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IB Diploma Core Concepts & Definitions

Essential mathematical rules, chemistry principles, and physics equations for IB Diploma HL/SL students.

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De Moivre's Theorem

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De Moivre's Theorem

For any complex number

z = r (cos θ + i sin θ)

and integer

n

: $

z^{n} = r^{n} (cos (n θ) + i sin (n θ)) = r^{n} e^{in θ}

Euler's Identity

e^{iπ} + 1 = 0

Vector Dot Product and Perpendicular Condition

u \cdot v = ∣ u ∣∣ v ∣ cos θ

\vec{u} \perp \vec{v} \iff \vec{u} \cdot \vec{v} = 0$.

Vector Cross Product Magnitude

∣ u \times v ∣ = ∣ u ∣∣ v ∣ sin θ

$ Equal to the area of the parallelogram formed by $\vec{u}

an d

\vec{v}$.

Integration by Parts Formula

\int u \frac{d v}{d x} d x = uv - \int v \frac{d u}{d x} d x

Continuous Random Variable Variance

Var (X) = E (X^{2}) - [E (X)]^{2} = \int_{- \infty}^{\infty} x^{2} f (x) d x - μ^{2}

Binomial Distribution Mean and Variance

For

X \sim B (n, p)

: $

E (X) = n p, Var (X) = n p (1 - p)

Poisson Distribution Property

For

X \sim Po (λ)

: $

E (X) = Var (X) = λ

Maclaurin Series Expansion

f (x) = f (0) + f^{'} (0) x + \frac{f ^{''} ( 0 )}{2 !} x^{2} + \dots + \frac{f ^{(n)} ( 0 )}{n !} x^{n} + \dots

First Law of Thermodynamics

Q = Δ U + W

w h er e

Q$ is heat added, $\Delta U$ is change in internal energy, and $W$ is work done by the system ($W = P\Delta V$).

De Broglie Wavelength

λ = \frac{h}{p} = \frac{h}{m v}

$ connecting particle momentum to matter wave wavelength.

Heisenberg's Uncertainty Principle

Δ x Δ p \geq \frac{h}{4 π} = \frac{ℏ}{2}

Radioactive Decay Law

N (t) = N_{0} e^{- λ t} with half-life T_{1/2} = \frac{ln 2}{λ}

Gibbs Free Energy Equation

Δ G = Δ H - T Δ S

$ A reaction is spontaneous under standard conditions if $\Delta G < 0$.

Arrhenius Equation for Rate Constant

k = A e^{- \frac{E _{a}}{R T}} ⟹ ln k = ln A - \frac{E _{a}}{R T}

Le Chatelier's Principle

If a chemical system at equilibrium experiences a change in concentration, temperature, or pressure, the equilibrium shifts in a direction that counteracts the imposed change.

SN1 vs. SN2 Organic Mechanisms

SN1: Two-step, carbocation intermediate, tertiary substrates favored, racemization. SN2: One-step concerted, primary substrates favored, inversion of stereochemistry.

Beer-Lambert Law

A = ϵc l

A b sor ban ce

A$ is directly proportional to molar absorptivity $\epsilon

, co n ce n t r a t i o n

c$, and path length $l$.

Standard Hydrogen Electrode (SHE)

Assigned potential

E^{\circ} = 0.00 V

at all temperatures. Hydrogen gas at

1 bar

bubbling over platinum in

1 mol dm^{- 3} H^{+}

solution.

Simple Harmonic Motion Acceleration

a = - ω^{2} x

$ Acceleration is proportional to displacement from equilibrium and directed toward equilibrium.

Doppler Effect for Moving Source

f^{'} = f (\frac{v}{v \pm v _{s}})

$ Minus sign for approaching source (higher frequency), plus sign for receding source.

Electromagnetic Induction: Faraday's Law

E = - N \frac{d Φ _{B}}{d t}

$ Induced EMF equals negative rate of change of magnetic flux.

Boyle's Law vs. Charles's Law

Boyle's:

P_{1} V_{1} = P_{2} V_{2}

(

T

constant). Charles's:

\frac{V _{1}}{T _{1}} = \frac{V _{2}}{T _{2}}

(

P

constant).

Hess's Law of Constant Heat Summation

The total enthalpy change for a chemical reaction is independent of the pathway taken from initial reactants to final products.

Bohr Model Hydrogen Energy Levels

E_{n} = - \frac{13.6 eV}{n ^{2}}

w h er e

n = 1, 2, 3, \dots$ is the principal quantum number.

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