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Mathematical typesetting: equations, derivations, matrices, commutative diagrams. Renders LaTeX math in the browser.
Chemistry — Half-life by Reaction Order
How half-life depends on order and concentration.
Chemistry & PhysicsChemistry — Real Gases
Compressibility factor and the van der Waals equation.
Chemistry & PhysicsChemistry — Half-Equivalence Point
Why pH equals pKa at half-neutralisation.
Chemistry & PhysicsChemistry — Polyprotic Acids
Stepwise dissociation of a diprotic acid.
Chemistry & PhysicsChemistry — Bond Order (MO Theory)
Bond order from molecular-orbital occupancy.
Chemistry & PhysicsMath — Linear Programming
Optimising a linear objective under constraints.
MathematicsMath — Proof by Contradiction
The structure of a proof by contradiction.
MathematicsMath — Partial Derivatives
Gradient, total differential and critical points.
MathematicsMath — Markov Chains
Transition matrices and the steady-state distribution.
MathematicsBiology — Bicarbonate Buffer
The blood pH buffer and its physiological control.
Life SciencesBiology — Polygenic Inheritance & Epistasis
Continuous variation and modified dihybrid ratios.
Life SciencesBiology — Protein Quantification
Colorimetric assays and the standard curve.
Life SciencesBiology — Ecological Efficiency
Energy transfer between trophic levels.
Life SciencesPhysics — Binary Orbits & Barycentre
Two-body orbits about a common centre of mass.
Chemistry & PhysicsPhysics — Heat Capacity
Specific and molar heat capacities and their ratio.
Chemistry & PhysicsPhysics — Superposition & Beats
Adding waves, beat frequency and interference conditions.
Chemistry & PhysicsPhysics — Conservation of Angular Momentum
Angular momentum conservation and its consequences.
Chemistry & PhysicsPhysics — Motors & Generators
Force, torque and induced EMF in rotating machines.
Chemistry & PhysicsChemistry — Salt Hydrolysis
The pH of salt solutions from their parent acid/base.
Chemistry & PhysicsChemistry — Catalysis & Activation Energy
How a catalyst speeds a reaction without changing ΔH.
Chemistry & PhysicsChemistry — Gas Volume Stoichiometry
Reacting gas volumes via molar volume and Avogadro's law.
Chemistry & PhysicsChemistry — Kw and Temperature
Temperature dependence of water's ionic product.
Chemistry & PhysicsMath — Modulus (Absolute Value) Equations
Solving and bounding modulus expressions.
MathematicsMath — Harmonic (R) Form
Writing a cosθ + b sinθ as a single sinusoid.
MathematicsMath — Implicit & Logarithmic Differentiation
Differentiating implicit relations and powers.
MathematicsMath — Cardinality & Countability
Sizes of sets and countable vs uncountable infinity.
MathematicsBiology — Types of Natural Selection
Directional, stabilising and disruptive selection.
Life SciencesBiology — Gene Flow & Migration
How migration changes allele frequencies.
Life SciencesBiology — Photosynthetic Efficiency
Energy conversion efficiency and limiting factors.
Life SciencesPhysics — Types of Friction
Static, kinetic and rolling friction.
Chemistry & PhysicsPhysics — Pulley Systems
Mechanical advantage of a block-and-tackle.
Chemistry & PhysicsPhysics — Surface Tension & Capillarity
Surface tension, Laplace pressure and capillary rise.
Chemistry & PhysicsPhysics — Nuclear Fission & Fusion
Energy-releasing nuclear reactions.
Chemistry & PhysicsPhysics — Rutherford Scattering
The experiment that revealed the nucleus.
Chemistry & PhysicsPhysics — Millikan's Oil-Drop Experiment
Measuring the quantised electron charge.
Chemistry & PhysicsPhysics — Mass Spectrometer
Separating ions by mass-to-charge ratio.
Chemistry & PhysicsPhysics — Lasers
Stimulated emission and laser properties.
Chemistry & PhysicsPhysics — Fibre Optics
Total internal reflection and numerical aperture.
Chemistry & PhysicsPhysics — Kirchhoff's Laws
Current and voltage laws for circuits.
Chemistry & PhysicsPhysics — Radioactive Activity
Activity, decay law and the becquerel.
Chemistry & PhysicsPhysics — Shell Theorem
Gravity inside and outside spherical bodies.
Chemistry & PhysicsChemistry — Orbital Hybridisation
Hybrid orbitals and the shapes they give.
Chemistry & PhysicsChemistry — Chirality & Optical Activity
Stereocentres, enantiomers and optical rotation.
Chemistry & PhysicsChemistry — Carbocation Stability
The stability order of carbocations.
Chemistry & PhysicsChemistry — Aromaticity (Hückel)
The criteria for an aromatic ring.
Chemistry & PhysicsChemistry — Complex Ions
Coordination complexes and stability constants.
Chemistry & PhysicsChemistry — Solubility Rules
Which ionic compounds dissolve in water.
Chemistry & PhysicsChemistry — Oxidation State Trends
Maximum and variable oxidation states.
Chemistry & PhysicsChemistry — Disproportionation
Simultaneous oxidation and reduction of one species.
Chemistry & PhysicsChemistry — Standard Enthalpy Definitions
The key standard enthalpy changes.
Chemistry & PhysicsChemistry — Determining Reaction Order
Reading order from initial-rate data.
Chemistry & PhysicsMath — Lagrange Multipliers
Constrained optimisation.
MathematicsMath — Line Integrals
Scalar and vector line integrals.
MathematicsMath — Divergence & Curl
Vector differential operators and identities.
MathematicsMath — Power Series
Power series and radius of convergence.
MathematicsMath — The Heat Equation
A parabolic PDE solved by separation of variables.
MathematicsMath — Gram–Schmidt Process
Building an orthonormal basis.
MathematicsMath — Singular Value Decomposition
Factorising any matrix into U Σ Vᵀ.
MathematicsMath — Gradient Descent
Iterative minimisation along the negative gradient.
MathematicsMath — Bayesian Inference
Posterior, likelihood and prior.
MathematicsMath — Game Theory
Equilibria and strategies.
MathematicsBiology — Chemiosmosis
ATP synthesis driven by a proton gradient.
Life SciencesBiology — Fermentation
Anaerobic regeneration of NAD⁺.
Life SciencesBiology — Thermoregulation
Maintaining body temperature by negative feedback.
Life SciencesBiology — Types of Mutation
Point and chromosomal mutations.
Life SciencesBiology — Transcription
Synthesising and processing mRNA.
Life SciencesBiology — Translation
Decoding mRNA into a polypeptide.
Life SciencesBiology — CRISPR–Cas9
Programmable genome editing.
Life SciencesBiology — Stem Cell Potency
Degrees of stem-cell potency.
Life SciencesPhysics — Spherical Mirrors
The mirror equation, focal length and magnification.
Chemistry & PhysicsPhysics — Refraction
Refractive index, Snell's law and apparent depth.
Chemistry & PhysicsPhysics — Dispersion
Why a prism splits white light.
Chemistry & PhysicsPhysics — Stellar Parallax
Distance from parallax angle; the parsec.
Chemistry & PhysicsPhysics — Standard Candles
Measuring cosmic distances from luminosity.
Chemistry & PhysicsPhysics — Antimatter
Pair production and annihilation.
Chemistry & PhysicsPhysics — Particle Conservation Laws
Quantities conserved in particle interactions.
Chemistry & PhysicsPhysics — Quark Content of Hadrons
Quark composition of common particles.
Chemistry & PhysicsPhysics — MRI & Larmor Precession
Nuclear magnetic resonance imaging.
Chemistry & PhysicsPhysics — PET Scanning
Positron emission tomography.
Chemistry & PhysicsPhysics — Relative Velocity
Combining velocities and river-crossing.
Chemistry & PhysicsPhysics — Vertical Circular Motion
Tension and minimum speed in a vertical circle.
Chemistry & PhysicsChemistry — Clausius–Clapeyron Equation
Vapour pressure as a function of temperature.
Chemistry & PhysicsChemistry — Gibbs Phase Rule
Degrees of freedom of a system at equilibrium.
Chemistry & PhysicsChemistry — Eyring Equation
Transition-state theory of reaction rates.
Chemistry & PhysicsChemistry — Molar Conductivity
Conductivity and Kohlrausch's law.
Chemistry & PhysicsChemistry — Third Law & Absolute Entropy
Entropy at absolute zero and standard entropies.
Chemistry & PhysicsChemistry — Complexometric Titration
EDTA titration of metal ions.
Chemistry & PhysicsChemistry — Ideal & Non-ideal Solutions
Raoult's law, total pressure and deviations.
Chemistry & PhysicsChemistry — Standard Electrode Potentials
Predicting feasibility from cell potentials.
Chemistry & PhysicsChemistry — Gravimetric Analysis
Quantifying an analyte by mass of a precipitate.
Chemistry & PhysicsChemistry — Mass & Charge Balance
Exact treatment of equilibrium concentrations.
Chemistry & PhysicsMath — The Wave Equation (PDE)
Separable solutions and standing-wave modes.
MathematicsMath — Laplace's Equation
Harmonic functions and steady states.
MathematicsMath — Cauchy's Theorem & Residues
Contour integration in the complex plane.
MathematicsMath — Systems of Linear ODEs
Eigenvalue solutions and stability.
MathematicsMath — Numerical Solution of ODEs
Euler and Runge–Kutta stepping.
MathematicsMath — Lagrange Interpolation
Fitting a polynomial through given points.
MathematicsMath — Moment Generating Functions
MGFs, moments and the PGF.
MathematicsMath — Random Walks & Diffusion
Mean-square displacement of a random walk.
MathematicsMath — KKT Conditions
Optimality with inequality constraints.
Mathematics