Question

Consider two solutions, solution \(\mathrm{A}\) and solution \(\mathrm{B} .\left[\mathrm{H}^{+}\right]\) in solution \(\mathrm{A}\) is 250 times greater than that in solution B. What is the difference in the pH values of the two solutions?

Short answer

The difference between the pH values of solutions A and B is approximately 2.60.

Step-by-step solution

Write down the given information

The concentration of \(\mathrm{H}^+\) ions in solution A is 250 times greater than that in solution B. Let \([\mathrm{H}^+]_\mathrm{B}\) be the concentration of \(\mathrm{H}^+\) ions in solution B. Then, \([\mathrm{H}^{+}]_\mathrm{A} = 250\cdot [\mathrm{H}^{+}]_\mathrm{B}\)

Write down the formulas for pH values of A and B

We will use the formula, pH \(= -\log_{10}[\mathrm{H}^+]\), to find the pH values of solutions A and B: $$ \begin{aligned} \text{pH}_\mathrm{A} &= -\log_{10}[\mathrm{H}^{+}]_\mathrm{A}\\ \text{pH}_\mathrm{B} &= -\log_{10}[\mathrm{H}^{+}]_\mathrm{B} \end{aligned} $$

Substitute the given information into the equations and simplify

Substitute the given concentration relationship into pH equations: $$ \begin{aligned} \text{pH}_\mathrm{A} &= -\log_{10}(250\cdot [\mathrm{H}^{+}]_\mathrm{B})\\ \text{pH}_\mathrm{B} &= -\log_{10}[\mathrm{H}^{+}]_\mathrm{B} \end{aligned} $$ Use the logarithmic property: \(\log_{10}(ab) = \log_{10}(a) + \log_{10}(b)\), we get: $$ \begin{aligned} \text{pH}_\mathrm{A} &= -\log_{10}(250) - \log_{10}[\mathrm{H}^{+}]_\mathrm{B}\\ \text{pH}_\mathrm{B} &= -\log_{10}[\mathrm{H}^{+}]_\mathrm{B} \end{aligned} $$

Calculate the difference between the pH values

To find the difference between the pH values of solutions A and B, subtract the equation for \(\text{pH}_\mathrm{B}\) from the equation for \(\text{pH}_\mathrm{A}\): $$ \begin{aligned} \text{Difference in pH} = \text{pH}_\mathrm{A} - \text{pH}_\mathrm{B} &= (-\log_{10}(250) - \log_{10}[\mathrm{H}^{+}]_\mathrm{B}) - (-\log_{10}[\mathrm{H}^{+}]_\mathrm{B}) \\ &= -\log_{10}(250) \end{aligned} $$ Now, calculate the value of \(-\log_{10}(250)\): $$ \begin{aligned} \text{Difference in pH} &= -\log_{10}(250) \\ &\approx 2.60 \end{aligned} $$ So, the difference between the pH values of solutions A and B is approximately 2.60.

Key concepts

pH scale

The pH scale is a measure of the acidity or basicity of an aqueous solution. It ranges from 0 to 14, with 7 being neutral. Solutions with a pH less than 7 are acidic, while those with a pH greater than 7 are basic or alkaline.

Understanding the pH scale is crucial in fields like chemistry, biology, and environmental science, as it affects the chemical behavior of substances. For example, the pH of a solution can influence enzyme activity in biological organisms or the corrosiveness of a solution in industrial settings.

To assist students, we should explain that each unit of pH represents a tenfold difference in hydrogen ion concentration. This means a solution with a pH of 4 is ten times more acidic than a solution with a pH of 5. Highlighting this fact can help clarify the significance of a pH difference in the exercise being considered.

Hydrogen ion concentration

Hydrogen ion concentration, often represented by \( [\mathrm{H}^+] \), is a measure of the number of hydrogen ions present in a solution. It is directly related to acidity, with higher concentrations indicating greater acidity. The concentration of hydrogen ions in a solution is inversely related to its pH; as \( [\mathrm{H}^+] \) increases, the pH decreases.

For educational content, it's important to emphasize that even a slight change in hydrogen ion concentration can result in a significant shift in pH. When comparing two solutions as in the exercise, understanding that a 250-fold increase in \( [\mathrm{H}^+] \) concentration leads to a pH difference is key to grasping the concept of pH calculation.

Incorporating this relationship into our explanation will help students visualize the impact of \( [\mathrm{H}^+] \) on the overall pH of a solution and thus better understand the steps involved in the pH calculation.

Logarithmic properties

The logarithmic properties play a critical role in understanding how the pH scale works. The pH value is calculated using a logarithmic scale based on powers of 10. This helps represent a wide range of hydrogen ion concentrations in a more manageable way. The formula used to calculate pH from hydrogen ion concentration is \( \text{pH} = -\log_{10}[\mathrm{H}^+] \).

When explaining this concept, highlight that the use of logarithms compresses the scale, so large changes in ion concentration translate to smaller, more readable changes in pH value. For example, by understanding that \( \log_{10}(ab) = \log_{10}(a) + \log_{10}(b) \), students can simplify the process of finding pH differences between two solutions.

By applying logarithmic properties, as shown in the exercise's step-by-step solution, students can learn to break down complex-sounding calculations into a sequence of more straightforward steps, aiding their understanding of pH calculation in various scientific contexts.