Problem 13 - Analytical Chemistry

Question

What is the \(\mathrm{pH}\) of a solution for which the concentration of \(\mathrm{H}^{+}\) is \(6.92 \times 10^{-6} \mathrm{M}\) ? What is the \(\left[\mathrm{H}^{+}\right]\) in a solution whose \(\mathrm{pH}\) is 8.923 ?

Step-by-Step Solution

Answer

The pH is approximately 5.160 and the \([\mathrm{H}^+]\) is approximately \(1.19 \times 10^{-9}\) M.

Step 1: Understanding pH calculation

The pH of a solution is calculated using the formula: \[ \mathrm{pH} = -\log \left( \left[\mathrm{H}^+\right] \right) \]where \([\mathrm{H}^+]\) is the hydrogen ion concentration.

Step 2: Calculate pH from given \([\mathrm{H}^+]\)

Substitute the given concentration of hydrogen ions into the pH formula:\[ \mathrm{pH} = -\log \left( 6.92 \times 10^{-6} \right) \]Using a calculator, this evaluates to:\[ \mathrm{pH} \approx 5.160 \]

Step 3: Understanding \([\mathrm{H}^+]\) calculation

To find the concentration \([\mathrm{H}^+]\), we use the inverse of the pH calculation:\[ \left[\mathrm{H}^+\right] = 10^{- ext{pH}} \]

Step 4: Calculate \([\mathrm{H}^+]\) from given pH

Substitute the given pH value to find the hydrogen ion concentration:\[ \left[\mathrm{H}^+\right] = 10^{-8.923} \]Using a calculator, this evaluates to:\[ \left[\mathrm{H}^+\right] \approx 1.19 \times 10^{-9} \mathrm{M} \]

Key Concepts

Understanding Hydrogen Ion Concentration

In chemistry, the hydrogen ion concentration is a critical measurement for determining the acidity or basicity of a solution. The concentration is denoted as \( [\mathrm{H}^+] \), and is measured in moles per liter (M). It directly influences the pH level of a solution.
The ionic concentration gives us insight into the behavior and reactivity of the solution. For example, a higher concentration of hydrogen ions indicates a more acidic environment, while a lower concentration suggests a more basic or alkaline setting.

• Acidic solutions have \( [\mathrm{H}^+] > 1 \times 10^{-7} \ \mathrm{M} \).
• Neutral solutions have \( [\mathrm{H}^+] = 1 \times 10^{-7} \ \mathrm{M} \).
• Basic solutions have \( [\mathrm{H}^+] < 1 \times 10^{-7} \ \mathrm{M} \).

Understanding this concept is foundational for any pH-related calculations you will perform in chemistry.

Role of Logarithmic Functions in Chemistry

Logarithmic functions play a vital role in chemistry, particularly when it comes to calculating pH. The basic formula used is: \( \mathrm{pH} = -\log \left( \left[\mathrm{H}^+\right] \right)\).
This means that pH is determined by taking the negative base 10 logarithm of the hydrogen ion concentration. This function allows us to handle the vast differences in ion concentrations in a manageable way. By converting it to a logarithmic scale, we can describe the acidity or basicity of solutions across a wide range of ion concentrations with simplicity and precision.

• Logarithmic scales are beneficial because they allow us to express very large or very small numbers succinctly.
• The logarithm base 10 is specifically used in chemistry as it is intuitive for scaling concentrations.

This method helps chemists and scientists make more accurate comparisons and assessments of chemical behavior in solutions.

Inverse Calculations in pH

In pH calculations, sometimes you need to reverse the process and determine the hydrogen ion concentration from a given pH value. This is referred to as an inverse calculation. It is quite straightforward:
\[ \left[\mathrm{H}^+\right] = 10^{-\mathrm{pH}} \] This formula lets you calculate the concentration of hydrogen ions by using the pH as an exponent with 10 as the base.
Inverse calculations are crucial for deducing the ion concentration when only the pH is known, providing necessary information for further chemical analysis or experimentation.

• When \(\mathrm{pH} = 7\), the solution is neutral, with \( [\mathrm{H}^+] = 1\times10^{-7} \ \mathrm{M} \).
• Lower \(\mathrm{pH}\) values correspond to higher \( [\mathrm{H}^+] \), indicating acidity.
• Higher \(\mathrm{pH}\) values correspond to lower \( [\mathrm{H}^+] \), indicating alkalinity.

This concept is integral in fields like biochemistry and environmental science, where understanding the pH and its implications are necessary for various applications.