Advanced A level theoretical chemistry - acid-base equilibrium

Part 5.2 The theory of the pH scale, self–ionisation of water and K_w

[Author ©  Dr Phil Brown PhD: Doc Brown's exam revision notes suitable for A level chemistry students of advanced pre–university/college advanced level theoretical–physical chemistry courses:  acid–base equilibrium revision notes on pH scale and self–ionisation of water [updated April 29th 2026 *]

ALL my advanced A level theoretical chemistry revision study notes

Full index for Part 5 acid-base equilibria and calculations

5.0 Basic notes and equations on acids, bases, salts, uses of acid–base titrations

5.1 Equilibria:  Lewis and Bronsted–Lowry acid–base theories

5.2 Self–ionisation of water and pH scale (this page)

5.3 Strong acids – examples and pH calculations

5.4 Weak acids – examples & pH, K_a and pK_a calculations

5.5 Strong bases – examples and pH calculations

5.6 Weak bases – examples and pH, K_b and pK_b calculations

5.2 The pH scale and the self–ionisation of water

What is the pH scale? What exactly is the pH of a solution?

How do we measure the pH of a solution?

Why does covalent liquid water contain a tiny percentage of ions?

What is K_w, the self–ionisation constant for water?

• 5.2.1 Despite being essentially covalent, the highly polar water molecule does undergo a minute amount of self–ionisation.

  • 2H₂O_(l) H₃O⁺_(aq) + OH^–_(aq)

    • ΔH = +57.1 kJ mol^–1, at 25^oC, 298K, ionisation is endothermic.

  • Kc =	[H3O+(aq)] [OH–(aq)]
    	––––––––––––––––––––
    	[H2O(l)]2

  • However, since the concentration of water is effectively constant for dilute aqueous solutions,

    • and the density of water is 1g cm^–3, so the molarity of water in water is ~1000/18 = 55.5 moles per dm³, i.e. 55.5M in itself!

    • This is huge compared to the very small molarity of hydrogen ions or hydroxide ions in dilute acid solutions of dilute alkali solutions.

  • the equilibrium expression is simplified to:

    • K_w = [H₃O⁺_(aq)] [OH^–_(aq)] and equals 1.0 x 10^–14 mol² dm^–6 at 298 K.

    • and K_w is called the ionic product of water and its value will increase with increase in temperature as the self–ionisation of water is an endothermic process.

    • The hydrated proton [H₃O]⁺ is referred to as the hydronium ion or a types of oxonium ion.

    • However, many texts, including mine, often just refer to it as the hydrogen ion, but watch out in exams! be precise!

  • pK_w = –log(K_w) = 14, so please note ...

    1. log₁₀, log or lg means to logarithm to base 10

    2. pX means –log₁₀(X/units of X) and allows a wide range of values to be expressed in a simpler numerical scale and an increase/decrease of 1 pX unit is equal to factorial decrease/increase of 10 of the value of X. (see pH table below)

       • Note that X = 10^–pX

    3. pH = –log[H⁺_(aq)/mol dm^–3], which is the formal definition of pH, also ...

       • pOH = –log[OH^–_(aq)/mol dm^–3]

       • and pK_w = pH + pOH

       • and these will be explained in more detail later and a reminder that in associated calculations [x] means concentration of x in mol dm^–3.

       • Note that mathematically [H⁺_(aq)] = 10^–pH

    4. Later you will also come across in weak acid/base quantitative chemistry ...

       • pK_a = –log(K_a/mol dm^–3) and pK_b = –log(K_b/mol dm^–3).

• 5.2.2 Historically the H of pH is shorthand for the hydrogen ion, H⁺ and pH is a mathematical function of its concentration.

  • The p was used to mean power/potential in terms of H⁺ ion concentration, and it is mathematically –log to the base 10 of a concentration, which in this case is for the H⁺ ion concentration.

  • It is the – sign in the mathematical definition which means that the higher the acid/H⁺ ion concentration is, the lower the pH.

  • Note:

    • (i) The scale was devised to give a more 'reasonable' number system because of the huge range of concentrations possible that can have measurable chemical effects e.g. 10^–14 to 10¹ (means pH 14 to pH–1).

    • (ii) You can even talk about the pCl of seawater, which is a function of the concentration of the chloride ion, Cl–, from the salts in seawater and there are special electrodes that can measure pH, pCl or p of any other ion.

  • The pH of a solution is defined as minus log to the base 10 of the hydrogen ion concentration in mol dm^–3.

  • pH = –log( [H₃O⁺_(aq)]/mol dm^–3), and in the 'anti–log' format, [H₃O⁺_(aq)] = 10^–pH.

    • (NOT e^–pH, so get to know your calculator functions!)

  • log maybe shown on your calculator as log₁₀ or just lg.

    • (NOT natural logarithms log_e or ln)

  • pH = –log( [OH^–_(aq)]/mol dm^–3), and in the 'anti–log' format, [OH^–_(aq)] = 10^–pOH.

  • Therefore: pH + pOH = pK_w

  • In pure water [H₃O⁺_(aq)] = [OH^–_(aq)] at pH 7, but if anything is dissolved to form either hydrogen ions or hydroxide ions, then the pH will change e.g.

    • if [H₃O⁺_(aq)] > [OH^–_(aq)] then pH <7, acidic,

    • if [H₃O⁺_(aq)] < [OH^–_(aq)] then pH >7, alkaline.

• 5.2.3 Using the K_w expression the relative molarities of hydrogen ions and hydroxide ions in aqueous solution at various pH's can be calculated and are shown in the below. In terms of pH and molar concentrations ...

  • acidic: pH <7, [H⁺] > 10^–7, [H⁺_(aq)] > [OH^–_(aq)], [OH^–_aq)] < 10^–7 mol dm^–3

  • neutral: pH 7, [H⁺_(aq)] = [OH^–_(aq)] = 10^–7 mol dm^–3 (at 25^oC, 298K)

  • alkaline–basic: pH >7, [H⁺_(aq)] < 10^–7, [OH^–_(aq)] > [H⁺_(aq)], [OH^–_(aq)] > 10^–7 mol dm^–3

• 

• This means a change of 1 unit in the pH is equal to a ten–fold change in concentration of either the hydrogen/oxonium ion or hydroxide ion e.g.

  • A decrease of 1 pH unit means that [H⁺_(aq)] rises by a factor of 10, and [OH^–_(aq)] decreases by a factor of 10.

  • An increase of 1 pH unit means that [H⁺_(aq)] decreases by a factor of 10, and [OH^–_(aq)] rises by a factor of 10.

  • A change of 2 pH units is equal to factorial decrease/increase of 100 etc. check this out in the table above where the molarities are expressed as powers of 10. Incidentally please note that 10 = 1 x 10¹, 1 = 1 x 10⁰, 0.1 = 1 x 10^–1 and 10^–2 = 1 x 10^–2 etc. but I've used the briefest numerical description to fit the table across the page.

  • Note that for the self–ionisation: 2H₂O_(l) H₃O⁺_(aq) + OH^–_(aq)

    • ΔH = +57.1 kJ mol^–1, at 25^oC, 298K, an endothermic reaction, which means that increasing temperature, increases its acidity, i.e. the pH falls, but not a lot! though technically hot water is an extremely weak acid.

• So, the pH is dependent on the relative concentrations of the H⁺_(aq) and the OH^–_(aq) concentrations.

  • a high H⁺_(aq) concentration means a low pH and low OH^–_(aq) concentration, usually strong acid

  • lower H⁺_(aq) concentration means higher pH and higher OH^–_(aq) concentration, less acid

  • a high OH^–_(aq) concentration means a high pH and low H⁺_(aq) concentration, usually strong base/alkali

  • lower OH^–_(aq) concentration means lower pH and higher H⁺_(aq) concentration, less alkaline

• In general: pH 1–2 strong acids, pH 3–6 weak acids, pH 7 neutral, pH 8–11 weak base/alkali, pH 12–14 strong base/alkali

• Neutralisation ionically is: H⁺_(aq) + OH^–_(aq) ==> H₂O_(l) (exothermic)

  • The pH of a solution, or determining the neutralisation point, can be measured with ...

    • ... an indicator colour comparison card or indicator added to the solution and compared with a colour versus pH calibration chart to give an approximate value, and

    • a pH meter which is calibrated with 'buffer solutions' of exactly known pH.

  • When mixing an acid and alkali the neutralisation end–point can also be determined by

    • the point of maximum temperature rise

    • –

• 5.24 Acid and bases in non–aqueous media – liquid ammonia – another case of self–ionisation

  • Liquid ammonia can self–ionise just like water (need low temperatures and high pressure!).

  • 2NH₃ NH₄⁺ + NH₂^–

  • One ammonia molecule acts as the acid and another acts as the acid.

  • The ammonium ion, NH₄⁺, is the conjugate acid.

  • The amide ion, NH₂^–, is the conjugate base.

  • –

WHAT NEXT?

Advanced Equilibrium Chemistry Notes Part 1. Equilibrium, Le Chatelier's Principle–rules * Part 2. K_c and K_p equilibrium expressions and calculations * Part 3. Equilibria and industrial processes * Part 4 Partition between two phases, solubility product K_sp, common ion effect, ion–exchange systems * Part 5. pH, weak–strong acid–base theory and calculations * Part 6. Salt hydrolysis, acid–base titrations–indicators, pH curves and buffers * Part 7. Redox equilibria, half–cell electrode potentials, electrolysis and electrochemical series * Part 8. Phase equilibria–vapour pressure, boiling point and intermolecular forces watch out for sub–indexes to multiple sections or pages

Chemical Equilibrium Notes Parts 5 & 6 Index