To solve this problem, we need to determine the concentration of hydroxide ions ([tex]\( \text{OH}^- \)[/tex]) in a solution with a given pH. Let's proceed step-by-step:
1. Understand the Relationship Between pH and pOH:
The pH and pOH of a solution are related through the following equation:
[tex]\[ \text{pH} + \text{pOH} = 14 \][/tex]
2. Find the Value of pOH:
Given that the pH is 4.20, we can calculate the pOH as follows:
[tex]\[ \text{pOH} = 14 - \text{pH} \][/tex]
[tex]\[ \text{pOH} = 14 - 4.20 \][/tex]
[tex]\[ \text{pOH} = 9.80 \][/tex]
3. Calculate the Concentration of [tex]\( \text{OH}^- \)[/tex] Ions:
The concentration of hydroxide ions ([tex]\( [\text{OH}^-] \)[/tex]) is calculated using the relationship:
[tex]\[ [\text{OH}^-] = 10^{-\text{pOH}} \][/tex]
Substituting the value of pOH:
[tex]\[ [\text{OH}^-] = 10^{-9.80} \][/tex]
4. Evaluate the Exponent:
Solving [tex]\( 10^{-9.80} \)[/tex], we get:
[tex]\[ [\text{OH}^-] = 1.584893192461111 \times 10^{-10} \][/tex]
Therefore, the concentration of [tex]\( \text{OH}^- \)[/tex] ions is approximately:
[tex]\[ 1.6 \times 10^{-10} \, M \][/tex]
Looking at the given choices, the correct answer is:
[tex]\[ \boxed{1.6 \times 10^{-10} \, M} \][/tex]
So, the correct option is:
[tex]\[ \text{E.} \quad 1.6 \times 10^{-10} \, M \][/tex]
