ABSTRACT
Name Symbol Definition SI unit Space and Time cartesian space coordinates x, y, z m spherical polar coordinates r, θ, φ m, 1, 1 generalized coordinate q, qi (varies) position vector r r = xi + yj + zk m length l m special symbols: height h breadth b thickness d, δ distance d radius r diameter d path length s length of arc s area A, As, S m2 volume V, (υ) m3 plane angle α, β, γ, θ, φ… α = s/r rad, 1 solid angle ω, Ω ω = A/r2 sr, 1 time t s period T T = t/N s frequency v, f v = 1/T Hz circular frequency, angular frequency ω ω = 2piv rad s-1, s-1 characteristic time interval, relaxation time,
time constant τ, T τ = |dt/dlnx| s angular velocity ω ω= dφ/dt rad s-1, s-1 velocity υ, u, w, c, r˙ υ = dr/dt m s-1
speed υ, u, w, c v = |υ| m s-1 acceleration a, (g) a = dυ/dt m s-2
Classical Mechanics mass m kg reduced mass µ µ = m1m2/(m1 + m2) kg density, mass density ρ ρ = m/V kg m-3 relative density d d = ρ/ρ 1 surface density ρA, ρS ρA = m/A kg m-2 specific volume υ υ = V/m = 1/ρ m3 kg-1 momentum p p = mv kg m s-1 angular momentum, action L L = r × p J s moment of inertia I, J I = Σmiri2 kg m2 force F F = dp/dt = ma N torque, moment of a force T, (M) T = r × F N m energy E J potential energy Ep, V, Φ Ep = ∫F ∙ ds J kinetic energy Ek, T, K Ek = 1/2mv2 J work W, w W = ∫F ∙ ds J Hamilton function H H(q, p) = T (q, p) + V(q) J Lagrange function L L(q, q˙) = T (q, q˙) – V (q) J pressure p, P p = F/A Pa, N m-2 surface tension γ, σ y = dW/dA N m-1, J m –2 weight G, (W, P) G = mg N gravitational constant G F = Gm1 m2/r2 N m2 kg –2 normal stress σ σ = F/A Pa shear stress τ τ = F/A Pa linear strain, relative elongation ε, e ε = ∆l/l 1 modulus of elasticity, Young’s modulus E E = σ/ε Pa shear strain γ γ = ∆x/d 1 shear modulus G G = τ/γ Pa volume strain, bulk strain θ θ = ∆V/V0 1 bulk modulus, compression modulus K K = – V0(dp/dV) Pa viscosity, dynamic viscosity η, µ τx,z = η(dvx/dz) Pa s fluidity φ φ = 1/η m kg –1 s kinematic viscosity v v = η/ρ m2 s-1 friction coefficient µ, (f ) Ffrict = µFnorm 1 power P P = dW/dt W sound energy flux P, P a P = dE/dt W acoustic factors reflection factor ρ ρ = Pr/P0 1 acoustic absorption factor αa, (α) αa = 1 – ρ 1 transmission factor τ τ = Ptr/P0 1 dissipation factor δ δ = αa – τ 1
Electricity and Magnetism quantity of electricity, electric charge Q C charge density ρ ρ = Q/V C m-3 surface charge density σ σ = Q/A C m-2 electric potential V, φ V = dW/dQ V, J C-1 electric potential difference U, ∆V, ∆φ U = V2 – V1 V electromotive force E E = ∫(F/Q) ∙ ds V electric field strength E E = F/Q = – grad V V m-1 electric flux Ψ Ψ = ∫D ∙ dA C electric displacement D D = εE C m-2 capacitance C C = Q/U F, C V-1 permittivity ε D = εE F m-1 permittivity of vacuum ε0 ε0 = µ0-1 c0-2 F m-1 relative permittivity εr εr = ε/ε0 1 dielectric polarization (dipole moment per
volume) P P = D – ε0E C m-2
electric susceptibility χe χe = εr – 1 1
electric current I I = dQ/dt A electric current density j, J I = ∫j ∙ dA A m-2 magnetic flux density, magnetic induction B F = Qv × B T magnetic flux Φ Φ = ∫B ∙ dA A m-2 magnetic field strength H B = µH A m-2 permeability µ B = µH N A-2, H m-1 permeability of vacuum µ0 H m-1 relative permeability µr µr = µ/µ0 1 magnetization (magnetic dipole moment
per volume) M M = B/µ0 – H A m-1 magnetic susceptibility χ, κ, (χm) χ = µr – 1 1 molar magnetic susceptibility χm χm = Vmχ m3 mol-1 magnetic dipole moment m, µ Ep = – m ∙ B A m2, J T-1 electrical resistance R R = U/I Ω conductance G G = 1/R S loss angle δ δ = (pi/2) + φI –φU 1, rad reactance X X = (U/I)sin δ Ω impedance (complex impedance) Z Z = R + iX Ω admittance (complex admittance) Y Y = 1/Z S susceptance B Y = G + iB S resistivity ρ ρ = E/j Ω m conductivity κ, γ, σ κ = 1/ρ S m-1 self-inductance L E = – L(dI/dt) H mutual inductance M, L1 2 E1 = L1 2(dI2 /dt) H magnetic vector potential A B = ∇ × A Wb m-1 Poynting vector S S = E × H W m-2
Quantum Mechanics momentum operator p˙ p˙ = – ih∇ m-1 J s kinetic energy operator Tˆ Tˆ = –(h2/2m)∇2 J
Hamiltonian operator Ĥ Ĥ = Tˆ + V J wavefunction, state function Ψ, ψ, φ Ĥψ = Eψ (m-3/2) probability density P P = ψ*ψ (m-3) charge density of electrons ρ ρ = – eP (C m-3) probability current density S S = – iћ(ψ*∇ψ – ψ∇ψ*)/2me (m-2 s-1) electric current density of electrons j j = – eS (A m-2) matrix element of operator  Aij, 〈i|Â|j〉 Aij = ∫ψi*Âψjdτ (varies) expectation value of operator  〈A〉, Ā 〈A〉 = ∫ψ*ÂΨdτ (varies) hermitian conjugate of  † (†)ij = (Aji)* (varies)
commutator of  and Bˆ [Â, Bˆ ], [Â, Bˆ ]– [ Bˆ ] =  Bˆ – Bˆ  (varies) anticommutator [Â, Bˆ ]+ [Â, Bˆ ]+ =  Bˆ + Bˆ  (varies) spin wavefunction α; β 1 coulomb integral HAA HAA = ∫ψA*ĤψAdτ J resonance integral HAB HAB = ∫ψA*ĤψBdτ J overlap integral SAB SAB = ∫ψA*ψBdτ 1
Atoms and Molecules nucleon number, mass number A 1 proton number, atomic number Z 1 neutron number N N = A – Z 1 electron rest mass me kg mass of atom, atomic mass ma, m kg atomic mass constant mu mu = ma(12C)/12 kg mass excess ∆ ∆ = ma – Amu kg elementary charge, proton charge e C Planck constant h J s Planck constant/2π ћ ћ = h/2pi J s Bohr radius a0 a0 = 4piε0ћ2/mee2 m Hartree energy Eh Eћ = ћ2/mea02 J Rydberg constant R∞ R∞ = Eh/2hc m-1 2
ionization energy Ei J electron affinity Eea J dissociation energy Ed, D J from the ground state D0 J from the potential minimum De J principal quantum number (H atom) n E = – hcR/n2 1 angular momentum quantum numbers see under Spectroscopy magnetic dipole moment of a molecule m,µ Ep = – m ∙ B J T-1 magnetizability of a molecule ξ m = ξB J T-2 Bohr magneton µB µB = eћ/2me J T-1 nuclear magneton µN µN = (me/mp)µB J T-1 magnetogyric ratio (gyromagnetic ratio) γ γ = µ/L C kg-1 g factor g 1 Larmor circular frequency ωL ωL = (e/2m)B s –1 Larmor frequency vL vL = ωL/2pi Hz longitudinal relaxation time T1 s transverse relaxation time T2 s electric dipole moment of a molecule p, µ Ep = – p ∙ E C m quadrupole moment of a molecule Q; Θ Ep = 1/2Q: V˝ = 1/3Θ: V˝ C m2 quadrupole moment of a nucleus eQ eQ = 2 ∙ 〈ΘZZ〉 C m2 electric field gradient tensor q qαβ = – ∂2V/∂α∂β V m-2 quadrupole interaction energy tensor χ χαβ = eQqαβ J electric polarizability of a molecule α p (induced) = αE C m2 V-1 activity (of a radioactive substance) A A = – dNB/dt Bq decay (rate) constant, disintegration (rate)
constant λ A = γNB s –1 half life t1/2, T1/2 s mean life τ s level width Γ Γ = ħ/τ J disintegration energy Q J cross section (of a nuclear reaction) σ m2
Spectroscopy total term T T = Etot/hc m-1 transition wavenumber ~v, (v) ~v = T´ – T˝ m-1 transition frequency v v = (E´ – E˝)/h Hz electronic term Te Te = Ee/hc m-1 vibrational term G G = Evib/hc m-1 rotational term F F = Erot/hc m-1
spin orbit coupling constant A Ts.o. = A〈 Lˆ ∙ Ŝ〉 m-1 principal moments of inertia IA; IB; IC IA ≤ IB≤ IC kg m2 rotational constants, in wavenumber A B C; ; Ã = h/8pi2cIA m-1 in frequency A; B; C A = h/8pi2IA Hz inertial defect ∆ ∆ = IC – IA – IB kg m2 asymmetry parameter κ
κ = − −
−
( ) ( )
2B A C A C
centrifugal distortion constants, S reduction DJ ; DJK ; DK; d1 ; d2 m-1 A reduction ∆J ; ∆JK ; ∆K; δJ ; δK m-1 harmonic vibration wavenumber ωe ; ωr m-1 vibrational anharmonicity constant ωexe ; xrs ; gu’ m-1 vibrational quantum numbers υr ; lt 1 Coriolis zeta constant ζrsα 1 angular momentum quantum numbers see additional information below degeneracy, statistical weight g, d, β 1 electric dipole moment of a molecule p, µ Ep = – p ∙ E C m transition dipole moment of a molecule M, R M = ∫ψ´pψ˝dτ C m molecular geometry, interatomic distances, equilibrium distance re m zero-point average distance r m
ground state distance r0 m substitution structure distance rs m vibrational coordinates, internal coordinates Ri , ri , θj , etc. (varies) symmetry coordinates Si (varies) normal coordinates mass adjusted Qr kg1/2 m dimensionless qr 1 vibrational force constants, diatomic f, (k) f = ∂2V/∂r2 J m-2 polyatomic, internal coordinates fij fij = ∂2V/∂ri∂rj (varies) symmetry coordinates Fij Fij = ∂2V/∂Si∂Sj (varies) dimensionless normal coordinates φrst…, krst… m-1 nuclear magnetic resonance (NMR), magnetogyric ratio γ γ = µ/Iħ C kg-1 shielding constant σA BA = (1 – σA)B 1 chemical shift, δ scale δ δ = 106(v – v0)/v0 1 (indirect) spin-spin coupling constant JAB Ĥ/h = JAB ÎA ∙ ÎB Hz direct (dipolar) coupling constant DAB Hz longitudinal relaxation time T1 s transverse relaxation time T2 s electron spin resonance, electron
paramagnetic resonance (ESR, EPR), magnetogyric ratio γ γ = µ/sħ C kg-1 g factor g hν = gµBB 1 hyperfine coupling constant, in liquids a, A Ĥhfs /h = aŜ ∙ Î Hz in solids T Ĥhfs /h = Ŝ ∙ T ∙ Î Hz
Angular momentum Operator symbol Quantum number symbol
Total Z-axis z-axis electron orbital Lˆ L ML Л one electron only lˆ l ml λ electron spin Ŝ S MS Σ one electron only ŝ s ms σ
electron orbital + spin Lˆ + Ŝ Ω=Л+∑ nuclear orbital (rotational) Rˆ R KR, kR nuclear spin Iˆ I MI internal vibrational
spherical top lˆ l(lζ) Kl other ĵ, pˆi l(lζ)
sum of R + L(+ j) Nˆ N K, k sum of N + S Ĵ J MJ K, k
sum of J + I Fˆ F MF Electromagnetic Radiation
Name Symbol Definition SI unit wavelength λ m speed of light in vacuum c0 m s-1 in a medium c c = c0/n m s-1 wavenumber in vacuum v~ v~ = v/c0 = 1/nλ m-1 wavenumber (in a medium) σ σ = 1/λ m-1 frequency v v = c/λ Hz circular frequency, pulsatance ω ω = 2piv s-1, rad s –1 refractive index n n = c0/c 1
Planck constant/2π ħ ħ = h/2pi J s radiant energy Q, W J radiant energy density ρ, w ρ = Q/V J m-3 spectral radiant energy density in terms of frequency ρv , wv ρ = dρ/dv J m-3 Hz-1
in terms of wavenumber ρ v vw, ρ ρ νν =d d/ J m-2
in terms of wavelength ρλ , wλ ρλ = dρ/dλ J m-4 Einstein transition probabilities spontaneous emission Anm dNn/dt = – AnmNn s-1
stimulated emission Bnm dNn/dt = – ρ ( )nm v v × BnmNn s kg-1 stimulated absorption Bmn dNn/dt = – ρ ( )nm v v BmnNm s kg-1 radiant power, radiant energy per time Φ, P Φ = dQ/dt W radiant intensity I I = dΦ/dΩ W sr-1 radiant exitance (emitted radiant flux) M M = dΦ/dAsource W m-2 irradiance, (radiant flux received) E, (I) E = dΦ/dA W m-2 emittance ε ε = M/Mbb 1 Stefan-Boltzmann constant σ Mbb = σT4 W m-2 K-4 first radiation constant c1 c1 = 2pihc02 W m2 second radiation constant c2 c2 = hc0/k K m transmittance, transmission factor τ, T τ = Φtr /Φ0 1 absorptance, absorption factor α α = Φabs /Φ0 1 reflectance, reflection factor ρ ρ = Φrefl /Φ0 1 (decadic) absorbance A A = –lg(1 – αi) 1 napierian absorbance B B = –ln(1 – αi) 1 absorption coefficient (linear) decadic a, K a = A/l m-1 (linear) napierian α α = B/l m-1 molar (decadic) ε ε = a/c = A/cl m2 mol-1 molar napierian κ κ = α/c = B/cl m2 mol-1
absorption index k k = α/4pi v 1 complex refractive index nˆ nˆ = n + ik 1
molar refraction R, Rm R n n
Vm= −
+
( ) ( )
2 1 2
m3 mol-1
angle of optical rotation α 1, rad
Solid State lattice vector R, R0 m fundamental translation vectors for the
crystal lattice a1 ; a2 ; a3 , a; b; c R = n1a1 + n2 a2 + n3 a3 m
(circular) reciprocal lattice vector G G ∙ R = 2pim m-1 (circular) fundamental translation vectors
for the reciprocal lattice b1 ; b2 ; b3 , a*; b*; c* ai ∙ bk = 2piδik m-1
lattice plane spacing d m Bragg angle θ nλ = 2d sin θ 1, rad order of reflection n 1 order parameters short range σ 1 long range s 1 Burgers vector b m particle position vector r, Rj m equilibrium position vector of an ion R0 m displacement vector of an ion u u = R – R0 m Debye-Waller factor B, D 1 Debye circular wavenumber qD m-1 Debye circular frequency ωD s-1 Grüneisen parameter γ, Γ γ = αV/κCv 1
Madelung constant α, M E N z z e
= +α
piε
_ 2
0 04 1
density of states NE NE = dN(E)/dE J-1 m-3 (spectral) density of vibrational modes Nω, g Nω = dN(ω)/dω s m-3
resistivity tensor ρik E = ρ ∙ j Ω m conductivity tensor σik σ = ρ-1 S m-1 thermal conductivity tensor λik Jq = – λ ∙ grad T W m-1 K-1 residual resistivity ρR Ω m relaxation time τ τ = l/υF s Lorenz coefficient L L = λ/σT V2 K-2 Hall coefficient AH , RH E = ρ ∙ j + RH(B × j) m3 C-1 thermoelectric force E V Peltier coefficient Π V Thomson coefficient µ, (τ) V K-1 work function Φ Φ = E∞ – EF J number density, number concentration n, (p) m-3 gap energy Eg J donor ionization energy Ed J acceptor ionization energy Ea J Fermi energy EF , εF J circular wave vector, propagation vector k, q k = 2pi/λ m-1 Bloch function uk(r) ψ(r) = uk(r) exp(ik · r) m-3/2 charge density of electrons ρ ρ(r) = – eψ*(r)ψ(r) C m-3 effective mass m* kg mobility µ µ = υdrift/E m2 V-1 s-1 mobility ratio b b = µn/µp 1 diffusion coefficient D dN/dt = – DA(dn/dx) m2 s-1
diffusion length L L D= τ m characteristic (Weiss) temperature θ, θw K Curie temperature TC K Néel temperature TN K
Statistical Thermodynamics number of entities N 1 number density of entities, number
concentration n, C n = N/V m-3 Avogadro constant L, NA mol-1 Boltzmann constant k, kB J K-1 gas constant (molar) R R = Lk J K-1 mol-1 molecular position vector r (x, y, z) m molecular velocity vector c(cx , cy , cz), u(ux , uy , uz ) c = dr/dt m s-1 molecular momentum vector p(px , py , pz ) p = mc kg m s-1 velocity distribution function (Maxwell) f(cx ) f(cx ) = (m/2pikT)1/2 × exp
(– mcx2/2kT) m-1 s
speed distribution function (MaxwellBoltzmann) F(c)
F(c) = (m/2pikT)3/2 × 4pic2exp (– mc2/2kT) m-1 s
average speed c , u , 〈c〉, 〈u〉 c = ∫cF(c)dc m s-1 generalized coordinate q (m) generalized momentum p p = ∂L/∂q (kg m s-1) volume in phase space Ω Ω = (1/h)∫pdq 1 probability P 1 statistical weight, degeneracy g, d, W, ω, β 1 density of states ρ(E) ρ(E) = dN/dE J-1 partition function, sum over states, for a single molecule q, z q g kTi
i = exp( – / )i ε∑ 1
for a canonical ensemble (system, or assembly) Q, Z 1
microcanonical ensemble Ω 1 grand (canonical ensemble) Ξ 1 symmetry number σ, s 1 reciprocal temperature parameter β β = 1/kT J-1 characteristic temperature Θ K
General Chemistry number of entities (e.g. molecules, atoms,
ions, formula units) N 1
amount (of substance) n nB = NB /L mol Avogadro constant L, NA mol-1 mass of atom, atomic mass ma, m kg mass of entity (molecule, or formula unit) mf , m kg atomic mass constant mu mu = ma(12C)/12 kg molar mass M MB = m/nB kg mol-1 relative molecular mass (relative molar
mass, molecular weight) Mr Mr,B = mB /mu 1
molar volume Vm Vm ,B = V/nB m3 mol-1 mass fraction w wB = mB /Σmi 1 volume fraction φ φB = VB /ΣVi 1 mole fraction, amount fraction, number
fraction x, y xB = nB /Σni 1 (total) pressure p, P Pa partial pressure pB pB = yB p Pa mass concentration (mass density) γ, ρ γB = mB /V kg m-3 number concentration, number density of
entities C, n CB = NB /V m-3 amount concentration, concentration c cB = nB /V mol m-3 solubility s sB = cB (saturated solution) mol m-3 molality (of a solute) m, (b) mB = nB /mA mol kg-1 surface concentration Γ ΓB = nB/A mol m-2 stoichiometric number v 1 extent of reaction, advancement ξ ∆ξ = ∆nB /vB mol degree of dissociation α 1
Chemical Thermodynamics heat q, Q J work w, W J internal energy U ∆U = q+ w J enthalpy H H = U + pV J thermodynamic temperature T K Celsius temperature θ, t θ/ºC = T/K – 273.15 ºC entropy S dS ≥ dq/T J K-1 Helmholtz energy (Helmholtz function) A A = U – TS J Gibbs energy (Gibbs function) G G = H – TS J Massieu function J J = – A/T J K-1 Planck function Y Y = – G/T J K-1 surface tension γ, σ γ = (∂G/∂As )T ,p J m-2, N m-1 molar quantity X Xm Xm = X/n (varies) specific quantity X x x = X/m (varies) pressure coefficient β β = (∂p/∂T)v Pa K-1 relative pressure coefficient αp αp = (1/p)(∂p/∂T)V K-1 compressibility, isothermal κT κT = – (1/V)(∂V/∂p)T Pa-1 isentropic κS κS = – (1/V)(∂V/∂p)S Pa-1 linear expansion coefficient αl αl = (1/l)(∂l/∂T) K-1 cubic expansion coefficient α, αV , γ α = (1/V)(∂V/∂T)p K-1 heat capacity, at constant pressure Cp Cp = (∂H/∂T)p J K-1 at constant volume CV CV = (∂U/∂T)V J K-1 ratio of heat capacities γ, (κ) γ = Cp/CV 1 Joule-Thomson coefficient µ, µJT µ = (∂T/∂p)H K Pa-1 second virial coefficient B pVm = RT(1 + B/Vm + …) m3 mol-1 compression factor (compressibility factor) Z Z = pVm /RT 1 partial molar quantity X XB, (X B´ ) XB = (∂X/∂nB )T, p, nj ≠ B (varies) chemical potential (partial molar Gibbs
energy) µ µB = (∂G/∂nB ) T, p, nj ≠ B J mol
absolute activity λ λ = exp (µ/RT) 1
standard chemical potential µ φ , µo J mol-1
standard reaction Gibbs energy (function) ∆rG
φ ∆ = ∑rG* ν µB B
affinity of reaction A, (A) A G p T= − ∂ ∂ = −∑( / ) ,ξ ν µB B B
J mol-1
standard reaction enthalpy ∆rH
φ ∆ = ∑rH HνB B
J mol-1
standard reaction entropy ∆rS
φ ∆ = ∑rS SνB B
J mol-1 K-1
equilibrium constant K φ , K K φ = exp( – ∆rG
φ /RT) 1 equilibrium constant,
pressure basis Kp K pp B=∏B Bν PaΣv
concentration basis Kc K cc =∏B B Bν (mol m-3)Σv
molality basis Km K mm =∏B B Bν (mol kg-1)Σv
fugacity f, p~ f p p TB B B B= →λ λ lim
( / )0 Pa
fugacity coefficient φ φB = fB /pB 1
activity and activity coefficient referenced to Raoult’s law, (relative) activity
a a
=
−
exp
*µ µ 1
activity coefficient f fB = aB/xB 1 activities and activity coefficients referenced
to Henry’s law, (relative) activity, molality basis am µ µa
−
φ 1
concentration basis ac a RTc , exp *B B B=
−
µ µ φ 1
mole fraction basis ax a RTx , exp *B B B=
−
µ µ φ
activity coefficient, molality basis γm am, B = γm, BmB/m
φ 1 concentration basis γc ac, B = γc, BcB/c
φ 1 mole fraction basis γx ax, B = γx, BxB 1 ionic strength, molality basis Im , I Im = ½ ΣmBzB2 mol kg-1 concentration basis Ic , I Ic = ½ ΣcBzB2 mol m-3 osmotic coefficient, molality basis φm φm = (µA* – µA )/(RTMAΣmB) 1 mole fraction basis φx φx = (µA – µA *)/(RT1nxA) 1
osmotic pressure Π Π = cBRT (ideal dilute
solution) Pa
(i) Symbols used as subscripts to denote a chemical process or reaction These symbols should be printed in roman (upright) type, without a full stop (period).
