To understand the relationship between pH and pOH, we need to consider the properties of water and the ion product constant for water ([tex]\(K_w\)[/tex]).
1. Ion Product of Water ([tex]\(K_w\)[/tex]): At 25°C, the ion product constant for water is [tex]\(K_w = [H^+][OH^-] = 1.0 \times 10^{-14}\)[/tex].
2. Definitions:
- pH: A measure of the hydrogen ion concentration, defined as [tex]\( pH = -\log[H^+] \)[/tex].
- pOH: A measure of the hydroxide ion concentration, defined as [tex]\( pOH = -\log[OH^-] \)[/tex].
3. Relationship:
- Considering the ion product constant for water, we have: [tex]\( [H^+] \times [OH^-] = 1.0 \times 10^{-14} \)[/tex].
If we take the negative logarithm of both sides of this equation, we get:
[tex]\[
-\log([H^+]) + (-\log([OH^-])) = -\log(1.0 \times 10^{-14})
\][/tex]
Simplifying this, we use the properties of logarithms:
[tex]\[
pH + pOH = 14
\][/tex]
Given this relationship, one can derive the equations provided in the options:
A. [tex]\( pH = pOH - 14 \)[/tex]
- This would imply that the numbers do not sum to 14, which contradicts the established relationship since we already know that [tex]\( pH + pOH \)[/tex] should equal 14.
B. [tex]\( pH = 14 + pOH \)[/tex]
- This suggests that the sum of pH and pOH would be greater than 14, which is not consistent with the actual relationship.
C. [tex]\( pOH = 14 - pH \)[/tex]
- This is consistent because if we subtract pH from 14, we are left with the value of pOH (i.e., [tex]\( pH + pOH = 14 \)[/tex]).
D. [tex]\( pOH = 14 + pH \)[/tex]
- Similar to option B, this would imply pOH is greater than 14 (assuming positive pH values), which would not satisfy the core relationship we have.
Based on the proper derivation, the correct answer is:
C. [tex]\( pOH = 14 - pH \)[/tex]
