Final answer:
A material with a pH of 5 has 100 times more hydrogen ions than one with a pH of 7, and a material with a pH of 13 has 10,000 times fewer hydrogen ions than one with a pH of 9 due to the logarithmic nature of the pH scale.
Step-by-step explanation:
A material with a pH of five has 100 times as many hydrogen ions as one with a pH of 7. This is because the pH scale is logarithmic, meaning each whole pH value below 7 is ten times more acidic (or has ten times more hydrogen ions) than the next higher whole value. Therefore, moving from pH 7 to pH 5 involves two whole pH units: from pH 7 to pH 6 is a tenfold increase in hydrogen ion concentration, and moving from pH 6 to pH 5 is another tenfold increase, resulting in a total of 100 times more hydrogen ions (10 x 10).
Similarly, a material with a pH of 13 has 10,000 times fewer hydrogen ions than one with a pH of 9. Again, because of the logarithmic nature of the pH scale, an increase of each whole pH value above 7 means the solution is ten times less acidic (or possesses ten times fewer hydrogen ions) than the next lower whole value. Thus, moving from pH 9 to pH 13 passes through four whole pH units: each step is a tenfold decrease in hydrogen ion concentration, leading to 10 x 10 x 10 x 10, which equals 10,000 times fewer hydrogen ions.