To calculate the pH of a neutral aqueous solution at [tex]\(0^{\circ} C\)[/tex] given that the [tex]\(K_w\)[/tex] of water at this temperature is [tex]\(0.12 \times 10^{-14}\)[/tex]:
1. Understand the ion product of water ( [tex]\(K_w\)[/tex] ):
The ion product of water ([tex]\(K_w\)[/tex]) expresses the relationship between the concentrations of hydrogen ions ([tex]\([H^+]\)[/tex]) and hydroxide ions ([tex]\([OH^-]\)[/tex]) in water. For pure water or a neutral solution, the concentrations of hydrogen and hydroxide ions are equal.
2. Express the relationship between [tex]\([H^+]\)[/tex] and [tex]\(K_w\)[/tex]:
Since [tex]\([H^+]\)[/tex] equals [tex]\([OH^-]\)[/tex] in a neutral solution, and their product is [tex]\(K_w\)[/tex],
[tex]\[
[H^+][OH^-] = K_w
\][/tex]
For a neutral solution:
[tex]\[
[H^+] = [OH^-]
\][/tex]
Thus,
[tex]\[
[H^+]^2 = K_w
\][/tex]
3. Calculate the concentration of hydrogen ions, [tex]\([H^+]\)[/tex]:
[tex]\[
[H^+] = \sqrt{K_w}
\][/tex]
Given [tex]\(K_w = 0.12 \times 10^{-14}\)[/tex],
[tex]\[
[H^+] = \sqrt{0.12 \times 10^{-14}}
\][/tex]
The value of [tex]\([H^+]\)[/tex] is:
[tex]\[
[H^+] \approx 3.4641016151377544 \times 10^{-8}
\][/tex]
4. Calculate the pH from the hydrogen ion concentration:
The pH is the negative logarithm (base 10) of the hydrogen ion concentration:
[tex]\[
pH = -\log_{10}([H^+])
\][/tex]
Using the calculated hydrogen ion concentration:
[tex]\[
pH = -\log_{10}(3.4641016151377544 \times 10^{-8})
\][/tex]
The value of the pH is approximately:
[tex]\[
pH \approx 7.4604093769761874
\][/tex]
So, the pH of a neutral aqueous solution at [tex]\(0^{\circ} C\)[/tex] given the [tex]\(K_w\)[/tex] value is:
[tex]\[
pH \approx 7.4604
\][/tex]
