Foundations

An equilibrium product constrains dissolution of a specified solid phase.

Electrode potentials are specified relative to a reference half-cell.

Reversible electrochemical potential depends on activities and temperature.

A mechanism proposes steps that account for an overall chemical transformation.

Many rate constants follow an exponential temperature model over a limited range.

Reaction enthalpies can be combined because enthalpy is a state function.

Phase diagrams describe equilibrium states under stated conditions.

Quantitative analysis needs a validated relation between response and analyte amount.

Differential interactions with mobile and stationary phases separate mixture components.

Absorbance can be proportional to path length and absorbing-species concentration.

Characteristic structural groups organize organic compounds and recurring reactivity.

The same molecular formula can describe different connectivity or spatial arrangements.

Addition, elimination, substitution and rearrangement classify net structural changes.

Curved arrows record proposed changes in electron-pair bonding.

For the canonical oscillator, a=√(mω/(2ħ))x+ip/√(2mħω) and its adjoint satisfy [a,a†]=1.

In the long-wavelength electric-dipole approximation, atomic transition strength depends on the dipole matrix element.

For a self-adjoint Hamiltonian bounded below, every normalized trial state with a finite energy expectation satisfies ⟨ψ_trial|H|ψ_trial⟩≥E₀.

For a simple compressible equilibrium system, U(S,V,{N_i}) has differential dU=T dS−p dV+Σ_i μ_i dN_i.

The Legendre transforms H=U+pV and F=U−TS change the natural state variables: dH=T dS+V dp and dF=−S dT−p dV at fixed composition.

Equality of mixed derivatives of smooth equilibrium potentials gives (∂T/∂V)_S=−(∂p/∂S)_V, (∂T/∂p)_S=(∂V/∂S)_p, (∂S/∂V)_T=(∂p/∂T)_V and (∂S/∂p)_T=−(∂V/∂T)_p.

For a fixed amount of simple compressible matter, C_V=(∂U/∂T)_V and C_p=(∂H/∂T)_p.

Thermal equilibrium is transitive, permitting a common temperature scale.

For canonical Z=Σ_i exp(−βE_i), F=−k_BT lnZ, U=−∂_β lnZ and S=k_B(lnZ+βU).

For a system exchanging energy and particles with reservoirs at T and μ, p_i,N=exp[−β(E_i,N−μN)]/Ξ and Ξ=Σ_N,i exp[−β(E_i,N−μN)].

In a classical canonical model, each independent quadratic term in an unrestricted coordinate or momentum contributes k_BT/2 to the mean energy.

Pure phases coexist at common T and p when their molar Gibbs energies are equal.

Macroscopic electric polarization P gives bound volume charge ρ_b=−∇·P and surface charge σ_b=P·n at a material boundary.

A damped bound-electron oscillator gives ε_r(ω)=1+Ne²/[ε₀m(ω₀²−ω²−iγω)] for the exp(−iωt) convention.

A Bravais lattice consists of translations R=Σ_i n_i a_i with integer n_i and independent primitive vectors.

Reciprocal vectors obey a_i·b_j=2πδ_ij and G=Σ_i h_i b_i with integer h_i.