To determine the difference in the concentration of hydrogen ions between two brands with pH values of 4.5 and 5.0, we will use the relationship between pH and hydrogen ion concentration [tex]\([H^+]\)[/tex] given by the formula:
[tex]\[ \text{pH} = -\log_{10}[H^+] \][/tex]
1. Convert the pH values to hydrogen ion concentrations.
For a pH of 4.5:
[tex]\[ [H^+]_1 = 10^{-\text{pH}} = 10^{-4.5} \][/tex]
For a pH of 5.0:
[tex]\[ [H^+]_2 = 10^{-\text{pH}} = 10^{-5.0} \][/tex]
Hence:
[tex]\[ [H^+]_1 = 3.1622776601683795 \times 10^{-5} \][/tex]
[tex]\[ [H^+]_2 = 1.0 \times 10^{-5} \][/tex]
2. Determine the relative difference in hydrogen ion concentration between the two pH levels by dividing the hydrogen ion concentration of the vinegar with pH 5.0 by the hydrogen ion concentration of the vinegar with pH 4.5.
[tex]\[ \frac{[H^+]_2}{[H^+]_1} = \frac{1.0 \times 10^{-5}}{3.1622776601683795 \times 10^{-5}} \][/tex]
3. Simplify the calculation:
[tex]\[ \frac{1.0 \times 10^{-5}}{3.1622776601683795 \times 10^{-5}} = 0.31622776601683794 \][/tex]
This indicates the hydrogen ion concentration in the vinegar with pH 5.0 is approximately [tex]\(0.316\)[/tex] times that with pH 4.5, which matches the answer [tex]\(3.2 \times 10^{-1}\)[/tex].
Therefore, the approximate difference in the concentration of hydrogen ions between the two brands of vinegar is:
[tex]\[ \boxed{3.2 \times 10^{-1}} \][/tex]
