On the pH scale, a decrease in pH by 3 units implies what change in the hydronium ion concentration, $\left[H_3{O}^{+}\right]$?

• A. $\left[H_3{O}^{+}\right]$ increases by a factor of ${10}^3$.
• B. $\left[H_3{O}^{+}\right]$ increases by a factor of $3$.
• C. $\left[H_3{O}^{+}\right]$ decreases by a factor of ${10}^3$.
• D. $\left[H_3{O}^{+}\right]$ remains unchanged.

Answer: (A)

Verified step by step guidance

1. Recall that pH is defined as the negative logarithm (base 10) of the hydronium ion concentration: \(\mathrm{pH} = -\log_{10} [\mathrm{H}_3\mathrm{O}^+]\).
2. Understand that a decrease in pH means the solution is becoming more acidic, which corresponds to an increase in the hydronium ion concentration.
3. Express the change in hydronium ion concentration when pH changes by 3 units: if the initial pH is \(\mathrm{pH}_1\) and the final pH is \(\mathrm{pH}_2 = \mathrm{pH}_1 - 3\), then
4. \[\mathrm{pH}_1 = -\log_{10} [\mathrm{H}_3\mathrm{O}^+]_1\]
5. \[\mathrm{pH}_2 = -\log_{10} [\mathrm{H}_3\mathrm{O}^+]_2 = \mathrm{pH}_1 - 3\]
6. Rearrange the equation to find the ratio of the hydronium ion concentrations:
7. \[-\log_{10} [\mathrm{H}_3\mathrm{O}^+]_2 = -\log_{10} [\mathrm{H}_3\mathrm{O}^+]_1 - 3\]
8. Multiply both sides by -1:
9. \[\log_{10} [\mathrm{H}_3\mathrm{O}^+]_2 = \log_{10} [\mathrm{H}_3\mathrm{O}^+]_1 + 3\]
10. Use logarithm properties to combine terms:
11. \[\log_{10} [\mathrm{H}_3\mathrm{O}^+]_2 = \log_{10} \left( [\mathrm{H}_3\mathrm{O}^+]_1 \times 10^3 \right)\]
12. Therefore, the hydronium ion concentration increases by a factor of \$10^3$ when the pH decreases by 3 units.