Problem 8 - Analytical Chemistry

Question

Calculate the hydroxide ion concentration of the following solutions: (a) \(2.6 \times 10^{-5} \mathrm{M} \mathrm{HCl}\) (b) \(0.20 \mathrm{M} \mathrm{HNO}_{3}\), (c) \(2.7 \times 10^{-9} \mathrm{M} \mathrm{HClO}_{4}\), (d) \(1.9 \mathrm{M} \mathrm{HClO}_{4}\)

Step-by-Step Solution

Answer

(a) \(3.85 \times 10^{-10} \ M\), (b) \(5.0 \times 10^{-14} \ M\), (c) \(3.7 \times 10^{-6} \ M\), (d) \(5.26 \times 10^{-15} \ M\).

Step 1: Understanding the Relationship between Hydrogen Ions and Hydroxide Ions

The relationship between hydrogen ion concentration \([H^+]\) and hydroxide ion concentration \([OH^-]\) in a solution is given by the water ion product, \(K_w\). For water at 25°C, \(K_w = 1.0 imes 10^{-14}\), thus: \[ [H^+][OH^-] = 1.0 \times 10^{-14} \] This means once we know the concentration of one, we can calculate the other.

Step 2: Calculating [OH-] from [H+] for Solution (a)

Given \( [HCl] = 2.6 \times 10^{-5} \, \text{M} \), and because \([HCl]\) is a strong acid, it completely dissociates: \[ [H^+] = 2.6 \times 10^{-5} \, \text{M} \] Using \(K_w\), find \( [OH^-] \): \[ [OH^-] = \frac{1.0 \times 10^{-14}}{2.6 \times 10^{-5}} = 3.85 \times 10^{-10} \, \text{M} \]

Step 3: Calculating [OH-] from [H+] for Solution (b)

For \( [HNO_3] = 0.20 \, \text{M} \), since it is also a strong acid, it dissociates completely: \[ [H^+] = 0.20 \, \text{M} \] Then, \( [OH^-] \) is found as: \[ [OH^-] = \frac{1.0 \times 10^{-14}}{0.20} = 5.0 \times 10^{-14} \, \text{M} \]

Step 4: Calculating [OH-] from [H+] for Solution (c)

For \( [HClO_4] = 2.7 \times 10^{-9} \, \text{M} \), it fully dissociates as well: \[ [H^+] = 2.7 \times 10^{-9} \, \text{M} \] Therefore, \[ [OH^-] = \frac{1.0 \times 10^{-14}}{2.7 \times 10^{-9}} = 3.7 \times 10^{-6} \, \text{M} \]

Step 5: Calculating [OH-] from [H+] for Solution (d)

Given \( [HClO_4] = 1.9 \, \text{M} \), fully dissociating gives: \[ [H^+] = 1.9 \, \text{M} \] Using \(K_w\), \( [OH^-] \) is: \[ [OH^-] = \frac{1.0 \times 10^{-14}}{1.9} = 5.26 \times 10^{-15} \, \text{M} \]

Key Concepts

Water Ion Product

The water ion product, often symbolized by \( K_w \), is a crucial element in understanding the relationship between hydrogen ions \( [H^+] \) and hydroxide ions \( [OH^-] \) in aqueous solutions. At 25°C, this equilibrium constant is \[ 1.0 \times 10^{-14} \]. This means that the product of the hydrogen ion concentration and the hydroxide ion concentration in pure water will always yield this value.
The formula used is:

• \([H^+][OH^-] = K_w = 1.0 \times 10^{-14}\)

By knowing either the \([H^+]\) or \([OH^-]\) concentration, the other can be easily calculated with the help of this fundamental relationship. Since the \(K_w\) is a constant, it is unaffected by changes in concentration; however, it is temperature-dependent.
This understanding provides a foundation for further calculations, especially when determining the concentrations of ions after strong acid dissociation.

Strong Acid Dissociation

Strong acids are unique in that they completely dissociate in water, meaning they break down entirely into their ions. For example, when hydrochloric acid (\(HCl\)) is added to water, it splits into \(H^+\) and \(Cl^-\) ions fully. This is because strong acids have a very high dissociation constant.This complete dissociation implies:

• The concentration of \(H^+\) ions in the solution is equal to the initial concentration of the strong acid added. Thus, if you have a 0.20 M solution of \(HNO_3\), it translates directly into 0.20 M of \(H^+\) ions.

Knowing this allows students to quickly determine \([H^+]\), which can then be used to find \([OH^-]\) using the water ion product \(K_w\).Understanding the behavior of strong acids in solution is essential for accurate calculation and analysis of hydrogen ion concentration, which is a stepping stone to many other concepts in chemistry.

Hydrogen Ion Concentration

Hydrogen ion concentration \([H^+]\) is a significant factor that influences the acidity of a solution. It is the direct result of the dissociation process of acids in a solution. The more hydrogen ions present, the more acidic the solution will be.For strong acids like \(HCl\), \(HNO_3\), and \(HClO_4\), complete dissociation helps in determining \([H^+]\) easily:

• The concentration of \(H^+\) is synonymous with the initial concentration of the acid.

This concentration can then be used in the water ion product equation to back-calculate \([OH^-]\). It is important to correctly identify whether an acid is strong or weak before assuming dissociation levels because weak acids do not fully dissociate, complicating the calculation.
Grasping the concept of hydrogen ion concentration is vital, as it is used in various other calculations and establishes the groundwork for understanding pH values.

pH and pOH Calculations

The concepts of pH and pOH are pivotal in assessing the acidity or basicity of a solution.Understanding pH:

• The pH is calculated using the formula \(\text{pH} = -\log[H^+]\).
• Low pH values indicate acidic solutions, while high values indicate basic solutions.

For pOH:

• The calculation is similar: \(\text{pOH} = -\log[OH^-]\).
• Just like pH, a lower pOH implies a basic solution, while a higher one indicates acidity.

Moreover, the relationship between pH and pOH is straightforward: \(\text{pH} + \text{pOH} = 14\) at room temperature. This derives from the formula \([H^+][OH^-] = 1.0 \times 10^{-14}\), connecting them through the water ion product.
By understanding these calculations, students can quickly determine the acidity and its corresponding properties, thus gaining insight into the behavior of various solutions.