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

Factoring rewrites a polynomial as a product, such as $x^{2} - 5 x + 6$ $= (x - 2) (x - 3)$ . Set the product equal to zero and each factor gives one small equation, whose solutions are where the graph meets the x-axis.

The quadratic formula and the discriminant

The quadratic formula $x = (- b \pm \sqrt{b^{2} - 4 a c}) / (2 a)$ solves any equation $a x^{2} + b x + c = 0$ with $a \neq 0$ . The discriminant $b^{2} - 4 a c$ 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 $y = a \cdot f (b (x - h)) + k$ , $h$ shifts the graph right by $h$ and $k$ shifts it up by $k$ . The number $a$ 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: $(f \circ g) (x) = f (g (x))$ , 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: $m = (y_{2} - y_{1}) / (x_{2} - x_{1})$ , 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 $y = m x + b$ , 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 $f (x) = a x^{2} + b x + c$ 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 $x = - b / (2 a)$ , 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 $a_{n} x^{n} + \dots + a_{1} x + a_{0}$ with whole-number powers. Its degree $n$ and leading coefficient $a_{n}$ 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, and 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). The factor theorem follows, since x − c is a factor exactly when P(c) = 0.

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, and to raise a power to a power, multiply. A negative exponent means a reciprocal, $x^{- n} = 1 / x^{n}$ , and a fractional exponent means a root, so x^(m/n) is the nth root of xᵐ.

Logarithm rules and exponential equations

A logarithm answers “what exponent?”: ${log}_{b} x = y$ means $b^{y} = x$ . 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 $A = A_{0} e^{k t}$ , interest compounded $n$ times a year follows $A = P {(1 + r / n)}^{n t}$ , 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 $b^{k}$ is $C (n, k) a^{n - k} b^{k}$ . 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 $cos 2 x = 1 - 2 {sin}^{2} x$ , 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.

The law of sines and the law of cosines

The law of sines and the law of cosines solve triangles that have no right angle. The law of cosines handles two sides with the angle between them, or all three sides. The law of sines handles any triangle in which you know a side, the angle opposite it and one more part. Two sides and an angle that is not between them can fit two triangles, one or none, so that case needs a test.

Complex numbers: arithmetic and polar form

A complex number $a + b i$ combines a real part $a$ and an imaginary part $b$ , where $i^{2} = - 1$ . 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 that never change the solutions, and the 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. A square matrix has an inverse exactly when its determinant is not zero.

Conic sections: parabolas, ellipses, hyperbolas

Circles, ellipses, parabolas and hyperbolas are the curves a plane cuts from a double cone, and each is the set of points at given distances from a center, a focus or a line. Completing the square turns a general second-degree 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

A limit is the value $f (x)$ gets close to as $x$ gets close to a number $a$ , from both sides, whether or not $f (a)$ exists. Substitute $a$ when the function is defined there. When substitution gives $0 / 0$ , rewrite the expression first and then substitute.

Continuity and the Intermediate Value Theorem

A function 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. The graph of a continuous function has no hole, jump or vertical asymptote. 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 to $y = f (x)$ at $x = a$ , and the instantaneous rate of change of $f$ there. It is the limit of the secant slopes $(f (a + h) - f (a)) / h$ as $h$ approaches $0$ , when that limit exists.

The power rule and basic derivative rules

To differentiate a polynomial or a sum of powers, use three rules. The power rule says the derivative of $x^{n}$ is $n x^{n - 1}$ . A constant factor stays in front, and a sum is differentiated term by term. Rewrite roots and fractions as powers of x first.

Derivatives of eˣ, ln x and trig functions

The derivative of $e^{x}$ is $e^{x}$ , and the derivative of $ln x$ is $1 / x$ for $x > 0$ . With $x$ in radians, the derivative of $sin x$ is $cos x$ and the derivative of $cos x$ is $- sin x$ . Each follows from the limit definition, and each combines with the sum, product and chain rules.

The product rule for derivatives

To differentiate a product $f (x) g (x)$ , use the product rule. The derivative is $f^{'} (x) g (x) + f (x) g^{'} (x)$ , one term for each factor changing while the other holds still. It is not $f^{'} (x) g^{'} (x)$ .

The quotient rule for derivatives

To differentiate a quotient $f (x) / g (x)$ , use the quotient rule. The derivative is $(f^{'} g - f g^{'}) / g^{2}$ , the bottom times the top’s derivative minus the top times the bottom’s derivative, all over the bottom squared. It holds wherever g(x) ≠ 0.

The chain rule for derivatives

The chain rule differentiates a function inside another function, such as $sin (x^{2})$ or ${(3 x + 1)}^{50}$ . Differentiate the outside function while leaving the inside untouched, then multiply by the derivative of the inside. In symbols, the derivative of f(g(x)) is f′(g(x))·g′(x).

Implicit differentiation

Implicit differentiation finds $d y / d x$ 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 $d y / d x$ , then solve for $d y / d x$ .

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 $d y = f^{'} (x) d x$ 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, so 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 $f (x) \to L$ as $x \to \infty$ or as $x \to - \infty$ , 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 $\infty / \infty$ 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 $\int f (x) d x = F (x) + C$ . 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 $Δ x = (b - a) / n$ , take a height $f (x_{i}^{*})$ in each piece, and add the products $f (x_{i}^{*}) \cdot Δ x$ . Left, right and midpoint sums choose different heights, and 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 $F (x) = \int_{a}^{x} f (t) d t$ , 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 that $d u = g^{'} (x) d x$ . Integrate in $u$ , then put $g (x)$ back in place of $u$ . For a definite integral, change the limits to values of u instead.

Area between two curves

The area between two curves is the integral of top minus bottom, $A = \int (f (x) - g (x)) d x$ from $a$ to $b$ , where $f (x) \geq g (x)$ on $[a, b]$ . Find $a$ and $b$ where the curves intersect, and test a point to see which curve is on top. 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 $π R^{2}$ or $π (R^{2} - r^{2})$ . 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. It rewrites the integral of $u d v$ as $u v$ minus the integral of $v d u$ , trading an integral you cannot do for one you can. It works 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, written as ln x times 1.

Trigonometric substitution

Trigonometric substitution handles integrals that contain $\sqrt{a^{2} - x^{2}}$ , $\sqrt{a^{2} + x^{2}}$ or $\sqrt{x^{2} - a^{2}}$ . Replace $x$ with $a sin θ$ , $a tan θ$ or $a sec θ$ , and an identity such as $1 - {sin}^{2} θ = {cos}^{2} θ$ turns the expression under the root into a perfect square. Integrate in θ, then return to x with a right triangle, or change the limits of a definite integral and stay in θ.

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, and 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 $d y / d x = g (x) h (y)$ , and it is solved by separation of variables. Divide by h(y), put dx on the other side, 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, and it converges when its partial sums approach a finite limit. A geometric series $a + a r + a r^{2} + \dots$ converges exactly when $∣ r ∣ < 1$ , and its sum is then $a / (1 - r)$ . Terms that do not shrink to 0 force divergence. 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 $n$ is the polynomial whose value and first $n$ derivatives at a center $a$ match a function’s. Near $a$ it approximates the function, and each added term improves the fit. Letting $n$ grow gives the Taylor series, called a Maclaurin series when centered at 0. 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, but 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.

The binomial distribution and expected value

A binomial random variable counts the successes in a fixed number $n$ of independent trials that each succeed with the same probability $p$ . The chance of exactly $k$ successes is $C (n, k) p^{k} {(1 - p)}^{n - k}$ . On average there are $n p$ successes, with variance np(1 − p), and when n is large enough a normal curve with that mean and standard deviation approximates the whole distribution.

Confidence intervals: the t interval for a mean

A confidence interval gives a range of plausible values for an unknown population mean. It is the sample mean plus or minus a margin of error, a critical value times the standard error $s / \sqrt{n}$ , 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 $H_{0}$ states the claim, such as $μ = 500$ , and the alternative $H_{a}$ states what you suspect instead. The $t$ statistic counts how many standard errors the sample mean lies from the claim. The p-value is the probability of a result at least that extreme if H₀ were true, and a small p-value is evidence against H₀.

The chi-square goodness-of-fit test

A chi-square goodness-of-fit test asks whether counts in categories match a claimed distribution, such as a fair die or a $3 : 1$ genetic ratio. Each category adds ${(observed - expected)}^{2} / expected$ to the statistic $χ^{2}$ . When the claim is right, χ² follows a chi-square distribution with k − 1 degrees of freedom, and a small p-value means the counts fit the claim poorly.

Linear regression, correlation and residuals

Linear regression fits the line of best fit, the least-squares line $\hat{y} = a + b x$ that makes the squared vertical misses from the points, the residuals, as small as possible. The slope $b$ 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

Dimensional analysis converts a quantity to new units by multiplying it by conversion factors, ratios equal to one, each arranged so that a unit you do not want cancels. The units left at the end tell you whether the setup was right.

Isotopes and average atomic mass

Atoms of one element all have the same number of protons, but they can differ in their number of neutrons. Those versions are isotopes, told apart by mass number. The atomic mass on the periodic table is the average of the isotope masses, each weighted by how common that isotope is in nature. That average, not the mass of any single atom, is what molar mass is built from.

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, $6.022 \times 10^{23}$ 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 $M_{1} V_{1} = M_{2} V_{2}$ .

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 $O_{2}$ for last, and recount at the end.

Stoichiometry and theoretical yield

Stoichiometry predicts how much product a reaction can form, or how much of a reactant it needs. Convert the 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. Find it by working out how much product each reactant could make on its own. The smaller amount is the theoretical yield, and $actual \div theoretical \times 100 %$ 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, $q = m c Δ T$ , 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 $Δ H$ , because enthalpy is a state function. Reversing an equation flips the sign of $Δ H$ 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: $E = h ν$ , where $h$ is Planck’s constant. Since $c = λ ν$ , $E = h c / λ$ , 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 the center 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, where 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, $P V = n R T$ , relates a gas’s pressure $P$ , volume $V$ , amount $n$ and absolute temperature $T$ through the gas constant $R = 0.08206 L \cdot atm / (mol \cdot K)$ . Solve it for the unknown, with the temperature in kelvin and every unit matching R.

General chemistry II

Enthalpy of solution and lattice energy

Dissolving a solid can warm or cool the water, and the enthalpy of solution, $Δ H (soln)$ , says which and by how much. Build it as a cycle: break the solid into separate particles, then let the water surround them. For an ionic solid, ΔH(soln) is minus the lattice energy plus the hydration enthalpies of the ions.

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

A dissolved solute lowers a solvent’s vapor pressure and freezing point, raises its boiling point and creates osmotic pressure, by amounts that depend on how many solute particles there are, not on what they are. Calculate each change, and run the equations backward to find a molar mass or a van ’t Hoff factor.

The Beer–Lambert law and absorbance

A colored solution lets through only a fraction of the light that enters it, and the more absorbing molecules the light meets, the less gets through. The absorbance, $A = - log T$ , grows in direct proportion to concentration, A = εbc, so a measured absorbance gives the concentration. In practice a calibration line from standard solutions does the conversion.

Finding rate laws from experimental data

A rate law, $rate = k {[A]}^{m} {[B]}^{n}$ , says how a reaction’s speed depends on each concentration. Find each order by changing one concentration at a time and comparing the initial rates, or by finding which plot of a concentration against time is straight, then solve for k.

Integrated rate laws and half-life

An integrated rate law gives a reactant’s concentration at any time from its starting concentration and $k$ . For a first-order reaction $[A] = {[A]}_{0} e^{- k t}$ , and the half-life, $0.693 / k$ , is the same however much is left, which is why radioactive decay is described by half-lives.

Activation energy from Arrhenius plots

Rate constants rise with temperature by an amount set by the activation energy, $E_{a}$ , the energy barrier a reaction must cross. Plot $ln k$ against $1 / T$ and the slope is $- E_{a} / R$ , or use the two-point Arrhenius equation when only two rate constants are known.

Potential energy diagrams for reactions

A potential energy diagram plots a reaction’s energy as the atoms move from reactants to products. The drop or rise between the two ends is $Δ H$ , and the climb from either end to the top of the hump is that direction’s activation energy. A catalyst lowers the hump without moving either end.

Reaction mechanisms and rate laws

A mechanism is a list of elementary steps that claims to show how a reaction happens. It is consistent with the data when its steps add up to the overall reaction and its slow step, with any intermediate replaced, predicts the observed rate law.

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: $pH = - log [H_{3} O^{+}]$ . A strong acid ionizes completely, so its concentration is $[H_{3} O^{+}]$ . A weak acid ionizes only partly, so $[H_{3} O^{+}]$ comes from $K_{a}$ 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 holds its $pH$ nearly steady when a little strong acid or base is added, because it holds a weak acid and its conjugate base in comparable amounts. Its $pH$ follows from the Henderson–Hasselbalch equation, $pH = p K_{a} + log ([A^{-}] / [HA])$ . Added acid or base is reacted first, in moles, and 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 only until its ions reach equilibrium with the solid, and the solubility product, $K_{sp}$ , is that equilibrium constant: the ion concentrations multiplied together, 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, and at constant temperature and pressure the Gibbs free energy change, $Δ G = Δ H - T Δ S$ , decides: a negative $Δ G$ means spontaneous as written. $Δ G °$ also fixes the equilibrium constant through ΔG° = −RT ln K, and for a mixture that is not at equilibrium ΔG = ΔG° + RT ln Q.

Cell potential, Nernst equation and electrolysis

A galvanic cell runs a redox reaction with oxidation at the anode and reduction at the cathode in separate compartments, so the electrons must flow through a wire. Its standard potential is $E °_{cell}$ $= E °_{cathode} - E °_{anode}$ , 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

A molecule can be written as a skeletal structure, a Lewis structure, a molecular formula, a SMILES string or a systematic name. All but the formula record which atom is bonded to which, so only the formula cannot tell isomers apart. None of these flat forms shows the molecule’s real shape in space.

SMILES: a structure written as a line of text

SMILES is a way to type a structure. Atoms are their symbols, neighbors in the string are bonded, a branch goes in parentheses, and a ring is closed by a pair of matching digits. Square brackets hold whatever is unusual, such as a charge, a stated number of hydrogens, an isotope or a stereocenter. One molecule has many valid strings, so programs settle on one canonical string when they need to compare.

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 two changes. The ending is -ene or -yne, and the chain is numbered to give the multiple bond the lowest number, written just before the ending, as in 2-methylpent-2-ene. Many textbooks still write the number in front, 2-methyl-2-pentene, and both are correct. When each carbon of a double bond has one group besides hydrogen, cis (same side) or trans (opposite sides) says how they are arranged, and E or Z says it for any double bond.

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, written just before the ending, as in pentan-2-ol and pentan-2-one. When a compound has two groups, the senior one takes the ending and the other becomes a prefix, as in 4-hydroxybutan-2-one. Aldehydes and acids have the group at carbon 1, so they need no number. An ester is named for the alkyl group on its oxygen, then the acid with -oate. Benzene rings take their groups as prefixes, with familiar names such as phenol.

Chirality and R/S configuration

A carbon with four different groups can be arranged in two ways that are mirror images and cannot be laid over each other. To tell them apart, rank the four groups by atomic number (the Cahn–Ingold–Prelog rules), look with the lowest-ranked group pointing away from you, and follow 1 → 2 → 3: clockwise is R, counterclockwise is S. In a wedge drawing, a solid wedge comes toward you and a hashed wedge goes away.

Cis/trans and E/Z alkene isomers

A carbon–carbon double bond does not rotate, so the groups on its two carbons stay on the same side or on opposite sides, which gives two different compounds. Cis and trans describe this when each carbon has one group besides hydrogen. With three or four groups they stop making sense, and E/Z uses the CIP priorities instead. Rank the two groups on each carbon. If the higher-ranked ones are on the same side the alkene is Z, and on opposite sides it is E.

Enantiomers, diastereomers and meso compounds

Stereoisomers share their connections and differ in arrangement. Mirror-image pairs are enantiomers, and stereoisomers that are not mirror images are diastereomers. A molecule with $n$ stereocenters has at most $2^{n}$ stereoisomers, fewer when symmetry makes some identical: a meso compound has stereocenters but is its own mirror image. Enantiomers share most physical properties but rotate polarized light in opposite directions by equal amounts, and an equal mixture of the two, a racemic mixture, does not rotate it. The excess of one enantiomer over the other follows from the measured rotation.

Biology

Introductory biology

Designing a controlled experiment

An experiment tests a hypothesis by changing one factor on purpose and measuring what happens. The independent variable is the factor the experimenter changes, the dependent variable is what is measured, and controlled variables are held the same so they cannot explain the result. A control group, treated the same except for the independent variable, gives the baseline. Random assignment and replication guard against confounders and chance, so a difference between groups can be credited to the treatment.

Magnification and scale bars in microscopy

Magnification tells how many times larger an image is than the real object. It is $image size \div actual size$ , with both lengths in the same unit. A scale bar is a line on the image labeled with the real length it stands for. Measuring it gives the magnification, and comparing any feature with it gives that feature’s actual size, even after the image is resized.

Cell structure: organelles and their jobs

Every cell has a plasma membrane, cytoplasm, ribosomes and DNA. Prokaryotic cells, the bacteria and archaea, are small and have no nucleus or other membrane-bound organelles, and their DNA lies in a region called the nucleoid. Eukaryotic cells are larger and divide their work among organelles. The nucleus holds the DNA, ribosomes and the rough ER make proteins, the Golgi apparatus modifies and ships them, and mitochondria make ATP. Plant cells add chloroplasts, a cell wall and a large central vacuole.

Membrane transport: diffusion, osmosis, pumps

A cell membrane is a phospholipid bilayer that lets some molecules through and holds others back. Small nonpolar molecules such as $O_{2}$ diffuse straight across it. Ions, and polar solutes such as glucose, need a channel or a carrier protein. Passive transport runs down a concentration gradient and costs the cell no energy. Active transport moves a solute up its gradient and needs energy, usually from ATP, as in the sodium-potassium pump. Water crosses by osmosis, toward the side with more dissolved solute.

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 $V_{\max}$ , 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, as a competitive inhibitor raises the apparent Km and a noncompetitive inhibitor lowers Vmax.

Cellular respiration: from glucose to ATP

Cellular respiration releases the energy stored in glucose in small steps and stores part of it as ATP. Glycolysis splits glucose into two pyruvate in the cytosol. In the mitochondrion each pyruvate becomes acetyl-CoA, and the citric acid cycle oxidizes it to CO₂. These stages make only 4 ATP per glucose directly. Most of the energy leaves them as electrons carried by NADH and FADH₂. The electron transport chain passes those electrons to oxygen and builds the proton gradient that drives ATP synthase, for about 30 to 32 ATP in all.

Photosynthesis and the Calvin cycle

Photosynthesis turns light energy into the chemical energy of sugar. In the thylakoid membranes of the chloroplast, the light-dependent reactions split water, release oxygen and make ATP and NADPH. In the stroma around them, the Calvin cycle spends that ATP and NADPH to fix carbon dioxide. The enzyme rubisco joins CO₂ to RuBP, the product is reduced to the three-carbon sugar G3P, and most of the G3P is rebuilt into RuBP so the cycle can continue.

Meiosis: from diploid cells to gametes

Meiosis makes gametes, sperm and eggs, with half the chromosome number of the parent cell. A diploid cell, $2 n$ , with pairs of homologous chromosomes copies its DNA and then divides twice. Meiosis I separates the homologous pairs, which halves the chromosome number. Meiosis II separates sister chromatids, as mitosis does. The four haploid cells, n, differ from each other and from the parent, because homologs swap segments by crossing over and each pair lines up independently of the others.

Punnett squares and inheritance probability

A Punnett square predicts the chance of each offspring genotype from the alleles its parents carry. Each parent passes one of its two alleles, each with probability $\frac{1}{2}$ , so Aa × Aa gives $\frac{1}{4}$ AA, $\frac{1}{2}$ Aa and $\frac{1}{4}$ aa. Under complete dominance that is 3 dominant to 1 recessive in phenotype. The same rules, multiplied gene by gene, predict two-gene crosses and the chances in a family pedigree.

Transcription and translation: DNA to protein

A gene’s information flows from DNA to RNA to protein. In transcription, RNA polymerase reads the template strand of a gene and builds a messenger RNA, mRNA, that matches the other strand, the coding strand, with U in place of T. In translation, a ribosome reads the mRNA three bases at a time, one codon per amino acid, from the start codon AUG to a stop codon. One changed base can change one amino acid, as in sickle-cell disease.

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 heating, cooling 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 and includes both.

Restriction digests and gel electrophoresis

A restriction enzyme cuts DNA wherever it finds its recognition sequence, so a digest turns one molecule into fragments of predictable lengths. Each fragment runs from one cut to the next, or to an end of a linear molecule, and the lengths add up to the whole. On an agarose gel shorter fragments travel farther, and fragments of the same length share one band.

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 $p^{2}$ , $2 p q$ and q². Those proportions stay the same every generation while the Hardy–Weinberg conditions hold. A sample that departs from them is evidence that at least one condition fails, such as random mating, or that selection, migration, mutation or drift is at work.

Phylogenetic trees: reading relatedness

A phylogenetic tree is a hypothesis about how organisms are related through common ancestry. Its tips are the taxa compared, each node is a common ancestor, and a clade is an ancestor with all of its descendants. Two taxa are more closely related when they share a more recent common ancestor, whatever their order along the tips. Trees are built from shared derived characters, and parsimony prefers the tree that needs the fewest evolutionary changes.

Neurons and the action potential

A resting neuron holds a membrane potential of about $- 70 mV$ , negative inside, kept by the sodium-potassium pump and by $K^{+}$ leaking out through open channels. A stimulus that depolarizes the membrane to threshold, about $- 55 mV$ , opens voltage-gated Na⁺ channels: Na⁺ rushes in and the inside swings to about +30 mV. The Na⁺ channels then inactivate and voltage-gated K⁺ channels open. K⁺ flows out and repolarizes the membrane, overshooting briefly below rest. The action potential is all-or-none, and it travels along the axon to the synapse.

Antibodies: how they recognize antigens

Antibodies are proteins, made by B cells, that bind specific antigens. Each antibody is a Y built from two identical heavy chains and two identical light chains joined by disulfide bonds. The tips of the arms are variable regions whose shape fits one epitope, a small part of the antigen. The stem, the Fc region, is constant within a class and links the antibody to the rest of the immune system. Binding neutralizes toxins and viruses, clumps pathogens together and marks them for destruction.

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, $l_{x}$ , 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. 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 $d N / d t = r N$ and $N (t) = N_{0} e^{r t}$ , and 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.

Trophic levels and energy flow: the 10% rule

A trophic level is a feeding step in a food chain, from producers to primary consumers (herbivores), secondary consumers and so on. At each step most of the energy is used in respiration and lost as heat, or is never eaten or digested, so only a small fraction becomes new biomass for the next level. The textbook figure is 10%, and measured averages are lower, about 6%. That is why energy pyramids narrow upward and food chains rarely have more than four or five levels.