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Chemistry: Atoms First 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

Hydronium-ion concentrations in real solutions span fourteen or more orders of magnitude — from about 10 M in concentrated strong acid down to 10⁻¹⁴ M or below in strong base. Working with such numbers directly is clumsy, so chemists compress the scale with a . The pH of a solution is defined as

pH = -log[H3O+]

where [H3O+] is the molar concentration of hydronium ion. A related quantity, , is defined the same way for hydroxide ion:

pOH = -log[OH−]

Because Kw = [H3O+][OH−] = 1.0 × 10-14 at 25 °C, the two scales are tied together by

pH + pOH = 14.00

This topic develops those definitions, the arithmetic of logarithms, and the practical meaning of the scale. Everything that follows in this chapter — acid strength, salt hydrolysis, buffers, titrations — reports its results in pH units.

Why this matters

The pH scale is the standard language for acidity in virtually every applied field:

  • Medicine and nursing: Blood pH is tightly regulated near 7.4; a shift of a few tenths of a unit (acidosis or alkalosis) is a medical emergency. Urine pH, gastric fluid, and IV fluid choices all rely on pH thinking.
  • Biology and agriculture: Enzyme activity peaks at a characteristic optimal pH — which is why soil pH determines which crops thrive and fermentation tanks are monitored continuously.
  • Industry and the environment: Water-treatment plants adjust pH to control corrosion and disinfection; lakes, aquariums, and pools are monitored on the same scale. One pH unit is a tenfold change in hydronium concentration, so small pH shifts are large concentration changes.

The college version

Core Concepts

Logarithms compress a wide range

A logarithm answers the question "ten to what power?" log(1.0 × 10-3) = -3 because 10-3 = 0.001. The negative log converts tiny concentrations into convenient positive numbers: [H3O+] = 1.0 × 10-7 M gives pH 7, and [H3O+] = 1.0 × 10-3 M gives pH 3. Each decrease of one pH unit is a tenfold increase in hydronium: pH 4 is ten times more acidic than pH 5 and a hundred times more acidic than pH 6 — surprising because the scale looks linear.

Significant figures in pH

When you take a logarithm, digits before the decimal only carry the power of ten; digits after the decimal carry the significant figures. A concentration with two significant figures, such as 3.6 × 10-4 M, must be reported as pH with two decimals (3.44), not two total digits; a pH of 5.25 (two decimals) likewise implies two significant figures in [H3O+] (5.6 × 10-6 M).

pOH and the pH + pOH = 14.00 relationship

pOH is defined exactly like pH but for hydroxide ion. The connection comes from taking -log of both sides of the Kw expression:

-logKw = -log([H3O+][OH−]) = -log[H3O+] - log[OH−]

Since -logKw = -log(1.0 × 10-14) = 14.00,

pH + pOH = 14.00

at 25 °C. So every aqueous solution can be described by either scale: acidic has pH < 7 and pOH > 7; basic has pH > 7 and pOH < 7; neutral has both at 7.00. "Neutral means pH 7" is only strictly true at 25 °C, because Kw changes with temperature.

Measuring pH

Three common tools measure pH. Indicators are weak acids or bases whose color depends on the protonated form present (litmus turns red below pH 4.5, blue above pH 8.3); pH paper gives a rough reading by color match. A uses a glass electrode whose voltage depends on [H3O+] and reports pH directly — accurate, but it must be calibrated with standard buffers.

Dilution and pH

Diluting a strong acid solution by a factor of ten raises its pH by exactly one unit, because the hydronium concentration falls tenfold — as long as the solution remains far more concentrated than pure water's own 10-7 M background. Dilution can never turn an acidic solution basic: the pH approaches 7 (pure water's neutral value) but never crosses it.

How It Works / Step-by-Step Process

Worked example 1: from concentration to pH

Problem. A solution has [H3O+] = 3.6 × 10-4 M. Calculate its pH and classify it.

Solution.

  1. Write the defining equation: pH = -log[H3O+] and substitute:
  2. Substitute:

pH = -log(3.6 × 10-4) = -(0.556 - 4) = 3.44

(The log of 3.6 × 10-4 splits into log3.6 + log10-4 = 0.556 - 4.)

  1. The concentration had two significant figures, so pH is reported with two decimals: 3.44. Because pH < 7, the solution is acidic.

Worked example 2: from hydroxide to pH via pOH

Problem. At 25 °C, [OH−] = 2.0 × 10-3 M. Find the pH.

Solution.

  1. Write the pOH definition: pOH = -log[OH−] and substitute:

pOH = -log(2.0 × 10-3) = 2.70

  1. Convert with the sum rule: pH = 14.00 - pOH = 14.00 - 2.70 = 11.30.

Dimensional analysis note: the logarithm acts on the numeric value of the molarity; the result is a pure number. The solution is basic (pH > 7), consistent with [OH−] well above 10-7 M.

Worked example 3: from pH back to concentration

Problem. The pH of a solution is 5.25. Find [H3O+].

Solution.

  1. Rearrange the definition: [H3O+] = 10-pH and substitute:

[H3O+] = 10-5.25 = 5.6 × 10-6 M

  1. The pH had two decimals, so the concentration gets two significant figures. Check: -log(5.6 × 10-6) = 5.25. ✓

Worked example 4: dilution with dimensional analysis

Problem. Exactly 10.0 mL of 0.010 M HCl is diluted to 100.0 mL. Find the new pH.

Solution.

  1. Use the dilution relationship M1V1 = M2V2 (moles of solute are unchanged), solving for the new molarity:

M2 = M1V1V2 = (0.010 M)(10.0 mL)100.0 mL = 1.0 × 10-3 M

Dimensional analysis: (M)(mL)/(mL) = M, so the result is a molarity. (HCl is a strong acid, so [H3O+] = 1.0 × 10-3 M.)

  1. Convert to pH: pH = -log(1.0 × 10-3) = 3.00. The tenfold dilution raised the pH from 2.00 to 3.00 — exactly one unit.

Common Confusions

Do Not ConfuseWithDifference
pH 4 vs pH 5Linear difference of oneEach pH unit is a tenfold change; pH 4 is 10× more acidic than pH 5.
Neutral = pH 7Neutral at any temperatureKw changes with temperature, so neutral pH is exactly 7.00 only at 25 °C.
Significant figures in pHDigits before the decimalIn a log, only digits after the decimal are significant; pH 3.44 has two significant figures.
Low pHLow acidityLow pH means HIGH [H3O+]; pH 2 is far more acidic than pH 9.
pOHpHpOH measures hydroxide, not hydronium; pH + pOH = 14.00, so low pH means high pOH.
Diluting an acidDiluting toward basicDilution moves pH only toward 7, never past it.
Eli, the EliExplains learning guide

Eli explains

The same idea, in plain words

Explain it like I’m 10

The pH scale is a number line from 0 to 14 telling how "sour" or "soapy" a liquid is: low numbers mean lots of hydronium (acidic, like lemon juice), high numbers mean lots of hydroxide (basic, like soap), and 7 is the middle, like pure water. Each step is a tenfold change, so pH 4 is ten times more acidic than pH 5. pOH is the same line measured from the other end, and the two ends always add up to 14.

Key takeaways

  • pH = -log[H3O+] and pOH = -log[OH−]; at 25 °C, pH + pOH = 14.00.
  • Acidic: pH < 7; neutral: pH = 7; basic: pH > 7 (at 25 °C). Each pH unit = tenfold change in [H3O+].
  • To go from pH back to concentration: [H3O+] = 10-pH.
  • Significant figures: decimal places in pH = significant figures in concentration.
  • Kw = 1.0 × 10-14 at 25 °C; its value (and therefore "neutral" pH) changes with temperature.
  • Diluting a strong acid by 10× raises pH by exactly 1; dilution never pushes pH past 7.
  • pH meters must be calibrated with buffers; indicators give only approximate values.

Check yourself

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

  1. Write the definitions of pH and pOH, and the relationship between them at 25 °C.

    Show answer

    pH = -log[H3O+], pOH = -log[OH−], and pH + pOH = 14.00 at 25 °C.

  2. A solution has [H3O+] = 1.0 × 10-9 M. What are its pH and pOH? Acidic, basic, or neutral?

    Show answer

    pH = -log(1.0 × 10-9) = 9.00, so pOH = 14.00 - 9.00 = 5.00. Basic.

  3. The pH of a solution is 8.70. What is [OH−]? (Hint: start with pOH.)

    Show answer

    pOH = 14.00 - 8.70 = 5.30, so [OH−] = 10-5.30 = 5.0 × 10-6 M.

  4. How many times more acidic is a solution at pH 3 than one at pH 6?

    Show answer

    One thousand times (three pH units = 103 in concentration).

  5. Why must a pH value of 7.00 be qualified with a temperature?

    Show answer

    Because Kw depends on temperature, the neutral pH shifts from 7.00 as temperature changes.

Keep learning

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

Study tools & related lessonsKey vocabulary · Related

Key vocabulary

pH
-log[H3O+]; the standard measure of acidity
pOH
-log[OH−]; the basicity twin of pH
logarithm
The exponent to which 10 must be raised to give a number
neutral solution
[H3O+] = [OH−], so pH = pOH = 7.00 at 25 °C
indicator
A dye whose color changes with pH
pH meter
An electrode-based instrument reporting pH directly

Sources & references

  1. openstax.org — Chemistry Atoms First 2e

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

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