Question

Consider two solutions, solution \(\mathrm{A}\) and solution B. \(\left[\mathrm{H}^{+}\right]\) in solution \(\mathrm{A}\) is 500 times greater than that in solution \(\mathrm{B}\). What is the difference in the \(\mathrm{pH}\) values of the two solutions?

Short answer

The difference in the pH values of solution A and solution B is approximately 2.70.

Step-by-step solution

Recall the definition of pH

The pH of a solution is defined as the negative logarithm (base 10) of the hydrogen ion concentration (\(\mathrm{H}^{+}\)): \[ \mathrm{pH} = -\log_{10}\left[\mathrm{H}^{+}\right] \]

Represent the concentrations of \(\mathrm{H}^{+}\) in both solutions

Given that the concentration of \(\mathrm{H}^{+}\) in solution A is 500 times greater than that of solution B, we can represent the concentration in solution A as: \[ [\mathrm{H}^{+}]_\mathrm{A} = 500 \cdot [\mathrm{H}^{+}]_\mathrm{B} \]

Calculate the pH of each solution

The pH values of solution A and solution B can be found using the formula given in Step 1: \[ \mathrm{pH}_\mathrm{A} = -\log_{10}\left[\mathrm{H}^{+}\right]_\mathrm{A} = - \log_{10}\left(500 \cdot [\mathrm{H}^{+}]_\mathrm{B}\right)\] \[ \mathrm{pH}_\mathrm{B} = -\log_{10}\left[\mathrm{H}^{+}\right]_\mathrm{B} \]

Find the difference in pH values

To find the difference in pH values of the two solutions, subtract the pH value of solution B from the pH value of solution A: \[ \Delta \mathrm{pH} = \mathrm{pH}_\mathrm{A} - \mathrm{pH}_\mathrm{B} \] Substitute the expressions for \(\mathrm{pH}_\mathrm{A}\) and \(\mathrm{pH}_\mathrm{B}\) found in Step 3: \[ \Delta \mathrm{pH} = - \log_{10}\left(500 \cdot [\mathrm{H}^{+}]_\mathrm{B}\right) - (-\log_{10}\left[\mathrm{H}^{+}\right]_\mathrm{B}\) \] Using the logarithm properties, we can solve for the difference in pH values: \[ \Delta \mathrm{pH} = -\log_{10}(500)+\log_{10}\left[\mathrm{H}^{+}\right]_\mathrm{B}+ \log_{10}\left[\mathrm{H}^{+}\right]_\mathrm{B} \] Since the last two terms cancel each other out, we have: \[ \Delta \mathrm{pH} = -\log_{10}(500) \] Now, calculate the numerical value: \[ \Delta \mathrm{pH} \approx -2.70\] Thus, the difference in the pH values of the two solutions is approximately 2.70.