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Chapter 1Mathematical Foundations Units, SI & Dimensional Analysis Carrying units through every calculation
SI base units, prefixes and the unit-factor method that turns 'which number do I multiply?' into bookkeeping you can trust.
Gen Chem Analytical
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Chapter 1Mathematical Foundations Significant Figures & Rounding How many digits actually mean something
Sig-fig rules for each operation, rounding conventions, and why your answer can't be more precise than your data.
Gen Chem Analytical
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Chapter 1Mathematical Foundations Ratios, Proportion & Percentages Scaling, dilutions and composition
Direct and inverse proportion, percentage composition, and the dilution factor — the arithmetic of the whole wet lab.
Gen Chem Analytical
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Chapter 1Mathematical Foundations Logarithms & Exponentials The maths behind pH, decay and rate constants
Logs turn multiplication into addition and compress huge ranges — exactly what pH, absorbance and Arrhenius need. Fully interactive.
Gen Chem Analytical Thermo / Kinetics
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Chapter 1Mathematical Foundations Sum & Product Notation (Σ, Π) Reading sigma and pi like a chemist
Compact notation for sums and products that appears the moment you meet partition functions, averages and equilibria.
Thermo / Kinetics Quantum
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Chapter 2Algebra & Functions Rearranging Equations Make any symbol the subject
The single most-used skill in chemistry: confidently solving a formula for the quantity you actually want.
Gen Chem Organic Analytical +2
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Chapter 2Algebra & Functions Indices, Powers & Roots The rules of exponents
Multiplying, dividing and nesting powers — including fractional and negative indices that show up in rate laws and equilibrium expressions.
Gen Chem Thermo / Kinetics
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Chapter 2Algebra & Functions Quadratics & Equilibria ICE tables and the quadratic formula
When the small-x approximation breaks, equilibrium problems become quadratics. Here's how to solve them cleanly and pick the valid root.
Gen Chem Analytical
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Chapter 2Algebra & Functions Simultaneous Equations Solving several unknowns at once
Substitution and elimination for systems of equations — the engine behind Hess's-law cycles and multi-component mixtures.
Gen Chem Analytical
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Chapter 2Algebra & Functions Straight-Line Graphs & Linearization y = mx + c, and how to force data onto it
Gradient and intercept, and the chemist's trick of transforming curves (logs, reciprocals) into straight lines you can fit by eye.
Gen Chem Analytical Thermo / Kinetics
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Chapter 2Algebra & Functions Partial Fractions Splitting awkward fractions apart
Breaking a ratio of polynomials into simple pieces — the key that unlocks integrating second-order rate laws.
Thermo / Kinetics
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Chapter 3Geometry & Trigonometry Geometry: Area, Volume & Molecular Shape From beakers to bond angles
Areas and volumes of the shapes chemistry actually uses, plus the geometry of tetrahedra and octahedra behind VSEPR.
Gen Chem Organic
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Chapter 3Geometry & Trigonometry Trigonometry & Radians Angles, waves and the unit circle
Bond angles, diffraction and every wave in chemistry speak trigonometry — and they speak it in radians.
Organic Quantum
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Chapter 3Geometry & Trigonometry Polar & Spherical Coordinates Describing orbitals in (r, θ, φ)
Atomic orbitals are naturally round, so they're described in polar and spherical coordinates, not x–y–z.
Quantum
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Chapter 4Limits & Differentiation Limits & Continuity What happens as you approach a value
The idea of a limit underpins every derivative and integral — and explains behaviour 'at infinite dilution' or 'as t → ∞'.
Thermo / Kinetics Quantum
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Chapter 4Limits & Differentiation Differentiation & Rates of Change The slope of a curve is a reaction rate
Drag a tangent along a concentration–time curve and watch the instantaneous rate appear. The single most useful idea in physical chemistry, built from the ground up.
Thermo / Kinetics Quantum
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Chapter 4Limits & Differentiation Chain, Product & Quotient Rules Differentiating the functions chemistry throws at you
The three rules for differentiating combinations of functions — essential once exponentials and composite expressions appear.
Thermo / Kinetics Quantum
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Chapter 4Limits & Differentiation Stationary Points & Optimization Finding maxima, minima and transition states
Set the derivative to zero to locate peaks and troughs — the maths of equilibrium geometries and reaction-energy profiles.
Thermo / Kinetics Quantum
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Chapter 4Limits & Differentiation Partial Derivatives Changing one variable at a time
Thermodynamics lives on surfaces of many variables; partial derivatives let you hold the rest fixed and vary just one.
Thermo / Kinetics Quantum
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Integration, Area & Triple Integrals Adding up infinitesimal slices — in 1D and 3D
Watch rectangles shrink into the exact area under a curve, see why that area is the integrated rate law and p–V work, then take it to 3D with the triple integral that normalises a wavefunction. Fully interactive.
Thermo / Kinetics Analytical Quantum
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Integration Techniques Substitution and integration by parts
The two work-horse methods for integrals that don't yield to standard forms — needed throughout kinetics and quantum.
Thermo / Kinetics Quantum
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Definite Integrals in Chemistry Work, normalization and expectation values
Putting integration to work: the area that is p–V work, the condition that normalises a wavefunction, and average values in quantum mechanics.
Thermo / Kinetics Quantum
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Chapter 6Differential Equations First-Order Differential Equations Separating variables to find rate laws
The reason e^{−kt} appears everywhere: solving d[A]/dt = −k[A] by separating the variables and integrating.
Thermo / Kinetics
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Chapter 6Differential Equations Second-Order Differential Equations Oscillators and the Schrödinger equation
Equations with a second derivative describe vibrations and the particle-in-a-box — the gateway to quantum mechanics.
Quantum Thermo / Kinetics
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Chapter 7Series & Approximations
Sequences & Series Adding up infinitely many terms
Arithmetic, geometric and infinite series — the maths that sums a partition function and an energy-level ladder.
Thermo / Kinetics Quantum
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Chapter 7Series & Approximations Taylor & Maclaurin Series Turning any function into a polynomial
Approximate eˣ, ln(1+x) and more as polynomials — the basis of nearly every 'for small x…' simplification in physical chemistry.
Thermo / Kinetics Quantum
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Chapter 7Series & Approximations
Binomial & Small-x Approximations (1 + x)ⁿ ≈ 1 + nx
The approximation that justifies the equilibrium 'small-x' shortcut and the Debye–Hückel limiting law.
Gen Chem Thermo / Kinetics
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Vectors, Dot & Cross Products Direction matters
Dipole moments, lattice vectors and angular momentum are all vectors — quantities with both size and direction.
Quantum Organic
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Complex Numbers & the Argand Plane Where i = √−1 earns its keep
Drag a point around the Argand plane and watch Euler's formula e^{iθ} = cos θ + i sin θ tie rotation, waves and wavefunctions together. Fully interactive.
Quantum Analytical
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Chapter 10Matrices & Linear Algebra Matrices, Determinants & Inverses The algebra of arrays
Adding, multiplying and inverting matrices — the bookkeeping that represents symmetry operations and linear transformations.
Quantum
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Chapter 10Matrices & Linear Algebra Eigenvalues & Eigenvectors The heart of quantum chemistry
Build a Hückel matrix and solve for its eigenvalues to get molecular-orbital energies — diagonalization is what computational chemistry does. Fully interactive.
Quantum
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Chapter 11Probability & Statistics Errors & Propagation of Uncertainty How uncertainty travels through a calculation
Absolute vs relative error and the rules for combining uncertainties — so your final result carries an honest ± value.
Analytical Gen Chem
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Chapter 11Probability & Statistics Mean, Standard Deviation & Variance Describing a set of repeats
The summary statistics behind every repeated measurement — mean, spread, and the difference between sample and population.
Analytical
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Chapter 11Probability & Statistics The Normal Distribution & Peak Shapes Bell curves, from error to spectroscopy
The Gaussian describes random error, the Maxwell–Boltzmann speed spread and the shape of a spectroscopic peak.
Analytical Quantum
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Chapter 11Probability & Statistics Linear Regression & Calibration The best straight line through your data
Drag real calibration points and watch least-squares find the best-fit line, R², and the unknown concentration. Fully interactive.
Analytical
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Chapter 11Probability & Statistics Statistical Tests (t, Q & F) Is the difference real?
Confidence intervals and the t-, Q- and F-tests that decide whether results agree, whether to reject an outlier, and whether two methods differ.
Analytical
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Chapter 11Probability & Statistics
Probability & Boltzmann Statistics Counting microstates
Basic probability and combinatorics lead to the Boltzmann distribution — how energy spreads over states and why entropy is k ln W.
Thermo / Kinetics Quantum