pH, pOH, and the Auto-ionization of Water (25 °C)
This calculator converts between hydronium concentration \([H_3O^+]\), hydroxide concentration \([OH^-]\),
pH and pOH at \(25^\circ\text{C}\) using \(K_w = 1.0\times10^{-14}\).
It also classifies solutions as acidic, neutral, or basic from the computed pH (not from the raw input).
1) Core definitions
\[
\mathrm{pH} = -\log_{10}[H_3O^+],\qquad
\mathrm{pOH} = -\log_{10}[OH^-]
\]
The ionic product of water at \(25^\circ\text{C}\) is
\[
K_w = [H_3O^+][OH^-] = 1.0\times10^{-14},
\]
which implies the well-known relation
\[
\mathrm{pH} + \mathrm{pOH} = \mathrm{p}K_w = 14.00 .
\]
2) Conversions the tool performs
- Given \([H_3O^+]\):
\(\displaystyle \mathrm{pH} = -\log_{10}[H_3O^+]\),
\(\displaystyle [OH^-] = \dfrac{K_w}{[H_3O^+]}\),
\(\displaystyle \mathrm{pOH} = -\log_{10}[OH^-] = 14.00 - \mathrm{pH}\).
- Given \([OH^-]\):
\(\displaystyle \mathrm{pOH} = -\log_{10}[OH^-]\),
\(\displaystyle [H_3O^+] = \dfrac{K_w}{[OH^-]}\),
\(\displaystyle \mathrm{pH} = 14.00 - \mathrm{pOH}\).
- Inverse forms (from logarithms back to concentrations):
\(\displaystyle [H_3O^+] = 10^{-\mathrm{pH}}\),\;
\(\displaystyle [OH^-] = 10^{-\mathrm{pOH}}\).
3) Classification rule
The calculator decides the type from the computed pH:
- Acidic: \(\mathrm{pH} < 7\)
- Neutral: \(\mathrm{pH} \approx 7\) (within a small numerical tolerance)
- Basic: \(\mathrm{pH} > 7\)
Note: Very dilute inputs can be counter-intuitive. For example, entering
\([H_3O^+] < 10^{-7}\,\mathrm{M}\) (less than pure water) yields \(\mathrm{pH} > 7\) and the solution is basic.
The tool always bases the label on the resulting pH, avoiding “acidic/basic” decisions from the raw input alone.
4) Diagram interpretation
- The pink column tracks \([H_3O^+]\) (downward arrow indicates increasing acidity as \([H_3O^+]\) grows).
- The center columns are the linear pH and pOH scales (0–14). Diamond markers show the calculated values; the dashed line at 7 marks neutrality.
- The blue column tracks \([OH^-]\) (upward arrow indicates increasing basicity as \([OH^-]\) grows).
5) Assumptions and limits
- Temperature: Uses \(K_w=1.0\times10^{-14}\) at \(25^\circ\text{C}\). At other \(T\), \(K_w\) changes and so does \(\mathrm{p}K_w\).
- Activities vs concentrations: The relations above are exact in terms of activities. At moderate ionic strength this tool approximates activities by molarities.
- Very dilute solutions: When acid/base is extremely dilute, water’s own auto-ionization (\([H_3O^+]\approx 10^{-7}\,\mathrm{M}\))
can dominate; the tool’s classification is still derived from the computed pH.
6) Quick example
If \([H_3O^+] = 3.2\times10^{-5}\,\mathrm{M}\), then
\(\mathrm{pH} = -\log_{10}(3.2\times10^{-5}) \approx 4.49\),
\([OH^-] = \dfrac{1.0\times10^{-14}}{3.2\times10^{-5}} \approx 3.1\times10^{-10}\,\mathrm{M}\),
\(\mathrm{pOH} \approx 9.51\), so the solution is acidic.
