Nanochemistry — IUPAC-VII applied-chemistry at the 1–100 nm grain where quantum-size effects and high surface-to-volume ratios dominate over bulk-like behaviour. Foundations: the surface-to-volume ratio for a sphere is A/V = 3/R —…
nanochemistry
Sphere surface-to-volume ratio A/V = 3/R
Surface-to-volume ratio for a sphere is A/V = 3/R — inverse-linear in radius. At R = 1 nm, A/V = 3 nm⁻¹: ~30% of atoms in a 1-nm…
Quantum confinement (1D box): E_n = n²π²ℏ²/(2 m a²) ; E_n ∝ 1/a² (size scaling)
Quantum confinement: when the system dimension a approaches the de Broglie wavelength, electronic levels discretise. Infinite 1D square…
Scherrer equation: crystallite size t = Kλ/(β cos θ)
Scherrer equation (Scherrer 1918): for a crystallite of mean size t, an ideal narrow X-ray source at wavelength λ produces a Bragg peak of…
A/V at R=1 ⇒ 3; scale law A/V ∝ 1/R (SymPy-simplified sphere ratio)
Sympy-exact witness of the sphere surface-to-volume ratio. Canonical oracle pins: SA_V_form = '3/R' (after sp.simplify), SA_V_at_R1 = '3'…
E_1(a=1) = π²/2 ; E_1(a=2) = π²/8 ; halving ratio = 4 (1/a² scaling)
Sympy-exact witness of the 1/a² quantum-confinement scaling. Unit natural units (ℏ = m = 1, n = 1): E_1(a=1) = π²/2, E_1(a=2) = π²/8,…
Scherrer at (K, λ, β, θ) = (1, π, π, 0): t = 1 (unit crystallite-size canonical)
Sympy-exact witness of the Scherrer crystallite-size relation at a clean unit-parameter point. Setup: K=1 (dimensionless shape factor),…