Question

Consider two solutions, solution A and solution B. [H \(\left.^{+}\right]\) in solution A is 25 times greater than that in solution \(B\). What is the difference in the pH values of the two solutions?

Short answer

The difference in the pH values of the two solutions is approximately 1.40.

Step-by-step solution

Write the equation for the hydrogen ion concentration of both solutions

Given that the concentration of hydrogen ions in solution A is 25 times greater than in solution B, we can write the equation: \[ [H^+ _A] = 25[H^+ _B] \]

Find the pH of both solutions using the pH formula

Now, we'll use the pH formula to write the pH values of both solutions. The pH formula is: \[ pH = - \log[H^+] \] So for solution A, the pH is: \[pH_A = -\log[H^+ _A] \] And for solution B, the pH is: \[ pH_B = -\log[H^+ _B] \]

Substitute the hydrogen ion concentration equation into the pH equations

We can replace the hydrogen ion concentration in the equations of step 2 to solve for the pH values of both solutions. We know that: \[ [H^+ _A] = 25[H^+ _B] \] So the pH equations become: \[ pH_A = -\log(25[H^+ _B]) \] \[ pH_B = -\log[H^+ _B] \]

Calculate the difference in pH values

We want to find the difference between the pH values of the two solutions, which can be written as: \[ \Delta pH = pH_A - pH_B \] Now, we can substitute the pH equations from step 3 into the difference equation: \[ \Delta pH = (-\log(25[H^+ _B])) - (-\log[H^+ _B]) \]

Use the logarithm properties to simplify the equation and find the difference

Apply the logarithm property, \(\log{a * b} = \log{a} + \log{b}\), to simplify the equation: \[ \Delta pH = (-\log25 -\log[H^+ _B]) - (-\log[H^+ _B]) \] The -\log[H^+ _B] terms cancel each other out, so we have: \[ \Delta pH = -\log25 \] Now, use the calculator to find -\log(25): \[ \Delta pH \approx 1.40 \] So the difference between the pH values of the two solutions is approximately 1.40.

Key concepts

Hydrogen Ion Concentration

Hydrogen ion concentration is an important concept in understanding the acidity or basicity of a solution. It refers to the amount of hydrogen ions, denoted as \(H^+\), present in a solution. These ions are what cause a substance to be considered acidic. The concentration is typically measured in moles per liter (M), and the more hydrogen ions present, the more acidic the solution is. For example:

• A solution with a high \(H^+\) concentration will have a low pH value and is considered acidic.
• Conversely, a solution with a low \(H^+\) concentration will have a high pH value and is more basic.

To compare two solutions, like in our exercise, understanding the relationship between their hydrogen ion concentrations is crucial. Solution A has a hydrogen ion concentration 25 times that of Solution B, indicating that it is significantly more acidic.

Logarithm Properties

Logarithmic functions are vital in working with pH calculations. The pH scale itself is a logarithmic scale derived from the concentration of hydrogen ions. The formula to calculate pH is given by:\[ pH = - \log[H^+] \] In this formula, the logarithm base is 10, and it helps us transform a potentially complicated small number into a manageable figure. The properties of logarithms assist in simplifying expressions. For instance, the property \(\log(a \times b) = \log a + \log b\) is often used to break down complex logarithmic expressions.In our exercise, this logarithmic property helps simplify the expression \(\log(25[H^+ _B])\) which allows us to see the pH difference as \(-\log 25\), showing how powerful these properties are in simplifying and solving such equations.

pH Difference

The pH scale ranges from 0 to 14, where lower values correspond to more acidic conditions, and higher values indicate more basic conditions. The pH difference between two solutions tells us how much more acidic or basic one solution is compared to another.In the exercise, we found the difference in pH between Solutions A and B. We do this by subtracting the pH of Solution B from the pH of Solution A: \[\Delta pH = pH_A - pH_B \] Through the simplification process using logarithm properties, we determined \(\Delta pH\) to be approximately 1.40. This means that Solution A is noticeably more acidic than Solution B, just as you'd expect with its 25 times greater hydrogen ion concentration. Knowing how to calculate and interpret \(\Delta pH\) is essential for comparing the acidic or basic nature of different solutions, especially in chemistry and environmental science.