,Useful relations Series expansions
x2 x3
At 298.15 K ex = 1+ x + + +
2! 3!
RT 2.4790 kJ mol−1 RT/F 25.693 mV x2 x3
(RT/F) ln 10 59.160 mV kT/hc ln(1 + x ) = x − + −
207.225 cm−1 2 3
kT 25.693 meV Vm 2.4790 × 10−2 1 1
m3 mol−1 = 1− x + x 2 − = 1+ x + x 2 +
1+ x 1− x
24.790 dm3 mol−1
x3 x5 x2 x4
sin x = x − + − cos x = 1− + −
Selected units* 3! 5! 2! 4!
1N 1 kg m s−2 1J 1 kg m2 s−2
1 Pa 1 kg m−1 s−2 1W 1 J s−1 Derivatives; for Integrals, see the Resource
section
1V 1 J C−1 1A 1 C s−1
1T 1 kg s−2 A−1 1P 10−1 kg m−1 s−1 d(f + g) = df + dg d(fg) = f dg + g df
1S 1 Ω−1 = 1 A V−1 f 1 f df df dg
d = df − 2 dg = for f = f ( g (t ))
g g g dt dg dt
* For multiples (milli, mega, etc), see the Resource section
∂y ∂x ∂ y ∂x ∂z
Conversion factors ∂ x = 1/ ∂ y ∂ x ∂ z ∂ y = −1
z z z y x
θ/°C = T/K − 273.15*
dx n deax d ln(ax ) 1
= nx n−1 = aeax =
1 eV 1.602 177 × 10−19 J 1 cal 4.184* J dx dx dx x
96.485 kJ mol−1
∂ g ∂h
8065.5 cm−1 df = g ( x , y )dx + h( x , y )dy is exact if =
∂ y x ∂x y
1 atm 101.325* kPa 1 cm−1 1.9864 × 10−23 J
760* Torr Greek alphabet*
1D 3.335 64 × 10−30 C m 1Å 10−10 m*
Α, α alpha Ι, ι iota Ρ, ρ rho
* Exact value
Β, β beta Κ, κ kappa Σ, σ sigma
Mathematical relations Γ, γ gamma Λ, λ lambda Τ, τ tau
π = 3.141 592 653 59 … e = 2.718 281 828 46 … Δ, δ delta Μ, μ mu ϒ, υ upsilon
Ε, ε epsilon Ν, ν nu Φ, ϕ phi
Logarithms and exponentials
Ζ, ζ zeta Ξ, ξ xi Χ, χ chi
ln x + ln y + … = ln xy… ln x − ln y = ln(x/y) Η, η eta Ο, ο omicron Ψ, ψ psi
a ln x = ln xa ln x = (ln 10) log x Θ, θ theta Π, π pi Ω, ω omega
= (2.302 585 …) log x * Oblique versions (α, β, …) are used to denote physical
exeyez…. = ex+y+z+… ex/ey = ex−y observables.
(ex)a = eax e±ix = cos x ± i sin x
, 18
PERIODIC TABLE OF THE ELEMENTS VIII
VIIA
Group 1 2 1 H 13 14 15 16 17 2 He
helium
I II
Period 1 hydrogen
III IV V VI VII 4.00
1.0079
1s1 1s2
IA IIA IIIA IVA VA VIA VIIA
3 Li 4 Be 5 B 6 C 7 N 8 O 9 F 10 Ne
lithium beryllium boron carbon nitrogen oxygen fluorine neon
2 6.94 9.01 10.81 12.01 14.01 16.00 19.00 20.18
2s1 2s2 2s22p1 2s22p2 2s22p3 2s22p4 2s22p5 2s22p6
11 Na 12 Mg 13 Al 14 Si 15 P 16 S 17 Cl 18 Ar
sodium magnesium aluminium silicon phosphorus sulf ur chlorine argon
3 22.99 24.31 3 4 5 6 7 8 9 10 11 12 26.98 28.09 30.97 32.06 35.45 39.95
3s1 3s2 3s23p1 3s23p2 3s23p3 3s23p4 3s23p5 3s23p6
IIIB IVB VB VIB VIIB VIIIB IB IIB
19 K 20 Ca 21 Sc 22 Ti 23 V 24 Cr 25 Mn 26 Fe 27 Co 28 Ni 29 Cu 30 Zn 31 Ga 32 Ge 33 As 34 Se 35 Br 36 Kr
potassium calcium scandium titanium vanadium chromium manganese iron cobalt nickel copper zinc gallium germa nium arsenic selenium bromine krypton
4 39.10 40.08 44.96 47.87 50.94 52.00 54.94 55.84 58.93 58.69 63.55 65.41 69.72 72.64 74.92 78.96 79.90 83.80
Period
4s1 4s2 3d14s2 3d24s2 3d34s2 3d54s1 3d54s2 3d64s2 3d74s2 3d84s2 3d104s1 3d104s2 4s24p1 4s24p2 4s24p3 4s24p4 4s24p5 4s24p6
37 Rb 38 Sr 39 Y 40 Zr 41 Nb 42 Mo 43 Tc 44 Ru 45 Rh 46 Pd 47 Ag 48 Cd 49 In 50 Sn 51 Sb 52 Te 53 I 54 Xe
5 rubidium st rontium yttrium zirconium niobium molybdenum technetium ruthenium rhodium palladium silver cadmium indium tin antimony tellurium iodine xenon
85.47 87.62 88.91 91.22 92.91 95.94 (98) 101.07 102.90 106.42 107.87 112.41 114.82 118.71 121.76 127.60 126.90 131.29
5s1 5s2 4d15s2 4d25s2 4d45s1 4d55s1 4d55s2 4d75s1 4d85s1 4d10 4d105s1 4d105s2 5s25p1 5s25p2 5s25p3 5s25p4 5s25p5 5s25p6
55 Cs 56 Ba 57 La 72 Hf 73 Ta 74 W 75 Re 76 Os 77 Ir 78 Pt 79 Au 80 Hg 81 Tl 82 Pb 83 Bi 84 Po 85 At 86 Rn
caesium barium lanthanum hafnium tantalum tungsten rhenium osmium iridium platinum gold mercury thallium lead bismuth polonium astatine radon
6 138.91
132.91 137.33 178.49 180.95 183.84 186.21 190.23 192.22 195.08 196.97 200.59 204.38 207.2 208.98 (209) (210) (222)
6s1 6s2 5d16s2 5d26s2 5d36s2 5d46s2 5d56s2 5d66s2 5d76s2 5d96s1 5d106s1 5d106s2 6s26p1 6s26p2 6s26p3 6s26p4 6s26p5 6s26p6
87 Fr 88 Ra 89 Ac 104 Rf 105Db 106 Sg 107 Bh 108 Hs 109 Mt 110 Ds 112Cn 113 Nh 114 Fl 115 Mc 116 Lv
111 Rg 112 117 Ts 118 Og
francium radium act inium rutherfordium dubnium seaborgium bohrium hassium meitnerium darmstadtium roentgenium copernicium nihonium flerovium moscovium livermorium tennessine oganesson
7 (223) (226) (261) (262) (263) (262) (265) (266)
(227) (271) (272) ? ? ? ? ? ? ?
7s1 7s2 6d17s2 6d27s2 6d37s2 6d47s2 6d57s2 6d67s2 6d77s2 6d87s2 6d97s2 6d107s2 7s27p1 7s27p2 7s27p3 7s27p4 7s27p5 7s27p6
58 Ce 59 Pr 60 Nd 61 Pm 62 Sm 63 Eu 64 Gd 65 Tb 66 Dy 67 Ho 68 Er 69 Tm 70 Yb 71 Lu
cerium praseodymium neodymium promethium samarium europium gadiolinium terbium dysprosium holmium erbium thulium ytterbium Lanthanoids
lutetium
6 140.12 140.91 144.24 (145) 150.36 151.96 157.25 158.93 162.50 164.93 167.26 168.93 173.04 174.97 (lanthanides)
Numerical values of molar
masses in grams per mole (atomic 4f15d16s2 4f36s2 4f46s2 4f56s2 4f66s2 4f76s2 4f75d16s2 4f96s2 4f106s2 4f116s2 4f126s2 4f136s2 4f146s2 5d16s2
weights) are quoted to the number
of significant figures typical of 90 Th 91 Pa 92 U 93 Np 94 Pu 95 Am 96 Cm 97 Bk 98 Cf 99 Es 100Fm 101Md 102 No 103 Lr
most naturally occurring samples. thorium protactinium uranium neptunium plutonium americium curium berkelium californium einsteinium fermium mendelevium nobelium lawrencium Actinoids
7 232.04 231.04 238.03 (237) (244) (243) (247) (247) (251) (252) (257) (258) (259) (262) (actinides)
6d27s2 5f26d17s2 5f36d17s2 5f46d17s2 5f67s2 5f77s2 5f76d17s2 5f97s2 5f107s2 5f117s2 5f127s2 5f137s2 5f147s2 6d17s2
, FUNDAMENTAL CONSTANTS
Constant Symbol Value
Power of 10 Units
Speed of light c 2.997 924 58* 10 8
m s−1
Elementary charge e 1.602 176 634* 10−19 C
Planck’s constant h 6.626 070 15 10 −34
Js
ħ = h/2π 1.054 571 817 10−34 Js
Boltzmann’s constant k 1.380 649* 10 −23
J K−1
Avogadro’s constant NA 6.022 140 76 1023 mol−1
Gas constant R = NAk 8.314 462 J K−1 mol−1
Faraday’s constant F = NAe 9.648 533 21 10 4
C mol−1
Mass
Electron me 9.109 383 70 10−31 kg
Proton mp 1.672 621 924 10−27 kg
Neutron mn 1.674 927 498 10 −27
kg
Atomic mass constant mu 1.660 539 067 10−27 kg
Magnetic constant μ0 1.256 637 062 10 −6
J s2 C−2 m−1
(vacuum permeability)
Electric constant 0
1/
0c 2 8.854 187 813 10−12 J−1 C2 m−1
(vacuum permittivity)
4πε0 1.112 650 056 10−10 J−1 C2 m−1
Bohr magneton μB = eħ/2me 9.274 010 08 10 −24
J T−1
Nuclear magneton μN = eħ/2mp 5.050 783 75 10−27 J T−1
Proton magnetic moment μp 1.410 606 797 10 −26
J T−1
g-Value of electron ge 2.002 319 304
Magnetogyric ratio
Electron γe = gee/2me 1.760 859 630 1011 T−1 s−1
Proton γp = 2μp/ħ 2.675 221 674 108 T−1 s−1
Bohr radius a0 = 4πε0ħ /e me 2 2
5.291 772 109 10 −11
m
Rydberg constant R
mee 4 /8h3c
02 1.097 373 157 10 5
cm−1
hcR ∞ /e 13.605 693 12 eV
Fine-structure constant α = μ0e2c/2h 7.297 352 5693 10−3
α−1 1.370 359 999 08 102
Stefan–Boltzmann constant σ = 2π k /15h c
5 4 3 2
5.670 374 10−8 W m−2 K−4
Standard acceleration of free fall g 9.806 65* m s−2
Gravitational constant G 6.674 30 10 −11
N m2 kg−2
* Exact value. For current values of the constants, see the National Institute of Standards and Technology (NIST) website.
x2 x3
At 298.15 K ex = 1+ x + + +
2! 3!
RT 2.4790 kJ mol−1 RT/F 25.693 mV x2 x3
(RT/F) ln 10 59.160 mV kT/hc ln(1 + x ) = x − + −
207.225 cm−1 2 3
kT 25.693 meV Vm 2.4790 × 10−2 1 1
m3 mol−1 = 1− x + x 2 − = 1+ x + x 2 +
1+ x 1− x
24.790 dm3 mol−1
x3 x5 x2 x4
sin x = x − + − cos x = 1− + −
Selected units* 3! 5! 2! 4!
1N 1 kg m s−2 1J 1 kg m2 s−2
1 Pa 1 kg m−1 s−2 1W 1 J s−1 Derivatives; for Integrals, see the Resource
section
1V 1 J C−1 1A 1 C s−1
1T 1 kg s−2 A−1 1P 10−1 kg m−1 s−1 d(f + g) = df + dg d(fg) = f dg + g df
1S 1 Ω−1 = 1 A V−1 f 1 f df df dg
d = df − 2 dg = for f = f ( g (t ))
g g g dt dg dt
* For multiples (milli, mega, etc), see the Resource section
∂y ∂x ∂ y ∂x ∂z
Conversion factors ∂ x = 1/ ∂ y ∂ x ∂ z ∂ y = −1
z z z y x
θ/°C = T/K − 273.15*
dx n deax d ln(ax ) 1
= nx n−1 = aeax =
1 eV 1.602 177 × 10−19 J 1 cal 4.184* J dx dx dx x
96.485 kJ mol−1
∂ g ∂h
8065.5 cm−1 df = g ( x , y )dx + h( x , y )dy is exact if =
∂ y x ∂x y
1 atm 101.325* kPa 1 cm−1 1.9864 × 10−23 J
760* Torr Greek alphabet*
1D 3.335 64 × 10−30 C m 1Å 10−10 m*
Α, α alpha Ι, ι iota Ρ, ρ rho
* Exact value
Β, β beta Κ, κ kappa Σ, σ sigma
Mathematical relations Γ, γ gamma Λ, λ lambda Τ, τ tau
π = 3.141 592 653 59 … e = 2.718 281 828 46 … Δ, δ delta Μ, μ mu ϒ, υ upsilon
Ε, ε epsilon Ν, ν nu Φ, ϕ phi
Logarithms and exponentials
Ζ, ζ zeta Ξ, ξ xi Χ, χ chi
ln x + ln y + … = ln xy… ln x − ln y = ln(x/y) Η, η eta Ο, ο omicron Ψ, ψ psi
a ln x = ln xa ln x = (ln 10) log x Θ, θ theta Π, π pi Ω, ω omega
= (2.302 585 …) log x * Oblique versions (α, β, …) are used to denote physical
exeyez…. = ex+y+z+… ex/ey = ex−y observables.
(ex)a = eax e±ix = cos x ± i sin x
, 18
PERIODIC TABLE OF THE ELEMENTS VIII
VIIA
Group 1 2 1 H 13 14 15 16 17 2 He
helium
I II
Period 1 hydrogen
III IV V VI VII 4.00
1.0079
1s1 1s2
IA IIA IIIA IVA VA VIA VIIA
3 Li 4 Be 5 B 6 C 7 N 8 O 9 F 10 Ne
lithium beryllium boron carbon nitrogen oxygen fluorine neon
2 6.94 9.01 10.81 12.01 14.01 16.00 19.00 20.18
2s1 2s2 2s22p1 2s22p2 2s22p3 2s22p4 2s22p5 2s22p6
11 Na 12 Mg 13 Al 14 Si 15 P 16 S 17 Cl 18 Ar
sodium magnesium aluminium silicon phosphorus sulf ur chlorine argon
3 22.99 24.31 3 4 5 6 7 8 9 10 11 12 26.98 28.09 30.97 32.06 35.45 39.95
3s1 3s2 3s23p1 3s23p2 3s23p3 3s23p4 3s23p5 3s23p6
IIIB IVB VB VIB VIIB VIIIB IB IIB
19 K 20 Ca 21 Sc 22 Ti 23 V 24 Cr 25 Mn 26 Fe 27 Co 28 Ni 29 Cu 30 Zn 31 Ga 32 Ge 33 As 34 Se 35 Br 36 Kr
potassium calcium scandium titanium vanadium chromium manganese iron cobalt nickel copper zinc gallium germa nium arsenic selenium bromine krypton
4 39.10 40.08 44.96 47.87 50.94 52.00 54.94 55.84 58.93 58.69 63.55 65.41 69.72 72.64 74.92 78.96 79.90 83.80
Period
4s1 4s2 3d14s2 3d24s2 3d34s2 3d54s1 3d54s2 3d64s2 3d74s2 3d84s2 3d104s1 3d104s2 4s24p1 4s24p2 4s24p3 4s24p4 4s24p5 4s24p6
37 Rb 38 Sr 39 Y 40 Zr 41 Nb 42 Mo 43 Tc 44 Ru 45 Rh 46 Pd 47 Ag 48 Cd 49 In 50 Sn 51 Sb 52 Te 53 I 54 Xe
5 rubidium st rontium yttrium zirconium niobium molybdenum technetium ruthenium rhodium palladium silver cadmium indium tin antimony tellurium iodine xenon
85.47 87.62 88.91 91.22 92.91 95.94 (98) 101.07 102.90 106.42 107.87 112.41 114.82 118.71 121.76 127.60 126.90 131.29
5s1 5s2 4d15s2 4d25s2 4d45s1 4d55s1 4d55s2 4d75s1 4d85s1 4d10 4d105s1 4d105s2 5s25p1 5s25p2 5s25p3 5s25p4 5s25p5 5s25p6
55 Cs 56 Ba 57 La 72 Hf 73 Ta 74 W 75 Re 76 Os 77 Ir 78 Pt 79 Au 80 Hg 81 Tl 82 Pb 83 Bi 84 Po 85 At 86 Rn
caesium barium lanthanum hafnium tantalum tungsten rhenium osmium iridium platinum gold mercury thallium lead bismuth polonium astatine radon
6 138.91
132.91 137.33 178.49 180.95 183.84 186.21 190.23 192.22 195.08 196.97 200.59 204.38 207.2 208.98 (209) (210) (222)
6s1 6s2 5d16s2 5d26s2 5d36s2 5d46s2 5d56s2 5d66s2 5d76s2 5d96s1 5d106s1 5d106s2 6s26p1 6s26p2 6s26p3 6s26p4 6s26p5 6s26p6
87 Fr 88 Ra 89 Ac 104 Rf 105Db 106 Sg 107 Bh 108 Hs 109 Mt 110 Ds 112Cn 113 Nh 114 Fl 115 Mc 116 Lv
111 Rg 112 117 Ts 118 Og
francium radium act inium rutherfordium dubnium seaborgium bohrium hassium meitnerium darmstadtium roentgenium copernicium nihonium flerovium moscovium livermorium tennessine oganesson
7 (223) (226) (261) (262) (263) (262) (265) (266)
(227) (271) (272) ? ? ? ? ? ? ?
7s1 7s2 6d17s2 6d27s2 6d37s2 6d47s2 6d57s2 6d67s2 6d77s2 6d87s2 6d97s2 6d107s2 7s27p1 7s27p2 7s27p3 7s27p4 7s27p5 7s27p6
58 Ce 59 Pr 60 Nd 61 Pm 62 Sm 63 Eu 64 Gd 65 Tb 66 Dy 67 Ho 68 Er 69 Tm 70 Yb 71 Lu
cerium praseodymium neodymium promethium samarium europium gadiolinium terbium dysprosium holmium erbium thulium ytterbium Lanthanoids
lutetium
6 140.12 140.91 144.24 (145) 150.36 151.96 157.25 158.93 162.50 164.93 167.26 168.93 173.04 174.97 (lanthanides)
Numerical values of molar
masses in grams per mole (atomic 4f15d16s2 4f36s2 4f46s2 4f56s2 4f66s2 4f76s2 4f75d16s2 4f96s2 4f106s2 4f116s2 4f126s2 4f136s2 4f146s2 5d16s2
weights) are quoted to the number
of significant figures typical of 90 Th 91 Pa 92 U 93 Np 94 Pu 95 Am 96 Cm 97 Bk 98 Cf 99 Es 100Fm 101Md 102 No 103 Lr
most naturally occurring samples. thorium protactinium uranium neptunium plutonium americium curium berkelium californium einsteinium fermium mendelevium nobelium lawrencium Actinoids
7 232.04 231.04 238.03 (237) (244) (243) (247) (247) (251) (252) (257) (258) (259) (262) (actinides)
6d27s2 5f26d17s2 5f36d17s2 5f46d17s2 5f67s2 5f77s2 5f76d17s2 5f97s2 5f107s2 5f117s2 5f127s2 5f137s2 5f147s2 6d17s2
, FUNDAMENTAL CONSTANTS
Constant Symbol Value
Power of 10 Units
Speed of light c 2.997 924 58* 10 8
m s−1
Elementary charge e 1.602 176 634* 10−19 C
Planck’s constant h 6.626 070 15 10 −34
Js
ħ = h/2π 1.054 571 817 10−34 Js
Boltzmann’s constant k 1.380 649* 10 −23
J K−1
Avogadro’s constant NA 6.022 140 76 1023 mol−1
Gas constant R = NAk 8.314 462 J K−1 mol−1
Faraday’s constant F = NAe 9.648 533 21 10 4
C mol−1
Mass
Electron me 9.109 383 70 10−31 kg
Proton mp 1.672 621 924 10−27 kg
Neutron mn 1.674 927 498 10 −27
kg
Atomic mass constant mu 1.660 539 067 10−27 kg
Magnetic constant μ0 1.256 637 062 10 −6
J s2 C−2 m−1
(vacuum permeability)
Electric constant 0
1/
0c 2 8.854 187 813 10−12 J−1 C2 m−1
(vacuum permittivity)
4πε0 1.112 650 056 10−10 J−1 C2 m−1
Bohr magneton μB = eħ/2me 9.274 010 08 10 −24
J T−1
Nuclear magneton μN = eħ/2mp 5.050 783 75 10−27 J T−1
Proton magnetic moment μp 1.410 606 797 10 −26
J T−1
g-Value of electron ge 2.002 319 304
Magnetogyric ratio
Electron γe = gee/2me 1.760 859 630 1011 T−1 s−1
Proton γp = 2μp/ħ 2.675 221 674 108 T−1 s−1
Bohr radius a0 = 4πε0ħ /e me 2 2
5.291 772 109 10 −11
m
Rydberg constant R
mee 4 /8h3c
02 1.097 373 157 10 5
cm−1
hcR ∞ /e 13.605 693 12 eV
Fine-structure constant α = μ0e2c/2h 7.297 352 5693 10−3
α−1 1.370 359 999 08 102
Stefan–Boltzmann constant σ = 2π k /15h c
5 4 3 2
5.670 374 10−8 W m−2 K−4
Standard acceleration of free fall g 9.806 65* m s−2
Gravitational constant G 6.674 30 10 −11
N m2 kg−2
* Exact value. For current values of the constants, see the National Institute of Standards and Technology (NIST) website.
