Problem 24 - Chemistry Matter and Change

Question

Calculate the \(\mathrm{pH}\) of solutions having the following ion concentrations at 298 \(\mathrm{K} .\) \begin{equation} \text { a. }\left[\mathrm{H}^{+}\right]=1.0 \times 10^{-2} M \quad \text { b. }\left[\mathrm{H}^{+}\right]=3.0 \times 10^{-6} \mathrm{M} \end{equation}

Step-by-Step Solution

Answer

a. pH = 2, b. pH \( \approx \) 5.52

Step 1: Understanding pH Formula

The pH of a solution can be calculated using the formula: \[ \mathrm{pH} = -\log_{10}[\mathrm{H}^+] \] where \([\mathrm{H}^+]\) is the concentration of hydrogen ions in the solution.

Step 2: Calculate pH for Part (a)

Given \([\mathrm{H}^+] = 1.0 \times 10^{-2} \ M\). Using the formula, we calculate: \[ \mathrm{pH} = -\log_{10}(1.0 \times 10^{-2}) \] \[ \mathrm{pH} = -(-2) = 2 \] Therefore, for part (a), the pH is 2.

Step 3: Calculate pH for Part (b)

Given \([\mathrm{H}^+] = 3.0 \times 10^{-6} \ M\). Using the formula, we calculate: \[ \mathrm{pH} = -\log_{10}(3.0 \times 10^{-6}) \] \[ \mathrm{pH} \approx 5.52 \] Therefore, for part (b), the pH is approximately 5.52.

Key Concepts

Hydrogen Ion Concentration

At the heart of pH calculation is the concentration of hydrogen ions, often denoted as \([\mathrm{H}^+]\). This concentration is a measure of how many hydrogen ions are present in a solution. Whether it's a large or small number of ions, it influences the acidity or basicity of the solution.
The hydrogen ion concentration is typically expressed in molarity (\(M\)), which is moles of hydrogen ions per liter of solution. For example, when you see \([\mathrm{H}^+] = 1.0 \times 10^{-2}\ M\), it means there are 0.01 moles of hydrogen ions in one liter of solution.
Understanding \([\mathrm{H}^+]\) is essential because it is directly used in the formula to calculate pH. By determining \([\mathrm{H}^+]\), you can assess how acidic or basic the solution is, aiding in various chemical and biological processes.

Logarithmic Function

The pH scale is based on a logarithmic function, involving the base 10 logarithm. This function transforms the potentially unwieldy numbers representing \([\mathrm{H}^+]\) into more manageable figures we use to understand acidity and basicity.
The formula \( \mathrm{pH} = -\log_{10}[\mathrm{H}^+] \) clearly shows the relationship. It means that every step on the pH scale represents a tenfold change in hydrogen ion concentration.
For example:

• If \([\mathrm{H}^+]\) is \(10^{-2}\), then \( \mathrm{pH} \) is 2.
• If \([\mathrm{H}^+]\) is \(10^{-6}\), then \( \mathrm{pH} \) is 6.

This function compresses large ranges of concentration into a simple scale from 0 to 14, making it easier to compare acidity levels. Remember, a lower pH indicates higher acidity, while a higher pH signifies a more basic solution.

Acidic Solutions

Solutions are considered acidic when they have a higher concentration of hydrogen ions compared to hydroxide ions.
Acidic solutions have pH values less than 7. As seen from the exercise, a \( \mathrm{pH} \) of 2 indicates a strong acid. This is because the hydrogen ion concentration is high, specifically \( 1.0 \times 10^{-2}\ M \).
Acidic solutions can be found in everyday life:

• Lemon juice (with a pH around 2) is an example of a natural acidic solution.
• Vinegar, another acidic solution, typically has a pH of around 2-3.

Knowing the acidity of solutions can be important in various contexts, from food science to biological systems where pH can affect enzyme activity and metabolic processes.

Solution Chemistry

Solution chemistry involves studying how substances dissolve in a solvent, forming a homogeneous mixture. Understanding this concept is crucial, as it lays the foundation for pH calculations and helps predict how molecules behave in a solution.
In a solution, the solute (like an acid) is dissolved in the solvent (usually water in a biological or simple laboratory setting). This produces hydrogen ions in the case of an acidic solution. The process involves:

• The solute dispersing at the molecular level.
• Solvent molecules surrounding solute particles, often leading to dissociation in the case of acids, releasing \([\mathrm{H}^+]\).

Understanding these processes aids in grasping why hydrogen ion concentration changes, influencing pH. By comprehensively understanding solution chemistry, one can predict the behavior of acids in solutions, their interactions, and how they might affect chemical reactions.