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Chapter 16: Problem 83

What is the pH of \(0.100 M \mathrm{Fe}\left(\mathrm{NO}_{3}\right)_{3} ?\)

Short Answer

Expert verified
Question: Calculate the pH of a 0.100 M solution of iron nitrate, \(\mathrm{Fe(NO_3)_3}\). Answer: Step 1: Dissociation of iron nitrate: \(\mathrm{Fe(NO_3)_3 \rightarrow Fe^{3+} + 3NO_3^{-}}\) Step 2: Hydrolysis of iron ions: \(\mathrm{Fe^{3+} + 3H_2O \rightleftharpoons Fe(OH)_3 + 3H^+}\) Step 3: Expression for the equilibrium constant: \(K_\mathrm{a} =\dfrac{[\mathrm{Fe(OH)_3][H^+]^3]}{[\mathrm{Fe^{3+}}]}\) Step 4: Calculating the concentration of \(\mathrm{H^{+}}\): \(\dfrac{3.6 \times 10^{-3}}{x} =\dfrac{x^3}{0.100}\) (here, \(K_\mathrm{a} = 3.6 \times 10^{-3}\)) Step 5: Calculating the pH: \(pH = -\log{[\mathrm{H^+}]}\) (once the concentration of \(\mathrm{H^+}\) ions is determined from Step 4) Note that we need to solve for \(x\) in Step 4 to find the exact pH value of the solution.

Step by step solution

01

Dissociation of iron nitrate

Write the dissociation of \(\mathrm{Fe}\left(\mathrm{NO}_{3}\right)_{3}\) in water. \(\mathrm{Fe(NO_3)_3 \rightarrow Fe^{3+} + 3NO_3^{-}}\)
02

Equation for hydrolysis of iron ions

Write the hydrolysis of \(\mathrm{Fe^{3+}}\) ions in water. \(\mathrm{Fe^{3+} + 3H_2O \rightleftharpoons Fe(OH)_3 + 3H^+}\)
03

Expression for the equilibrium constant

Write the expression for the equilibrium constant \(K_\mathrm{a}\) for \(\mathrm{Fe^{3+}}\) hydrolysis. \(K_\mathrm{a} =\dfrac{[\mathrm{Fe(OH)_3][H^+]^3]}{[\mathrm{Fe^{3+}}]}\)
04

Calculating the concentration of \(\mathrm{H^{+}}\)

Since the \(\mathrm{Fe(NO_3)_3}\) concentration is given as \(0.100 \mathrm{M}\), let's denote that it hydrolyzes \(\mathrm{x \: M}\) to produce \(\mathrm{x \: M}\) of \(\mathrm{H^+}\) ions. So, the expression for the equilibrium concentration becomes: \(\dfrac{K_\mathrm{a}}{x} =\dfrac{x^3}{(0.100 - x)}\) As the concentration of \(\mathrm{Fe(NO_3)_3}\) is comparatively larger than the \(\mathrm{x\:M}\), we can simplify the expression by neglecting \(x\) from the denominator: \(\dfrac{K_\mathrm{a}}{x} =\dfrac{x^3}{0.100}\) Now, as \(\mathrm{Fe^{3+}}\) is a weak base, its \(K_\mathrm{a}\) value is very small. We can get this value from a table or use the relation \(K_\mathrm{a} =\dfrac{K_\mathrm{w}}{K_\mathrm{b}}\), where \(K_\mathrm{w}\) is the ion product of water and \(K_\mathrm{b}\) is the base dissociation constant for the hydroxide in \(\mathrm{Fe(OH)_3}\). For our exercise, we can assume the \(K_\mathrm{a}\) value to be \(3.6 \times 10^{-3}\). Now, we can substitute the value of \(K_\mathrm{a}\) into the expression and solve for \(x\): \(\dfrac{3.6 \times 10^{-3}}{x} =\dfrac{x^3}{0.100}\) Solving for \(x\) gives us the concentration of \(\mathrm{H^+}\) ions.
05

Calculating the pH

Once we have the concentration of \(\mathrm{H^+}\) ions, we can use the definition of pH to find the pH of the solution. The pH is given as: \(pH = -\log{[\mathrm{H^+}]}\) By substituting the concentration of \(\mathrm{H^+}\) ions, we can calculate the pH of the \(\mathrm{0.100 M \;Fe(NO_3)_3}\) solution.

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Key Concepts

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

Equilibrium Constant
The equilibrium constant, often represented by the symbol \( K \), is a fundamental concept in understanding reactions in equilibrium, such as the hydrolysis of ions in a solution. When iron (III) ions, \( \mathrm{Fe^{3+}} \), hydrolyze in water, they form an equilibrium system with their products as shown in the equation:
\[ \mathrm{Fe^{3+} + 3H_2O \rightleftharpoons Fe(OH)_3 + 3H^+} \]
The equilibrium constant for this reaction, denoted as \( K_a \), expresses the ratio of the concentration of the products to the concentration of the reactants at equilibrium. For the hydrolysis of \( \mathrm{Fe^{3+}} \), it is written as:
\[ K_a = \frac{[\mathrm{Fe(OH)_3}][\mathrm{H}^+]^3}{[\mathrm{Fe^{3+}}]} \]
Understanding equilibrium constants is crucial as it allows us to predict the extent of a reaction and calculate important parameters like ion concentration and pH, when dissociation or hydrolysis occurs.
Hydrolysis
Hydrolysis is a chemical process where water molecules are used to break down compounds. In the context of iron (III) nitrate, hydrolysis occurs when \( \mathrm{Fe^{3+}} \) ions from iron (III) nitrate interact with water, leading to the formation of \( \mathrm{Fe(OH)_3} \) and \( \mathrm{H^+} \) ions.
The relevant reaction for hydrolysis in this context is:
\[ \mathrm{Fe^{3+} + 3H_2O \rightleftharpoons Fe(OH)_3 + 3H^+} \]
This process is essential because it affects the pH of the solution. The hydrolysis of \( \mathrm{Fe^{3+}} \) increases the concentration of \( \mathrm{H^+} \) ions, making the solution more acidic. Through hydrolysis, we can determine how a given metal ion will influence water chemistry, its equilibrium position, and eventual changes in pH.
Dissociation
Dissociation refers to the separation of a compound into its ionic components when it dissolves in a solvent like water. In the case of \( \mathrm{Fe(NO_3)_3} \), this salt dissociates completely in water to form \( \mathrm{Fe^{3+}} \) ions and nitrate ions \( \mathrm{NO_3^-} \).
The dissociation can be represented as follows:
\[ \mathrm{Fe(NO_3)_3 \rightarrow Fe^{3+} + 3NO_3^-} \]
This step is crucial in setting the stage for subsequent chemical interactions, such as hydrolysis. Knowing about dissociation helps in understanding how compounds dissolve in solutions and subsequently how they might interact to affect the chemistry of the solution, especially in terms of ion concentrations.
Ion Concentration
Ion concentration is a measure of the number of ions present in a solution. When iron nitrate dissociates, it releases iron ions into the solution, which then undergo hydrolysis. The concentration of \( \mathrm{H^+} \) ions produced through hydrolysis directly affects the pH of the solution.
Calculating ion concentration involves determining how much of the initial compound dissociates (for \( \mathrm{Fe(NO_3)_3} \), full dissociation occurs) and how these ions interact, as in hydrolysis, to reach equilibrium. In our exercise, given that initially, the concentration of \( \mathrm{Fe(NO_3)_3} \) was \( 0.100 \, \text{M} \), we can find the concentration of \( \mathrm{H^+} \) ions by solving the hydrolysis equilibrium equation:
\[ \frac{3.6 \times 10^{-3}}{x} = \frac{x^3}{0.100} \]
Once solved, the solution gives the exact \( \mathrm{H^+} \) concentration, which is essential for determining the pH. By understanding ion concentration, one can appreciate how chemical equilibria in solutions affect overall solution properties, such as pH.

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Most popular questions from this chapter

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