Equilibrium: Fundamentals, Acids and Bases, and Solubility

This is Unit Ten of Chemistry, Part Three: buffers and solubility. Part One built the machinery — the equilibrium constant K, the reaction quotient Q, and the ICE table that solves for an unknown equilibrium concentration. Part Two applied that machinery to acids and bases, arriving at Ka for a weak acid, at the pH calculations that follow from it, and at the titration curve. This part spends all of it twice: once on acid-base systems built to hold a pH steady, and once on solids that barely dissolve. Two new constants appear, Ksp and Kf, and neither is a new kind of number — each is the same equilibrium constant named for the particular equilibrium it describes. The through-line is the comparison of Q against K, and by the end you will be able to say why a buffer resists pH change and what it must contain; say what buffer capacity measures that the acid-to-base ratio does not; say what Ksp describes and what it says nothing about; say why a common ion lowers solubility while a ligand raises it; and predict whether mixing two solutions produces a precipitate.

Four entries. A buffer is a weak acid and its conjugate base kept in the same solution, and the four entries here are what that arrangement does, what it requires, the equation that prices it in one line, and the limit that equation cannot see from inside itself.

Both directions covered.

Chem 1887 Buffers: How They Resist pH Change

A buffer holds a weak acid and its conjugate base together, so added hydronium is consumed by the base and added hydroxide by the acid. A buffered solution's pH barely moves where pure water swings sharply. It resists pH change; it does not prevent it.

Half a pair is not a buffer.

Chem 1890 Buffer Composition Requirement

An effective buffer needs comparable amounts of both the weak acid and its conjugate base, because running out of either removes its ability to absorb more of one kind of addition. Acetic acid with sodium acetate buffers well; acetic acid alone does not.

One line, if you already know the concentrations.

Chem 1893 Henderson-Hasselbalch Equation Application

Take the negative log of Ka to get pKa, then add the log of the ratio of conjugate base to acid. Thirty hundredths molar acetate over twenty hundredths molar acetic acid, pKa four point seven four, gives about four point nine two. The equation is the weak-acid equilibrium expression rearranged, which is why it skips the ICE table. Inverting the ratio flips the sign of the log term and gives the wrong pH outright.

How much before it gives.

Chem 1896 Buffer Capacity

Buffer capacity is how much strong acid or base a buffer absorbs before its pH moves substantially, and it grows with the conjugate pair's total concentration. An ideal one-to-one ratio at low total concentration still has weak capacity — the ratio sets the pH, the amounts set how long it holds.

Two worked problems close the acid-base half of this unit. The first builds a buffer, stresses it, and stresses it again until it fails, showing that the Henderson-Hasselbalch equation reports its own breakdown rather than needing a separate warning bolted on. The second returns to the titration curve Part Two plotted and calculates the pH at four points along it. That curve is worth restating before we use it: its equivalence point is the calculated moment where acid and base have exactly reacted, distinct from the endpoint, which is where an indicator visibly changes colour; and at exactly half the equivalence volume, a weak acid's curve reads pKa directly off the pH axis.

Built, then pushed past its limit.

Chem 1899 Capstone: Buffer pH Calculation

Four tenths of a mole each of acetic acid and sodium acetate in one litre give equal moles, so pH equals pKa, four point seven four — and a full ICE table returns the identical answer, because Henderson-Hasselbalch is that equilibrium expression rearranged. Adding four hundredths of a mole of hydrochloric acid converts that much acetate to acid, and the new ratio gives four point six five, a modest drop where pure water would collapse several units. A further four tenths of a mole consumes every remaining acetate, and the equation returns the log of zero — undefined. Capacity is the equation announcing its own limit.

One curve, four different tools.

Chem 1903 Capstone: Titration Curve pH-at-Stages Synthesis

Twenty-five millilitres of tenth-molar acetic acid titrated with tenth-molar sodium hydroxide, equivalence at twenty-five millilitres. Before any titrant, acetic acid's weak-acid ICE table alone gives about two point eight seven. At half-equivalence, acid and conjugate base are equimolar and Henderson-Hasselbalch returns pKa, four point seven four. At equivalence only acetate remains, and its hydrolysis run as a weak-base ICE table gives a basic pH near eight point seven. Past equivalence, excess hydroxide dominates and the strong-base shortcut gives about twelve. No single equation spans the curve. Recognising which of four already-built tools each stage calls for is the skill.

The unit turns from acids to solids, and the machinery does not change. A sparingly soluble solid sitting in contact with its own dissolved ions is an equilibrium like any other, with an equilibrium constant of its own. Four entries: what that constant is, how a solubility is read from it, what happens when the solution already holds one of the ions, and how that case is calculated.

The constant for dissolving.

Chem 1907 Solubility Product Ksp Defined

Ksp is the equilibrium constant for a sparingly soluble solid dissolving into its ions — the product of ion concentrations raised to their coefficients, with the solid omitted, as heterogeneous equilibria always omit it. For silver chloride, Ksp is silver-ion concentration times chloride-ion concentration. It describes a solid at its solubility limit and says nothing about an unsaturated solution.

Solve for x, and x is the answer.

Chem 1910 Molar Solubility from Ksp

Build an ICE table where dissolution produces x moles per litre of each ion according to its coefficient, substitute into the Ksp expression, and solve for x — which is the molar solubility itself. Silver chloride at Ksp one point eight times ten to the minus ten gives x about one point three times ten to the minus five molar. For a salt like silver chromate, the ion carrying coefficient two enters as two x.

The solution already holds one ion.

Chem 1913 Common-Ion Effect on Solubility

A solution already holding one of a solid's ions raises Q above Ksp, and the equilibrium shifts back toward undissolved solid — so the solid dissolves less. Silver chloride dissolves less in sodium chloride solution than in pure water. This is not a new rule beyond Le Châtelier; it is the same Q-against-K stress applied to a solubility equilibrium.

The Initial row is not zero.

Chem 1916 Molar Solubility with a Common Ion Present

Write the ICE table with the common ion's existing concentration already in the Initial row, substitute into Ksp, and solve for x — usually with the small-x approximation, since suppression is heavy enough that x barely moves the total. Silver chloride in tenth-molar sodium chloride gives x about one point eight times ten to the minus nine molar, thousands of times below plain water. Omitting the pre-existing concentration badly overstates the result.

Three entries on the reverse question. Rather than asking how much solid dissolves, these ask whether a solid appears at all when two solutions are mixed — and if several could appear, which one appears first. The test is the Q-against-K comparison this unit opened with, unchanged.

Compute Q, compare to Ksp.

Chem 1919 Precipitation Prediction via Q vs. Ksp

Mixing two solutions forms a precipitate if the resulting ion concentrations give Q greater than Ksp. Q of five times ten to the minus eight against a Ksp of one point eight times ten to the minus ten predicts a precipitate. The standard error is forgetting that mixing dilutes each ion before Q is computed, which turns a correct comparison into a wrong call.

A reaction type, named.

Chem 1922 Precipitation as a Reaction Type

A precipitation reaction exchanges ions between two soluble compounds and forms a product whose ions exceed its Ksp, dropping out as a solid. Silver nitrate and sodium chloride exchange partners; silver and chloride exceed silver chloride's Ksp and precipitate, while sodium and nitrate stay dissolved. Not every ion exchange precipitates — one occurs only where some combination's Q exceeds its Ksp.

Least soluble goes first.

Chem 1925 Selective Precipitation

Raising a reagent's concentration gradually precipitates the least soluble compound first, while the other ions stay dissolved — which separates them. Adding chloride slowly to a solution of silver and lead ions precipitates silver chloride well before enough accumulates for lead chloride. The separation depends on a real gap between the two Ksp values; near-identical ones will not separate cleanly.

Two entries and a capstone. A fourth constant joins Ka, Kb and Ksp, and it follows the same pattern each of those did: a formation constant named for the equilibrium it measures. Then the unit's closing problem takes one salt and runs it through every case this part has built.

A metal ion, and its ligands.

Chem 1928 Complex Ion Formation and Formation Constant Kf

A complex ion forms when a metal ion, acting as a Lewis acid, bonds several Lewis-base ligands, and Kf measures how far that equilibrium favours the complex. Silver with two ammonia molecules has a large Kf, so the complex predominates once ammonia is present. A large Kf mirrors a large Ka or Ksp — it describes the equilibrium's extent, not its speed.

Remove the ion, and more dissolves.

Chem 1931 Complexation's Effect on Solubility

A ligand that forms a strong complex with a dissolved metal ion pulls its free concentration down, which drops Q below Ksp and lets more solid dissolve. Silver chloride, nearly insoluble in water, dissolves readily once excess ammonia forms the silver-ammonia complex. The solid did not become more soluble — complexation removed the free ion that the solubility equilibrium was measuring, which is the opposite manoeuvre to the common-ion effect on the same equation.

One salt, four added species.

Chem 1935 Capstone: Ksp, Common-Ion, and Complexation Synthesis

Silver chloride, Ksp one point eight times ten to the minus ten. In pure water an ICE table gives a molar solubility near one point three times ten to the minus five. In fifty-thousandths molar sodium chloride, the common ion drops that to about three point six times ten to the minus nine. Mixing equal volumes of two dilute solutions halves each concentration and still gives a Q above Ksp, so a precipitate forms. Adding excess ammonia forms the silver-ammonia complex, pulls free silver down, drops Q below Ksp, and the solid redissolves. One Ksp value, four added species, four outcomes, one comparison.

Five things to carry out of this part, and out of the unit. A buffer resists pH change because it holds a weak acid and its conjugate base together, so an addition of either kind is consumed by one of the pair — and it needs comparable amounts of both, because half a pair is not a buffer. Buffer capacity is a separate property from the ratio: the ratio sets where the pH sits, the pair's total concentration sets how much addition it absorbs before that pH moves. Ksp is the equilibrium constant for a sparingly soluble solid dissolving into its ions, describing that solid at its solubility limit — it says nothing about a solution not yet saturated. A common ion lowers solubility by raising Q above Ksp and pushing the equilibrium back toward solid; a ligand raises solubility because complexation pulls the free metal ion out of solution and drops Q below Ksp. Same equation, opposite directions, and in neither case did the solid itself change. And mixing two solutions precipitates when the diluted ion concentrations give a Q greater than Ksp — the comparison this whole unit was built on. Ka, Kb, Ksp and Kf name equilibrium constants of four particular kinds of equilibrium, not four sorts of quantity. Unit Eleven takes up thermodynamics and electrochemistry.