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IMAT Chemistry · Acids, Bases and pH

The Water Equilibrium and the pH Scale

Water always holds a tiny, fixed product of H⁺ and OH⁻ ions; pH and pOH turn those very small concentrations into simple numbers on a scale of tens.

All Acids, Bases and pH notes

Why this matters

Every pH calculation in the chapter rests on this page. The older papers asked for the pH of a given hydrogen ion concentration, used Kw to turn a hydroxide concentration into a pH, and compared the hydrogen ion concentrations of two acids whose pH values differ by two.

Concept 1 of 3: The ionic product of water, Kw

Even pure water is not just molecules. A very small fraction of molecules pass a proton to a neighbour, making one H3O+\mathrm{H_3O^+} and one OH−\mathrm{OH^-}. The product of the two concentrations is fixed at a given temperature, so if one goes up, the other must go down. Adding acid does not remove the hydroxide ions; it only makes them rarer.

Definition

  • Water ionises: 2H2O⇌H3O++OH−\mathrm{2H_2O \rightleftharpoons H_3O^+ + OH^-}, often written H2O⇌H++OH−\mathrm{H_2O \rightleftharpoons H^+ + OH^-}.
  • Kw, the ionic product of water, is [H+][OH−]=1.0×10−14 mol2 L−2[\mathrm{H^+}][\mathrm{OH^-}] = 1.0 \times 10^{-14}\ \text{mol}^2\,\text{L}^{-2} at 25 °C, in pure water and in every aqueous solution.
  • Neutral means [H+]=[OH−][\mathrm{H^+}] = [\mathrm{OH^-}]. At 25 °C both are 1.0×10−7 mol/L1.0 \times 10^{-7}\ \text{mol/L}.
  • Acidic means [H+]>[OH−][\mathrm{H^+}] > [\mathrm{OH^-}]; basic (alkaline) means [OH−]>[H+][\mathrm{OH^-}] > [\mathrm{H^+}].
  • The ionisation is endothermic, so Kw grows with temperature. At 50 °C Kw is about 5.5×10−145.5 \times 10^{-14}, so neutral water there has a pH of about 6.6, and it is still neutral.

Ionic product of water at 25 °C

Kw=[H+][OH−]=1.0×10−14K_w = [\mathrm{H^+}][\mathrm{OH^-}] = 1.0 \times 10^{-14}
  • [H+][\mathrm{H^+}]hydrogen ion concentration, in mol/L
  • [OH−][\mathrm{OH^-}]hydroxide ion concentration, in mol/L

Worked example

A cleaning solution at 25 °C has [OH−]=2.0×10−3 mol/L[\mathrm{OH^-}] = 2.0 \times 10^{-3}\ \text{mol/L}. Find [H+][\mathrm{H^+}] and say whether the solution is acidic or basic.
Practice this conceptself-check · 3 quick reps

pH below 7 does not always mean acidic

The neutral point is pH 7 only at 25 °C. Kw increases with temperature, so hot pure water has a pH below 7 but is still neutral, because [H+]=[OH−][\mathrm{H^+}] = [\mathrm{OH^-}]. Neutral is defined by equal concentrations, not by the number 7.

Concept 2 of 3: pH as a logarithmic scale of hydrogen ion concentration

Hydrogen ion concentrations run from about 1 mol/L down to 10−1410^{-14} mol/L: awkward numbers. pH keeps only the power of ten and drops the minus sign. So each step of one pH unit is a factor of ten in [H+][\mathrm{H^+}], and a lower pH means more hydrogen ions.

Definition

  • pH =−log⁡10[H+]= -\log_{10}[\mathrm{H^+}], so [H+]=10−pH[\mathrm{H^+}] = 10^{-\text{pH}}.
  • At 25 °C: pH < 7 is acidic, pH = 7 is neutral, pH > 7 is basic.
  • A difference of nn pH units is a factor of 10n10^n in [H+][\mathrm{H^+}]. pH 3 has 100 times the [H+][\mathrm{H^+}] of pH 5.
  • No calculator in IMAT: learn log⁡2≈0.30\log 2 \approx 0.30, log⁡3≈0.48\log 3 \approx 0.48, log⁡5≈0.70\log 5 \approx 0.70. Then −log⁡(2×10−4)=4−0.30=3.70-\log(2 \times 10^{-4}) = 4 - 0.30 = 3.70.
  • pH can be below 0 (very concentrated strong acid) or above 14, though IMAT rarely goes there.

pH

pH=−log⁡10[H+][H+]=10−pH\text{pH} = -\log_{10}[\mathrm{H^+}] \qquad [\mathrm{H^+}] = 10^{-\text{pH}}
  • [H+][\mathrm{H^+}]hydrogen ion concentration, in mol/L

Worked example

Find the pH of a solution with [H+]=5.0×10−4 mol/L[\mathrm{H^+}] = 5.0 \times 10^{-4}\ \text{mol/L}, and the [H+][\mathrm{H^+}] of a solution of pH 9.
Practice this conceptself-check · 3 quick reps

pH is a scale of tens, and lower means more acid

pH 2 is not twice as acidic as pH 4: it has 100 times the hydrogen ion concentration. And because of the minus sign, the solution with the smaller pH number has the larger [H+][\mathrm{H^+}]. IMAT options often offer the ratio of the pH values or the reversed direction.

Concept 3 of 3: pOH and the rule pH + pOH = 14

pOH does for hydroxide ions what pH does for hydrogen ions. Taking the log of Kw turns the product into a sum, so the two numbers always add up to 14 at 25 °C. For a base it is easier to find pOH first and then subtract from 14.

Definition

  • pOH =−log⁡10[OH−]= -\log_{10}[\mathrm{OH^-}].
  • At 25 °C, pH+pOH=14\text{pH} + \text{pOH} = 14 (this is −log⁡Kw-\log K_w, sometimes written pKw\text{p}K_w).
  • A high pOH means few hydroxide ions, so a low pOH goes with a high pH.

pOH and pH at 25 °C

pOH=−log⁡10[OH−]pH+pOH=14\text{pOH} = -\log_{10}[\mathrm{OH^-}] \qquad \text{pH} + \text{pOH} = 14
  • [OH−][\mathrm{OH^-}]hydroxide ion concentration, in mol/L

Worked example

A solution at 25 °C has [OH−]=1.0×10−4 mol/L[\mathrm{OH^-}] = 1.0 \times 10^{-4}\ \text{mol/L}. Find its pOH, its pH and its [H+][\mathrm{H^+}].
Practice this conceptself-check · 3 quick reps

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

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