The pH and pOH Scales

Core Concept

Because [H₃O⁺] in water spans many orders of magnitude (from ~10⁰ M in strong acid to ~10⁻¹⁴ M in strong base), chemists compress it onto a log scale. pH = −log[H₃O⁺] and pOH = −log[OH⁻]. The "p" means "negative base-10 logarithm," so every unit step on the pH scale corresponds to a factor of ten in acidity. At 25 °C the two scales are tied together by pH + pOH = 14.00, a direct consequence of Kw.

Key Ideas

• pH = −log[H₃O⁺] and pOH = −log[OH⁻]; both are unitless.
• Logarithmic scale: pH 3 is ten times more acidic than pH 4, and 100 times more acidic than pH 5.
• The 25 °C rule: pH + pOH = 14.00; derived by taking −log of both sides of Kw.
• Inverse logs: [H₃O⁺] = 10^(−pH) and [OH⁻] = 10^(−pOH).
• Acid–base–neutral boundaries (25 °C): acidic pH < 7, neutral pH = 7, basic pH > 7.
• Significant figures in logs: the number of decimal places in pH equals the number of significant figures in the concentration (the integer part just locates the decimal point).

Equations and Variables

• pH = −log[H₃O⁺]
• pOH = −log[OH⁻]
• [H₃O⁺] = 10^(−pH)
• [OH⁻] = 10^(−pOH)
• pH + pOH = 14.00 (25 °C)
• pKw = pH + pOH (general, any temperature)

How It Works

1. Take any [H₃O⁺], such as 2.5 × 10⁻⁴ M.
2. Compute its negative logarithm: pH = −log(2.5 × 10⁻⁴) = 3.60.
3. To go backward, raise 10 to the negative power: [H₃O⁺] = 10^(−3.60) = 2.5 × 10⁻⁴ M.
4. Because Kw fixes [OH⁻] = Kw/[H₃O⁺], taking logs of Kw gives pH + pOH = pKw = 14.00 at 25 °C — so once you have pH, pOH is just 14.00 − pH.
5. The sign convention (the minus) makes smaller pH mean more acidic, which is the number line students actually use.

Worked Example

A soft drink has [H₃O⁺] = 3.2 × 10⁻³ M. Find its pH and pOH, and classify it.

pH = −log(3.2 × 10⁻³) = 2.49

pOH = 14.00 − 2.49 = 11.51

Because pH (2.49) < 7, the drink is acidic. As a check, [OH⁻] = 10^(−11.51) = 3.1 × 10⁻¹² M, and (3.2 × 10⁻³)(3.1 × 10⁻¹²) ≈ 1.0 × 10⁻¹⁴ ✓.

How it works

1. Take any [H₃O⁺], such as 2.5 × 10⁻⁴ M.
2. Compute its negative logarithm: pH = −log(2.5 × 10⁻⁴) = 3.60.
3. To go backward, raise 10 to the negative power: [H₃O⁺] = 10^(−3.60) = 2.5 × 10⁻⁴ M.
4. Because Kw fixes [OH⁻] = Kw/[H₃O⁺], taking logs of Kw gives pH + pOH = pKw = 14.00 at 25 °C — so once you have pH, pOH is just 14.00 − pH.
5. The sign convention (the minus) makes smaller pH mean more acidic, which is the number line students actually use.

Common confusions

• "pH 6 → pH 7 is a one-unit, linear change." — It is a ten-fold change in [H₃O⁺]; the scale is logarithmic, not linear.
• "pH + pOH = 14 always." — Only at 25 °C; the general relation is pH + pOH = pKw.
• "A solution with pH = 7 must be neutral." — Only if the temperature is 25 °C.
• "Lower [H₃O⁺] means lower pH." — Opposite: lower [H₃O⁺] means higher pH (the minus sign inverts the relationship).
• "pH can't be negative." — It can; a 10 M strong acid has pH = −1.

Quick review

• pH = −log[H₃O⁺], pOH = −log[OH⁻], both unitless log scales.
• Reverse: [H₃O⁺] = 10^(−pH).
• At 25 °C: pH + pOH = 14.00.
• Logarithmic scale: each unit is ×10.
• Acidic < 7, neutral = 7, basic > 7 (25 °C).