Problem 25 - Chemistry Matter and Change

Question

Calculate the \(\mathrm{pH}\) of aqueous solutions with the following \([\mathrm{H}+]\) at 298 \(\mathrm{K}\). \begin{equation} \text { a. }\left[\mathrm{H}^{+}\right]=0.0055 M \quad \text { b. }\left[\mathrm{H}^{+}\right]=0.000084 \mathrm{M} \end{equation}

Step-by-Step Solution

Answer

The pH values are 2.26 for 0.0055 M and 4.08 for 0.000084 M.

Step 1: Understanding pH

The pH of a solution is a measure of its hydrogen ion concentration, and it is calculated using the formula: \[ pH = - ext{log}_{10} ext{[H}^+ ext{]} \] where \(\text{[H}^+\text{]}\) is the concentration of hydrogen ions in moles per liter (M). We will use this formula for each provided concentration.

Step 2: Calculate pH for Part a

Given \(\text{[H}^+\text{]} = 0.0055\,M\). Plug this concentration into the pH formula: \[ pH = -\log_{10}(0.0055) \] Calculate the pH: \[ pH = 2.26 \] This represents the pH of the solution for part a.

Step 3: Calculate pH for Part b

Given \(\text{[H}^+\text{]} = 0.000084\,M\). Use the pH formula to find the pH: \[ pH = -\log_{10}(0.000084) \] Calculate the pH: \[ pH = 4.08 \] This represents the pH of the solution for part b.

Step 4: Compile the Results

Now that we have calculated the pH for both parts a and b, we can summarize the results: - For \(\text{[H}^+] = 0.0055\,M\), \(pH = 2.26\).- For \(\text{[H}^+] = 0.000084\,M\), \(pH = 4.08\).

Key Concepts

Hydrogen Ion Concentration

Hydrogen ion concentration is a measure of the number of hydrogen ions, denoted as \([H^+]\), present in a solution. It is typically expressed in terms of molarity, which is moles of hydrogen ions per liter of solution (M). This concentration is one of the key factors determining the acidity of a solution. Higher hydrogen ion concentration means more acidic solutions, whereas lower concentration signifies a more basic or alkaline environment.
\[ \text{[H}^+\text{]} = \text{molarity of hydrogen ions} \]
In the exercises given, the concentration of hydrogen ions informs us of how concentrated or diluted an acidic solution is. For instance, in part a, \([H^+] = 0.0055\,M\) indicates a fairly concentrated acidic solution compared to part b, \([H^+] = 0.000084\,M\), which is less concentrated.

pH Formula

The pH formula is a way to numerically express the acidity or basicity of a solution. The formula is expressed as:
\[ pH = -\log_{10}([H^+]) \]
This formula involves taking the negative base 10 logarithm of the hydrogen ion concentration. The result gives us a pH value, which is a number usually between 0 and 14. This logarithmic scale allows us to easily compare how acidic or basic different solutions are.
For example, in the original exercise, the pH of the solution with \([H^+] = 0.0055\,M\) is calculated as:
- \( pH = -\log_{10}(0.0055) \approx 2.26 \)
This means the solution is quite acidic, as pH values under 7 indicate acidity.

Logarithms in Chemistry

Logarithms play a crucial role in chemistry, particularly when working with the pH scale. The pH scale uses logarithms because acids and bases can exist in very high or very low concentrations, spanning several orders of magnitude. Logarithms can compress this range into a more manageable scale from 0 to 14.
The logarithm function used for pH is base 10, which is common in scientific calculations because it simplifies the mathematical processing of very large or small numbers by reducing them to manageable figures. For instance:

• If \(x = 10^2\), then \(\log_{10}(x) = 2\).
• Conversely, if \([H^+] = 0.0055\,M\), \(pH = -\log_{10}(0.0055)\).

Thus, logarithms help make understanding and using the pH scale feasible.