GaCl}_3\) dissolves in water, it releases these ions, which undergo hydrolysis, behaving as weak acids. This reaction contributes to slight shifts in pH.Key aspects of gallium ions:
  • They are trivalent, meaning they have a charge of \( +3 \) when dissolved, which significantly affects their behavior in solution.
  • Their hydrolytic nature highlights their importance in understanding chemical equilibria when calculating the pH of solutions containing \( ext{GaCl}_3\).
By appreciating how gallium ions interact with water, we gain insight into their role in the acidity of a solution, emphasizing their hydrolysis capability despite their weak acidic strength.

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Most popular questions from this chapter

The \(\mathrm{pH}\) of the contents of the human stomach can be as low as \(1.0 .\) Calculate the value of \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) in the stomach when the \(\mathrm{pH}=1.0\).

The acid-dissociation constant at \(25^{\circ} \mathrm{C}\) for the equilibrium described by $$ \begin{aligned} \left[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{5+}(a q)+& \mathrm{H}_{2} \mathrm{O}(l) \leftrightharpoons \\ & \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\left[\mathrm{Fe}(\mathrm{OH})\left(\mathrm{H}_{2} \mathrm{O}\right)_{\xi}\right]^{2+}(a q) \end{aligned} $$ is \(K_{\mathrm{a}}=6.7 \times 10^{-3} \mathrm{M} .\) Calculate the \(\mathrm{pH}\) of a \(0.20-\mathrm{M}\) solution of \(\mathrm{Fe}\left(\mathrm{NO}_{3}\right)_{3}(a q)\) at \(25^{\circ} \mathrm{C}\).

Identify which of the following species are Br酶nsted-Lowry acids and which are Bronsted-Lowry bases in water. In each case give the chemical formula for the conjugate member of the conjugate acid-base pair. (a) \(\mathrm{ClCH}_{2} \mathrm{COOH}(a q)\) (b) \(\mathrm{NH}_{3}(a q)\) (c) \(\mathrm{ClO}^{-}(a q)\) (d) \(\mathrm{HCOO}^{-}(a q)\) (e) \(\mathrm{HN}_{\mathrm{s}}(a q)\) (f) \(\mathrm{NO}_{2}^{-}(a q)\)

Determine whether each of the following substances is an Arrhenius acid, a Br贸nsted-Lowry acid, or a Lewis acid (it is possible for each to be of more than one type): (a) \(\mathrm{HCl}(a q)\) (b) \(\mathrm{AlCl}_{3}(a q)\) (c) \(\mathrm{BCl}_{3}(a q)\)

Calculate the \(\mathrm{pH}\) and the concentrations of the other species in a \(0.10-\mathrm{M}\) aqueous solution of chloroacetic acid, \(\mathrm{ClCH}_{2} \mathrm{COOH}(a q)\), given that \(K_{\mathrm{a}}=\) \(1.4 \times 10^{-5} \mathrm{M}\) at \(25^{\circ} \mathrm{C}\)
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