What is the [tex]\left[ OH ^{-} \right][/tex] in a solution that has a [tex]\left[ H _3 O ^{+} \right] = 5.0 \times 10^{-2} \, M[/tex]?

A) [tex]2.0 \times 10^{-12} \, M[/tex]
B) [tex]1.0 \times 10^{-14} \, M[/tex]
C) [tex]1.0 \times 10^{-12} \, M[/tex]
D) [tex]5.0 \times 10^{-2} \, M[/tex]
E) [tex]2.0 \times 10^{-13} \, M[/tex]



Answer :

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]