The maths behind your chemistry degree — taught by the chemistry that needs it
Every topic is ordered like a maths course but framed by where you actually meet it: logs for pH, differentiation for reaction rates, matrices for quantum. Start with the fully interactive differentiation lesson, then explore the rest of the map.
Free & open to every ChemGenius userUnits, 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.
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.
Ratios, Proportion & Percentages
Scaling, dilutions and composition
Direct and inverse proportion, percentage composition, and the dilution factor — the arithmetic of the whole wet lab.
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.
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.
Rearranging Equations
Make any symbol the subject
The single most-used skill in chemistry: confidently solving a formula for the quantity you actually want.
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.
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.
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.
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.
Partial Fractions
Splitting awkward fractions apart
Breaking a ratio of polynomials into simple pieces — the key that unlocks integrating second-order rate laws.
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.
Trigonometry & Radians
Angles, waves and the unit circle
Bond angles, diffraction and every wave in chemistry speak trigonometry — and they speak it in radians.
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.
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 → ∞'.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Sequences & Series
Adding up infinitely many terms
Arithmetic, geometric and infinite series — the maths that sums a partition function and an energy-level ladder.
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.
Binomial & Small-x Approximations
(1 + x)ⁿ ≈ 1 + nx
The approximation that justifies the equilibrium 'small-x' shortcut and the Debye–Hückel limiting law.
Vectors, Dot & Cross Products
Direction matters
Dipole moments, lattice vectors and angular momentum are all vectors — quantities with both size and direction.
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.
Matrices, Determinants & Inverses
The algebra of arrays
Adding, multiplying and inverting matrices — the bookkeeping that represents symmetry operations and linear transformations.
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.
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.
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.
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.
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.
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.
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.