Chemical Atlas

An element's atoms come in isotopes — same proton count, different neutron counts, so different masses. The standard atomic weight on the periodic table is the abundance-weighted average of those isotope masses, $\bar{A} = \sum_i f_i A_i$, where each $f_i$ is an isotope's fractional abundance and $A_i$ its mass. The averaging arithmetic is exact; the abundances and isotope masses are sourced measurements (CIAAW). That single averaged number is the atomic weight every molar mass in this Atlas sums.

Choosing the smallest whole-number coefficients that make every element's atom count — and the total charge — equal on both sides. ChemKernel finds them by exact arithmetic — the one set of whole-number coefficients that conserves every atom and the total charge, re-verified independently — so the coefficients are derived rather than guessed. For the calcium-phosphate lesson that gives $3\,\text{CaCl}_2 + 2\,\text{Na}_3\text{PO}_4 \rightarrow \text{Ca}_3(\text{PO}_4)_2 + 6\,\text{NaCl}$.

A buffer holds both halves of a weak-acid equilibrium at once — a weak acid $\mathrm{HA}$ and a supply of its conjugate base $\mathrm{A^-}$ (from a salt). The extra $\mathrm{A^-}$ is a common ion: by Le Chatelier it drives $\mathrm{HA} \rightleftharpoons \mathrm{H^+} + \mathrm{A^-}$ back to the left, so the acid barely ionizes and the pH settles near $\mathrm{p}K_a$ instead of the low value the acid alone would give. Solving the mass-action law $K_a = [\mathrm{H^+}][\mathrm{A^-}]/[\mathrm{HA}]$ for $[\mathrm{H^+}]$ and taking $-\log_{10}$ gives Henderson–Hasselbalch, $\mathrm{pH} = \mathrm{p}K_a + \log_{10}\dfrac{[\mathrm{A^-}]}{[\mathrm{HA}]}$ — not a new law, just $K_a$ in logarithmic form. When the acid and base concentrations are equal the ratio is 1 and $\mathrm{pH} = \mathrm{p}K_a$. A buffer resists change because adding a little $\mathrm{H^+}$ or $\mathrm{OH^-}$ only shifts the large $[\mathrm{A^-}]/[\mathrm{HA}]$ ratio slightly.

The standard cell potential is the voltage a galvanic cell drives at standard conditions: $E^\circ_\text{cell} = E^\circ_\text{cathode} - E^\circ_\text{anode}$, read straight off the table of standard reduction potentials (see standard reduction potential). It is intensive — it does not scale when you multiply the equation through, because it is energy per unit charge. A positive $E^\circ_\text{cell}$ means the reaction is spontaneous as written; the two are linked by $\Delta G^\circ = -nFE^\circ$, where $n$ is the moles of electrons (see electron ledger) and $F$ the Faraday constant. So $E^\circ_\text{cell} > 0 \Leftrightarrow \Delta G^\circ < 0$: the cell runs on its own and can do electrical work. Away from standard conditions the Nernst equation corrects $E$. The relation is exact inside the standard-state model; $E^\circ$ is a sourced measured value.

A reversible reaction ($\rightleftharpoons$) does not stop when a reactant runs out — it settles where the forward and reverse rates match, a dynamic balance with both directions still running. At that point the reaction quotient $Q = \dfrac{[\text{products}]^{\nu}}{[\text{reactants}]^{\nu}}$ equals the equilibrium constant $K$ (a large $K$ favors products, a small $K$ favors reactants). The bookkeeping is the ICE table — Initial, Change, Equilibrium — which is nothing but the species ledger $c_i = c_{i,0} + \nu_i\,x$ in concentrations: every row is a straight line in one extent $x$. The difference from a reaction run to completion is only how far $x$ goes — here it is the value that makes $Q = K$, found and its residual checked, not the limiting-reagent minimum.

Atoms are neither created nor destroyed in a reaction, so once the equation is balanced every element's atom count is equal on both sides. This is the constraint the balancer solves and the atom-balance check enforces at build time; it is also why the ledger's coefficients $\nu_i$ must come from a balanced equation. For each element $E$, the signed contributions of all species sum to zero.

Solving by carrying the units: multiply a quantity by conversion factors written as fractions equal to 1, chosen so the unwanted unit cancels and the wanted one survives. If the units come out right, the setup is right. Volume $\rightarrow$ moles $\rightarrow$ mass is just $\text{mL}\times\frac{1\,\text{L}}{1000\,\text{mL}}\times\frac{\text{mol}}{\text{L}}\times\frac{\text{g}}{\text{mol}}$, with every unit but grams cancelling. ChemKernel's units engine checks the cancellation on every generated problem.

When a soluble ionic compound dissolves, it separates into free ions rather than staying intact: $\text{Na}_3\text{PO}_4(aq) \rightarrow 3\,\text{Na}^+(aq) + \text{PO}_4^{3-}(aq)$. Treating a strong electrolyte as completely dissociated is a disclosed idealization — exact inside the model, not proven — and it is what lets the complete-ionic equation show every dissolved salt as its separate ions.

How strongly an atom pulls the electrons of a bond toward itself, on Pauling's fitted 0–4 scale (fluorine tops it at 3.98). It is not a directly measured quantity — the scale is constructed from bond energies — and the noble gases carry no value at all. Its main job is comparison: the DIFFERENCE between two bonded atoms' electronegativities (ΔEN) estimates bond polarity, from evenly shared (small ΔEN) through polar covalent to ionic (large ΔEN) — a guideline with exceptions, applied in the Valence Table's bonding mode.

One number, $\xi$ (in moles), that says how far a reaction has run. Every species' amount is a straight line in it: $n_i = n_{i,0} + \nu_i\,\xi$, with $\nu_i$ negative for reactants and positive for products. Stoichiometry, limiting reagents, leftovers, ICE tables, and reaction rates ($d\xi/dt$) are all just views of this single ledger.

The half-life $t_{1/2}$ is the time for a reactant to fall to half its concentration. Its concentration dependence is the fingerprint of the order: for a first-order reaction the integrated law $[\mathrm{A}] = [\mathrm{A}]_0 e^{-kt}$ gives $t_{1/2} = \ln 2 / k \approx 0.693/k$ — independent of the starting concentration, so the reactant keeps halving on the same clock (1.000 → 0.500 → 0.250 → …, each step the same $t_{1/2}$). That constancy is the classic test for first order (and the reason radioactive decay, also first order, has a fixed half-life). For second order $t_{1/2}$ grows as the reactant depletes; for zero order it shrinks — so a constant half-life is not a general fact but a first-order one.

A half-reaction shows one half of a redox process with its electrons written explicitly: the oxidation half loses them ($\mathrm{Zn} \rightarrow \mathrm{Zn^{2+}} + 2\,e^-$, at the anode), the reduction half gains them ($\mathrm{Cu^{2+}} + 2\,e^- \rightarrow \mathrm{Cu}$, at the cathode). Each half is balanced for both atoms and charge (the electrons carry the charge). To combine them into the overall reaction you scale each half so the electrons cancel — the number transferred is $n = \operatorname{lcm}$ of the two electron counts (see electron ledger). Splitting a reaction this way is what lets you read a cell's voltage off a table of standard potentials, one half at a time (see standard reduction potential).

An integrated rate law solves the rate law for concentration as a function of time — it turns a rate into a prediction. The form depends on the order: zero order gives $[\mathrm{A}] = [\mathrm{A}]_0 - kt$ (a straight line that reaches zero at a finite time, half-life $t_{1/2} = [\mathrm{A}]_0/2k$, which shrinks); first order gives $[\mathrm{A}] = [\mathrm{A}]_0 e^{-kt}$ (exponential decay, $t_{1/2} = \ln 2/k$, constant — the signature); second order gives $\frac{1}{[\mathrm{A}]} = \frac{1}{[\mathrm{A}]_0} + kt$ (asymptotic, $t_{1/2} = 1/k[\mathrm{A}]_0$, which grows). Each linearizes a different plot — $[\mathrm{A}]$, $\ln[\mathrm{A}]$, or $1/[\mathrm{A}]$ versus $t$ — which is how the order is read from data. The order is a disclosed model (regime-2) and $k$ is sourced (regime-3), but the law that follows is exact, so a constant half-life is not a general fact — it is the fingerprint of first order (see half life).

The attractions between whole molecules — distinct from the covalent bonds within one. They decide the physical properties: melting and boiling points, whether a substance is a gas, liquid, or solid at room temperature, and what dissolves in what. Three kinds, and which a molecule has follows from its verified structure: London dispersion forces act between all molecules (fleeting, induced dipoles from moving electrons) and are the only force for a nonpolar molecule; dipole–dipole forces act additionally between polar molecules (the permanent dipoles attract); and hydrogen bonding — the strongest of the three for small molecules — acts when a molecule has an H bonded directly to N, O, or F (a small, strongly electron-pulling atom leaves that H very positive). The dominant force is the strongest kind present, ranked hydrogen bonding > dipole–dipole > dispersion. The evidence is in the boiling points of similar-sized hydrides: CH₄ (dispersion only) boils at −161.5 °C, NH₃ (hydrogen bonding) at −33.3 °C, and H₂O (hydrogen bonding, with two donors and two acceptors) all the way up at 100 °C — same rough size, wildly different boiling points, because the force holding the molecules together is stronger. CO₂ (nonpolar, dispersion only) doesn't even form a liquid at 1 atm: it sublimes at −78.5 °C. The presence of each force is machine-derived from the structure (polarity, and the H-on-N/O/F test); the strength ordering is the sourced rule, and one caveat is disclosed — dispersion grows with molecular size and polarizability, so for large molecules it can outweigh a small dipole.

The energy needed to remove the most loosely held electron from a gas-phase atom: $\mathrm{X(g)} \rightarrow \mathrm{X^+(g)} + e^-$. First ionization energies generally climb across a period and fall down a group — the same pull that shrinks atoms holds their electrons — but the measured values keep two famous dips per period (the s→p boundary and the first paired p electron), which is why ordering drills must be answered from data, not from the slogan. Low ionization energy is what makes metals cation-formers.

A bookkeeping of a molecule's valence electrons into bonding pairs (shared) and lone pairs (unshared). It is an electron ledger, the exact counterpart of the species ledger: start from the total valence electrons $V = \sum(\text{group electrons}) - \text{charge}$, then place them so that every atom reaches a full shell — a duet for hydrogen, an octet for the rest — with $2\times(\text{bonds}) + 2\times(\text{lone pairs}) = V$ (electrons are conserved, none invented). Each atom's formal charge $= V_{\text{atom}} - (\text{nonbonding electrons}) - \tfrac{1}{2}(\text{bonding electrons})$, and the formal charges must sum to the molecular charge. Every one of these is exact integer arithmetic — ChemKernel refuses a structure that fails to conserve electrons, complete an octet, or sum its formal charges to the charge. The octet model itself (that atoms seek eight) is the disclosed assumption; the accounting on top of it is machine-checked.

The reactant that runs out first, capping how far a reaction can proceed. Read it off the species ledger $n_i = n_{i,0} + \nu_i\,\xi$: the limiting reagent is the one whose available amount, divided by its stoichiometric coefficient, is smallest — it hits zero at the maximum extent $\xi_{\max}$. Volume or mass alone never decides it; moles relative to the coefficients do.

The mass of one mole of a substance, in g/mol — the sum of its atoms' standard atomic weights. It converts between mass and moles, $n = m/\mathcal{M}$, so it turns a ledger amount into a mass on the balance. The arithmetic is exact; the atomic weights it sums are sourced data (CIAAW).

Concentration as amount of solute per litre of solution: $M = n/V$, in mol/L. It is the bridge from something you can measure — a volume — to moles: multiply a volume in litres by a molarity and the litres cancel, leaving the moles that feed the ledger.

Valence-shell electron-pair repulsion: the electron domains around a central atom — each bond (single, double, or triple counts as one) and each lone pair — spread out as far apart as possible, and that fixes the shape. Count the domains: 2 → linear (180°), 3 → trigonal planar (120°), 4 → tetrahedral (109.5°). Lone pairs keep the electron geometry but bend the visible molecular shape (a 4-domain atom with one lone pair is trigonal pyramidal, with two is bent) and squeeze the angle a little. Shape then decides polarity: add up the bond dipoles as vectors — if the molecule is symmetric enough that they cancel (linear CO2, tetrahedral CH4) it is nonpolar; if they do not (bent H2O, pyramidal NH3) it is polar. Polar bonds do not force a polar molecule; the geometry has the final say. The domain count is machine-derived from the Lewis structure; the geometry names and the ideal angles are the sourced convention, and the net-polarity conclusion is a disclosed model.

The reaction with the spectators removed — only the species that actually change. Start from the complete ionic equation (every dissolved strong electrolyte shown as its free ions), cancel whatever appears unchanged on both sides, and re-check that atoms and charge still balance. What remains is the real chemical event: for the precipitation lesson, $\text{Ca}^{2+} + \text{CO}_3^{2-} \rightarrow \text{CaCO}_3(s)$.

The rules for turning a formula into a name and back. An ionic compound is named cation-then-anion: the metal keeps its element name — or, for a variable-charge metal, its charge as a Roman numeral (the Stock system, $\text{iron(III)}$) — and the nonmetal or polyatomic ion takes its -ide/-ate name. The formula is the reverse and is pure charge balance: the two ions' charges cross to whatever subscripts make the compound neutral, so $\text{Fe}^{3+}$ and $\text{SO}_4^{2-}$ give $\text{Fe}_2(\text{SO}_4)_3$. The names are sourced conventions; the assembled formula is machine-checked (charge crossover, verified neutral).

An oxidation number is the charge an atom would have if every bond were fully ionic — a bookkeeping device that makes electron transfer countable. It is assigned by a rule hierarchy (a free element is 0; a monatomic ion is its charge; fluorine −1; a group-1 metal +1, group-2 +2; hydrogen +1; oxygen −2), and the one remaining element is fixed by the requirement that the numbers sum to the species' charge — that sum is exact integer accounting, machine-checked. An element whose oxidation number rises is oxidized (it loses electrons); one that falls is reduced. This is what completes the free-element redox flag (see redox): the numbers say not just that electrons moved but how many. The rules are a sourced convention (a few molecules — peroxides, hydrides — take named exceptions); the accounting on top of them is a theorem.

The fraction of the theoretical maximum a reaction actually delivers, as a percentage: $\text{percent yield} = \dfrac{\text{actual}}{\text{theoretical}} \times 100\%$. The theoretical yield is the product mass predicted by mass stoichiometry from the limiting reagent — the ledger at maximum extent. The actual yield is what is measured; it falls short because of side reactions, incomplete conversion, and losses in handling. Yield cannot exceed 100% (that would break conservation of mass); a value above it signals an impure product or a measurement error, not extra matter.

A property that varies systematically with position on the periodic table: covalent radius shrinks left → right and grows top → bottom; first ionization energy and electronegativity do the opposite. The usual explanation is effective nuclear charge — across a period, protons are added while electrons enter the same shell, pulling everything tighter; down a group, each row adds a screened outer shell. That story is useful and widely taught, but it is an interpretation laid over measured data, and the data keeps exceptions the slogan hides (boron ionizes below beryllium, oxygen below nitrogen). On the Valence Table the trend lenses color the sourced values; the pattern panels tell the story and name its exceptions.

pH compresses the hydronium concentration onto a log scale: $\mathrm{pH} = -\log_{10}[\mathrm{H^+}]$. Because it is a logarithm, every whole pH unit is a tenfold change in $[\mathrm{H^+}]$ — pH 3 is ten times more acidic than pH 4. At 25 °C pure water is pH 7 (neutral); below 7 is acidic, above 7 basic, tied together by $[\mathrm{H^+}][\mathrm{OH^-}] = K_w = 10^{-14}$. On a log scale the decimal places are the significant figures: a two-significant-figure $[\mathrm{H^+}]$ fixes pH to two decimals. Reading pH backwards, $[\mathrm{H^+}] = 10^{-\mathrm{pH}}$.

A group of atoms bonded together that carries a single net charge and moves through a reaction as one unit — carbonate $\text{CO}_3^{2-}$, phosphate $\text{PO}_4^{3-}$, nitrate $\text{NO}_3^-$. Its charge is a sourced convention; its atom composition is machine-checked. In charge balance the whole group counts as one charged block, which is why the neutral salt $\text{Ca}_3(\text{PO}_4)_2$ keeps the phosphate in parentheses.

A polyprotic acid has more than one ionizable proton and gives them up in stages, each a separate equilibrium with its own constant: $\mathrm{H_3PO_4} \rightleftharpoons \mathrm{H^+} + \mathrm{H_2PO_4^-}$ ($K_{a1}$), then $\mathrm{H_2PO_4^-} \rightleftharpoons \mathrm{H^+} + \mathrm{HPO_4^{2-}}$ ($K_{a2}$), then $\mathrm{HPO_4^{2-}} \rightleftharpoons \mathrm{H^+} + \mathrm{PO_4^{3-}}$ ($K_{a3}$). Successive constants fall steeply — typically $K_{a1} \gg K_{a2} \gg K_{a3}$, each roughly $10^5$ smaller — because pulling a proton off an already-negative ion is far harder. That separation is what makes the arithmetic simple: the first ionization sets $[\mathrm{H^+}]$ almost entirely, so the pH is essentially the first stage's, and each later stage is the same ICE solve run on the concentrations the stage before it left. A useful consequence for a diprotic-or-more acid: the second ionization's product — the twice-deprotonated species, here $\mathrm{HPO_4^{2-}}$ — sits at a concentration $\approx K_{a2}$, because after stage 1 $[\mathrm{H^+}] \approx [\mathrm{H_2PO_4^-}]$ and the second-stage mass-action law collapses to $K_{a2} \approx [\mathrm{HPO_4^{2-}}]$.

A reaction in which mixing two solutions produces an insoluble solid — the precipitate — that drops out of solution. Whether a solid forms, and which one, is read from the solubility rules; the amount that forms is then pure ledger. In the lesson, $\text{Ca}^{2+}$ and $\text{CO}_3^{2-}$ meet and leave solution as solid $\text{CaCO}_3$, while the spectator ions stay dissolved.

A rate law relates the reaction rate to concentrations: $\text{rate} = k[\mathrm{A}]^m[\mathrm{B}]^n$. The exponents are the reaction orders — the overall order is $m+n$ — and, crucially, they are found by experiment, not read off the balanced-equation coefficients (a common trap). The proportionality constant $k$ is the rate constant (a sourced, temperature-dependent datum). A first-order reaction, rate $= k[\mathrm{A}]$, is the simplest: its concentration decays exponentially and its half-life is constant (see half life). The rate law is a disclosed model — regime-2; $k$ is regime-3 — while the integrated form that follows from it is exact given the model.

The heat released or absorbed by a reaction at constant pressure, per mole of reaction: $\Delta H_{\text{rxn}}$. A negative value is exothermic (heat flows out); a positive value is endothermic. Because enthalpy is a state function, $\Delta H_{\text{rxn}}$ depends only on the substances, not the path — Hess's law — so it can be assembled from tabulated standard enthalpies of formation $\Delta H_f^\circ$: $\Delta H_{\text{rxn}} = \sum \nu\,\Delta H_f^\circ(\text{products}) - \sum \nu\,\Delta H_f^\circ(\text{reactants})$, where an element in its standard state has $\Delta H_f^\circ = 0$ by definition. The tabulated value is per mole of reaction; the heat an actual run delivers is $q = \Delta H_{\text{rxn}}\cdot\xi$, scaled by the extent $\xi$ the limiting reagent allows — the energy is the extent ledger's other column.

The reaction quotient $Q$ has the same algebraic form as the equilibrium constant $K$ — products over reactants, each raised to its coefficient, $Q = \prod_i [\text{species}_i]^{\nu_i}$ — but evaluated at any composition, not only at equilibrium. Comparing $Q$ to $K$ tells you which way the reaction must move to reach equilibrium: $Q < K$ shifts forward (toward products), $Q > K$ shifts reverse, $Q = K$ is already there. For a dissolving salt the test is $Q$ against $K_{sp}$: mix two solutions, compute $Q = [\text{cation}]^a[\text{anion}]^b$ at the instant of mixing, and if $Q > K_{sp}$ the solution is supersaturated and a precipitate forms — the quantitative form of the Phase-0 'will it precipitate?' rules. $Q$ is the snapshot; $K$ is the destination.

The reaction rate is how fast the reaction extent advances — the time derivative of the species ledger. For a reactant A with coefficient $a$, the extent-normalized rate is $-\tfrac{1}{a}\tfrac{d[\mathrm{A}]}{dt}$ (divided by $a$ so one number describes the whole reaction); it is the same $d\xi/dt$ that stoichiometry, thermochemistry, and equilibrium all view differently — here watched in time. Two conventions coexist, differing by the factor $a$: that extent-normalized rate, and the rate of disappearance of a chosen species, $-\tfrac{d[\mathrm{A}]}{dt}$. The kinetics lessons here follow one reactant, so their measured rate law $\text{rate}=k[\mathrm{A}]^n$ and its integrated form use the per-species rate $-\tfrac{d[\mathrm{A}]}{dt}=k[\mathrm{A}]^n$ — the tabulated $k$ is that species' disappearance constant. A rate is positive and changes as the reaction proceeds (usually slowing as reactants deplete). How the rate depends on concentration is not read off the balanced equation — it is the empirically measured rate law (see rate law). Equilibrium is the limit where the forward and reverse rates become equal, so the ledger stops moving.

A redox (reduction–oxidation) reaction transfers electrons from one species to another. The two always come paired: whatever is oxidized (loses electrons — its oxidation number rises) hands them to whatever is reduced (gains electrons — its oxidation number falls), so there is no oxidation without reduction. You can spot redox without any numbers when a free element becomes combined or vice versa (a neutral element must change oxidation state) — the reaction classifier already flags this. Assigning oxidation numbers (see oxidation number) then makes the transfer quantitative: how many electrons move, $n$. Combustion, single-replacement, corrosion, respiration, and every battery are redox. Splitting the electron flow into two half-reactions is the whole of electrochemistry (see electron ledger).

For a sparingly soluble salt at saturation, $\mathrm{M}_a\mathrm{X}_b(s) \rightleftharpoons a\,\mathrm{M}^{n+} + b\,\mathrm{X}^{m-}$, the solubility product is $K_{sp} = [\mathrm{M}^{n+}]^a[\mathrm{X}^{m-}]^b$. The dissolving solid is a pure phase (activity 1), so it does not appear — that exclusion is the whole move. The equilibrium's extent is the molar solubility $s$ (mol dissolved per litre): $[\mathrm{M}^{n+}] = a s$, $[\mathrm{X}^{m-}] = b s$, so $K_{sp} = (a s)^a (b s)^b$ — a polynomial in $s$ (a cubic for a 1:2 salt), solved for $s$. A smaller $K_{sp}$ means less dissolves, but $K_{sp}$ values compare solubility directly only for salts of the same ion ratio. Comparing $K_{sp}$ to the reaction quotient $Q$ predicts precipitation: $Q > K_{sp}$ means a solid forms (the Phase-0 precipitation rules, quantified).

A short, ordered list of empirical rules that predict whether an ionic compound dissolves in water — with exceptions. They are tabulated facts, not theorems: nitrates and group-1 salts are soluble; most carbonates are not, except those of group-1 cations and ammonium. ChemKernel applies them in a fixed precedence and cites the governing rule, so a precipitate is classified, never asserted.

An ion that appears unchanged on both sides of the complete-ionic equation — present before and after, never consumed. Cancelling the spectators is exactly what turns the complete-ionic equation into the net ionic one. They do not vanish: $\text{Na}^+$ and $\text{Cl}^-$ stay dissolved while $\text{Ca}^{2+}$ and $\text{PO}_4^{3-}$ leave solution as solid — they simply sit the reaction out.

A standard reduction potential $E^\circ$ is the voltage a reduction half-reaction ($\mathrm{M}^{n+} + n\,e^- \rightarrow \mathrm{M}$) drives at standard conditions, measured against the standard hydrogen electrode (SHE), which is defined as exactly 0 V. A more positive $E^\circ$ means a stronger pull on electrons — that couple is reduced (it is the cathode); the couple with the lower $E^\circ$ is forced to run backwards as oxidation (the anode). Only differences are physical: the zero point (the SHE) is an arbitrary reference, so a single $E^\circ$ is meaningful only relative to another. The values are a sourced, tabulated measured dataset (regime-3); the cell voltage built from them, $E^\circ_\text{cell} = E^\circ_\text{cathode} - E^\circ_\text{anode}$, is exact given the table (see cell potential).

The mole-accounting of a balanced reaction: the coefficients $\nu_i$ fix the ratios in which species are consumed and produced. Every change is a multiple of one extent — $\Delta n_i = \nu_i\,\xi$ — so a mole ratio like "3 mol $\text{Ca}^{2+}$ per 1 mol $\text{Ca}_3(\text{PO}_4)_2$" is just $\nu$ read off the equation. It is the mole ratio, never the raw masses or volumes, that converts one species into another.

Water is both a weak acid and a weak base to itself: two molecules trade a proton, $2\,\mathrm{H_2O} \rightleftharpoons \mathrm{H_3O^+} + \mathrm{OH^-}$, so even pure water carries a trace of ions. Their product is the ion-product constant of water, $K_w = [\mathrm{H^+}][\mathrm{OH^-}] = 1.0\times10^{-14}$ at 25 °C. This is the bridge between acids and bases: whatever raises $[\mathrm{H^+}]$ must lower $[\mathrm{OH^-}]$ to keep the product at $K_w$, and vice versa. Taking $-\log_{10}$ of both sides turns the product into a sum, $\mathrm{pH} + \mathrm{pOH} = \mathrm{p}K_w = 14.00$ — so measuring a weak base's $[\mathrm{OH^-}]$ (from its $K_b$ equilibrium) immediately gives its pH: $\mathrm{pH} = 14.00 - \mathrm{pOH}$. In pure water $[\mathrm{H^+}] = [\mathrm{OH^-}] = 10^{-7}\,\text{M}$, i.e. pH 7 — the definition of neutral.

A common ion is a species already present in solution that also appears in an equilibrium — shared between a dissolved salt and the reaction. By Le Chatelier's principle, adding more of a product ion drives the equilibrium back toward reactants, so the reaction proceeds less. Two headline cases, one idea: a buffer adds the conjugate base $\mathrm{A^-}$ to a weak acid $\mathrm{HA} \rightleftharpoons \mathrm{H^+} + \mathrm{A^-}$, suppressing ionization so the pH holds near $\mathrm{p}K_a$; a common ion in a saturated solution adds (say) $\mathrm{F^-}$ to $\mathrm{CaF_2(s)} \rightleftharpoons \mathrm{Ca^{2+}} + 2\,\mathrm{F^-}$, suppressing dissolution so far less solid dissolves than in pure water. Nothing new is happening — the equilibrium constant is unchanged. It is only that the mass-action expression is now satisfied at a smaller reaction extent, because one product started above zero. Solubility, then, is not a fixed property of a salt: it depends on what is already dissolved.

Electrochemistry is the species ledger with electrons as the tracked quantity — electron bookkeeping plus free energy per charge. The reaction extent is measured in **moles of electrons transferred, $n$**: the electrons lost by the species oxidized are exactly the electrons gained by the species reduced, so charge is conserved and the two half-reactions close (see half reaction). It is the same conserved-but-changing ledger as extent of reaction, now counting electrons instead of atoms. Two quantities hang off $n$: the charge moved, $nF$ (with the Faraday constant $F$ = charge per mole of electrons), and the free energy, $\Delta G^\circ = -nFE^\circ$ — the bridge from the cell voltage to thermochemistry (see cell potential).

A titration adds a base of known concentration to an acid and tracks the pH; the plot of pH against added volume is the titration curve. For a weak acid titrated by a strong base it has four parts, each just the species ledger solved by whichever equilibrium dominates: at the start it is the pure weak acid; adding base converts some acid to its conjugate base, giving a buffer whose pH creeps up slowly — flattest at the half-equivalence point, where equal amounts of acid and conjugate base make $\mathrm{pH} = \mathrm{p}K_a$ (the curve reads off $\mathrm{p}K_a$); at the equivalence point all the acid has become its conjugate base, a weak base, so the pH jumps steeply and lands above 7 (basic — not 7, which is the strong-acid/strong-base case); past equivalence the excess strong base sets the pH. The equivalence volume comes from stoichiometry ($n_\text{acid} = n_\text{base}$); the pH at every point comes from the equilibrium.