To find the concentration of hydroxide ions [tex]\([\text{OH}^-]\)[/tex] in a solution when the concentration of hydronium ions [tex]\([\text{H}_3\text{O}^+]\)[/tex] is given as [tex]\(5.0 \times 10^{-2} \, \text{M}\)[/tex], we use the ion-product constant of water [tex]\((K_w)\)[/tex].
1. Step 1: Recall the ion-product constant for water:
The ion-product constant for water at 25°C is:
[tex]\[
K_w = [\text{H}_3\text{O}^+][\text{OH}^-] = 1.0 \times 10^{-14}
\][/tex]
2. Step 2: Write down the given information:
[tex]\[
[\text{H}_3\text{O}^+] = 5.0 \times 10^{-2} \, \text{M}
\][/tex]
3. Step 3: Rearrange the [tex]\(K_w\)[/tex] expression to solve for [tex]\([\text{OH}^-]\)[/tex]:
[tex]\[
[\text{OH}^-] = \frac{K_w}{[\text{H}_3\text{O}^+]}
\][/tex]
4. Step 4: Substitute the known values into the rearranged equation:
[tex]\[
[\text{OH}^-] = \frac{1.0 \times 10^{-14}}{5.0 \times 10^{-2}}
\][/tex]
5. Step 5: Perform the division:
[tex]\[
[\text{OH}^-] = \frac{1.0 \times 10^{-14}}{5.0 \times 10^{-2}} = 2.0 \times 10^{-13} \, \text{M}
\][/tex]
Therefore, the concentration of hydroxide ions [tex]\([\text{OH}^-]\)[/tex] in the solution is [tex]\(2.0 \times 10^{-13} \, \text{M}\)[/tex].
The correct answer is:
[tex]\[
\boxed{2.0 \times 10^{-13} \, \text{M}}
\][/tex]
