To determine the concentration of hydronium ions [tex]\([H_3O^+]\)[/tex] in a solution given the concentration of hydroxide ions [tex]\([OH^-]\)[/tex], we need to use the ion-product constant for water at 25°C. This constant, [tex]\(K_w\)[/tex], is the product of the concentrations of [tex]\(H_3O^+\)[/tex] and [tex]\(OH^-\)[/tex] ions in water, and it is always [tex]\(1 \times 10^{-14}\)[/tex] M[tex]\(^2\)[/tex].
We're given:
[tex]\[
[OH^-] = 1 \times 10^{-12} \, \text{M}
\][/tex]
The ion-product constant is:
[tex]\[
K_w = [H_3O^+][OH^-] = 1 \times 10^{-14} \, \text{M}^2
\][/tex]
To find the concentration of [tex]\(H_3O^+\)[/tex], we rearrange the equation to solve for [tex]\([H_3O^+]\)[/tex]:
[tex]\[
[H_3O^+] = \frac{K_w}{[OH^-]}
\][/tex]
Substituting the known values:
[tex]\[
[H_3O^+] = \frac{1 \times 10^{-14} \, \text{M}^2}{1 \times 10^{-12} \, \text{M}}
\][/tex]
Simplifying the fraction:
[tex]\[
[H_3O^+] = \frac{1 \times 10^{-14}}{1 \times 10^{-12}} = 1 \times 10^{-2} \, \text{M}
\][/tex]
So, the concentration of [tex]\(H_3O^+\)[/tex] in the solution is:
[tex]\[
[H_3O^+] = 1 \times 10^{-2} \, \text{M}
\][/tex]
Therefore, the correct answer is:
[tex]\[
1 \times 10^{-2} \, \text{M}
\][/tex]
