From observations to proton transfer

An acid is a chemical species that donates a proton, $H^+$, to another species in a Brønsted–Lowry reaction. A base is a chemical species that accepts a proton, $H^+$, from another species in a Brønsted–Lowry reaction. This definition is more useful than the older “sour/slippery/litmus” description because it explains what is moving in the reaction.

The older Arrhenius idea still has value: acids form $H^+$ in water and bases form $OH^-$ in water. Its limitation is that it is tied to aqueous solutions and misses bases such as ammonia. Brønsted–Lowry theory is broader: in $NH_3(g) + HCl(g) \to NH_4Cl(s)$, HCl donates $H^+$, so HCl is the acid; $NH_3$ accepts $H^+$, so $NH_3$ is the base. This is why the definition of acid has evolved over time: better models explain more observations with fewer awkward exceptions.

$H^+$ and $H_3O^+$ in water

A bare proton is far too charge-dense to wander freely through water. In aqueous solution it attaches to water to form $H_3O^+$. In equations, IB accepts both $H^+(aq)$ and $H_3O^+(aq)$: $H^+(aq)$ is convenient shorthand, while $H_3O^+(aq)$ reminds us what is actually present.

For example, hydrogen chloride reacting with water can be written as either:

$HCl(aq) \to H^+(aq) + Cl^-(aq)$

or, more explicitly,

$HCl(aq) + H_2O(l) \to H_3O^+(aq) + Cl^-(aq)$.

Base and alkali

An alkali is a base that is soluble in water and produces hydroxide ions in aqueous solution. Sodium hydroxide is both a base and an alkali. Copper(II) oxide is a base, because it reacts with acids, but it is not an alkali because it is not water-soluble. Keep that distinction tidy: all alkalis are bases, but not all bases are alkalis.

To deduce the acid and base in a reaction, look for the species that loses $H^+$ and the species that gains $H^+$. In $OH^-(aq) + H^+(aq) \to H_2O(l)$, $OH^-$ is the Brønsted–Lowry base because it accepts a proton.

Conjugates differ by one $H^+$

A conjugate acid–base pair is a pair of chemical species that differ by exactly one proton. The acid member has one more $H^+$ than the base member. When an acid donates $H^+$, it becomes its conjugate base; when a base accepts $H^+$, it becomes its conjugate acid.

For the reaction

$$HCN(aq) + H_2O(l) \rightleftharpoons H_3O^+(aq) + CN^-(aq)$$

there are two conjugate pairs: $HCN/CN^-$ and $H_3O^+/H_2O$. HCN donates $H^+$ to become $CN^-$, while $H_2O$ accepts $H^+$ to become $H_3O^+$.

To write a conjugate base, remove one $H^+$ and reduce the charge by 1: $H_2PO_4^- \to HPO_4^{2-}$. To write a conjugate acid, add one $H^+$ and increase the charge by 1: $SO_4^{2-} \to HSO_4^-$.

This is especially useful for polyatomic anions. Common conjugate acids include $NO_3^- \to HNO_3$, $CO_3^{2-} \to HCO_3^-$, $HCO_3^- \to H_2CO_3$, $PO_4^{3-} \to HPO_4^{2-}$, and $CH_3COO^- \to CH_3COOH$. Notice that you add only one proton at a time; $H_2SO_4$ and $SO_4^{2-}$ are not a conjugate pair because they differ by two protons.

Amphiprotic species

An amphiprotic species is a chemical species that can both donate a proton and accept a proton. Water is the classic example:

$H_2O(l) \rightleftharpoons H^+(aq) + OH^-(aq)$ water acting as an acid

$H_2O(l) + H^+(aq) \rightleftharpoons H_3O^+(aq)$ water acting as a base

Water can even react with itself:

$H_2O(l) + H_2O(l) \rightleftharpoons H_3O^+(aq) + OH^-(aq)$.

Hydrogencarbonate, $HCO_3^-$, is another favourite. As an acid:

$HCO_3^-(aq) + H_2O(l) \rightleftharpoons CO_3^{2-}(aq) + H_3O^+(aq)$

As a base:

$HCO_3^-(aq) + H_2O(l) \rightleftharpoons H_2CO_3(aq) + OH^-(aq)$.

Amphiprotic and amphoteric are not identical

An amphoteric species is a substance that can react with both acids and bases. All amphiprotic species are amphoteric, but not all amphoteric species are amphiprotic. Zinc oxide, for instance, reacts with acids and alkalis but has no hydrogen atom to donate, so it is amphoteric but not amphiprotic.

Metal oxides tend to be basic, non-metal oxides tend to be acidic, and oxides near the metal–non-metal divide can be amphoteric. This helps explain why sulfur and nitrogen oxides cause acid rain: they are non-metal oxides that form acids after reaction with water and oxygen in the atmosphere.

Amino acids give a nice biological example. In water, an amino acid can exist as a zwitterion, with $-NH_3^+$ and $-COO^-$ groups in the same species. In acid it accepts $H^+$ at the carboxylate end; in base it donates $H^+$ from the ammonium end.

Why a logarithmic scale is useful

Aqueous hydrogen ion concentrations often span many powers of ten, so we use a logarithmic scale. The relationship is:

$pH = -\log_{10}[H^+]$, where $pH$ is the acidity scale value (dimensionless) and $[H^+]$ is the equilibrium concentration of hydrogen ions in aqueous solution ($\mathrm{mol\ dm^{-3}}$).

Rearranging gives:

$[H^+] = 10^{-pH}$, where $[H^+]$ is the equilibrium concentration of hydrogen ions in aqueous solution ($\mathrm{mol\ dm^{-3}}$) and $pH$ is the acidity scale value (dimensionless).

A decrease of 1 pH unit means $[H^+]$ is ten times larger. pH 2 is not “a bit more acidic” than pH 3; it has ten times the hydrogen ion concentration.

Acidic, neutral and basic at $298\ \mathrm{K}$

At $298\ \mathrm{K}$, a neutral aqueous solution has equal hydrogen and hydroxide ion concentrations. Acidic, neutral and basic solutions are recognized by comparing $[H^+]$ and $[OH^-]$, not just by memorising pH numbers:

Measuring pH

Universal indicator gives an estimate because its colour is matched to a chart. It is quick and cheap, but subjective and usually only precise to about one pH unit. A pH probe gives a more precise digital reading and is better when you need continuous data, small pH changes, coloured solutions, or a titration curve. The probe must be calibrated, usually with buffer solutions of known pH.

A sketch of pH against $[H^+]$ is a decreasing logarithmic curve: it falls steeply at very small $[H^+]$ and then becomes flatter as $[H^+]$ increases. This shape is the whole reason pH is convenient — it compresses a huge concentration range into manageable numbers.

Water ionizes slightly

Water is mostly molecular, but a tiny fraction ionizes:

$H_2O(\text{l}) \rightleftharpoons H^+(\text{aq}) + OH^-(\text{aq})$

For dilute aqueous solutions, this equilibrium is summarized by:

$K_p = [H^+][OH^-]$, where $K_p$ is the ion product constant of water ($\mathrm{mol^2\,dm^{-6}}$), $[H^+]$ is the equilibrium concentration of hydrogen ions ($\mathrm{mol\,dm^{-3}}$) and $[OH^-]$ is the equilibrium concentration of hydroxide ions ($\mathrm{mol\,dm^{-3}}$).

The usual symbol is written as $K_w$. At 298 K, $K_w = 1.00 \times 10^{-14}\ \mathrm{mol^2\,dm^{-6}}$. This value is given in the data booklet.

Because the product $[H^+][OH^-]$ is fixed at a given temperature, $[H^+]$ and $[OH^-]$ have an inverse relationship. Increase $[H^+]$, and $[OH^-]$ must decrease. That is Le Châtelier’s principle in action: adding acid pushes the water ionization equilibrium left.

In pure water at 298 K, $[H^+] = [OH^-]$, so each is $1.00 \times 10^{-7}\ \mathrm{mol\,dm^{-3}}$. If the temperature rises, water ionizes to a greater extent, so $K_w$ increases. That is why “neutral pH” is exactly 7 only under the usual 298 K assumption.

To classify a solution, compare the two ion concentrations: acidic if $[H^+] > [OH^-]$, neutral if equal, and basic if $[H^+] < [OH^-]$.

Strength means extent of ionization

A strong acid is an acid that ionizes completely in aqueous solution. A weak acid is an acid that ionizes only partially in aqueous solution. Similarly, a strong base is a base that ionizes completely or dissociates completely to produce the basic species in water, while a weak base is a base that reacts only partially with water or acids to accept protons.

Use a single arrow for strong acids and bases, and an equilibrium arrow for weak ones:

$HCl(aq) \to H^+(aq) + Cl^-(aq)$

$CH_3COOH(aq) \rightleftharpoons H^+(aq) + CH_3COO^-(aq)$

The strong acids you are expected to know are $HCl$, $HBr$, $HI$, $HNO_3$ and $H_2SO_4$. Group 1 hydroxides, such as $LiOH$, $NaOH$ and $KOH$, are strong bases.

Strong/weak is not concentrated/dilute

A concentrated solution is a solution that contains a large amount of solute per unit volume. A dilute solution is a solution that contains a small amount of solute per unit volume. Strength describes ionization; concentration describes how much was dissolved. A dilute strong acid is still strong. A concentrated weak acid is still weak.

Compares strength with concentration using strong and weak acid examples.

Direction of acid–base equilibria

Acid–base equilibria favour the side with the weaker acid and weaker base — the weaker conjugates. A strong acid has a very weak conjugate base; for example, $Cl^-$ barely acts as a base in water. A weak acid has a conjugate base that is more willing to accept $H^+$.

For hydrogen halides, acid strength increases down group 17 because the $H-X$ bond becomes longer and weaker, so proton donation becomes easier: $HF$ is weak, while $HCl$, $HBr$ and $HI$ are strong.

In the laboratory, equal-concentration strong and weak acids can be distinguished by pH, electrical conductivity, and rate of reaction with a reactive metal or carbonate. The strong acid gives a lower pH, higher conductivity and faster gas production because it has a higher concentration of mobile ions at the start.

Neutralization patterns

A neutralization reaction is a proton transfer reaction in which an acid reacts with a base to form a salt, often with water as a product. A salt is an ionic compound formed when the replaceable hydrogen ion of an acid is replaced by a cation, or when an acid reacts with a base.

Core equations you should be able to formulate:

The net ionic equations show the proton transfer most clearly:

$H^+(aq) + OH^-(aq) \to H_2O(l)$

$CO_3^{2-}(aq) + 2H^+(aq) \to CO_2(g) + H_2O(l)$

$HCO_3^-(aq) + H^+(aq) \to CO_2(g) + H_2O(l)$.

Bases in this topic include ammonia, amines, soluble carbonates and soluble hydrogencarbonates. Organic acids, such as carboxylic acids, behave as weak acids; for example:

$CH_3COOH(aq) + NH_3(aq) \rightleftharpoons CH_3COO^-(aq) + NH_4^+(aq)$.

Parent acid and parent base of a salt

To identify the parent acid and base, split the salt into ions, then add $H^+$ to the anion and $OH^-$ to the cation as needed. For $K_2SO_4$, the ions are $K^+$ and $SO_4^{2-}$; the parent base is $KOH$ and the parent acid is $H_2SO_4$. For ammonium salts, the parent base may be written as ammonia, $NH_3$, or ammonium hydroxide, $NH_4OH$.

Salts formed in neutralization can often be separated by crystallization if they are soluble: gently evaporate some water, allow crystals to form, then filter and dry them. If an insoluble salt is made, filtration is the natural separation step.

Neutralization reactions are exothermic because the bonds and attractions formed in the products, especially O–H bonds in water and ionic attractions in the salt solution or solid, are energetically favourable compared with the bonds broken. If an acid reacts with a metal to release hydrogen gas, that reaction is not just acid–base; it is also a redox reaction because electrons are transferred to $H^+$ to form $H_2$.

Strong acid–strong base curves

A pH curve is a graph that shows how the pH of a reaction mixture changes as titrant volume is added during a titration. For this SL statement, only monoprotic strong acid–strong base neutralizations are assessed.

For a strong acid analyte titrated with a strong base, the curve starts at low pH, rises slowly at first, then rises very steeply near the equivalence point, and finally levels off at high pH when base is in excess. For a strong base analyte titrated with a strong acid, the same shape is inverted.

The equivalence point is the point in a titration where reactants have been mixed in the exact stoichiometric ratio shown by the balanced equation. That is why it is also called the stoichiometric point. In a strong acid–strong base titration at 298 K, the equivalence point is at pH 7 because the salt formed does not hydrolyse significantly.

The y-intercept tells you the initial pH of the analyte before any titrant is added. The steep vertical region marks rapid pH change near equivalence. The flattening region tells you the pH is now controlled mainly by excess titrant.

In titration calculations, the amount of solute is found from $n = cV$, where $n$ is amount of substance (mol), $c$ is amount concentration ($\mathrm{mol\ dm^{-3}}$) and $V$ is solution volume ($\mathrm{dm^3}$). At equivalence, use the balanced equation ratio to connect the amount of acid to the amount of base. For a $1:1$ monoprotic reaction such as $HCl + NaOH \to NaCl + H_2O$, the amounts of HCl and NaOH are equal at equivalence.

The hydroxide version of pH

pOH is a logarithmic scale value that describes hydroxide ion concentration in aqueous solution. It is defined by:

$pOH = -\log_{10}[OH^-]$, where $pOH$ is the basicity scale value (dimensionless) and $[OH^-]$ is the equilibrium concentration of hydroxide ions (mol dm$^{-3}$).

Rearranging gives:

$[OH^-] = 10^{-pOH}$, where $[OH^-]$ is the equilibrium concentration of hydroxide ions (mol dm$^{-3}$) and $pOH$ is the basicity scale value (dimensionless).

At 298 K, combining the pH, pOH and $K_w$ relationships gives:

$pH + pOH = 14.00$, where $pH$ is the acidity scale value (dimensionless) and $pOH$ is the basicity scale value (dimensionless).

So you can move between all four quantities: $[H^+]$, $[OH^-]$, pH and pOH. For example, if pH is known, find pOH by $14.00 - pH$, then find $[OH^-]$ using $10^{-pOH}$. If $[OH^-]$ is known, find pOH first, then pH.

Conversion steps linking [H⁺], pH, pOH and [OH⁻] at 298 K.

Dissociation constants measure weak strength

For a weak acid, written generally as $HA$:

$HA(aq) \rightleftharpoons H^+(aq) + A^-(aq)$

$K_a = [H^+][A^-]/[HA]$, where $K_a$ is the acid dissociation constant ($\mathrm{mol\ dm^{-3}}$ for this expression), $[H^+]$ is the equilibrium concentration of hydrogen ions ($\mathrm{mol\ dm^{-3}}$), $[A^-]$ is the equilibrium concentration of conjugate base ($\mathrm{mol\ dm^{-3}}$), and $[HA]$ is the equilibrium concentration of undissociated acid ($\mathrm{mol\ dm^{-3}}$).

For a weak base, written generally as $B$:

$B(aq) + H_2O(l) \rightleftharpoons BH^+(aq) + OH^-(aq)$

$K_b = [BH^+][OH^-]/[B]$, where $K_b$ is the base dissociation constant ($\mathrm{mol\ dm^{-3}}$ for this expression), $[BH^+]$ is the equilibrium concentration of conjugate acid ($\mathrm{mol\ dm^{-3}}$), $[OH^-]$ is the equilibrium concentration of hydroxide ions ($\mathrm{mol\ dm^{-3}}$), and $[B]$ is the equilibrium concentration of unprotonated base ($\mathrm{mol\ dm^{-3}}$).

Water is omitted from the expression because it is the solvent and its concentration is effectively constant in dilute solution.

Larger $K_a$ means a stronger weak acid. Larger $K_b$ means a stronger weak base. The p-values reverse the comparison:

$pK_a = -\log_{10}K_a$, where $pK_a$ is the negative logarithmic acid strength value (dimensionless) and $K_a$ is the acid dissociation constant.

$pK_b = -\log_{10}K_b$, where $pK_b$ is the negative logarithmic base strength value (dimensionless) and $K_b$ is the base dissociation constant.

So: lower $pK_a$ means stronger acid; lower $pK_b$ means stronger base. This is a very common place for students to flip the logic — remember that p-values are negative logs.

How dissociation constants and p-values compare weak acid and base strength.

Deriving the conjugate relationship

For the conjugate pair $HA/A^-:$

$HA(aq) \rightleftharpoons H^+(aq) + A^-(aq)$

$K_a = [H^+][A^-]/[HA]$

and

$A^-(aq) + H_2O(l) \rightleftharpoons HA(aq) + OH^-(aq)$

$K_b = [HA][OH^-]/[A^-]$

Multiplying the two expressions cancels $[HA]$ and $[A^-]$, leaving:

$K_a \times K_b = K_w$, where $K_a$ is the acid dissociation constant of an acid, $K_b$ is the base dissociation constant of its conjugate base, and $K_w$ is the ion product constant of water.

At 298 K, this also gives $pK_a + pK_b = 14.00$ for a conjugate acid–base pair. Do not apply this to unrelated acids and bases.

Calculations without quadratics

The syllabus does not expect quadratic equations. When $K_a$ or $K_b$ is very small, the extent of ionization is small, so the equilibrium concentration of the weak acid or base is usually taken as approximately equal to its initial concentration.

For a weak acid of initial concentration $C$, if $x = [H^+]$ at equilibrium, then often:

$K_a \approx x^2/C$, where $x$ is the equilibrium concentration of $H^+$ formed by the weak acid $(\text{mol dm}^{-3})$ and $C$ is the initial concentration of the weak acid $(\text{mol dm}^{-3})$.

For a weak base:

$K_b \approx y^2/C_b$, where $y$ is the equilibrium concentration of $OH^-$ formed by the weak base $(\text{mol dm}^{-3})$ and $C_b$ is the initial concentration of the weak base $(\text{mol dm}^{-3})$.

State the approximation when you use it. The chemical reason is simple: weak acids and weak bases mostly remain un-ionized.

Hydrolysis of salt ions

Hydrolysis is a reaction in which an ion reacts with water to form its conjugate acid or conjugate base, changing $[H^+]$ or $[OH^-]$. A salt solution is not automatically neutral. Its pH depends on whether its ions are conjugates of weak acids or weak bases.

The rule I use in class is: ions from strong parents usually do nothing; ions from weak parents hydrolyse.

Salt solution pH from parent acid/base strengths and ion hydrolysis.

Examples you should be comfortable writing:

Ammonium ion, acidic:

$$NH_4^+(aq) + H_2O(l) \rightleftharpoons NH_3(aq) + H_3O^+(aq)$$

Carboxylate ion, basic:

$$RCOO^-(aq) + H_2O(l) \rightleftharpoons RCOOH(aq) + OH^-(aq)$$

Carbonate ion, basic:

$$CO_3^{2-}(aq) + H_2O(l) \rightleftharpoons HCO_3^-(aq) + OH^-(aq)$$

Hydrogencarbonate ion can act both ways, so compare the relevant strengths:

$$HCO_3^-(aq) + H_2O(l) \rightleftharpoons H_2CO_3(aq) + OH^-(aq)$$ $$HCO_3^-(aq) + H_2O(l) \rightleftharpoons CO_3^{2-}(aq) + H_3O^+(aq)$$

If the ion produces $H_3O^+$, it lowers pH. If it produces $OH^-$, it raises pH. The acidity of hydrated transition metal ions and $Al^{3+}(aq)$ is outside this topic, so do not bring it into these salt-pH predictions.

Four curve families

For monoprotic acid–base titrations, the curve shape depends on whether the acid and base are strong or weak.

Strong acid with strong base: low initial pH, no buffer region, large vertical jump, equivalence at pH 7, then high-pH flattening if base is in excess.

Weak acid with strong base: higher initial pH than a strong acid of the same concentration, a buffer region before equivalence, a smaller vertical jump, and equivalence above pH 7 because the salt anion hydrolyses to form $OH^-$.

Strong acid with weak base: the mirrored idea — equivalence below pH 7 because the salt cation hydrolyses to form $H_3O^+$.

Weak acid with weak base: buffer behaviour can appear on both sides, but there is no sharp vertical section, so the equivalence point is hard to locate accurately.

Half-equivalence points

The half-equivalence point is the point in a titration where exactly half of the original weak acid or weak base has been neutralized. For a weak acid titrated with strong base, the buffer contains equal concentrations of $HA$ and $A^-$ at half-equivalence, so $pH = pK_a$.

For a weak base titrated with strong acid, the buffer contains equal concentrations of $B$ and $BH^+$ at half-equivalence, so $pOH = pK_b$.

The buffer region is the gently sloping part of the curve where both members of a weak conjugate pair are present in appreciable amounts. When collecting data for a pH curve, add smaller volumes of titrant where pH changes quickly — especially near the equivalence point — so you do not skip over the important features.

Indicator equilibria

An acid–base indicator is a weak acid or weak base whose conjugate acid and conjugate base have different colours. For a weak acid indicator, write:

$HInd(aq) \rightleftharpoons H^+(aq) + Ind^-(aq)$

$K_\text{Ind} = \frac{[H^+][Ind^-]}{[HInd]}$, where $K_\text{Ind}$ is the indicator acid dissociation constant ($mol\ dm^{-3}$ for this expression), $[Ind^-]$ is the equilibrium concentration of the deprotonated indicator form ($mol\ dm^{-3}$), and $[HInd]$ is the equilibrium concentration of the protonated indicator form ($mol\ dm^{-3}$).

At low pH, high $[H^+]$ shifts the equilibrium left, so the $HInd$ colour dominates. At high pH, lower $[H^+]$ shifts the equilibrium right, so the $Ind^-$ colour dominates. The colour change occurs over a range, usually centred close to the indicator $pK_a$.

Universal indicator is a mixture of several acid–base indicators that gives a sequence of colours over a wide pH range. It is useful for estimating pH, but not for precise titration end points.

Acid–base indicators and redox indicators are similar because both use a visible colour change to signal a stage in a titration. The difference is the underlying equilibrium: acid–base indicators change colour through proton transfer, while redox indicators change colour through electron transfer or, in some titrations, the titrant is self-indicating.

End point versus equivalence point

The end point is the point in a titration where the indicator shows its chosen colour change. The equivalence point is the point where acid and base have reacted in the exact stoichiometric ratio. Good titration technique tries to make these two points coincide as closely as possible, but they are not the same idea.

An appropriate indicator has a transition range within the steep part of the pH curve around the equivalence point. Use the salt identity to decide the approximate equivalence pH:

• strong acid + strong base $\to$ neutral salt $\to$ equivalence near pH 7;
• weak acid + strong base $\to$ basic salt $\to$ equivalence above pH 7;
• strong acid + weak base $\to$ acidic salt $\to$ equivalence below pH 7;
• weak acid + weak base $\to$ no sharp pH change $\to$ ordinary indicators are poor choices.

For example, if the salt at equivalence is sodium ethanoate, the parent acid is weak ethanoic acid and the parent base is strong sodium hydroxide, so the equivalence point is above pH 7. An indicator changing in the basic region, such as phenolphthalein, is suitable. If the salt is ammonium chloride, the parent acid is strong hydrochloric acid and the parent base is weak ammonia, so an acidic-range indicator is preferred.

Use only a few drops of indicator. Indicators are weak acids or bases themselves, so adding too much introduces a small but real systematic error.

What a buffer contains

A buffer solution is an aqueous solution that resists pH change when small amounts of acid or alkali are added. It must contain appreciable amounts of both members of a weak conjugate acid–base pair.

An acidic buffer

An acidic buffer is a buffer made from a weak acid and its conjugate base. A typical example is $CH_3COOH/CH_3COO^-$.

When acid is added:

$$H^+(aq) + CH_3COO^-(aq) \to CH_3COOH(aq)$$

When alkali is added:

$$OH^-(aq) + CH_3COOH(aq) \to CH_3COO^-(aq) + H_2O(l)$$

A basic buffer

A basic buffer is a buffer made from a weak base and its conjugate acid. A typical example is $NH_3/NH_4^+$.

When acid is added:

$$H^+(aq) + NH_3(aq) \to NH_4^+(aq)$$

When alkali is added:

$$OH^-(aq) + NH_4^+(aq) \to NH_3(aq) + H_2O(l)$$

Strong acids and strong bases cannot make useful buffers with their conjugates because their conjugates are too weak to remove added acid or alkali. A buffer works because added strong acid is converted into a weak acid, and added strong base is converted into a weak base. Le Châtelier’s principle describes the same behaviour: adding $H^+$ or $OH^-$ disturbs the conjugate equilibrium, and the buffer components react to oppose that disturbance.

A buffer has limited capacity. Once either conjugate component is used up, the pH changes rapidly.

Calculating buffer pH from the equilibrium constant

For an acidic buffer containing HA and $A^-$, start with:

$K_a = \frac{[H^+][A^-]}{[HA]}$

Rearrange:

$[H^+] = K_a\frac{[HA]}{[A^-]}$

Taking negative logarithms gives the Henderson–Hasselbalch form:

$pH = pK_a + \log_{10}([A^-]/[HA])$, where $pH$ is the acidity scale value (dimensionless), $pK_a$ is the negative logarithmic acid strength value (dimensionless), $[A^-]$ is the concentration of conjugate base in the buffer ($\mathrm{mol\ dm^{-3}}$), and $[HA]$ is the concentration of weak acid in the buffer ($\mathrm{mol\ dm^{-3}}$).

For a basic buffer, it is often neatest to calculate pOH first:

$pOH = pK_b + \log_{10}([BH^+]/[B])$, where $pOH$ is the basicity scale value (dimensionless), $pK_b$ is the negative logarithmic base strength value (dimensionless), $BH^+$ is the concentration of conjugate acid in the buffer ($\mathrm{mol\ dm^{-3}}$), and $[B]$ is the concentration of weak base in the buffer ($\mathrm{mol\ dm^{-3}}$).

Then convert using $pH + pOH = 14.00$ at 298 K.

The key idea is ratio. If $[A^-] = [HA]$, then $\log_{10}(1) = 0$, so $pH = pK_a$. If the conjugate base concentration is larger than the acid concentration, $pH$ is above $pK_a$; if it is smaller, $pH$ is below $pK_a$.

Dilution usually has little effect on buffer pH because both buffer concentrations are reduced by the same factor, leaving their ratio unchanged. But dilution does reduce buffer capacity: there are fewer moles of HA and $A^-$ available to remove added $OH^-$ or $H^+$. In the limit of extreme dilution, the solution tends towards the behaviour of water at that temperature.