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Chemistry 2e · Acid-Base Equilibria

pH and pOH

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On this page 8 sections
  1. In 30 seconds
  2. Why this matters
  3. The college version
  4. Eli explains
  5. Key takeaway
  6. Check yourself
  7. Study tools
  8. Sources & references

In 30 seconds

The pH scale is a compact way to report the concentration of hydronium ions, which in pure water spans many orders of magnitude. Because [H3O+] for common solutions ranges from about 10-14 M to 100 M, a linear scale would be unreadable. Instead, pH is defined as the negative base-10 logarithm:

pH = -log10[H3O+]

Similarly, reports the hydroxide concentration, and the two are tied together through the water ion product:

pOH = -log10[OH−],   pH + pOH = 14.00  (at 25 °C)

The second relationship follows directly from Kw = [H3O+][OH−] = 1.0 × 10-14: taking -log of both sides of the Kw expression gives pH + pOH = 14.00. This topic builds the definitions, the conversions between the four quantities ([H3O+], [OH−], pH, pOH), and how to classify solutions as acidic, basic, or neutral. The "p" notation (meaning "negative log of") reappears throughout chemistry — pKa, pKb, pKsp — so mastering it here pays off for the rest of the book.

Why this matters

pH is one of the most measured quantities in science and medicine:

  • Physiology: Blood pH is tightly regulated near 7.40 (normal arterial range roughly 7.35–7.45). A shift outside this range — acidosis or alkalosis — can be life-threatening because it changes enzyme activity, oxygen binding, and nerve function. Every pH calculation in this topic is the same math used to interpret blood gas reports.
  • Everyday chemistry: Swimming pools, aquariums, soil, and drinking water are monitored in pH units; one unit represents a tenfold change in acidity, so a pool at pH 7.0 is ten times more acidic than one at pH 8.0.
  • Pharmaceuticals: Drug stability and absorption depend on pH; many products are formulated as buffers (a later topic) to hold pH where the drug is stable and soluble.
  • The scale itself: Because pH is logarithmic, students who misread it think a change from 6 to 5 is "a little more acidic." It is actually a factor of ten. Getting comfortable with logarithms is the single most useful skill in this chapter.

The college version

Core Concepts

The logarithmic definitions

By definition, at 25 °C:

pH = -log10[H3O+],   pOH = -log10[OH−]

Each unit of pH corresponds to a tenfold change in [H3O+]. A solution with pH 3 has [H3O+] = 10-3 M; pH 4 has 10-4 M — ten times less acidic.

The "p" operation has a universal meaning: pX = −log₁₀(X). It converts tiny or huge numbers into convenient, roughly 0–14 numbers, and it converts multiplication into addition (that is why pH + pOH = pKw instead of pH × pOH = Kw).

The four-way conversion map

Given any one of the four quantities, all others follow:

  • [H3O+] to pH: pH = -log[H3O+]
  • pH to [H3O+]: [H3O+] = 10-pH
  • [OH−] to pOH: pOH = -log[OH−]
  • pOH to [OH−]: [OH−] = 10-pOH
  • Between the two ions: [H3O+][OH−] = Kw = 1.0 × 10-14
  • Between the two scales: pH + pOH = 14.00

A robust workflow: convert whatever you know into [H3O+] or [OH−] first, use Kw to get the other ion, then convert to the corresponding p-value. This avoids memorizing a dozen special cases.

Classification: acidic, basic, neutral

At 25 °C, pure water has [H3O+] = [OH−] = 1.0 × 10-7 M, so pH = pOH = 7.00.

  • Neutral: pH = 7.00
  • Acidic: pH < 7.00 (more H3O+ than OH−)
  • Basic: pH > 7.00 (more OH− than H3O+)

This classification is temperature-dependent because Kw changes with temperature (at higher temperatures Kw is larger, so neutral pH is below 7). At the 25 °C standard used in nearly all course problems, 7.00 is the boundary.

Significant figures in logarithms

The number of decimal places in a pH value equals the number of significant figures in the concentration. Example: [H3O+] = 1.0 × 10-7 M has two significant figures, so its pH is reported as 7.00 (two decimal places), not 7. The rule exists because the digits before the decimal in a logarithm encode the exponent (order of magnitude), which carries no significant-figure information.

Strong acids and bases: the simplifying shortcut

For a strong acid like HCl, ionization is complete: a 0.010 M solution of HCl has [H3O+] = 0.010 M directly (each HCl produces one H3O+), so pH = −log(0.010) = 2.00. For a strong base like NaOH, [OH−] equals the base concentration directly (0.010 M NaOH gives pOH = 2.00, pH = 12.00). No equilibrium calculation is needed — but be careful with bases that release two hydroxides (like Ba(OH)2), which produce twice the hydroxide concentration per mole.

How It Works / Step-by-Step Process

Worked example 1: concentration → pH → pOH (strong acid)

Problem. What is the pH and pOH of a 0.0025 M solution of HCl at 25 °C?

Solution.

  1. HCl is a strong acid; each formula unit releases one proton, so [H3O+] = 0.0025 M = 2.5 × 10-3 M.
  2. Apply the definition:

pH = -log(2.5 × 10-3) = -(-2.60) = 2.60

  1. Use the sum rule:

pOH = 14.00 - 2.60 = 11.40

  1. (Optional check) [OH−] = 10-11.40 = 4.0 × 10-12 M; verify with Kw: (2.5 × 10-3)(4.0 × 10-12) = 1.0 × 10-14. ✓ The two significant figures in 0.0025 match the two decimal places in pH 2.60.

Worked example 2: strong base with a two-hydroxide base

Problem. Find the pH of a 0.0050 M solution of Ba(OH)2 at 25 °C.

Solution.

  1. Ba(OH)2 is a strong base that releases two hydroxides per formula unit: [OH−] = 2 × 0.0050 = 0.010 M = 1.0 × 10-2 M. (Watch the stoichiometry — this is the classic trap.)
  2. Compute pOH:

pOH = -log(1.0 × 10-2) = 2.00

  1. Compute pH:

pH = 14.00 - 2.00 = 12.00

Dimensional analysis: molarity of Ba(OH)2 times (2 mol OH−/mol Ba(OH)2) = molarity of hydroxide — the factor of 2 comes from the balanced dissociation, Ba(OH)2 → Ba2+ + 2OH−.

Worked example 3: pH → concentration (inverse logarithm)

Problem. A solution has pH 5.42. Find [H3O+].

Solution.

  1. Invert the definition:

[H3O+] = 10-pH = 10-5.42

  1. Evaluate: 10-5.42 = 3.8 × 10-6 M.

Check the significant-figure rule in reverse: pH has two decimal places, so the concentration has two significant figures (3.8), matching. The solution is acidic (pH < 7), consistent with [H3O+] > 10-7 M.

Worked example 4: using Kw directly for a weak base scenario

Problem. A solution has [OH−] = 2.0 × 10-4 M. Find [H3O+], pH, and pOH at 25 °C.

Solution.

  1. From Kw:

[H3O+] = Kw[OH−] = 1.0 × 10-142.0 × 10-4 = 5.0 × 10-11 M

  1. Convert both:

pOH = -log(2.0 × 10-4) = 3.70,   pH = -log(5.0 × 10-11) = 10.30

  1. Verify with the sum rule: pH + pOH = 10.30 + 3.70 = 14.00. ✓ The solution is basic, as expected from [OH−] > [H3O+].

Common Confusions

Do Not ConfuseWithDifference
pH 6 vs pH 5 "slightly different"Tenfold differenceThe scale is logarithmic: pH 5 has 10 ×  the [H3O+] of pH 6.
pH 7 always neutralpH 7 neutral only at 25 °CKw changes with temperature, moving the neutral point; at 100 °C neutral pH is about 6.14.
Strong acid concentration = pHStrong acid concentration = [H3O+]The concentration equals [H3O+] (for one-proton acids); pH is the negative log of it.
Forgetting the factor of 2Ba(OH)₂ producing two OH−Polyhydroxide strong bases double (or triple) the hydroxide concentration — always write the dissociation.
pOH being "unnecessary"pOH being the mirror of pHBoth are needed; pH + pOH = 14 is the fastest check on any calculation.
Decimal places vs significant figures in pHReported pH digitsThe decimals of pH carry the significant figures of the concentration; the integer part only carries the exponent.
Eli, the EliExplains learning guide

Eli explains

The same idea, in plain words

Explain it like I’m 10

pH is like a "sourness meter" that uses a trick to handle really big numbers. Instead of writing 0.0000001 (too many zeros), scientists count the zeros and write 7. The more zeros, the less sour — that's why smaller pH numbers are more acidic. A change of one on the meter means ten times more (or less) sourness, like going from one scoop of lemon juice to ten scoops.

Key takeaways

  • pH = -log[H3O+]; pOH = -log[OH−]; at 25 °C, pH + pOH = 14.00.
  • [H3O+][OH−] = Kw = 1.0 × 10-14 at 25 °C — the master equation linking the four quantities.
  • Neutral at 25 °C: pH = 7.00. Acidic: pH < 7. Basic: pH > 7.
  • One pH unit = tenfold change in [H3O+].
  • Inversions: [H3O+] = 10-pH, [OH−] = 10-pOH.
  • Decimal places in pH = significant figures in concentration (e.g., 1.0 × 10-7 M → pH 7.00).
  • Strong acid: [H3O+] = acid concentration (per H+ released); strong base: [OH−] = base concentration (times hydroxide count).
  • pH is temperature-sensitive because Kw is; the 7.00 boundary assumes 25 °C.

Check yourself

6 review questions from the chapter. Try each one, then open the answer.

  1. What is the pH of a solution with [H3O+] = 1.0 × 10-8 M? Is it acidic or basic?

    Show answer

    pH = −log(1.0 × 10⁻⁸) = 8.00. Since pH > 7, the solution is basic (more OH− than H3O+).

  2. A solution has pH 3.25. What is [H3O+]?

    Show answer

    [H3O+] = 10-3.25 = 5.6 × 10-4 M.

  3. What is the pH of a 0.010 M solution of NaOH?

    Show answer

    NaOH releases one OH− per formula unit: [OH−] = 0.010 M, pOH = 2.00, pH = 14.00 − 2.00 = 12.00.

  4. How many times more acidic is pH 2 than pH 5?

    Show answer

    Three pH units, so 103 = 1000 times more acidic.

  5. If [OH−] = 1.0 × 10-6 M at 25 °C, find [H3O+], pH, and pOH.

    Show answer

    [H3O+] = Kw/[OH−] = 1.0 × 10-14/1.0 × 10-6 = 1.0 × 10-8 M; pOH = 6.00; pH = 14.00 − 6.00 = 8.00.

  6. Why does a pH value like 5.00 have two decimal places?

    Show answer

    The concentration has two significant figures (e.g., 1.0 × 10⁻⁵ M), and the number of decimal places in a logarithm equals the number of significant figures in the original number.

Keep learning

Ready to build on this? Continue to the next lesson.

Study tools & related lessonsKey vocabulary · Related

Key vocabulary

pH
-log10[H3O+]; a compact acidity scale
pOH
-log10[OH−]; the basicity counterpart
pKw
-logKw = 14.00 at 25 °C
logarithmic scale
A scale where each step multiplies by ten
acidic / basic / neutral
pH < 7 / pH > 7 / pH = 7 at 25 °C
hydronium ion (ceH3O+)
The actual form of the proton in water

Sources & references

  1. openstax.org — Chemistry 2e

This lesson was adapted from the open educational references above; their licenses and attributions are preserved. See Copyright & Licensing.

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