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Acetic acid is a weak acid that partly transfers protons to water, supplying hydronium and negative ions. It is therefore a weak electrolyte in water.

An increase in oxidation state records oxidation; a decrease records reduction. Thus ${\mathrm{Fe}}^{2+}\to {\mathrm{Fe}}^{3+}$ raises the iron oxidation state by one, while ${\mathrm{MnO}}_4^{-}\to {\mathrm{Mn}}^{2+}$ lowers the manganese oxidation state from $+7$ to $+2$. Five iron ions supply the five-electron inventory required per permanganate ion.

In an acidic chemical solution, the complete half-reactions areFirst balance oxygen with four water molecules, then hydrogen with eight protons, and finally electric charge with five electrons. Multiply the iron half-reaction by five and add; the exchanged electrons cancel:Both sides contain one manganese, four oxygen, eight hydrogen and five iron chemical atoms, with net electric charge $+17e$. The ${\mathrm{H}}^{+}$ notation represents the acidic aqueous proton inventory; it does not imply bare protons moving independently of water. These half-reactions and their sum are the acidic balance used in the documented permanganate-iron titration.^[ref]

The coefficients predict amounts consumed, not a collision of five ions or an elementary reaction. Bulk chemical solution also contains compensating counterions, which are omitted from this net chemical equation. The hydrogen and oxygen oxidation states are unchanged; balancing their inventories remains essential even though they do not account for the exchanged electrons.

An action potential is a brief regenerative excursion of membrane potential, produced by changing electrical resistance and conductance through voltage-gated ion channels. In a typical fast neuronal action potential, sodium entry raises membrane potential, activating further sodium channels. This feedback produces the rapid rising phase. Inactivation of sodium channels and increased conductance to potassium then lower membrane potential again; persisting potassium conductance can produce an undershoot below the resting level. See Purves and colleagues: the ionic basis of action potentials.

For an illustrative electrical calculation, write inward sodium electric current as $I_{\mathrm{Na}}=g_{\mathrm{Na}}(E_{\mathrm{Na}}-V)$. At $V=-50\mathrm{mV}$, taking $E_{\mathrm{Na}}=60\mathrm{mV}$ and $g_{\mathrm{Na}}=10\mathrm{nS}$ gives $I_{\mathrm{Na}}=1.1\mathrm{nA}$ inward. A potassium current $I_{\mathrm{K}}=g_{\mathrm{K}}(E_{\mathrm{K}}-V)$ with $E_{\mathrm{K}}=-80\mathrm{mV}$ and $g_{\mathrm{K}}=20\mathrm{nS}$ gives $-0.6\mathrm{nA}$, outward. These illustrative values expose opposing driving forces; a waveform requires the conductances to evolve with membrane potential and time, as in the Hodgkin-Huxley model.

An action potential propagates along an axon because local electric currents bring neighboring regions of cell membrane into the regime where their voltage-gated ion channels regenerate the signal. It is a traveling pattern of excitation, not a packet of sodium ions traversing the entire axon. Regeneration allows an action potential to travel without the passive attenuation of a small local voltage change. See Ion Channels and the Electrical Properties of Membranes.

The neuronal refractory period constrains how soon another action potential can follow. Under comparable conditions, successive action potentials have roughly stereotyped shapes, making event count and timing useful for neural coding; waveform details can nevertheless change with channel state and conditions. The leaky integrate-and-fire model abstracts each action potential to a threshold event and reset. Its reset jump is a mathematical rule, not the actual regenerative waveform produced by sodium and potassium currents.

Active transport moves a substance uphill in its electrochemical gradient by coupling the transfer to a downhill process. Some carrier proteins use ATP hydrolysis; others couple uphill transfer of one solute to downhill transfer of another solute, often an ion. The latter mechanism uses a stored electrochemical gradient and need not consume ATP directly at that carrier protein. Maintaining the driving electrochemical gradient still requires a source of energy. See Alberts et al.: carrier proteins and active membrane transport.

For the neutral 8:2 example, B-to-A transfer costs $RT\ln 4\simeq 1.38629RT$ per mole. As an explicitly assumed bookkeeping example, suppose a tightly coupled cycle supplies a downhill free energy change of $-2RT$ per mole of transferred solute. The total is then $\Delta G_{\mathrm{cycle}}=RT\ln 4-2RT\simeq -0.61371RT$. This makes forward cycling energetically possible under those assumptions; it specifies neither a molecular mechanism nor a rate. Merely adding a carrier protein cannot supply the missing free energy. Conversely, as the transported concentration ratio rises, the required uphill cost rises, so a fixed driving process has a finite thermodynamic limit.

ATP is a nucleotide consisting of adenosine, itself adenine joined to ribose, carrying three phosphate groups. From the ribose outward, these groups are labelled $\alpha$, $\beta$, and $\gamma$. ATP connects energy-supplying processes to energy coupling for synthesis and active transport; the same carrier can be regenerated and reused. ^[ref].

The net hydrolysis commonly used in energy coupling isHere ${\mathrm{P}}_i$ denotes inorganic phosphate. At buffered pH $7$, this biochemical shorthand groups the appropriate protonation states into pools and leaves proton exchange with the buffer solution implicit. An explicit, charge-balanced choice is ${\mathrm{ATP}}^{4-}+{\mathrm{H}}_2\mathrm{O}⟶{\mathrm{ADP}}^{3-}+{\mathrm{HP}{\mathrm{O}}_4}^{2-}+{\mathrm{H}}^{+}$, without implying these are the only chemical species present. See the IUBMB recommendations for biochemical thermodynamics.

The quoted textbook biochemical standard state value is approximatelyThis convention uses fixed pH $7$, temperature $298.15\mathrm{K}$ ($25{}^{\circ }\mathrm{C}$), reference pressure $100\mathrm{kPa}$ ($1\mathrm{bar}$), unit reference thermodynamic activities for dissolved reactants and reaction products, and water thermodynamic activity approximated by one. The prime indicates the fixed-pH biochemical convention. See OpenStax Biology 2e, 6.4: standard ATP hydrolysis value and Biology 2e, 6.2: biochemical standard conditions. This is a reference value, rather than a universal value inside a biological cell; precise tables must also specify chemical solution conditions and any magnesium ion binding convention. Different reference conventions and data treatments give different tabulated values, as discussed in Li et al., 2010: a database of thermodynamic quantities for biochemical reactions.

An exergonic reaction has a negative change in Gibbs free energy, so the overall ATP hydrolysis is favourable under those reference conditions. Breaking a chemical bond by itself costs energy; forming new chemical bonds can release energy. The net change compares all reactants with all reaction products, including their interactions with water and the entropy contribution. Calling the phosphate connection a "high-energy bond" describes a large favourable change in Gibbs free energy on hydrolysis, rather than energy released by bond breaking alone. See 7.5 Strengths of Ionic and Covalent Bonds and Cooper, The Cell: ATP and metabolic energy.

Composition changes the driving force. At the same fixed pH, temperature, pressure, and binding convention, write the molar reaction change asHere $R$ is the molar gas constant, $T$ the absolute temperature, and $a$ a dimensionless thermodynamic activity for a biochemical pool. Constant water thermodynamic activity is absorbed into the reference term; fixed pH accounts for buffered protons. The shorthand $\Delta G$ means transformed molar reaction Gibbs free energy change, ${\Delta }_r{G}^{\prime }$, rather than a finite sample's total change. The natural logarithm acts on a dimensionless reaction quotient, never on concentrations with units. See the IUBMB biochemical thermodynamics recommendations and the general equilibrium constant relation.

For a stipulated ideal dilute chemical solution, let $a_i=[i]/c^{\circ }$, with $c^{\circ }=1\mathrm{mol}{\mathrm{L}}^{-1}$. Choose $[\mathrm{ATP}]=5\mathrm{mmol}{\mathrm{L}}^{-1}$, $[\mathrm{ADP}]=0.5\mathrm{mmol}{\mathrm{L}}^{-1}$, and $[{\mathrm{P}}_i]=1\mathrm{mmol}{\mathrm{L}}^{-1}$, ignoring activity coefficient corrections and magnesium binding. These are illustrative inputs, not a claim about a particular biological cell. ThenandIncreasing $Q$ tenfold raises $\Delta G$ by $RT\ln 10\approx 5.708\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ at this temperature. Thus the ATP to ADP ratio and available inorganic phosphate matter, even with the same standard state value. These are local molar predictions at the stipulated composition, rather than a constant total yield as an isolated sample reacts.

Amount of substance measures a count of specified entities, such as chemical atoms, molecules, ions, or electrons, using the mole as its SI base unit. It differs from mass: equal amounts of substance of different chemical substances contain equal entity counts but generally have different masses. If $N$ entities are present, $n=N/N_{\mathrm{A}}$, where $N_{\mathrm{A}}$ is the Avogadro constant. The specified entity matters: one mole of water molecules contains two moles of hydrogen chemical atoms. ^[ref]

An anion is an ion with negative net electric charge, for example ${\mathrm{Cl}}^{-}$ or hydroxide.

A stream of positive ions in a gas discharge. Historical canal rays pass through holes in the cathode; collisions can also produce moving neutral chemical atoms.

In Mass spectra and isotopes by Francis William Aston, printed pp. 10–12, Fig. 1, Francis William Aston describes successive electric and magnetic deflections in one plane, with opposite senses. This geometry focuses a range of ion speeds; it does not require the straight-through cancellation of a velocity selector.

The reason compensation is possible follows from the Lorentz force. In a short region of length ${\ell }_E$, a small electric deflection angle has magnitude ${\theta }_E\simeq qE{\ell }_E/(m{v}^2)$: the transverse acceleration acts for a time ${\ell }_E/v$, and the deflection angle is the acquired transverse velocity divided by $v$. A short magnetic region of length ${\ell }_B$ gives ${\theta }_B\simeq qB{\ell }_B/(mv)$, since the magnetic force already contains one factor of speed. Slower ions therefore receive a disproportionately larger electric deflection.

An original local approximation makes the focusing idea explicit. At fixed mass-to-charge ratio, replace each region by a small angular kick and include the drift distances in two positive constants $a$ and $b$. With $t=v/v_0$, write the detector coordinate asChoosing the geometry and fields so that $b=2a$ eliminates the first-order speed sensitivity. Indeed,A fractional speed error $\epsilon =t-1$ then contributes only at order ${\epsilon }^2$ in this approximation. This is a local thin-kick explanation, not Aston's exact pole geometry, focal-plane equation, or a prediction of its measured mass resolution. Finite slit widths, initial directions, and field boundaries still matter. Unlike a velocity selector, focusing can collect nearby speeds into a narrow image instead of discarding all but a narrow accepted range.

An atomic basis of a crystal is the motif of chemical species $s_j$ and relative positions $r_j$ attached identically to every lattice point $R$ of a Bravais lattice $\Lambda$. The ideal periodic structure is the labeled setThe primitive lattice translation vectors generate $R$ by integer combinations; the atomic basis lists what is repeated at $R$. These are different meanings of the word basis. In particular, the offsets $r_j$ need not be linearly independent vectors, and one of them may be zero. A primitive unit cell represents $N$ motif sites, even though it represents only one lattice point.

Changing the origin, choosing another primitive unit cell, or moving a motif site by a Bravais lattice translation changes the coordinates used to describe the same infinite crystal structure. The grouping into a motif does not, by itself, identify bonded molecules. In solid sodium chloride, grouping one sodium cation and one chlorine anion into an atomic basis does not create a collection of discrete NaCl molecules; the ionic crystal has an extended network of equivalent neighboring ions.

The atomic number $Z$ is the number of protons in an atomic nucleus, and fixes the chemical element. It is a count, with no units. An atomic nucleus with six protons is always carbon, regardless of how many neutrons accompany them or how many electrons surround it. See the IUPAC Gold Book, atomic number.

Three independent bookkeeping questions become easy once their symbols are kept separate. Let $N$ count neutrons, $n_e$ count electrons, $A$ be the mass number, and $Q$ be the net electric charge. A proton carries electric charge $+e$ and an electron carries $-e$, where $e$ is the positive elementary charge; a neutron carries zero net electric charge. HenceThe reasoning is just counting: add protons and neutrons for $A$, and subtract negative electron contributions from positive proton contributions for $Q$. A neutral chemical atom has $Q=0$, so $n_e=Z$. For a positively charged ion, fewer electrons remain. Notation ${}_Z^A\mathrm{X}$ gives the mass number at upper left and atomic number at lower left, with $\mathrm{X}$ standing for the chemical element; a right superscript such as $+$ specifies the electric charge in units of $e$. See Flowers et al., Chemistry 2e, atomic structure and symbolism.

For neutral carbon-12, ${}_6^{12}\mathrm{C}$, the atomic number gives six protons, the subtraction $N=12-6=6$ gives six neutrons, and neutrality gives six electrons. Neutral carbon-13, ${}_6^{13}\mathrm{C}$, instead has $N=13-6=7$ neutrons, with the same six protons and six electrons. Thus these two isotopes occupy the same carbon box in the periodic table.

For neutral sodium of mass number $23$, ${}_{11}^{23}\mathrm{Na}$, the counts are eleven protons, $23-11=12$ neutrons, and eleven electrons. Removing one electron gives the ion ${}_{11}^{23}{\mathrm{Na}}^{+}$: $Q/e=1$, so $n_e=11-1=10$. Both its atomic number and its twelve neutrons stay unchanged. This is ionization, not conversion into the chemical element neon, even though a neutral neon chemical atom also has ten electrons.

This bookkeeping is useful whenever a symbol must be translated into a physical inventory. Mass number answers how many protons plus neutrons there are; electric charge answers the imbalance between protons and electrons; atomic number alone answers which chemical element it is. Keeping those questions distinct prevents an isotope or an ion from being mistaken for a different chemical element.

Attosecond streaking uses a synchronized reference electric field to change a photoelectron's outgoing momentum. Scanning relative delay produces an energy-versus-delay distribution. A short emission interval samples only a small part of the reference cycle; a longer interval samples a range of kicks and can broaden or reshape the distribution.

For an explicit classical limit, use the electric dipole approximation and choose the magnetic vector potential $A(t)$ along the detection axis with $E(t)=-dA/dt$ and $A(\infty )=0$. Neglect the residual ion's Coulomb force. An electron released at $t_i$ with mechanical momentum $p_0$ obeysIndeed, integrating $\dot{p}=-eE=e\dot{A}$ from release to the end of the field gives the stated minus sign. The useful clock variable here is $A$, rather than the instantaneous $E$ itself. If $A$ is nearly linear across a narrow emission window with temporal standard deviation ${\sigma }_t$, and initial momentum is sharp, thenThis local mapping fails near a zero of $E$, where first-order timing sensitivity vanishes. A real inversion must include the incident bandwidth, initial photoelectron distribution, collection angles and target–field response.

For an original dimensionless example in atomic units, choose $p_0=2$, $m=e=1$ and $A(t_i)=0.10$. Then $p_f=1.9$ and $K_f=1.805$, compared with initial kinetic energy $2$. Reversing $A$ gives $K_f=2.205$: the shifts are asymmetric because of the $A^2/2$ term. A measurement that keeps only the term linear in $A$ would miss $0.005$ in both cases.

Base pairing is the recognition and association of compatible nucleobases. In canonical Watson–Crick geometry, an adenine–thymine base pair has two hydrogen bonds, while a guanine–cytosine base pair has three. These counts refer to those specific geometries and usual chemical forms; they are not a definition of every possible base pair. RNA can also contain other pairings, and canonical adenine–uracil pairing replaces adenine–thymine pairing. See Clark, Douglas and Choi, Biology 2e: DNA structure.

For this illustrative duplex, the six base pairs are A–T, C–G, G–C, T–A, A–T, and C–G. Thus three contribute two hydrogen bonds each and three contribute three each:Each dotted connector in the diagram counts one hydrogen bond; coloured boxes denote whole nucleobases, not individual chemical atoms. The solid rails denote the covalent bond scaffold. Separating the two strands disrupts base pairing and base stacking without requiring the backbone to be cut.

The count of fifteen hydrogen bonds is not fifteen identical contributions to a duplex's Gibbs free energy. Base stacking, interactions with water and ions, temperature, and the order of neighbouring base pairs also matter. Knowing the fraction of guanine–cytosine base pairs alone does not fix a nucleic acid double helix's shape or stability. See Brown, Genomes: pairing and stacking interactions. What makes base pairing especially useful is its selectivity: an exposed DNA strand constrains the partner's nucleic acid sequence, giving DNA replication a chemical recognition rule.

In 1990, Wayne M. Itano, Daniel J. Heinzen, J. J. Bollinger and David J. Wineland reported inhibited transitions in about 5,000 ${}^9{\mathrm{Be}}^{+}$ ions held in a cylindrical Penning trap. Trapped magnesium ions, illuminated at 280 nm, kept the beryllium ions cold. A 320.7 MHz radio frequency field coupled two ground-state hyperfine structure sublevels during a 256 ms pulse adjusted for complete transfer. Up to 64 intervening 313 nm pulses addressed an optical transition from level 1: scattering distinguished it from level 2. Afterward, calibrated fluorescence counts over 100 ms measured the final quantum state population. This ensemble readout did not select ions that remained in the initial quantum state at every check; departure followed by return also contributes. The apparatus made inhibition by repeated quantum measurements experimentally accessible.^[ref]

For point ions of charge $\pm e$ in potential $\psi$, equilibrium electrochemical potentials give $n_{\pm }=n_0\exp [\mp e\psi /(k_{B}T)]$, with $n_0$ the reservoir density of each species. The charge density is $-2e{n}_0\sinh [e\psi /(k_{B}T)]$. A negative wall enriches positive counterions and depletes negative co-ions.

A bond-dissociation energy specifies the energy needed for a stated dissociation into separated fragments. An energy from the lowest vibrational state, an energy from the bottom of a potential energy well, and a finite-temperature molar dissociation enthalpy require different conventions; do not silently mix them in an electronegativity formula.

Which fragments, in which quantum states, and at which temperature? For a neutral gaseous molecule with a specified two-electron covalent bond, the usual thermochemical convention is homolysis,Each fragment receives one of the two bonding electrons; the dots denote unpaired electrons. The final fragments are isolated from each other, with their own relaxed molecular geometry and specified electronic quantum states. The energy cost includes this relaxation, rather than merely stretching the initial chemical bond while freezing everything else. For a neutral parent, heterolysis instead produces ${\mathrm{A}}^{+}(g)+{\mathrm{B}}^{-}(g)$, or the opposite assignment of electric charges; those are different final states and different energies. A value for gaseous neutral fragments cannot be transferred to ions, a chemical solution, or a solid without accounting for those changed conditions.^[ref]

The distinction is useful beyond terminology. Pauling electronegativity compares consistently defined bond-dissociation energies; photodissociation can reveal a specified ground-state threshold; a thermal chemical reaction balance requires compatible enthalpies. A threshold is also different from an activation energy: it compares initial and final quantum states, whereas a barrier can obstruct the path between them.

A Born-Haber cycle uses Hess's law to infer a lattice energy from measurable enthalpy differences and isolated-species data. The motivating difficulty is that separating a mole of an ionic crystal into mutually separated gaseous ions is not a convenient direct calorimetry experiment. Instead, compare two accounting routes between the same elemental reference states and the same solid. Because enthalpy is a state function, the signed differences close exactly when all inputs describe compatible states. These routes do not assert a reaction mechanism or a temporal sequence of reaction intermediates.^[ref]

For one mole of sodium chloride formula units, the overall formation reaction isThe stoichiometry requires half a mole of diatomic chlorine molecules, supplying one mole of chlorine chemical atoms. A half coefficient counts an amount of substance; it does not describe half of an individual molecule. Write each auxiliary chemical reaction with its own sign:Here $S$ is the sublimation enthalpy, $B$ the enthalpy of atomization of chlorine per mole of chemical atoms, and $I$ the molar ionization enthalpy represented by the ionization energy input in elementary treatments. If the bond-dissociation energy table gives the compatible molar enthalpy for ${\mathrm{Cl}}_2(g)\to 2\mathrm{Cl}(g)$, use half that number for $B$. Both ions in the last step are gaseous, and no aqueous hydration of ions belongs in that step.

Adding these equations cancels the gaseous chemical atoms, the gaseous ions and the free electron, leaving precisely the overall formation reaction. ThusThe closed traversal returns from the solid to the elemental reference states, adding $-F$; therefore $S+B+I+A+L_{\mathrm{form}}-F=0$. Notice that the signed attachment term $A$ is negative. Substituting a positive spectroscopic electron affinity for $A$ would reverse the attachment contribution.

The diagram deliberately uses invented, internally consistent inputs in $\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$: $S=100$, $B=120$, $I=500$, $A=-350$ and $F=-400$. Preparation of separated gaseous ions raises the enthalpy by $370$, so $L_{\mathrm{form}}=-770$ and $L_{\mathrm{diss}}=+770$. The closure check is $100+120+500-350-770+400=0$. These numbers are not measured sodium chloride data and do not replace the source's $769\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ example in lattice energy. Every level retains one sodium nucleus, one chlorine nucleus and zero total electric charge; the ionization level explicitly retains the free electron.

Assume an ideal dilute aqueous chemical solution with just one monoprotic acid pair $\mathrm{HA}/{\mathrm{A}}^{-}$, self-ionization of water, and other ions that balance electric charge without undergoing further chemical reactions. Hold the analytical concentration $C=[\mathrm{HA}]+[{\mathrm{A}}^{-}]$ constant. Keep the acid dissociation constant $K_a$ and the self-ionization of water constant $K_w$ dimensionless, consistently with pH:Thus $H$ denotes normalized hydrogen ion concentration, not a dimensional concentration. The chemical equilibrium and conserved analytical concentration giveAdding chemical base converts weak acid to conjugate base, increases free hydroxide, and decreases free hydrogen ion concentration. Balancing electric charge, including the added reagent's accompanying ions, gives the signed dose relative to any initial chemical equilibrium at $p_0$:Differentiate with respect to pH, using $\mathrm{d}H/\mathrm{d}p=-(\ln 10)H$. The result isThe first term is the monoprotic acid pair's contribution, ${\beta }_{\mathrm{pair}}$; the remaining terms are the free hydrogen ion and hydroxide contributions, ${\beta }_{\mathrm{water}}$. Each has units $\mathrm{mol}{\mathrm{L}}^{-1}$ per pH. Equivalently, with dimensional $H_c=[{\mathrm{H}}^{+}]$, $K_{a,c}=c^{\circ }{K}_a$ and $K_{w,c}=(c^{\circ }{)}^2{K}_w$, the expression is $(\ln 10)(C{K}_{a,c}{H}_c/(K_{a,c}+H_c{)}^2+H_c+K_{w,c}/H_c)$. Mixing the dimensional and dimensionless conventions would give incorrect units.

For $r=K_a/H=10^{\mathrm{pH}-\mathrm{p}{K}_a}$,because $(1+r{)}^2-4r=(1-r{)}^2\ge 0$. Equality occurs at $r=1$: equal weak acid and conjugate base concentrations, or pH equal to pKa. This is the exact maximum of the pair contribution in this model. The full buffer capacity need not have its maximum there: free hydrogen ions and hydroxide make it grow toward the acidic and alkaline extremes. When these contributions are small, the central peak lies close to pKa.

The cathode is the electrode at which the net electrode chemical reaction is reduction: the reacting chemical species gains electrons. This definition is independent of terminal sign. During Daniell cell discharge, ${\mathrm{Cu}}^{2+}$ cations accept electrons arriving at the copper cathode and become neutral copper. The cathode is then the positive terminal. In an externally driven electrolytic cell, the source's negative terminal supplies electrons to the negative cathode. See Pingarrón et al., §3.4: cathode and Daniell cell.

An ion carrying electric current need not undergo reduction at the cathode. For example, in the ideal Daniell cell charge-balance picture, potassium cations arriving from a salt bridge remain dissolved while ${\mathrm{Cu}}^{2+}$ cations are consumed. Follow the half-reaction to identify the reacting species; follow electric charge conservation to account for the other moving ions.

A cation is an ion with positive net electric charge, for example ${\mathrm{Na}}^{+}$ or hydronium.