Learn the concept. Work through the problem.

Solving linear equations

To solve a linear equation, do the same operation to both sides until the variable stands alone. Clear any fractions, expand brackets, collect the variable terms on one side and divide by the coefficient.

Factoring quadratics and finding zeros

Rewrite a polynomial as a product, then use its factors to understand where its graph meets the axis.

The quadratic formula and the discriminant

The quadratic formula x = (−b ± √(b² − 4ac))/(2a) solves any equation ax² + bx + c = 0 with a ≠ 0. The discriminant b² − 4ac tells you, before you solve, whether there are two, one or no real solutions.

Solving rational equations

To solve an equation with the variable in a denominator, first list the values that make a denominator zero. Multiply every term by the least common denominator, solve, and reject any answer that is an excluded value.

Solving radical equations

To solve an equation with a square root, isolate the root, square both sides and solve. Squaring can create answers that do not work, so substitute every candidate into the original equation and keep only those that satisfy it.

Linear inequalities and interval notation

Solve a linear inequality like an equation, with one extra rule: multiplying or dividing both sides by a negative number reverses the inequality sign. The solution is usually a whole interval of numbers, written as an inequality, on a number line or in interval notation.

Absolute value equations and inequalities

To solve an absolute value equation, isolate the absolute value, then split |A| = c into two equations, A = c or A = −c, and solve both. For an inequality with c > 0, |A| < c becomes the single interval −c < A < c, and |A| > c becomes two pieces, A < −c or A > c.

Function transformations and their graphs

A transformation moves or reshapes a graph without changing its basic shape. In y = a·f(b(x − h)) + k, h shifts the graph right by h, k shifts it up by k, a stretches it vertically by |a| and flips it if a < 0, and b scales horizontal distances by 1/|b| and flips the graph if b < 0.

Composite and inverse functions

Composition applies one function to the output of another: (f ∘ g)(x) = f(g(x)), with g applied first. An inverse function undoes a function, so f⁻¹(f(x)) = x. Only a one-to-one function, one whose graph passes the horizontal line test, has an inverse; to find it, solve y = f(x) for x, then swap the names of the variables.

Slope and equations of lines

The slope of a line measures how steep it is: m = (y₂ − y₁)/(x₂ − x₁), the change in y divided by the change in x between any two of its points. A line with slope m and y-intercept b has the equation y = mx + b, and the line through (x₁, y₁) with slope m is y − y₁ = m(x − x₁). Parallel lines have equal slopes, and perpendicular lines have slopes whose product is −1.

Quadratic functions: vertex form and graphs

A quadratic function f(x) = ax² + bx + c graphs as a parabola that opens upward when a > 0 and downward when a < 0. Its turning point, the vertex, lies on the axis of symmetry x = −b/(2a), and the vertex form f(x) = a(x − h)² + k shows the vertex (h, k) directly. The vertex gives the function’s minimum or maximum value, which is why quadratics answer questions about the greatest height or the largest revenue.

Polynomial functions: zeros and end behavior

A polynomial function is a sum of terms aₙxⁿ + … + a₁x + a₀ with whole-number powers. Its degree n and leading coefficient aₙ decide its end behavior, the direction of the graph far to the left and right. Its real zeros are its x-intercepts: at a zero of odd multiplicity the graph crosses the x-axis, and at a zero of even multiplicity it touches the axis and turns back. A polynomial of degree n has at most n real zeros and at most n − 1 turning points.

Polynomial division and the remainder theorem

Dividing a polynomial P(x) by a divisor D(x) gives a quotient Q(x) and a remainder R(x) of lower degree than D, with P(x) = D(x)Q(x) + R(x). Long division works for any divisor; synthetic division is a shortcut for divisors of the form x − c. The remainder theorem says that the remainder on dividing by x − c is P(c), so x − c is a factor exactly when P(c) = 0: the factor theorem.

Rational functions: asymptotes and holes

A rational function is a quotient of polynomials, f(x) = p(x)/q(x). Factor first: a factor that cancels leaves a hole, and a zero of the denominator that remains gives a vertical asymptote. Comparing the degrees of p and q gives the horizontal asymptote, or a slant asymptote, that describes the end behavior.

Exponent rules and rational exponents

To multiply powers with the same base, add the exponents; to divide them, subtract; to raise a power to a power, multiply. A negative exponent means a reciprocal, x⁻ⁿ = 1/xⁿ, and a fractional exponent means a root: x^(m/n) is the nth root of xᵐ.

Logarithm rules and exponential equations

A logarithm answers “what exponent?”: log_b x = y means bʸ = x. Logarithms turn products into sums, quotients into differences and powers into multiples, which is why rewriting an exponential equation in logarithmic form, or taking the logarithm of both sides, solves it.

Exponential growth and decay

A quantity grows or decays exponentially when it changes by the same factor over equal time intervals. Continuous growth follows A = A₀e^(kt), interest compounded n times a year follows A = P(1 + r/n)^(nt), and a decaying sample halves every half-life T: A = A₀(1/2)^(t/T).

Systems of linear equations in two variables

A system of linear equations asks for the values that satisfy every equation at once. With two variables each equation is a line, and the solutions are where the lines meet: one point, no point (parallel lines) or infinitely many points (the same line). Substitution and elimination find them exactly, and a check in both original equations confirms them.

The binomial theorem and Pascal’s triangle

The binomial theorem expands (a + b)ⁿ into n + 1 terms: the term with bᵏ is C(n, k)aⁿ⁻ᵏbᵏ. The coefficients C(n, k) are row n of Pascal’s triangle, and they count the ways to choose b from k of the n factors.

Precalculus

The unit circle, radians and exact trig values

The unit circle has radius 1 and its center at the origin. An angle θ in standard position ends at the point (cos θ, sin θ), so sine and cosine are coordinates and tangent is their ratio. Radians measure angles by arc length, and a reference angle with the signs of its quadrant gives the exact values at the special angles.

Trigonometric identities and equations

A trigonometric identity is an equation that holds at every angle where both sides are defined. The Pythagorean, sum and double-angle identities, such as cos 2x = 1 − 2sin²x, rewrite an expression into a more useful form: to simplify it, to prove another identity or to solve an equation. A trigonometric equation such as 2 sin x = 1, unlike an identity, holds only at particular angles, and usually at infinitely many of them.

Complex numbers: arithmetic and polar form

A complex number a + bi combines a real part a and an imaginary part b, where i² = −1. Complex numbers add and multiply like binomials, and dividing uses the conjugate. Plotted as points, they have a length, the modulus, and an angle, the argument: multiplying multiplies the lengths and adds the angles, which is how De Moivre’s theorem finds powers and roots.

Matrix row reduction, determinants and inverses

A matrix is a rectangular array of numbers. Writing a linear system as an augmented matrix lets you solve it with three row operations, none of which changes the solutions: Gaussian elimination reaches row echelon form, and reduced row echelon form shows the answer directly. Row reduction also shows when a system has no solution, or a free variable and infinitely many. For a square matrix, a nonzero determinant means an inverse exists.

Conic sections: parabolas, ellipses, hyperbolas

Circles, ellipses, parabolas and hyperbolas are the curves a plane cuts from a double cone, and each has a second-degree equation in x and y. Completing the square turns the general equation into a standard form that shows the center and radius of a circle, and the vertices, foci and asymptotes of the other conics.

Arithmetic and geometric sequences

A sequence is an ordered list of numbers, given by an explicit formula for the nth term or by a recursive formula that builds each term from the one before. An arithmetic sequence adds a common difference d at each step, and a geometric sequence multiplies by a common ratio r. Each kind has formulas for its nth term and for the sum of its first n terms.

Calculus I

Limits of functions and holes in graphs

Separate a function’s nearby behavior from its value at the point you are approaching.

Continuity and the Intermediate Value Theorem

A continuous function has no holes, jumps or vertical asymptotes: f is continuous at x = a when f(a) is defined, the limit of f(x) as x approaches a exists, and the two are equal. On a closed interval, continuity gives the Intermediate Value Theorem (IVT): the function takes every value between f(a) and f(b), which proves that an equation has a solution.

The limit definition of the derivative

The derivative f′(a) is the slope of the tangent line at x = a: the limit of the secant slopes (f(a + h) − f(a))/h as h approaches 0. When that limit exists, it is the instantaneous rate of change of f at a.

The power rule and basic derivative rules

To differentiate a polynomial or a sum of powers, use three rules: the power rule d/dx xⁿ = nxⁿ⁻¹, the constant multiple rule (a constant factor stays) and the sum rule (differentiate term by term). Rewrite roots and fractions as powers first.

The product rule for derivatives

The product rule says the derivative of f(x)g(x) is f′(x)g(x) + f(x)g′(x): each factor changes in turn while the other keeps its current value. It is not f′(x)g′(x).

The quotient rule for derivatives

The quotient rule gives the derivative of f/g: (f/g)′ = (f′g − fg′)/g², wherever g ≠ 0. The order in the numerator matters, because of the minus sign; simplify the numerator and keep the denominator squared.

The chain rule for derivatives

The chain rule differentiates a function inside another function, such as sin(x²) or (3x + 1)⁵⁰. Differentiate the outside function while leaving the inside untouched, then multiply by the derivative of the inside: the derivative of f(g(x)) is f′(g(x))·g′(x).

Derivatives of eˣ, ln x and trig functions

The derivative of eˣ is eˣ, the derivative of ln x is 1/x for x > 0, the derivative of sin x is cos x and the derivative of cos x is −sin x, with x in radians. Each follows from the limit definition, and each combines with the sum, product and chain rules.

Implicit differentiation

Implicit differentiation finds dy/dx when y is tied to x by an equation you have not solved for y. Differentiate both sides with respect to x, multiply the derivative of every y term by dy/dx (the chain rule), then solve for dy/dx.

Linear approximation and differentials

Near a point a, a differentiable function is close to its tangent line, so L(x) = f(a) + f′(a)(x − a) approximates f(x) for x near a. The differential dy = f′(x) dx uses the same idea to estimate how much the output changes when the input changes by a small amount dx.

The Mean Value Theorem and Rolle’s theorem

The Mean Value Theorem (MVT) says that a function’s average rate of change over an interval is matched, somewhere inside, by its instantaneous rate: some tangent line is parallel to the secant line. Rolle’s theorem is the case of equal endpoint values. The theorem explains why f′ = 0 means constant and f′ > 0 means increasing, and it bounds how much a function can change.

Curve sketching with derivatives

The first derivative tells you where a function increases or decreases, and its sign changes at critical points locate local maxima and minima. The second derivative tells you where the graph is concave up or down, and its sign changes locate inflection points. Together they give the graph’s shape.

Limits at infinity

A limit at infinity asks what value f(x) approaches as x grows without bound, or falls without bound. If f(x) → L as x → ∞ or as x → −∞, the line y = L is a horizontal asymptote of the graph. For a rational function, dividing the numerator and denominator by the highest power of x in the denominator shows the answer: 0, the ratio of the leading coefficients, or unbounded growth, depending on the degrees.

Optimization problems in calculus

To solve an optimization problem, write the quantity to maximize or minimize as a function of one variable, using the constraint to remove any other variable. Find the critical points where the derivative is zero, then compare the function’s values there and at the endpoints of the allowed interval.

L’Hôpital’s rule for indeterminate limits

L’Hôpital’s rule says that if f(x)/g(x) gives the indeterminate form 0/0 or ∞/∞ at a, its limit equals the limit of f′(x)/g′(x), provided that limit exists. Differentiate the top and bottom separately, not with the quotient rule, and check the form before every use.

Antiderivatives and indefinite integrals

An antiderivative of f is a function F whose derivative is f. On an interval, every antiderivative of f has the form F(x) + C, written ∫f(x) dx = F(x) + C. Find one by running the power rule and the other derivative rules backward, then check it by differentiating.

Riemann sums: left, right and midpoint

A Riemann sum approximates a definite integral by adding rectangle areas: split [a, b] into n pieces of width Δx = (b − a)/n, take a height f(xᵢ*) in each piece, and add the products f(xᵢ*)·Δx. Left, right and midpoint sums choose different heights; as n grows, all of them approach the integral.

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links derivatives and integrals. Part 1: if f is continuous and F(x) = ∫ₐˣ f(t) dt, then F′(x) = f(x). Part 2: if F is any antiderivative of a continuous f on [a, b], then the integral of f from a to b is F(b) − F(a).

Integration by substitution (u-substitution)

Integration by substitution reverses the chain rule. When an integrand contains an inner function g(x) and its derivative g′(x), set u = g(x), so du = g′(x) dx; integrate in u, then substitute back. For a definite integral, change the limits to u-values instead.

Area between two curves

To find the area between two curves, find where they intersect, decide which curve is on top, and integrate top minus bottom from a to b: A = ∫(f(x) − g(x)) dx, where f(x) ≥ g(x) on [a, b]. If the curves cross inside the interval, split the integral at each crossing so every piece is positive.

Calculus II

Volumes of revolution: disks, washers and shells

Turning a region about a line sweeps out a solid of revolution. Slice it perpendicular to the axis and each slice is a disk or a washer, with area πR² or π(R² − r²); slice it parallel to the axis and each piece is a cylindrical shell. Adding up the slices with an integral gives the volume.

Integration by parts: formula and examples

Integration by parts is the product rule run backward: the integral of u dv equals uv minus the integral of v du. It trades an integral you cannot do for one you can, when the integrand is a product in which one factor gets simpler when differentiated, such as x, and the other is easy to integrate, such as eˣ or cos x. It even integrates ln x, by treating it as ln x times 1.

Integration by partial fractions

Partial fraction decomposition rewrites a proper rational function P(x)/Q(x) as a sum of simpler fractions, one for each factor of the denominator. Each piece has a standard integral: A/(x − a) gives A ln|x − a|, a repeated factor gives a power of 1/(x − a), and an irreducible quadratic gives a logarithm and an arctangent. If the numerator’s degree is not lower than the denominator’s, divide first.

Improper integrals and convergence

An integral is improper when an endpoint is infinite or the integrand blows up somewhere on the interval. Replace the troublesome endpoint by a variable, integrate over the ordinary interval that results, and take the limit. If the limit is a finite number the integral converges to it; otherwise it diverges. The p-integrals ∫ from 1 to ∞ of dx/xᵖ, which converge exactly when p > 1, are the benchmarks for comparison.

Separable differential equations

A differential equation relates a function to its derivatives, and solving it means finding the function. A first-order equation is separable when it can be written dy/dx = g(x)h(y), and it is solved by separation of variables: divide by h(y), multiply by dx, and integrate each side with respect to its own variable. The result, with one constant of integration, is the general solution; an initial condition such as y(0) = 3 picks out one particular solution.

Infinite series and convergence tests

An infinite series adds infinitely many terms; it converges when its partial sums approach a finite limit. A geometric series converges exactly when |r| < 1. Terms that do not shrink to zero force divergence, and the integral, comparison, ratio and alternating series tests settle most of the series a course meets.

Taylor polynomials and Taylor series

A Taylor polynomial of degree n matches a function’s value and first n derivatives at a center a. Letting n grow gives the Taylor series; centered at 0 it is a Maclaurin series. Taylor’s remainder bounds the error of the polynomial, and the ratio test finds the interval where the series converges.

Introductory statistics

Mean, median and standard deviation

Three numbers describe most data sets. The mean is the balance point, the median is the middle value, and the standard deviation measures how far values typically sit from the mean. The median resists outliers; the mean and the standard deviation do not.

z-scores and the normal distribution

A z-score measures how many standard deviations a value lies above or below the mean. Standardizing puts different scales on one footing, and for data that follow a normal distribution it turns any value into a probability: the share of the distribution below or above it.

Confidence intervals: the t interval for a mean

A confidence interval gives a range of plausible values for an unknown population mean: the sample mean plus or minus a margin of error, a critical value times the standard error s/√n, so a larger sample size gives a narrower interval. When σ is unknown, the critical value comes from the t distribution with n − 1 degrees of freedom. The confidence level describes how often the method captures the true mean, not the chance that one computed interval is right.

Hypothesis testing: t-tests and p-values

A hypothesis test asks whether data are consistent with a claim. The null hypothesis states the claim, such as μ = 500; the alternative states what you suspect instead. The t statistic measures how far the sample mean falls from the claim in standard errors, and the p-value is the probability of a result at least that extreme if the null hypothesis were true.

The chi-square goodness-of-fit test

A goodness-of-fit test asks whether observed counts in categories match a claimed distribution, such as a fair die or a 3:1 genetic ratio. Each category contributes (observed − expected)²/expected, and the total χ² follows a chi-square distribution with k − 1 degrees of freedom when the model is right. A small p-value means the counts fit the model poorly.

Linear regression, correlation and residuals

Linear regression fits the line of best fit: the line that makes the squared vertical distances from the points, the residuals, as small as possible. The slope estimates how much y changes per unit of x. The correlation r, between −1 and 1, measures the strength and direction of a linear pattern, and r² gives the fraction of the variation in y that the line accounts for.

Chemistry

General chemistry I

Significant figures and rounding

Significant figures, or sig figs, are the digits in a measurement that carry meaning: every certain digit plus one estimated digit. When you multiply or divide, round the answer to the fewest significant figures in the data; when you add or subtract, round to the fewest decimal places.

Dimensional analysis and unit conversion

Use units to choose conversion factors, organize a calculation and check whether the answer makes sense.

The mole, molar mass and Avogadro’s number

To convert grams to moles, divide by the molar mass, the mass of one mole in g/mol, found by adding the atomic masses in the formula. To convert moles to particles, multiply by Avogadro’s number, 6.022 × 10²³ per mole.

Empirical and molecular formulas

To find an empirical formula, treat each element’s mass percent as grams in a 100 g sample, convert each mass to moles, and divide by the smallest amount to get the simplest whole-number ratio. The molar mass then tells you how many empirical units make one molecule.

Molarity and dilution calculations

Molarity, M, is the number of moles of solute per liter of solution: M = n/V. Diluting a solution adds solvent but no solute, so the moles of solute stay the same and M₁V₁ = M₂V₂.

Balancing chemical equations

To balance a chemical equation, change only the coefficients, never the subscripts, until every element has the same number of atoms on both sides. Balance the most complex substance first, leave free elements such as O₂ for last, and recount at the end.

Stoichiometry and theoretical yield

Stoichiometry predicts how much product a reaction can form. Convert each given mass to moles, use the mole ratio from the balanced equation, then convert back to grams; when two reactants are given, the one that runs out first sets the theoretical yield.

Limiting reactant and percent yield

The limiting reactant is the one that runs out first, so it sets how much product can form. To find it, work out how much product each reactant could make on its own: the smaller amount is the theoretical yield, and actual ÷ theoretical × 100% is the percent yield.

Calorimetry and specific heat (q = mcΔT)

Calorimetry measures heat through a temperature change. For a sample that stays in one phase, q = mcΔT, where c is the specific heat. In an insulated calorimeter the heat lost by one part is gained by the rest, so one measured temperature change gives an unknown specific heat or a reaction’s enthalpy.

Hess’s law and enthalpies of formation

Hess’s law says that if chemical equations add up to a target equation, their enthalpy changes add up to its ΔH, because enthalpy is a state function. Reversing an equation flips the sign of ΔH and multiplying it multiplies ΔH. Standard enthalpies of formation turn this into one formula: ΔH°rxn = Σ n ΔHf°(products) − Σ n ΔHf°(reactants).

Photon energy, wavelength and frequency

Light comes in photons, and each photon’s energy is set by its frequency: E = hν, where h is Planck’s constant. Since c = λν, E = hc/λ, so shorter wavelengths carry more energy. Multiplying by Avogadro’s number gives the energy of a mole of photons, the scale of chemical bond energies.

Lewis structures and formal charge

A Lewis structure shows how a molecule’s valence electrons are arranged as bonds and lone pairs. Count the valence electrons, join the atoms with single bonds, complete the outer atoms’ octets and put leftovers on the central atom; if it is still short, form double or triple bonds. Formal charge picks the best structure, and resonance describes molecules no single structure can.

VSEPR theory and molecular shape

VSEPR (valence-shell electron-pair repulsion) theory predicts a molecule’s shape from its Lewis structure. Count the electron domains around the central atom (a bond of any order counts once, and so does each lone pair), spread them as far apart as possible, then name the shape from the positions of the atoms alone.

The ideal gas law, PV = nRT

The ideal gas law, PV = nRT, relates a gas’s pressure P, volume V, amount n and absolute temperature T through the gas constant R = 0.08206 L·atm/(mol·K). Solve it for the unknown, with the temperature in kelvin and every unit matching R.

General chemistry II

Enthalpy of solution and lattice energy

Build an energy cycle for dissolving: separate the solute, separate the solvent and mix, or for an ionic solid, break the lattice and hydrate the ions. Then use the cycle to find ΔH(soln) and predict whether a solution warms or cools.

Molality, mass percent, mole fraction and ppm

Concentration can be stated several ways, each with its own denominator. Molarity counts moles of solute per liter of solution, and molality counts moles of solute per kilogram of solvent. Mass percent and parts per million compare the solute’s mass with the solution’s, and mole fraction compares a component’s moles with the total moles. Converting between them takes molar masses and, whenever a volume meets a mass, the solution’s density.

Colligative properties of solutions

Calculate freezing-point depression, boiling-point elevation, vapor pressure with nonvolatile or volatile solutes, and osmotic pressure, and use each to find a molar mass or a van ’t Hoff factor.

The Beer–Lambert law and absorbance

A solution that absorbs light lets through a fraction T of it, and its absorbance is A = −log T. The Beer–Lambert law says absorbance is proportional to concentration: A = εbc, where ε is the molar absorptivity at that wavelength and b the path length. A calibration line from standard solutions turns a measured absorbance into a concentration.

Finding rate laws from experimental data

Find each reactant’s order and the rate constant k from initial-rate data, then use integrated rate law plots and half-life to find the order and k from concentration–time data.

Activation energy from Arrhenius plots

Use rate constants measured at several temperatures to find the activation energy Ea and the frequency factor A from an Arrhenius plot, or Ea from two temperatures with the two-point form.

Potential energy diagrams for reactions

Read a potential energy diagram: where ΔH, the forward and reverse activation energies and the transition state are, how exothermic and endothermic reactions differ, and what a catalyst changes and what it cannot.

Reaction mechanisms and rate laws

Decide whether a proposed mechanism fits an observed rate law with a short if–then–else test: add the steps, write the slow step’s rate law, replace any intermediate, then compare.

Chemical equilibrium, K and ICE tables

At equilibrium the forward and reverse reactions run at the same rate, so no concentration changes. The equilibrium constant K, written from the balanced equation, fixes the ratio of products to reactants at a given temperature. The reaction quotient Q uses the concentrations present now: comparing Q with K tells you which way a mixture shifts, and an ICE table finds where it ends up.

Le Châtelier’s principle and equilibrium shifts

Le Châtelier’s principle predicts the effect of a disturbance on a system at equilibrium: the system shifts in the direction that partly counteracts the change. Adding a reactant or removing a product drives the reaction forward, and compressing a gas mixture favors the side with fewer moles of gas. Both work through the reaction quotient: the change makes Q differ from K, and the reaction runs until Q = K again. A change in temperature is different, because it changes K itself: heating favors the endothermic direction.

pH of strong and weak acids and bases

pH measures the hydronium-ion concentration on a logarithmic scale: pH = −log[H₃O⁺]. A strong acid ionizes completely, so its concentration is [H₃O⁺]. A weak acid ionizes only partly, so [H₃O⁺] comes from Ka and an ICE table. At 25 °C water’s ion product, Kw = 1.0 × 10⁻¹⁴, links acids and bases, and pH + pOH = 14.00.

Acid–base titration calculations

A titration adds a solution of known concentration, the titrant, to a measured volume of the analyte until the two have reacted in exactly the ratio of the balanced equation: the equivalence point. The moles of titrant used, n = cV, give the moles of analyte through the mole ratio, and dividing by the analyte’s volume gives its concentration. When a strong acid is titrated with a strong base, the pH rises slowly, jumps steeply near equivalence and is 7.00 at equivalence at 25 °C.

Buffers and the Henderson–Hasselbalch equation

A buffer contains a weak acid and its conjugate base in comparable amounts, so it absorbs small additions of strong acid or base with little change in pH. Its pH follows from the Henderson–Hasselbalch equation, pH = pKa + log([A⁻]/[HA]). The same chemistry shapes a weak-acid titration curve, whose half-equivalence point has pH = pKa.

Solubility product (Ksp) and molar solubility

A slightly soluble ionic solid dissolves until its ions reach equilibrium with the solid. The solubility product, Ksp, is that equilibrium constant: the product of the ion concentrations, each raised to its coefficient. From Ksp you can find the molar solubility, see how a common ion lowers it, and predict whether mixing two solutions forms a precipitate.

Gibbs free energy, entropy and spontaneity

A process is spontaneous when it can proceed without continuous outside help. At constant temperature and pressure, the Gibbs free energy change ΔG = ΔH − TΔS decides: a negative ΔG means spontaneous as written. ΔG° also fixes the equilibrium constant through ΔG° = −RT ln K, and for a nonstandard mixture ΔG = ΔG° + RT ln Q.

Cell potential, Nernst equation and electrolysis

In a galvanic cell, oxidation at the anode and reduction at the cathode happen in separate compartments, so the electrons flow through a wire. The standard cell potential is E°cathode − E°anode, and a positive value means the reaction is spontaneous, since ΔG° = −nFE°. The Nernst equation corrects E for nonstandard concentrations. In electrolysis an outside source drives a nonspontaneous reaction, and the charge counts the moles of electrons.

One molecule, five representations

Chemists write the same molecule in several ways. A skeletal structure shows the carbon framework, a Lewis structure shows every valence electron, a molecular formula counts the atoms, a SMILES string writes the connections as one line of text, and a systematic name spells out the parent chain and the groups on it. Each keeps some information and leaves the rest out, so no single drawing is the whole molecule.

Naming alkanes and haloalkanes

An alkane’s name is built from its longest continuous chain of carbons, which gives the root, such as hexane for six carbons. Each branch is an alkyl group, such as methyl or ethyl, and a halogen atom is a halo group, such as chloro. Number the chain from the end nearer the first branch, give each group its number, list the groups alphabetically and use di-, tri- or tetra- for identical groups.

Naming alkenes and alkynes

Alkenes and alkynes are named like alkanes with three changes: the parent chain is the longest chain that contains the double or triple bond, the ending is -ene or -yne, and the chain is numbered to give the multiple bond the lowest number, written before the name, as in 2-methyl-2-pentene. When each carbon of a double bond has one group besides hydrogen, cis (same side) or trans (opposite sides) says how they are arranged.

Naming organic compounds by functional group

A functional group decides a compound’s class and the ending of its name. The parent chain must contain the group, and it is numbered to give the group the lowest number: 2-pentanol, 2-pentanone. Aldehydes and acids have the group at carbon 1, so they need no number. Esters are named for the alkyl group on the oxygen, then the acid with -oate; ethers and amines are named for the groups on the oxygen or nitrogen. Benzene rings take their substituents as prefixes, with familiar names such as toluene and phenol.

Biology

Introductory biology

Magnification and scale bars in microscopy

Magnification tells how many times larger an image is than the real object: magnification = image size ÷ actual size, with both lengths in the same unit. A scale bar is a line drawn on the image and labeled with the real length it represents, so measuring it gives the magnification, and comparing any feature with it gives that feature’s actual size, even after the image is resized.

Enzyme kinetics: Michaelis–Menten and inhibition

Enzyme kinetics measures how fast an enzyme turns substrate into product. In the Michaelis–Menten model the initial rate v rises with the substrate concentration [S] and levels off at a maximum rate Vmax, when nearly every enzyme molecule is busy; Km is the substrate concentration that gives half of Vmax. Inhibitors change these two numbers in telltale ways: a competitive inhibitor raises the apparent Km, and a noncompetitive inhibitor lowers Vmax.

Punnett squares and inheritance probability

Connect parental alleles, possible gametes and offspring probabilities without confusing genotype with phenotype.

PCR: the polymerase chain reaction

The polymerase chain reaction (PCR) copies one chosen stretch of DNA millions of times. Two short primers match the two ends of the target, and each cycle of denaturation, annealing and extension can double the number of copies, so n cycles multiply the starting amount by up to 2ⁿ. The product, the amplicon, runs from one primer to the other.

Restriction digests and gel electrophoresis

Connect DNA cut positions to fragment lengths, then interpret the pattern those fragments make on a teaching gel.

Hardy–Weinberg equilibrium: p² + 2pq + q²

Hardy–Weinberg equilibrium is the baseline model of population genetics. For one gene with two alleles at frequencies p and q, where p + q = 1, random mating gives the genotypes AA, Aa and aa in the proportions p², 2pq and q², and those proportions stay the same every generation while the Hardy–Weinberg conditions hold. A sample that departs from them shows that at least one condition fails: mating may not be random, or selection, migration, mutation or drift may be at work.

Mark–recapture population estimates

Mark–recapture estimates how many animals live in an area without counting them all. Catch and mark M animals, release them to mix back in, then catch a second sample of C animals and count the R that carry marks. If marked animals make up the same fraction of the second catch as of the whole population, R/C = M/N, which gives the Lincoln–Petersen estimate N ≈ MC/R.

Survivorship curves and life tables

A survivorship curve follows a cohort, a group of individuals born at the same time, and plots the fraction still alive at each age, lₓ, usually on a logarithmic scale. Type I curves stay high and fall late, because most deaths come in old age; Type II curves fall as a straight line on the log scale, because a constant fraction dies at every age; and Type III curves plunge early, because most young die.

Exponential and logistic population growth

Population growth models predict how the number of individuals N changes over time. Exponential growth assumes a constant per-capita growth rate r, so dN/dt = rN and N(t) = N₀e^(rt): the population grows ever faster, without limit. Logistic growth adds a carrying capacity K, the largest population the environment can sustain, so growth slows as N approaches K and the curve is S-shaped.

Predator–prey cycles: the Lotka–Volterra model

Predator–prey models describe two populations that drive each other. In the Lotka–Volterra model, prey grow exponentially when there are no predators, predators die off when there is no prey, and each encounter removes prey and feeds predator growth. The populations cycle around an equilibrium, with each predator peak following a prey peak.