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

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: \[ [\mathrm{H}^{+}]_\mathrm{A} = 500 \cdot [\mathrm{H}^{+}]_\mathrm{B} \]

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

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.