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An acid donates protons and a base accepts them; pH is the negative logarithm of the hydrogen-ion concentration, so each unit of pH is a tenfold change in acidity.

The relatively weak, exposed pi bond in an alkene or alkyne is broken open by an addition reaction, in which an electrophile is attracted to the electron-rich pi bond, forms a bond to one carbon, and generates a cationic intermediate that a nucleophile then attacks, converting the double or triple bond into new single bonds to the added atoms.

The Langmuir isotherm models adsorption of a gas onto a solid surface as a dynamic equilibrium between molecules sticking to and leaving a finite number of identical surface sites, predicting that surface coverage rises linearly with pressure at low pressure but saturates as every site fills at high pressure, unlike the unbounded, ever-increasing coverage a simple proportional model would wrongly predict.

An aromatic ring like benzene has its electrons delocalised evenly around the whole ring rather than fixed in alternating single and double bonds, which makes it far more stable than a normal alkene and changes how it reacts.

The Arrhenius equation, k = A·e^(−Ea/RT), captures why reaction rates rise sharply with temperature: only molecules colliding with kinetic energy above the activation energy Ea can react, and the exponentially growing fraction of molecules meeting that threshold as temperature rises is what drives the rate constant's strong temperature dependence.

The carbon-oxygen double bond in a ketone or aldehyde is strongly polarised, leaving the carbon electron-poor and directly attractive to nucleophiles, which add across the pi bond to form a tetrahedral alkoxide intermediate — the same polarisation also lets the alpha carbon's C-H bonds ionise as a weak acid, opening a second, complementary reactivity mode called enolisation.

Carboxylic acid derivatives (acid chlorides, anhydrides, esters, and amides) share a carbonyl carbon bonded to a leaving group, and a nucleophile attacking that carbon forms the same tetrahedral intermediate seen in simple carbonyl addition, except the leaving group is then expelled to restore the carbonyl — an addition-elimination sequence whose overall rate is set by how good a leaving group each derivative carries.

A catalyst speeds up a reaction by opening an alternative reaction pathway with a lower activation energy, without being consumed itself and without changing the reaction's overall equilibrium position, and is classified as homogeneous when it shares the same phase as the reactants (as with enzymes in solution) or heterogeneous when it exists in a separate phase, most often as a solid surface that reactant molecules adsorb onto.

Atoms bond by transferring or sharing electrons, and where a bond sits on the ionic-to-covalent scale is set by the difference in electronegativity — which in turn predicts a substance's melting point, solubility and conductivity.

A reversible reaction reaches equilibrium when forward and reverse rates become equal, so concentrations stop changing while both reactions continue — and a disturbance shifts the balance in the direction that partly offsets it.

A colloidal particle suspended in a liquid picks up a surface charge, which attracts a cloud of counter-ions into a diffuse electric double layer around it; when the particle moves relative to the liquid — under an applied field (electrophoresis) or under gravity — only part of that cloud moves with it, so the electrokinetic (zeta) potential at the slipping plane, not the bare surface charge, is what actually governs how strongly like-charged particles repel each other and therefore whether the dispersion stays stable or coagulates.

A strong electrolyte dissociates essentially completely into ions in solution while a weak electrolyte reaches only partial dissociation equilibrium, and this distinction shows up directly in molar conductivity, which for a strong electrolyte declines only mildly with concentration due to ion-ion interactions but for a weak electrolyte drops sharply as concentration rises because the fraction actually dissociated falls.

Benzene's delocalised ring of electrons reacts with electrophiles by substitution rather than addition, because the intermediate formed by attack keeps a partial ring and the final step re-forms the aromatic system by losing a proton rather than gaining a second group, preserving the large stabilisation energy that addition would otherwise destroy.

A functional group is the specific arrangement of atoms in a molecule that reacts, so molecules with the same functional group undergo the same characteristic reactions regardless of how large or different the rest of the molecule is.

Hydrogen forms compounds across almost the entire periodic table, ranging from ionic hydrides with reactive metals to covalent hydrides with nonmetals, and passing electric current through water or a molten salt (electrolysis) drives the otherwise non-spontaneous decomposition reaction by forcing electrons onto cations at the cathode and pulling them from anions at the anode.

A mass spectrometer identifies molecules by ionising a sample, separating the resulting ions by their mass-to-charge ratio, and recording the abundance of each ratio, producing a mass spectrum whose fragmentation pattern acts as a structural fingerprint of the original molecule in addition to giving its precise molecular mass.

The electrochemical (activity) series ranks metals by their tendency to lose electrons and oxidise, predicting both which metal in a pair will corrode preferentially when they are in electrical contact and why a more reactive 'sacrificial' metal can protect a less reactive one from corrosion by oxidising in its place.

Molecular orbital theory builds bonding by combining atomic orbitals into new orbitals delocalised over the whole molecule, with bonding combinations lower in energy than the original atomic orbitals and antibonding combinations higher, while orbital hybridisation is a complementary bookkeeping device that mixes an atom's own orbitals into new shapes matching the geometry actually observed, such as the tetrahedral sp3 arrangement around a typical carbon atom.

A nucleophile is an electron-rich species that attacks an electron-poor carbon, displacing a leaving group; the reaction can happen in one concerted step or in two steps through a stable intermediate, and which pathway operates changes what predicts the outcome.

An element's oxide is basic if the element is strongly metallic, acidic if the element is strongly nonmetallic, and amphoteric for elements near the metal-nonmetal boundary, and the same trend continues into oxoacids — acids containing a central atom bonded to oxygen — whose acid strength increases with the electronegativity and oxidation state of that central atom.

Along the boundary between two phases of a pure substance on a phase diagram, the vapour pressure and temperature are locked together by the Clausius-Clapeyron equation, which follows from requiring both phases to have equal Gibbs energy at equilibrium and explains why boiling point rises with pressure and why the boundary's steepness is set by the transition's enthalpy.

Raoult's law states that an ideal solution's vapour pressure from each volatile component is that component's pure vapour pressure scaled by its mole fraction, and this simple dilution effect is the common origin of the colligative properties — boiling point elevation, freezing point depression, and osmotic pressure — which depend only on how many solute particles are dissolved, not on what they chemically are.

Reaction rate depends on how often reactant particles collide with enough energy and the right orientation, so concentration, temperature, surface area and catalysts all change it — while the equilibrium position stays untouched.

A covalent bond can break homolytically, with each atom keeping one electron to form two neutral radicals, or heterolytically, with both electrons going to one atom to form a cation and an anion, and which pathway a reaction follows determines whether the reactive species produced are radicals or the electrophiles and nucleophiles that dominate ionic organic chemistry.

A reaction's rate law expresses its rate as a product of a rate constant and the concentrations of reactants raised to experimentally determined powers — the reaction order in each species — which must be measured, not read off the balanced equation, since a reaction's order reflects its actual step-by-step mechanism rather than its overall stoichiometry.

In a redox reaction electrons transfer from one species to another: the donor is oxidised and the acceptor reduced, and because electrons cannot be created the two always happen together in matched amounts.

A balanced equation gives the fixed ratio in which substances react, in moles rather than in grams — so predicting how much product forms means converting to moles, applying the ratio, and converting back.

Group 14 spans carbon, whose ability to form strong bonds with itself and a wide range of other elements underlies the whole of organic chemistry, through silicon, whose comparably strong bonds to oxygen instead make silicate minerals the dominant chemistry of the solid Earth, down to increasingly metallic germanium, tin, and lead.

The halogens are the most reactive non-metal group because each atom needs only one more electron for a full outer shell, and their oxidising power falls steadily going down the group as that electron gets harder to attract.

The mole is the SI unit for counting particles, defined so that one mole of any substance contains the same number of entities — which lets a mass measured on a balance be converted into a count of atoms or molecules.

The Nernst equation relates an electrochemical cell's actual voltage to its standard potential and the concentrations (activities) of the species involved, showing that cell potential is not a fixed constant but shifts with concentration in a way that reaches zero exactly when the cell reaction reaches equilibrium.

Group 15 spans from nitrogen, a diatomic gas whose triple bond makes it exceptionally unreactive, through phosphorus and arsenic to the increasingly metallic antimony and bismuth, and nitrogen's own chemistry is dominated by an unusually wide range of oxidation states, from -3 in ammonia to +5 in nitric acid, reflecting how many different bonding arrangements its five valence electrons can support.

Ordering the elements by atomic number arranges them into groups whose members share an outer-electron configuration, so properties such as atomic radius, ionisation energy and reactivity vary in predictable trends across a period and down a group.

Transition metals fill an inner d-subshell rather than the outermost one, which lets them adopt several oxidation states, form coloured compounds, and bind surrounding molecules as coordination complexes held together by donated electron pairs.

A coloured or UV-absorbing solution absorbs light most strongly at wavelengths matching electronic transitions in its molecules, and the Beer-Lambert law states that absorbance is directly proportional to both the absorbing species' concentration and the path length of light through the sample, turning a simple absorbance measurement into a direct quantitative determination of concentration.

An algorithm is a finite, unambiguous sequence of steps that solves a class of problems; a program is one expression of an algorithm in a language, and the same algorithm can be written in many languages.

Conditionals let a program choose between paths and loops let it repeat work, and between them they turn a fixed list of instructions into something whose behaviour depends on its data.

A data structure organises values in memory so that particular operations — accessing an element, searching for one, inserting or removing one — can be performed efficiently, and choosing the right structure (an array for fast indexed access, a linked list for cheap insertion, and so on) is what actually determines whether an algorithm running on that data scales well as the amount of data grows.

Numerical methods solve equations that have no convenient closed-form solution by breaking the problem into many small, repeated steps a computer can execute, such as Euler's method for approximating a differential equation's solution as a sequence of small linear steps, at the cost of introducing approximation error that must be understood and controlled.

Not every solid is crystalline: an amorphous solid such as glass is cooled fast enough that its atoms freeze into a disordered, liquid-like arrangement without ever settling into a periodic lattice, while a liquid crystal occupies the opposite middle ground, retaining a liquid's ability to flow while still holding a partial, direction-dependent molecular order that ordinary liquids lack.

Diffraction, microscopy and spectroscopy each reveal a different scale of a material's structure — crystal geometry, real-space topology, or chemical bonding — and choosing the right method means matching the technique's probe to the length scale of the question.

Metals, ceramics, polymers and composites differ first in how their atoms bond, and that difference propagates into every property that decides where each one is used.

A composite material combines two or more distinct constituent materials — typically a reinforcing phase carrying most of the mechanical load and a surrounding matrix that binds it together and transfers stress — to achieve a combination of properties, such as high stiffness at low weight, that neither constituent could deliver alone.

A stressed material first deforms elastically, recovering its original shape completely once the stress is removed, but past a material-specific yield point it deforms plastically instead, undergoing a permanent shape change driven at the microscopic level by the irreversible motion of dislocations through the crystal lattice, and continued deformation eventually leads to fracture.

A polymer is a giant molecule built from many repeating small units (monomers) joined by polymerisation, and whether it softens reversibly on heating (thermoplastic, held together by chains that can slide past each other) or is permanently rigid once formed (thermoset, locked into a single covalently cross-linked network) is set entirely by the presence or absence of cross-links between chains.

A material's useful properties follow from its structure at several scales at once — atomic bonding, crystal arrangement, defects and microstructure — which is why the same element can be soft graphite or hard diamond.

Separating variables in Laplace's or the wave equation on a curved domain — a cylinder, a sphere, a quantum well — leaves behind a second-order ODE that isn't solved by sines, cosines or exponentials; each named family (Bessel for cylindrical symmetry, Legendre for spherical, Hermite for the quantum harmonic oscillator, Laguerre for the hydrogen atom's radial part) is the specific solution to one such equation, and together they form a complete orthogonal basis for expanding functions on that geometry, the same role sines and cosines play in ordinary Fourier series.

A general second-order linear PDE Au_xx + Bu_xy + Cu_yy + ⋯ = 0 is classified by the discriminant B² − 4AC in exact analogy with a conic section: negative means elliptic (like Laplace's equation, describing steady states), zero means parabolic (like the heat equation, describing diffusion), and positive means hyperbolic (like the wave equation, describing propagation) — and each class genuinely needs a different kind of boundary or initial condition to have a well-posed solution.

A handful of named distributions — binomial for counting successes in a fixed number of independent trials, Poisson for counting rare events over a continuum, normal for continuous measurement error, and Student's t for small-sample estimation — cover most of the random variables statistics actually deals with, and each is fixed by only one or two parameters.

Adjoining a number i whose square is −1 extends the real numbers to the complex numbers a + bi, in which every polynomial equation has a solution and arithmetic still works exactly as before.

A conformal map is a function of a complex variable that preserves angles between intersecting curves at every point where its derivative is nonzero — a consequence of holomorphy, not an extra assumption — which makes such maps the tool of choice for warping an awkward geometric region into a simple one (a disk, a half-plane) while keeping the local angular structure that a physical field's boundary conditions depend on.

The first derivative's sign locates where a function rises or falls and where it has local extrema; the second derivative's sign locates where the graph is convex or concave and where it has inflection points — together they turn an algebraic expression into an accurate sketch without plotting points.

The determinant of a square matrix is a single number giving the factor by which the transformation scales volume — zero exactly when the transformation collapses space and cannot be inverted.

Divergence measures how much a vector field spreads outward from a point, and curl measures how much it circulates around one — a fluid's velocity field has positive divergence at a source and non-zero curl wherever there is local rotation.

A system of first-order differential equations can be pictured as a flow on a plane, where every point is pushed in a direction and speed set by the equations; a fixed point is where that flow stops, and linearising the system there — via the eigenvalues of the Jacobian — classifies it as a stable node, unstable node, saddle, or spiral without ever solving the full nonlinear system.

An eigenvector of a matrix is a direction the transformation only stretches, never rotates, and the stretch factor is its eigenvalue — finding them reveals the axes along which a complicated transformation acts as simply as multiplication.

A differential equation relates a function to its own derivative; separable and linear first-order equations are the two types that can be solved by direct integration, giving a family of solutions narrowed by an initial condition.

A Fourier series represents a periodic function as an infinite sum of sines and cosines, built from the fact that these functions are mutually orthogonal over one period — the same orthogonality that gives each coefficient a closed-form integral, independent of every other term.

A function assigns exactly one output to each input; it can be undone by an inverse function only when it is one-to-one, which is why domains often have to be restricted before an inverse exists.

A function of a complex variable is differentiable in an unusually strong sense — holomorphic — exactly when its real and imaginary parts satisfy the Cauchy–Riemann equations, and that rigidity is what makes complex analysis far more powerful than its real counterpart.

A Green's function G(x, ξ) is the response of a linear operator to a single point impulse at ξ, built once from the boundary conditions alone — once it exists, the solution to that same operator acting on any source term f(x) is just the integral ∫G(x, ξ)f(ξ)dξ, turning every future problem with those boundary conditions into a lookup and an integral instead of a fresh derivation.

Hypothesis testing starts by assuming the null hypothesis is true, computes a test statistic measuring how far the sample departs from that assumption, and rejects the null only when that departure would be too unlikely to be explained by chance alone — it can disprove a null hypothesis but never proves one true, only fails to reject it.

An improper integral extends the definite integral to an infinite limit of integration or an integrand that is unbounded within the interval; it has a value only when the corresponding limit of ordinary integrals actually converges, not merely because the integrand shrinks.

An infinite series is the limit of its partial sums; it converges only when those sums settle on a value, and terms shrinking to zero is necessary for that but nowhere near sufficient.

A partial differential equation relates a function of several variables to its own partial derivatives, so a solution is a function rather than a number; the heat, wave and Laplace equations are the three archetypes worth recognising on sight, because each behaves qualitatively differently and models a different kind of physical process.

The Laplace transform rewrites a function of time as a function of a complex frequency variable s, turning derivatives into multiplication by s — so a linear differential equation with given initial conditions becomes an algebraic equation, solvable by ordinary algebra and then transformed back.

A Laurent series generalises a Taylor series by allowing negative powers of (z − z₀), which is exactly what's needed to represent a function around a point where it isn't holomorphic; the pattern of those negative-power terms classifies the singularity as removable (none), a pole of finite order (finitely many), or essential (infinitely many) — and only the essential case defeats every algebraic trick for evaluating a limit there.

The limit of a function at a point is the value its outputs approach as the input closes in, whether or not the function is even defined there — which is what makes it the right tool for describing instantaneous rates.

A line in space is fixed by a point and a direction vector, a plane by a point and a normal vector — and once both are written that way, questions about intersection, distance and angle become dot and cross products.

A matrix is a rectangular array of numbers that encodes a linear transformation; matrix multiplication composes those transformations, which is why it is not commutative.

A double or triple integral accumulates a quantity over a region of the plane or of space by integrating one variable at a time, with the limits of the inner integral describing the shape of the region.

An unconstrained local extremum of f(x, y) occurs only where every partial derivative vanishes, and the second-derivative (Hessian) test then tells a maximum from a minimum from a saddle point; when the search is restricted to a constraint g(x, y) = c, Lagrange's method replaces that condition with ∇f = λ∇g — the geometric fact that at a constrained extremum the level curve of f is tangent to the constraint curve.

A partial derivative measures how a function of several variables changes when one variable moves and the rest are held fixed, so a surface has a different slope in every direction and needs several derivatives to describe it.

A point estimate is a single number computed from a sample to stand in for an unknown population parameter, but it is almost never exactly right; an interval estimate reports a range around it — built from the sample's own variability — that would contain the true parameter in a stated proportion of repeated samples, which is the honest way to communicate how much a point estimate can be trusted.

Writing a complex number by its modulus and argument instead of its real and imaginary parts turns multiplication into multiplying lengths and adding angles, and Euler's formula compresses the whole idea into re^(iθ).

Probability assigns each outcome a number between 0 and 1; the rules for combining events depend on whether they are mutually exclusive and whether they are independent, which are different questions with different tests.

A random variable attaches a number to each outcome, and its distribution says how that number is spread; expectation and variance summarise the distribution, and the normal distribution arises whenever many small effects add together.

The residue theorem says the integral of a function around a closed contour equals 2πi times the sum of its residues — a single number attached to each isolated singularity enclosed by the path — turning integrals that are hard or impossible by real-variable methods into an algebra problem of finding a handful of residues.

A linear second-order differential equation with constant coefficients is solved by substituting a trial exponential solution, turning the differential equation into an algebraic characteristic equation whose roots determine whether the general solution is a sum of real exponentials, a single repeated exponential times a polynomial factor, or an oscillating exponential — exactly the mathematics behind damped and driven oscillators in physics.

Separation of variables guesses a solution to a partial differential equation as a product of single-variable functions, which turns the PDE into several ordinary differential equations linked by a shared constant; the resulting boundary-value ODE is a Sturm-Liouville problem, whose eigenvalues give the allowed separation constants and whose eigenfunctions form a complete, orthogonal basis for building the general solution as a series.

A system of linear equations has exactly one solution, none, or infinitely many — never any other count — and Gaussian elimination decides which case holds while producing the solution.

A tensor generalises vectors and matrices to an object with any number of indices that transforms in a fixed, predictable way under a change of coordinates; curvilinear systems like polar or spherical coordinates need this machinery because their basis vectors change direction from point to point, so gradient, divergence and curl each pick up extra terms that vanish in flat Cartesian coordinates.

The cross product of two vectors in space is a third vector perpendicular to both, whose length equals the area of the parallelogram they span and whose direction is fixed by the right-hand rule.

A definite integral is the limit of a sum of thin slices, giving an accumulated total such as area or work; the fundamental theorem of calculus evaluates it using any antiderivative.

The derivative of a function at a point is the limit of its difference quotient — the instantaneous rate of change, equal to the slope of the tangent line there.

The divergence theorem converts a volume integral of a field's divergence into a flux through the enclosing surface, and Stokes' theorem converts a surface integral of a field's curl into a circulation around the boundary curve — both are the fundamental theorem of calculus generalised to fields.

The dot product multiplies two vectors into a single number that measures how much they point the same way — it vanishes exactly when they are perpendicular, which makes it the standard test for a right angle.

Antidifferentiation reverses the derivative, recovering a family of functions that differ only by a constant — which is why every indefinite integral carries a +C.

A vector field assigns a vector to every point in space; the gradient of a scalar field is the vector field pointing in the direction of steepest increase, and it is always perpendicular to the field's level surfaces.

A vector space is any set whose elements can be added and scaled under the usual rules; a set of vectors is linearly independent when none is a combination of the others, and an independent spanning set is a basis whose size is the dimension.

A vector in space carries both a magnitude and a direction, is written in coordinates as an ordered triple, and adds component by component — so geometry in three dimensions becomes arithmetic.

In an alternating-current circuit, resistors, capacitors and inductors oppose current with an impedance that combines resistance with frequency-dependent reactance, so current and voltage can drift out of phase with each other.

An aerosol is solid or liquid particles suspended in air, and almost everything about one follows from its size: particles above a few micrometres settle out within minutes, those below a micrometre stay airborne for days and are carried by air currents and Brownian motion rather than gravity, and only the small ones reach deep into the lungs — which is why air quality limits are written for PM10 and PM2.5 rather than for mass alone. Measuring them means choosing which physical effect stands in for size: weighing what a filter collects, counting scattered-light pulses one particle at a time, or reading an attenuation, each of which answers a different question and can disagree with the others.

Ampère's law says the circulation of the magnetic field around any closed loop is proportional to the electric current passing through it, giving the magnetic-field analogue of Gauss's law and, once corrected by Maxwell, half of the electromagnetic wave equation.

Earth's atmosphere and its oceans are both fluids held in hydrostatic balance by gravity, so pressure and density fall off roughly exponentially with altitude in the air and rise with depth in the water; because air obeys the ideal gas law, a rising parcel expands and does work against the lower pressure around it, cooling adiabatically even with no heat exchanged — which is why the troposphere gets colder with altitude — while in the ocean, temperature and salinity together set the water's density and organise it into stable layers rather than a single well-mixed temperature.

An atom has a tiny, dense, positively charged nucleus made of protons and neutrons, surrounded by electrons; atoms of the same element with different neutron counts are called isotopes.

When atoms bond into a crystal, their discrete atomic energy levels broaden into continuous allowed energy bands separated by forbidden gaps, and whether a material conducts, semiconducts, or insulates is determined entirely by how these bands are filled with electrons and how large the gap is between the highest filled band and the next empty one.

A molecule's full Schrödinger equation couples every electron's motion to every nucleus's motion, which is intractable directly — the Born-Oppenheimer approximation exploits the huge mass difference between nuclei and electrons to solve them separately: electrons adjust to the nuclei's positions essentially instantly, giving a potential energy surface on which the much heavier, slower nuclei then vibrate and rotate, each motion quantized into its own ladder of discrete energy levels.

When two waves of slightly different frequency overlap, they alternately reinforce and cancel, producing a slow throbbing in loudness called beats, whose rate equals the plain difference between the two frequencies.

Along a streamline of a flowing fluid, the sum of pressure, kinetic energy per volume and gravitational potential energy per volume stays constant — where the fluid speeds up, its pressure must drop.