Formulas & shortcuts

Each element is either in the subset or out of it: 2 choices per element, 2ⁿ in total. For proper subsets remove the set itself; for non-empty proper subsets remove ∅ as well.

Union means “or”, intersection “and”, complement “not”. A complement is always taken in U; for sets of numbers with no U given, U = R. Always A ∩ B ⊆ A ⊆ A ∪ B, and both A ∩ B = A and A ∪ B = B mean A ⊆ B.

De Morgan’s laws: the complement turns a union into an intersection and an intersection into a union. “Not (A or B)” = “not A and not B”.

Adding |A| and |B| counts the common part twice, so it is subtracted once.

Add the singles, subtract the pairs, add back the triple: the centre is added 3 times and subtracted 3 times, so it must be added once more.

The set picture: a condition with a smaller solution set implies a condition with a larger one. “Small ⇒ large”: the smaller set is the sufficient condition, the larger set the necessary one (in Chinese textbooks 小范围推出大范围).

The only rule that reverses the sign: multiplying or dividing by a negative number. You may not multiply by an expression of unknown sign, such as x − 3, at all.

“Greater — outside the roots, less — between the roots” (when a > 0). If a < 0, first multiply the inequality by −1 and reverse the sign.

For “≥ 0 for every x”, D ≤ 0 (touching is allowed). For “< 0 for every x”, a < 0 and D < 0.

|x − c| < a: x is less than a away from c; |x − c| > a: x is more than a away from c. For |f(x)| < a solve the double inequality −a < f(x) < a.

A composite function (复合函数): the order matters, f(g(x)) ≠ g(f(x)) in general. If D(f) = [m, n], the domain of f(g(x)) is the solution set of m ≤ g(x) ≤ n.

Increasing + increasing is increasing; −f reverses the direction, and so does 1/f when f keeps one sign. For y = f(g(x)): if f and g change in the same direction the composite increases, if in opposite directions it decreases — 同增异减. The derivative gives the monotonicity of any function: “Applications of the derivative: monotonicity and extrema”.

odd ± odd = odd, even ± even = even; odd · odd = even, even · even = even, odd · even = odd — like the signs + and −. A polynomial with only even powers (and a constant) is even; with only odd powers, odd.

Changes inside the brackets act horizontally and “the other way round” (x − 2 moves right, 2x compresses); changes outside act vertically and “as written”. Chinese students remember the shifts as 左加右减，上加下减: “left add, right subtract; up add, down subtract”.

The graphs of f and f⁻¹ are symmetric about the line y = x. The most important pair, y = aˣ and y = logₐ x, is in “Logarithms and logarithmic functions”. Note that f⁻¹(x) is not 1/f(x).

If the even function decreases on [0, +∞), reverse the inequality: f(x₁) < f(x₂) ⇔ |x₁| > |x₂|.

In the first quadrant all power curves cross at (1, 1): “above 1, big exponents win; below 1, they lose” (2³ > 2², but 0.5³ < 0.5²). Parity depends on α = p/q in lowest terms: q odd, p even — even (x^(2/3)); q odd, p odd — odd (x^(3/5)); q even — defined only for x ≥ 0, neither (x^(3/4)).

For y = a^(kx + m) + n, the point where the exponent is 0 does not depend on a: kx + m = 0, y = 1 + n.

Equal bases ⇒ equal exponents, because y = aˣ is monotonic. In an inequality, a base above 1 keeps the sign and a base between 0 and 1 reverses it. Unlike logarithms, there is no domain step: aˣ is defined for every x. If the bases cannot be made equal (2ˣ = 5), the answer is a logarithm, x = log₂ 5 — see “Logarithms and logarithmic functions”.

“Same side ⇒ above 1”: 1.2^0.3 > 1 (both factors positive), 0.7^(−2) > 1 (both negative), 0.7^0.3 < 1 and 1.2^(−0.3) < 1 (opposite signs).

The definition and the basic identities. a^(logₐ N) = N is the logarithmic identity (对数恒等式): raising to a power and taking a logarithm undo each other. Chinese textbooks and CSCA write lg and ln.

In both cases the smaller argument must be positive; the larger one is then positive automatically. First write a number as a logarithm: 3 = log₂ 8, −1 = log_(1/3) 3.

For sin x the axes are x = π/2 + kπ and the centres (kπ, 0); for cos x the axes are x = kπ and the centres (π/2 + kπ, 0). Neighbouring axes are T/2 apart; an axis and the nearest centre are T/4 apart. The tangent curve has no axes, and its centres are (kπ/2, 0).

The other order — stretch first, shift second — also works, but then the shift is φ/ω, because sin(ωx + φ) = sin ω(x + φ/ω): a shift always acts on x itself, never on ωx.

For |a| < 1, sin x = a has two roots per period, symmetric about an axis x = π/2 + 2kπ, and cos x = a has two roots symmetric about x = 2kπ; tan x = a has one root in every period π.

Half the product of two sides and the sine of the angle between them. Together with the law of sines it also gives S = abc/(4R).

A acute: a < h — no triangle; a = h — one right triangle; h < a < b — two triangles; a ≥ b — one triangle. A right or obtuse: one triangle if a > b, none if a ≤ b.

It follows from cos C = (a² + b² − c²)/(2ab): the sign of the numerator is the sign of cos C. The triangle is acute if its largest angle is acute, so check only the longest side.

Terms whose positions have the same sum have the same sum (等差数列的性质). Both sides need the same number of terms: a₁ + a₉ = a₄ + a₆ = 2a₅, but a₁ + a₉ ≠ a₁₀. Consecutive blocks of k terms, Sₖ, S₂ₖ − Sₖ, S₃ₖ − S₂ₖ, again form an arithmetic sequence, with difference k²d.

The product version of the index property of arithmetic sequences. Consecutive blocks of k terms, Sₖ, S₂ₖ − Sₖ, S₃ₖ − S₂ₖ, form a geometric sequence with ratio qᵏ (when Sₖ ≠ 0). Also a₁a₂…a₂ₖ₋₁ = aₖ²ᵏ⁻¹: the product of an odd number of terms is the middle term to that power.

This works for every sequence, not only geometric ones. Sₙ − Sₙ₋₁ is the last term aₙ, but only for n ≥ 2, because S₀ is not part of the rule. Always check whether the formula for n ≥ 2 also gives a₁ = S₁; if not, the answer is piecewise.

Power and trigonometric functions. Special cases worth knowing by heart: (√x)′ = 1/(2√x), (1/x)′ = −1/x².

The chain rule for a composite function (复合函数): outer derivative × inner derivative. The Chinese school programme limits composite functions to the form f(ax + b), so expect this case first of all.

The equation of the tangent line at x₀ (point–slope form).

The physical meaning of the derivative: velocity is the rate of change of position, acceleration the rate of change of velocity (see «Kinematics: displacement, velocity, acceleration, free fall»).

A sufficient condition, not a necessary one: y = x³ increases on the whole line although y′(0) = 0. In a parameter question («f is increasing on I») require f′(x) ≥ 0 on I (zero only at isolated points).

The first-derivative test. f′(x₀) = 0 alone is not enough: y = x³ has y′(0) = 0, but y′ > 0 on both sides, so there is no extremum.

A function continuous on a closed interval always reaches its greatest value (最大值) and least value (最小值) — at a critical point or at an end. No sign chart is needed: just compare the numbers.

The Newton–Leibniz formula, the fundamental theorem of calculus (微积分基本定理). For f ≥ 0 on [a, b] the integral is the area under the curve; for a velocity v(t) ≥ 0 it is the distance travelled.

Perpendicular slopes are negative reciprocals of each other: 2 and −1/2, −3/4 and 4/3. The test needs both slopes to exist: a vertical line is perpendicular to every horizontal one, although it has no slope.

These conditions work for all lines, vertical ones included, so they are the safest choice in parameter questions. A parallel line keeps A and B and changes only C; a perpendicular line swaps A and B and changes one sign: Ax + By + C = 0 → Bx − Ay + C′ = 0.

This is the distance formula |PC| = √((x − a)² + (y − b)²) = r, squared. A point P lies inside, on or outside the circle when |PC| is less than, equal to or greater than r. Some English books write the centre as (h, k); CSCA and Chinese textbooks use (a, b).

In Chinese the three cases are 相交 (intersect), 相切 (tangent) and 相离 (separate). The algebraic check gives the same result: substituting the line into the circle gives a quadratic equation, and Δ > 0, Δ = 0 and Δ < 0 correspond to the three cases.

The perpendicular from the centre to a chord bisects it, so r, d and half the chord form a right triangle (see the drawing).

The first two formulas give the tangent at a point of the circle (“replace one x by x₀ and one y by y₀”). They work because the tangent is perpendicular to the radius at the point of contact; the same right angle gives |PT| by Pythagoras. From a point outside the circle there are exactly two tangents, from a point on it one, from a point inside none.

If two circles intersect, subtracting their general equations removes x² and y² and leaves a line through both common points: the line of the common chord (公共弦).

The end B₂(0, b) of the minor axis is equally far from both foci, and the two distances add up to 2a, so |B₂F₁| = |B₂F₂| = a. The right triangle OB₂F₂ has legs b and c and hypotenuse a: in an ellipse a is the largest of the three numbers.

For the chord: |AF₁| + |AF₂| = 2a and |BF₁| + |BF₂| = 2a, while |AF₁| + |BF₁| = |AB|. Adding gives |AB| + |AF₂| + |BF₂| = 4a, whatever the direction of the chord.

The distance from a point of the ellipse to a focus is called a focal radius (焦半径). Substituting y₀² = b²(1 − x₀²/a²) into |PF₂|² = (x₀ − c)² + y₀² gives (a − ex₀)², so the focal radius is linear in x₀: it is smallest, a − c, at the vertex nearest to the focus and largest, a + c, at the far vertex.

The same chord formula works for any curve, the parabola and the hyperbola included; |x₁ − x₂| comes from Vieta’s formulas without solving the equation.

On the branch nearer to F₂: |PF₁| − |PF₂| = 2a; on the branch nearer to F₁ it is the other way round. The closest a point of the hyperbola gets to a focus is c − a (at a vertex).

The chain of arrows: AB⃗ + BC⃗ + CD⃗ + … + YZ⃗ = AZ⃗ — the inner letters cancel. The usual laws hold: a⃗ + b⃗ = b⃗ + a⃗, λ(a⃗ + b⃗) = λa⃗ + λb⃗.

The midpoint of AB is ((x₁ + x₂)/2, (y₁ + y₂)/2). Equal vectors have equal coordinates — that is how the fourth vertex of a parallelogram is found.

Three points A, B, C lie on one line ⇔ AB⃗ ∥ AC⃗. Prefer x₁y₂ − x₂y₁ = 0 to the proportion x₁/x₂ = y₁/y₂: it also works when a coordinate is 0.

Square the length and expand it like (x ± y)² — this is the law of cosines in vector form (see “Solving triangles: the law of sines and the law of cosines”).

The coefficients are “crossed”: the closer D is to C, the larger the coefficient of AC⃗. For the midpoint t = ½; the coordinates of the centroid are the means of the vertices' coordinates.

Chinese textbooks: 实数 (b = 0), 虚数 (b ≠ 0), 纯虚数 (a = 0, b ≠ 0). Careful: many English books use “imaginary number” only for numbers of the form bi, while in the Chinese sense 3 + 2i is already 虚数. Complex numbers can be equal or unequal, but unless both are real they cannot be compared with < or >.

Quadrants: I — a > 0, b > 0; II — a < 0, b > 0; III — a < 0, b < 0; IV — a > 0, b < 0. A number on an axis lies in no quadrant.

Squares worth remembering: (a + bi)² = (a² − b²) + 2abi, (1 + i)² = 2i, (1 − i)² = −2i.

The powers go round the cycle i → −1 → −i → 1 with period 4, so four consecutive powers always add up to 0. Negative exponents too: i⁻¹ = −i, i⁻² = −1.

Unroll the lateral surface: a cylinder gives a rectangle 2πr × h; a cone gives a sector of radius l with arc length 2πr, whose central angle is 360° · r/l. For prisms and pyramids, simply add the areas of all faces.

Ask yourself: is the task finished after one choice (add) or only after all the steps (multiply)?

The last formula is inclusion–exclusion from “Sets and set operations”: the common part AB would otherwise be counted twice. Complementary events are always mutually exclusive, but not the other way round.

If A and B are independent, so are Ā and B, A and B̄, and Ā and B̄. Exactly one of two: P = p₁(1 − p₂) + (1 − p₁)p₂.

Estimates from a histogram: mean ≈ Σ midpoint × relative frequency; mode ≈ midpoint of the tallest bar; median = the point x where the area to the left of x is 0.5.

The median, the mode and the percentiles change like the mean (for a > 0); the range and the standard deviation are multiplied by |a|. The shift b never changes the spread.

E(X) is the long-run average value of X and D(X) measures how far X scatters around it. These are the formulas of the data lesson with the relative frequencies replaced by probabilities. Chinese textbooks write E(X) and D(X).

The same rules as for data (see «Numerical characteristics of data»): the shift b moves the expectation but never changes the variance.

B(1, p) is the two-point distribution. For a binomial variable D/E = 1 − p — a quick way to recover p and n.

Chinese textbooks give exactly these values (older books: 0.6826, 0.9544, 0.9974); CSCA items usually print them in the question. They hold for every normal distribution, whatever μ and σ are.

Same direction: the speeds subtract; opposite directions: they add. A gap L between two bodies closes in t = L / |v_AB|.

Area above the t-axis is positive displacement, area below it is negative. Displacement = the sum with signs; distance = the sum of the absolute values.

Hooke's law (胡克定律) holds within the elastic limit. The force always points back toward the natural length.

Friction depends on N and μ; it does not depend on the contact area, and kinetic friction does not depend on the speed.

The parallelogram rule (平行四边形定则). As θ grows from 0° to 180°, R decreases from F₁ + F₂ to |F₁ − F₂|.

Orthogonal decomposition (正交分解): replace a slanted force by two perpendicular components, then balance each axis separately.

A block on an incline with no other forces. If it slides down at constant speed, μmg cos θ = mg sin θ, so μ = tan θ, whatever the mass.

Newton's second law (牛顿第二定律). It is a vector law and holds at every instant: when the net force changes, the acceleration changes at the same moment. 1 N = 1 kg · m/s².

A block on a rough incline with no other forces. Sliding down, friction points up the slope and is subtracted; sliding up, both the gravity component and friction point down the slope and add up. If μ ≥ tan θ, a block at rest does not start to slide.

Gravity mg does not change in a lift; only the support force does.

Chinese textbooks write W = Fl cos α (l = displacement); the CSCA uses W for work and P for power, while some school textbooks use A and N. Only the component F cos α along the motion does work.

Works are scalars: add them as signed numbers, not as vectors.

The work of gravity depends only on the start and end heights, never on the path: going down it is positive and Ep decreases; going up it is negative and Ep increases.

The heat depends on the relative sliding distance, not on the distance measured from the ground. For a block sliding on a fixed floor, s_rel is simply the block's path.

Unlike work, impulse is not zero when the body stays in place: what counts is time, not distance.

Chinese: 动量定理 («momentum theorem»). It is Newton's second law in the form F = Δp/Δt: the impulse of the net force equals the change of the body's momentum.

Conditions: 1) the net external force is zero (a smooth horizontal surface: gravity and the normal force balance); 2) in a short collision or explosion the external forces are much smaller than the internal ones, so momentum is conserved approximately; 3) if the net external force is zero along one direction, only the momentum component along it is conserved.

On a belt the smaller wheel spins faster; on one axle the point farther from the axis moves faster.

For a pendulum the potential energy is mgh, but the picture is the same. The kinetic and the potential energy each repeat twice per oscillation, so they change with period T/2 (frequency 2f).

Charge is quantized: 1 C corresponds to 1/e = 6.25 × 10¹⁸ electrons.

A vector sum: same direction — add the magnitudes; opposite — subtract; perpendicular — E = √(E₁² + E₂²).

Uniform field only. The work is positive when the charge moves the way the field pushes it.

The field's work equals the decrease of potential energy: positive work → E_p falls; negative work (the charge is forced against the field) → E_p rises, like a stone being lifted.

Chinese textbooks put the signs of both q and U_AB into the formula; the sign of W then tells whether E_p falls (W > 0) or rises (W < 0). Many other textbooks write the same law as A = qU.

A particle of charge e accelerated from rest through U volts gains U electronvolts; a charge of 2e gains 2U eV.

Pure resistor: P = I²R = U²/R. Motor with coil resistance r_M: input power UI, heating power I²r_M, mechanical power UI − I²r_M.

If the angle α is measured between B and the normal to the surface, the same flux is Φ = BS cos α. Φ does not depend on the number of turns of a coil. When a loop is turned over (through 180°), the flux changes sign: ΔΦ = 2BS. Changes of flux drive the induced currents of the lesson “Electromagnetic induction: Faraday's law and Lenz's law”.

The force is greatest, F = BIL, when the wire is perpendicular to B, and zero when the wire is parallel to B. It is always perpendicular to both the wire and B.

v ⟂ B: F = qvB (maximum); v ∥ B: F = 0, and the particle moves on in a straight line. Direction: the left-hand rule with the four fingers along the velocity of a positive charge; for a negative charge (an electron) point them against the velocity.

Direction by the right-hand rule (右手定则): field lines enter the palm, the thumb points along v, the four fingers show the current in the rod. The rod is the source, so the fingers point to its higher-potential end (inside a source the current flows from − to +).

The form used in Chinese textbooks and in CSCA items (热力学第一定律). Every quantity is signed from the gas's point of view: whatever comes in is positive.

The same law in the school form used in Azerbaijan and Russia: the heat received goes partly into internal energy and partly into the work the gas does. Many English textbooks write ΔU = Q − W, where their W is the work done by the gas. Different letters, the same physics.

Moving right on the graph (expansion): the gas does work, W < 0. Moving left (compression): W > 0. A vertical segment (V constant) has zero area: no work.

Consequences: the angle between the incident and reflected rays is 2i; if the mirror turns by θ while the incident ray stays fixed, the reflected ray turns by 2θ. Rough surfaces scatter light in all directions (diffuse reflection, 漫反射), but every tiny piece still obeys i′ = i.

Since sin θ ≤ 1, only orders with k < d/λ can be seen. A longer wavelength is deviated more: in a grating spectrum red lies farthest out — the opposite of a prism, which bends violet most.

Malus's law. Unpolarised light loses half of its intensity in the first polariser, whatever the direction of its axis.

A shorter wavelength means a higher frequency — and, as the lesson «The photoelectric effect» shows, more energetic photons.

Lengths differ by 10⁵, so areas differ by 10¹⁰ and volumes by 10¹⁵. That is why most α particles pass without ever coming close to a nucleus.

A neutral atom has Z electrons. Only Z decides which chemical element a nucleus belongs to.

In every decay and every nuclear reaction the sum of the mass numbers and the sum of the charges are conserved. γ emission changes neither A nor Z: an excited daughter nucleus simply gives off its extra energy as a photon.

If Δm is in kilograms, multiply by c² to get joules; if it is in u, multiply by 931.5 to get MeV.

Converting one allotrope into another is a chemical change (a new substance forms), but not a redox reaction: the oxidation number of oxygen stays 0. It happens in lightning and in ozone generators.

Particle size is the essential difference between the three kinds of dispersion (分散质粒子直径). The Tyndall effect is only the method used to tell a colloid from a solution.

Preparing iron(III) hydroxide colloid (Fe(OH)₃胶体), the standard colloid of Chinese textbooks and CSCA items.

While a pure substance melts or boils, all the heat goes into pulling the particles apart, so the temperature stays constant — a flat step on the heating curve.

The criss-cross rule: x = b and y = a, then cancel any common factor. A polyatomic ion taken more than once keeps its brackets: Ca₃(PO₄)₂, not Ca₃P₂O₈.

The law of conservation of mass (质量守恒定律) written for atoms. Thermochemical and ionic equations obey it too, and an ionic equation must also balance the charge.

First balance the changing elements with these two numbers (cross-multiply to their LCM), then finish the rest — usually H and O — by inspection.

An ionic equation must balance both atoms and charge. 2Fe³⁺ + Fe → 3Fe²⁺ is right (+6 = +6); Fe³⁺ + Fe → 2Fe²⁺ has balanced atoms, but +3 ≠ +4.

Heat is proportional to the amount that reacts. Two definitions CSCA likes: the heat of combustion (燃烧热) refers to 1 mol of fuel burning completely to stable products (CO₂(g), H₂O(l)); the heat of neutralisation (中和热) of a strong acid and a strong base is 57.3 kJ per mole of H₂O formed.

Reactant bonds are broken and product bonds are formed — «broken minus formed». If the products’ bonds are stronger, ΔH < 0.

On dilution the amount (and the mass) of solute stays the same. V₁ and V₂ may both be in mL — the unit cancels.

c(H⁺) · c(OH⁻) = Kw holds in every aqueous solution, acidic or basic. Kw grows with temperature: near 100 °C it is of the order of 10⁻¹² (problems give Kw = 1 × 10⁻¹²), so neutral water has pH ≈ 6 there — still neutral, because c(H⁺) = c(OH⁻). Where Kw comes from, the ionisation of water, is part of “Theories of electrolyte solutions”.

Dilution divides c(H⁺) of an acid (or c(OH⁻) of a base) by 10ⁿ. Very dilute solutions approach pH 7 from their own side: water itself supplies 10⁻⁷ mol/L of H⁺ and of OH⁻.

The three typical reactions of basic oxides. Chinese textbooks write balanced equations with «=» instead of «→»; the meaning is the same.

The three typical reactions of acidic oxides; t° means heating.

Acid + base is neutralisation, covered in the next section. HNO₃ and concentrated H₂SO₄ react with metals without releasing H₂ — see «Common non-metals and their compounds».

Alkalis turn litmus blue and phenolphthalein pink. Insoluble bases do not change indicators, but they still react with acids.

It works for any strong acid and base and saves writing the equation. With masses, first convert to moles: n = m/M (see «Amount of substance: the mole»).

The same logic works for H₂SO₄ + NaOH (NaHSO₄ or Na₂SO₄) and for H₃PO₄ (NaH₂PO₄, Na₂HPO₄, Na₃PO₄). The properties of Na₂CO₃ and NaHCO₃ themselves are in «Common metals and their compounds».

Exchange reactions between acids, bases and salts (复分解反应, double decomposition) go only if a precipitate, a gas or water forms; the ionic view of this rule is in «Ionic reactions and tests for ions».

Al(OH)₃ dissolves only in strong bases: not in ammonia solution and not in carbonic acid. ZnO and Zn(OH)₂ react the same way (giving ZnCl₂ or Na₂[Zn(OH)₄]). Aluminium metal itself is in «Common metals and their compounds».

A one-step «oxide → base» or «oxide → acid» works only when the oxide reacts with water: CuO → Cu(OH)₂, Fe₂O₃ → Fe(OH)₃ and SiO₂ → H₂SiO₃ need two steps (make a salt first).

Chinese textbooks ask a lot about observations (colour, smoke, mist, flame): «smoke» (烟) is solid particles, «mist» (雾) is liquid droplets.

Mixing bleach with an acidic cleaner is dangerous: NaClO + 2HCl → NaCl + Cl₂↑ + H₂O. The laboratory preparation of Cl₂ is in «Preparation and identification of common gases», the chlor-alkali industry in «Industrial chemical processes».

The reverse reactions (Br₂ + 2KCl, I₂ + 2KBr) do not occur. The tests for Cl⁻, Br⁻ and I⁻ with AgNO₃ are in «Ionic reactions and tests for ions».

The four faces of SO₂. Decolourising KMnO₄ or bromine water is reduction, not bleaching; «bleaching» means only the magenta test.

Copper with concentrated H₂SO₄ needs heating; with dilute H₂SO₄ copper does not react at all.

NO and NO₂ are among the causes of smog and acid rain. The equilibrium 2NO₂ ⇌ N₂O₄ is treated in «Chemical equilibrium».

Ammonium salts are soluble white crystals. They decompose on heating (NH₄Cl → NH₃↑ + HCl↑, which recombine to NH₄Cl in the cool part of the tube — it looks like sublimation but is a chemical change; NH₄HCO₃ → NH₃↑ + CO₂↑ + H₂O), and with alkalis on warming they release NH₃: NH₄⁺ + OH⁻ → NH₃↑ + H₂O. That is why ammonium fertilisers must not be mixed with alkaline materials such as slaked lime or plant ash. The NH₄⁺ test is in «Ionic reactions and tests for ions», the preparation of NH₃ in «Preparation and identification of common gases».

In both cases the solution turns blue from Cu²⁺. When copper is in excess, the concentrated acid becomes dilute as it is used up and starts giving NO instead of NO₂.

Per mol of Ca(OH)₂: CO₂ ≤ 1 mol — the precipitate equals the CO₂; 1 < CO₂ < 2 mol — the precipitate is (2 − n(CO₂)) mol; CO₂ ≥ 2 mol — no precipitate is left.

In glass making at high temperature: Na₂CO₃ + SiO₂ → Na₂SiO₃ + CO₂↑ and CaCO₃ + SiO₂ → CaSiO₃ + CO₂↑.

The activity series of metals: activity decreases from left to right.

The «difference method» (差量法): the change in mass is proportional to the amount that has reacted.

Sodium peroxide releases oxygen — it is the oxygen source in breathing apparatus and submarines.

Thermal decomposition of sodium hydrogen carbonate — used to tell the two solids apart and to remove NaHCO₃ from Na₂CO₃.

Aluminium with acids and with alkalis.

NaOH added drop by drop to an Al³⁺ solution: the white precipitate first grows, then dissolves.

Conversions between Fe²⁺ and Fe³⁺: the total charge must be equal on both sides.

Electrochemical corrosion of steel (Chinese textbooks write Fe − 2e⁻ = Fe²⁺).

Put in the known values and solve for the one unknown. If the atoms of one element are in different states (Fe₃O₄, Na₂S₂O₃), you get their average, which may be a fraction.

Count per formula, not per atom: K₂Cr₂O₇ → 2Cr³⁺ takes 2 · 3 = 6 electrons. In an equation, multiply the per-formula changes up to their least common multiple.

The oxidising agent is stronger than the oxidation product, the reducing agent stronger than the reduction product. So two or three given reactions chain into a full order.

The factors come from the electrode reactions: Cu²⁺ + 2e⁻ → Cu, 2Cl⁻ → Cl₂↑ + 2e⁻, 2H₂O → O₂↑ + 4H⁺ + 4e⁻ (or 4OH⁻ → O₂↑ + 2H₂O + 4e⁻), Ag⁺ + e⁻ → Ag.

The charges on the right add up to zero. Acid salts: NaHCO₃ → Na⁺ + HCO₃⁻ (HCO₃⁻ is not split), but NaHSO₄ → Na⁺ + H⁺ + SO₄²⁻ in solution. Polyprotic weak acids ionise step by step: first H₂CO₃ ⇌ H⁺ + HCO₃⁻.

Together with the balance of atoms. In a redox ionic equation the electrons lost must also equal the electrons gained; if you check only the atoms, many wrong options look right.

General formulas of the hydrocarbon series (CₙH₂ₙ also fits cycloalkanes, CₙH₂ₙ₋₂ also fits dienes).

Substitution (取代反应), the characteristic reaction of alkanes.

Addition (加成反应) and addition polymerisation (加聚反应).

Reactions of benzene: two substitutions and one addition.

Sodium replaces only the hydrogen of the –OH group, never an H on carbon. The reaction is gentler than with water: the O–H hydrogen of ethanol is less active. The same ratio holds for –COOH: 2 –COOH ~ H₂.

Other oxidations of ethanol: it burns (C₂H₅OH + 3O₂ → 2CO₂ + 3H₂O), and strong oxidants — acidified K₂Cr₂O₇ (orange → green) or KMnO₄ (purple → colourless) — take it on to acetic acid.

In both reactions the aldehyde is oxidised (it is the reducing agent). They are the tests for the –CHO group; glucose, formic acid and formates give them too.

Na reacts with all four substances, NaOH only with the acids, and NaHCO₃ only with acids stronger than carbonic acid, that is with –COOH.

Naming: alcohol part + acid part «-ate» — CH₃COOC₂H₅ is ethyl acetate, HCOOCH₃ methyl formate. Esterification is a kind of substitution reaction.

An ester always splits at the same C–O bond: the acid part takes back –OH and the alcohol part takes back H.

Saponification (皂化反应) is the basic hydrolysis of a fat: 1 mol of fat uses 3 mol of NaOH and gives 1 mol of glycerol.

Alcoholic fermentation turns 1 mol of glucose into 2 mol of ethanol and 2 mol of CO₂.

Forming an ion changes only the electrons: a cation has lost q electrons, an anion has gained |q|. The nucleus (Z and A) stays the same.

The table value is a weighted average, which is why chlorine has 35.5 although no chlorine atom has a mass of 35.5. Turning it into grams per mole is the job of the lesson «Amount of substance: the mole».

Extra rules for main-group atoms: the outer shell never holds more than 8 electrons (the K shell 2), and 4s fills before 3d. That is why potassium is 2, 8, 8, 1 and not 2, 8, 9.

Ions with the same electron configuration (isoelectronic ions) are ordered by Z alone: O²⁻ > F⁻ > Na⁺ > Mg²⁺ > Al³⁺ — all have 10 electrons, and more protons pull them tighter.

For groups IVA–VIIA: |highest| + |lowest| = 8. Exceptions: fluorine has no positive oxidation number, and oxygen never reaches +6. Metals have no negative oxidation numbers.

Δχ = 0: non-polar covalent; 0 < Δχ < about 1.7: polar covalent (the larger Δχ, the more polar); Δχ > about 1.7 between a metal and a non-metal: mainly ionic. The border is only a guide: H–F (Δχ = 1.9) is a very polar covalent bond, and AlCl₃ (Δχ = 1.5) is a covalent compound.

C forms 4 bonds, N 3, O 2 and the halogens 1: as many as the electrons still missing from the octet. The remaining valence electrons stay as lone pairs (孤电子对): N has 1, O 2, Cl 3.

This is the formula of Chinese textbooks (中心原子上的孤电子对数). 2 domains → linear, 180°; 3 → planar, 120°; 4 → tetrahedral, 109.5°; lone pairs make the angles smaller.

The same ratio holds for Δc and, in one vessel, for Δn. Dividing any v(X) by its coefficient gives one number for the whole reaction, which is how rates written with different substances are compared.

An approximate rule (van ’t Hoff). Chinese textbooks state it in words: every 10 °C rise usually makes the rate 2–4 times larger. The time needed for the same change shrinks by the same factor: τ₂ = τ₁ / γ^((t₂ − t₁)/10).

On an energy diagram the reactants and the products are two levels, and the curve climbs a «hill» between them. E₁ is the height of the hill above the reactants, E₂ its height above the products. In an exothermic reaction the products lie lower and E₁ < E₂. A catalyst lowers the top of the hill, so E₁ and E₂ fall by the same amount and ΔH stays the same.

Q < K: there is «too little product», so v(forward) > v(reverse) until Q grows to K. Q > K: too much product, and the reverse reaction wins.

Kw holds in pure water and in every dilute aqueous solution — acidic, basic or neutral. Added H⁺ or OH⁻ (an acid or a base) suppresses the ionisation of water; ions that bind H⁺ or OH⁻ (hydrolysing salts) promote it.

The three balances of a Na₂CO₃ solution. Order of concentrations: c(Na⁺) > c(CO₃²⁻) > c(OH⁻) > c(HCO₃⁻) > c(H⁺) — OH⁻ comes from both hydrolysis steps and from water, so it exceeds HCO₃⁻.

Ksp compares solubilities directly only for salts of the same type (AgCl, AgBr and AgI are all 1 : 1). Ag₂CrO₄ has a smaller Ksp than AgCl but is more soluble. At 25 °C: Ksp(AgCl) = 1.8 × 10⁻¹⁰, Ksp(AgI) = 8.5 × 10⁻¹⁷, Ksp(BaSO₄) = 1.1 × 10⁻¹⁰.

Every reading is taken at the bottom of the meniscus with the eye level with it. The result is always written to 0.01 mL: 22.30 mL, not 22.3 mL.

The rule is «object left, weights right» (左物右码). If the rider is not used, swapping the pans does not change the result.

MnO₂ is a catalyst in the last two reactions: its mass and chemical properties are the same after the reaction, and it can be recovered by filtering. In the first reaction MnO₂ is a product. Collect O₂ over water (it is poorly soluble) or in an upright jar (M = 32 > 29). Test: a glowing splint relights.

Collect H₂ over water or in an inverted jar (M = 2). Crude zinc reacts faster than pure zinc, and a few drops of CuSO₄ solution speed the reaction up: copper deposited on the zinc forms tiny galvanic cells (see «Redox reactions»).

CO₂ is denser than air (M = 44) and fairly soluble in water, so it is usually collected in an upright jar. To check that the jar is full, hold a burning splint at its mouth (not inside): it goes out.

Without heating, KMnO₄ (or KClO₃) also gives Cl₂ with concentrated HCl, for example 2KMnO₄ + 16HCl(conc.) → 2KCl + 2MnCl₂ + 5Cl₂↑ + 8H₂O. Chlorine is toxic: the set-up must end with a tail-gas absorber.

Quick method without heating: drip concentrated ammonia solution onto solid CaO or NaOH. NH₃ is very soluble (≈ 700 volumes in 1 volume of water) and lighter than air: collect it only in a dry inverted vessel, with a loose plug of cotton wool at the mouth to stop air mixing in.

Why ordinary pressure, unlike ammonia synthesis? At atmospheric pressure the conversion of SO₂ is already over 90%, so costly high-pressure equipment would add little.

The relationship method (关系式法): link the first raw material with the final product through one conserved element and skip the intermediate equations. It assumes no losses — losses are included through the yield below.

Nitric acid from ammonia. Blowing extra air into the absorption tower lets all the nitrogen end up in the acid: 4NO₂ + O₂ + 2H₂O → 4HNO₃.

The chlor-alkali industry (氯碱工业) with an ion-exchange membrane cell. Chinese textbooks write electrode equations with «− 2e⁻» on the left and «=»: 2Cl⁻ − 2e⁻ = Cl₂↑.