School of Chemistry
Data and Formulae Booklet
Not To Be Removed From The Examination Room
 School of Chemistry University of Leeds 2010.
Index
Binomial series & Expansion ………...
29
Character Tables ……………………...
22
Angular Parts of Atomic Orbitals .. 12
Mulliken Labels …………………
Point Groups and their Symmetry
Operations ……………………….
Road Map for Systematically
Determining a Point Group ……...
Chemical Kinetics ……………………
22
Atomic Term Symbols …………. 13
Commutators and the Uncertainty
Principle ………………………… 9
Chromatography and Electrophoresis ..
23
Quantum Mechanics ………………….
9
Free Translational Motion ………
10
Harmonic Oscillator …………….
11
21
Hydrogen Atom …………………
12
Classical Mechanics ………………….
Conversion Table: Energy Units and
Related Quantities ……………………
Covalent Bond Lengths (table) ………
8
Operators ………………………..
9
Particle in a Box ………………...
10
11
Electrochemistry ……………………..
20
Enthalpy and Entropy of Fusion
and Vaporisation at Phase Transitions
(Table) ………………………………..
Rigid Rotor ……………………...
Schrödinger Equation and
Wavefunctions …………………..
Spherical Harmonics ……………
56
Gas Kinetic Theory …………………..
8
Redox Potentials at 298 K, vs.
Hydrogen Potential …………………...
Greek Alphabet ………………………
4
SI Derived Units with Special Names
and Symbols …………………………. 1-2
Intermolecular Forces …………………
15
Common non-SI units …………...
2
MAPLE ………………………………
32
Solution Equilibria …………………...
19
Mass Spectrometry …………………...
21
Spectrophotometry and Fluorimetry …
13
Mathematical Formulae ……………...
27
Statistics ……………………………...
25
Propagation of Errors …………...
26
24
14
4
5
9
12
Tunnelling Through a Barrier …... 10
7
Mathematical Symbols ………………. 31
Molecular Constants for selected
Diatomic Molecules (Table) …………
5
NMR …………………………………. 13
Periodic Table & Atomic Properties of
the Elements …………………………. 16-17
Phase Equilibria ……………………... 20
Thermodynamic Quantities (table) …..
Thermodynamics and Statistical
Mechanics ……………………………
Waves ………………………………...
Physical Constants …………………...
3
X-ray Crystallography ……………….. 15
Prefixes ……………………………….
4
Student’s t distribution table ……. 26
6
18
8
Names and Symbols for SI base units
Physical quantity
Length
symbol
l (lower case L)
base SI unit
metre
m
kg
Mass
m
kilogram
Time
t
second
s
Temperature
T
kelvin
K
Amount of
substance
Electric current
n
mole
mol
I
ampere
A
Luminous
intensity
Iv
candela
cd
SI derived Units for other Quantities
Physical quantity
Common symbol
Expression in base units
Volume
V
m3
Molar volume
Vm
m3 mol–1
Speed, velocity
s, v
m s–1
Angular velocity

s–1, rad s–1
Wavenumber
v
m–1
Acceleration
a
m s–2
Momentum
p
kg m s–1
(mass × velocity)
Energy
E
2 –2
kg m s
(mass × velocity2)
Force
F
kg m s–2
Density

kg m
Pressure
P
N m–2 = kg m–1 s–2
Surface tension

N m = kg s
Viscosity

Pa s = kg m–1s–1 (pressure × time)
Heat capacity
Entropy
C P, C V
S
(used as cm–1)
–1
(mass/volume)
–2
(force/area)
(force/ length)
J K–1 = kg m2 s–2 K–1
J K–1
C P, C V
J K–1 mol–1
Molar entropy
S
J K–1 mol–1
Molar energy
A, G, H
Molar heat capacity
(mass × acceleration)
–3
J mol–1 = kg m2 s–2 mol–1
1
SI Derived Units with Special Names and Symbols
Name of
SI unit
Physical quantity
Symbol for
SI unit
Expression in terms of SI
base units
hertz
Hz
s–1
force
newton
N
m kg s–2
pressure, stress
pascal
Pa
N m–2 = m–1 kg s–2
energy, work, heat
joule
J
Nm
= m2 kg s–2
power, radiant flux
watt
W
J s–1
= m2 kg s–3
coulomb
C
As
electric potential,
electromotive force
volt
V
J C–1
electric resistance
ohm

V A–1 = m2 kg s–3 A–2
electric conductance
siemen
S
–1
electric capacitance
farad
F
C V–1 = m–2 kg–1 s4 A2
magnetic flux density
tesla
T
V s m–2 = kg s–2 A–1
magnetic flux
weber
Wb
inductance
henry
H
frequency
electric charge
Vs
= m2 kg s–3 A–1
= m–2 kg–1 s3 A2
= m2 kg s–2 A–1
V A–1 s = m2 kg s–2 A–2
Common non–SI units
Angstrom (Å)
litre (L, l)
= 10
–10
= 1 dm
m
3
3
1 cm3 (cc)
= 1 mL
tonne
= 10 kg
Atmosphere (atm)
= 101.325 kPa
bar
= 105 Pa
1 atm
= 760 torr
Electron Volt (eV)
= 1.60218 ×10–19 J
Centipoise (cP)
= 10–3 Pa s
mm Hg
calorie
= 1 torr
= 4.184 J
The following units are deprecated
dyne (dyn)
= 10–5 N
erg
= 10–7 J
Gauss
= 104 T (tesla)
2
Selected Physical Constants
Speed of light in vacuum
c
2.99792458  108 m s–1 (defined value)
Permeability of vacuum
0
(410–7) =1.256637  10–6 H m–1
Permittivity of vacuum 1/(0c2)
0
8.854188  10–12 F m–1
Faraday constant
F
9.64853  104 C mol–1
Avogadro constant
NA
6.02214  1023 mol–1
Unified atomic mass unit
u
1.66054  10–27 kg
Boltzmann constant
kB
1.38065  10–23 J K–1
Gas constant
R
8.31447 J mol–1 K–1
Elementary charge
e
1.60218  10–19 C
Mass of electron
me
9.10938  10–31 kg
Mass of proton
mp
1.67262  10–27 kg
Mass of neutron
mn
1.67493  10–27 kg
Mass of hydrogen atom
mH
1.67343  10–27 kg
Proton–electron mass ratio
mp/me 1836.15
Fine structure constant e2/(20c)

7.2973510–3 (≈1/137)
Rydberg constant 2me2/(2h)
R
1.09737  107 m–1
R
13.6057 eV
RH
1.09678  107 m–1
h
6.62607  10–34 J s

1.054572  10–34 J s
a0
5.29177  10–11 m
Hartree energy 2R∞hc
Eh
27.2114 eV
Proton Magnetogyric ratio
p
26.7522107 rad T–1 s–1
for hydrogen
Planck constant
Bohr radius
/(4R∞)
3
Conversion Table: Energy Units and Related Quantities
J
J
kJ mol–1
eV
Hz
cm–1
1
6.0221020
6.2411018
1.5091033
5.0341022
1
1.03610–2
2.5061012
83.59
96.48
1
2.4181014
8.065103
1
3.33610–11
2.9981010
1
kJ mol–1 1.66110–21
eV
1.60210–19
Hz
6.62610–34 3.99010–13 4.13610–15
cm–1
1.98610–23
1.19610–2
1.24010–4
To convert 6 eV into cm–1, read along from eV in the left column and multiply by the number under
cm–1 in the top row, e.g.
6 eV= 6  8.065 103 cm–1/1 eV= 4.839104 cm–1
To convert kBT into cm–1 at T = 300 K
1.38110–23 J/K 300 K = (1.38110–23  300) J  5.0341022 cm–1/1J = 208.5 cm–1
Greek Alphabet

Normal text
a
b
g
d
e
z
h
q
i
k
l
m













alpha
beta
gamma
delta
epsilon
zeta
eta
theta
iota
kappa
lambda
mu

Normal text












n
x












nu
xi
omicron
pi
rho
sigma
tau
upsilon
phi
chi
psi
omega
o
p
r
s
t
u
f
c
y
w













Prefixes
z
a
f
p
n

m
c
d
k
M
G
T
P
E
Z
zepto
atto
femto
pico
nano
micro
milli
centi
deci
kilo
mega
giga
tera
peta
exa
zeta
–6
–3
–2
–1
3
6
9
12
15
18
1021
10
–21
10
–18
10
–15
10
–12
10
–9
10
10
10
10
10
10
10
10
10
10
4
Molecular Constants for selected Diatomic Molecules
v /cm–1
re /pm
Be /cm–1
H2
4401
74.14
60.853
432
D2
3115
74.15
30.444
439
HCl
2991
127.5
10.5934
432
OH
2720.9
96.99
10.01
423
HBr
2649
141.4
8.4649
366
N2
2358.6
109.8
1.99824
945
HI
2309
160.92
6.4264
298
CO
2170
112.8
1.9313
1080
NO
1904.03
115.08
1.7046
510
O2
1580
120.8
1.44563
498
PbH
1564.1
183.9
4.971
153
S2
725.68
188.9
0.2956
424
Cl2
559.7
198.8
0.244
243
I2
214.5
266.6
0.03737
151
Na2
159.2
307
0.1547
70.4
35
D0 /kJ mol–1
Data from Engel & Reid, Physical Chemistry & Herzberg, Spectra of Diatomic Molecules.
Covalent Bond Lengths
bond
r /pm
bond
r /pm
H–H
74
N–N
146
C–C
154
N=N
120
C=C
134
N≡N
110
C≡C
120
C–Cl
177
C=C (aromatic) 139
C–N
147
C–O
143
C=N
127
C=O
122
C≡N
116
C–H
110
C–S
182
5
Tables of Thermodynamic Quantities
o
Standard Enthalpies of Formation f H at 1 bar and 298 K unless otherwise stated, Enthalpies
o
of Combustion c H and Heat Capacities CP (at 298 K)
Name
f H o /
formula
kJ mole
–1
c H o /
kJ mole
–1
CP /
J K–1 mol–1
Ammonia (g)
NH3
–46.11
-
35.62
Benzene (liq)
C6H6
49.0
–3268
82.34
Ethane (g)
C2H6
–84.7
–1560
52.38
Ethanol (liq)
C2H5OH
–277.6
–1368
112.3
Fluorine (liq)
F2
–13.1 at 85.02 K
-
31.30 (gas)
C6H12O6
–1274
–2808
219.2
Hydrazine (g)
N2H4
50.6
-
49.6
Hydrogen (liq)
H2
–9.02 at 20.27 K
–286
28.84 (gas)
H2O2 (liq)
–187.9
-
43.1
Methane (g)
CH4
–74.81
–890
35.7
Methanol (l)
CH3OH
–238.7
–721
81.1
Nitrogen dioxide (g)
NO2
33.18
-
37.18
Nitrogen tetroxide (g)
N2O4
9.16
-
79.2
Oxygen (liq)
O2
12.99 at 90.18 K
-
29.38 (gas)
Octane (liq)
C8H8
–249.9
–5471
254.7
O3
142.7
-
39.2
NaCl
–411.2
-
50.5
Sucrose (s)
C12H22O11
-2222
–5645
427.6
Water (liq)
H2O
–285.8
-
36.2
Glucose (s)
Hydrogen peroxide
Ozone (g)
Salt (s)
Standard Enthalpy and Entropy of Fusion and Vaporisation at
Phase Transitions
Tf /K
 fus H o /
 fus S o /
kJ mole–1
J K–1mole–1
Tb /K
 vap H o /
 vap S o /
kJ mole–1
J K–1mole–1
Ar
83.81
1.188
14.17
87.29
6.506
74.53
C6H6
278.61
10.59
38.00
353.2
30.8
87.19
H2O
273.15
6.008
22.00
373.15
40.656
109.0
He
3.5
0.021
4.8
4.22
0.084
19.9
(at 8 K & 30 bar)
Data from Atkins & De Paula, Physical Chemistry & Engel & Reid, Physical Chemistry
6
Standard Redox Potentials at 298 K, vs. Hydrogen Potential
Couple or reduction half reaction
F2 +2e  2F
O3 +2H+ +2e  O2 +H2O
+2.076
Cl2 (g)+2e  2Cl
+1.3595
O2 +4H+ +4e  2H2O
+1.229
Cl2 (aq)+2e  2Cl
+0.954
Hg 2  2e  Hg
+0.851
Ag + +e  Ag
Q  2H  2e  QH2
+0.7996
+0.699
Cu   e  Cu
O2 +2H2O+4e  4OH
3
Strong oxidant
†
+0.521
+0.401
Fe  CN 6  e  Fe  CN 6
4
+0.358
Cu 2+ +2e  Cu
Hg2Cl2 +2e  2Hg(l )+2Cl
+0.342
+0.2412
in sat. KCl (SCE * )
Ag + +e  Ag (as Ag/AgCl electrode)
+0.225
in 1 mol/kg KCl
0
defined as zero
2H+ +2e  H2
NAD  H  2e  NADH
Pb2  2e  Pb
CO2  2H  2e  HCO2  H
#
–0.105
–0.126
–0.20
Cd2+ +2e  Cd
Zn 2+ +2e  Zn
PO34  2H2O  2e  HPO32  3OH
–0.403
–0.763
–1.05
V2+ +2e  V
Al3+ +3e  Al
Mg 2+ +2e  Mg
–1.175
–1.662
–2.372
Na + +e  Na
Li+ +e  Li
–2.714
–3.040
† Q = 1, 4-benzoquinone, QH2 dihydroquinone
#
E o (V)
+2.866
(–0.320, pH 7)
(–0.42, pH 7)
Strong reductant
* Saturated Calomel Electrode
NAD+ is nicotinamide adenine dinucleotide
7
Gas Kinetic Theory
Molar volume and concentration
Vm  V / n , c  n / V
Compression factor
Z
Ideal gas equation of state
PV  nRT
Distribution of molecular speeds
4  M 
f s 


  2 RT 
Root mean square speed
 3RT 
c

 M 
Mean speed
 8 RT 
c 

 M 
PV PVm

nRT RT
3/2
crel
Collision frequency
z
Mean free path

 3k T 
 B 
 m 
1/2
 8k T 
 B 
 m 
 8 RT 


  
2 cPN A
,
RT
c

z
 Ms 2 /  2 RT 
1/2
1/2
1/2
1/2
Mean relative speed
s2 e
, 
m1m2
m1  m2
  d2
(for identical molecules)
RT
k T
V
 B 
2 PN A
2 P
2 N A
Collision rate of gases on surfaces (per m2) Z 
P
 2 mk BT 1/2
Classical Mechanics
Velocity
v  dx / dt
Acceleration
a  dv / dt  d 2 x / dt 2
Momentum
p  mv
Force
F  ma  dp / dt   dVdx
Kinetic energy
K
p2 1 2
 mv
2m 2
Total Energy E  K  V
Waves
Electromagnetic waves v  c / 
Frequency
 s–1
Angular frequency
 rad s–1 and   2
Other waves v  v /  .
Frequency in wavenumbers   1/  cm–1 or  rad cm–1
Period 1/  sec or 2 /  sec
Wavevector k  2 / 
8
Quantum Mechanics
Photon Energy
E  h
de Broglie relation

Bohr condition
 E  h 
hc
 hcv

h
p
Operators
d
dx
Position
xˆ  x
Momentum
pˆ  i
Kinetic energy
2 d 2
Kˆ  
2m dx 2
Potential energy
Vˆ  V  x 
Hamiltonian
Hˆ  Kˆ  Vˆ
Commutators and the Uncertainty Principle
Operators  and B̂ then
ˆ ˆ  BA
ˆˆ .
 Aˆ , Bˆ   AB


Standard deviation of operator Â
Aˆ 
Aˆ 2  Aˆ
Uncertainty principle (general)
Aˆ Bˆ 
1  ˆ ˆ
A, B 
2
Uncertainty Principle (position – momentum)
xpx 

2
‘Time–energy uncertainty’ relation
 E t 
2

2
Angular momentum components
 lˆx , lˆy   ilˆz ,


 lˆz , lˆx   ilˆy ,


 lˆy , lˆz   ilˆx


Schrödinger Equation and Wavefunctions
Time dependent (TD) Schrödinger eqn.
  x, t 
Hˆ   x, t   i
t
Time independent (TI) Schrödinger eqn.
Ĥ  x   E  x 
Total wavefunction of TI systems
  x, t   eiEt /   x 
Wavefunction normalisation
N     x    x  dx Alternatively N 2     x  dx
Normalised if
   x    x  dx  1

2

9

Wavefunction orthogonality (x) and (x)
   x    x  dx  0
Overlap between wavefunctions (x) and (x)
S     x    x  dx


A     x  Aˆ   x  dx
Expectation value of any operator Â
Probability density
  x    x
Probability
Px0 x1     x dx
2
x1
x0
Free Translational Motion
Potential energy
V  x  0
Ek 
Total energy
2 2
k (mass m quantum number k)
2m
Wavefunction  k  Aeikx  Beikx  A   B   C cos  kx   D sin  kx 
Particle in a Box of length L
 2  n 
Total Energy En 


2m  L 
Potential energy
V  x  0
Wavefunction
 n  x 
Quantum numbers
n = 1, 2, 3, 4, …
2
joule
2
 n x 
sin 

L
 L 
Number of nodes N  n  1
Tunnelling Through a Square Barrier
Potential energy
0 if x  0

V  V0 if 0  x  L
0 if x  L

Wavefunctions
  A1eik1x  B1eik1x if x  0
V   k   2 / 2m if E  V
 0
2
0
Energy E  
2
V0   2   / 2m if E  V0
  A3eik3x if x  L
  A2eik2 x  B2eik2 x if 0  x  L and E  V0
  A2ei2 x  B2ei2 x if 0  x  L and E  V0
k
2mE

2
Reflection coefficient R  B1 / A1

2
2m V  E 

2
transmission coefficient T  1  R  A3 / A1
2
10
Harmonic Oscillator
Potential energy
1
V  x   kx2
2
Frequency

Total energy
1

Ev   v   hv joules
2

Quantum number
v = 0, 1, 2, 3…
Wavefunction
 v  N v H v  y  e y
Normalisation
N v  2 v v! 
k
rad s–1 or


v
1
2
k –1
s or


/2
1/2
1
2 c
Reduced mass  
Number of nodes
2
v
k
cm–1

mA mB
mA  mB
N v
where y  x /  and     /   k  
1/4
.
(v is quantum number, v frequency)
Hermite polynomials
H0  y   1
H1  y   2 y
H 3  y   8 y3  12 y
H 4  y   16 y 4  48 y 2  12
H 5  y   32 y 5  160 y3  120 y
H 6  y   64 y 6  480 y 4  720 y 2  120
H2  y   4 y2  2
Rigid Rotor
Energy
EJ 
2
J  J  1 joules
2I
Angular momentum quantum number
Degeneracy
J = 0, 1, 2, 3 …
Magnetic (Azimuthal or projection) quantum number
Rotational constant
B
gJ  2J 1
mz  0, 1, 2,   J

2
joules or B 
cm–1 where   h / 2
2I
4 cI
Moment of inertia
I   r 2 kg m2
Magnitude of angular momentum
J   J  J  1
Magnitude of z component
J z  mz 
Wavefunction
 j ,mz  ,    Y jmz  ,  
11
Hydrogen Atom
VeN  r   
Potential energy
Principal quantum number
En  
Total energy
hcRH
n
2
1 e
4 0 r
[ Hydrogenic Atom VeN  r   
n = 1, 2, 3…
joules
gn  n 2
Degeneracy
1 1
E  hcRH  2  2  where n1 < n2
 n1 n 2 
Transition energy
Orbital angular momentum quantum number
l  0, 1, 2,  , n  1
Magnetic (Azimuthal or projection) quantum number
ml  0, 1, 2,  , l
Number of nodes of radial wavefunction R is
N  n  l 1
Hamiltonian
Hˆ  Kˆ r  Vˆr  Kˆ  ,
Wavefunction
 n,l ,ml  r, ,   Rn,l  r  Ylml  , 
1 Z 2e
]
4 0 r
Spherical Harmonics Ylm  , 
l
Y00  ,    1/ 4
Y10  ,   
3
4
cos  
Y11  ,   
5
Y2  ,   
3
8
sin   e  i
15
 i
3 cos 2    1 Y21  ,   
sin   cos   e
16
8
0
2
Y2
 ,   
5
16
sin
2
  e 2i
Angular Parts of Atomic Orbitals
s  Y00
px 
d
Y
2
1
 Y2
0
z
2
1
1
1
 Y1

Y
2
py 
i
d xz 
1
d
x y
2
1
1
Y
2
2

1
2
 Y1
1
pz  Y1

d yz 
i
d xy 
i
1
 Y2
Y
2
1

2
2
0
2
 Y2

Y
2
1
2
 Y2

Y
2
2
2
 Y2

1
2
12
Atomic Term Symbols
2S 1
LJ
L = S, P, D, F, G, H, I  symbols represent numbers calculated using
L  l1  l2 , l1  l2 1 l1  l2 and where S  s1  s2 , s1  s2 1 s1  s2 and
J  L  S , L  S 1 L  S
Spectrophotometry and Fluorimetry
[c] l
[c] l
Beer’s Law may either be defined as Itrans  I 0e
or as Itrans  I010
I

A   log10  trans     c l Transmittance T  I trans / I 0
 I0 
kx
rate of production of x
Quantum yield of x
x 

rate of absorption
sum of all k ' s
Absorbance
If0
Stern-Volmer equations
I fQ
 1   0kQ Q
1
1
  kQ  Q 
Q 0
NMR
Nucleus
12
C
H
13
C
15
N
19
F
29
Si
31
P
103
Rh
195
Pt
2
H
14
N
11
B
23
Na
35
Cl
37
Cl
17
O
27
Al
10
B
1
Spin
quantum
number I
Magnetogyric ratio,
 / 107 rad T–1 s–1
Natural
Abundance
%
0
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1
1
3/2
3/2
3/2
3/2
5/2
5/2
3
0
+26.75
+6.73
–2.712
+25.18
–5.32
+11.32
–0.847
+5.84
+4.11
+1.934
+8.58
+7.08
+2.62
+2.18
–3.63
+6.97
+2.87
98.9
99.98
1.1
0.37
100.0
4.70
100.0
100.0
33.8
0.02
99.63
80.4
100.0
75.77
24.23
0.037
100.0
19.6
13
Spin quantum numbers { I, mz }, where I = 0, 1/2, 1, 3/2 …. mz  0, 1, 2,  I
Degeneracy
g  2I  1
Energy in field B
E   μ  B and in z direction
E    z B0   mz  B0 joule
Magnetic dipole moment  z  mz   joule/Tesla
Magnitude of spin angular momentum
Iˆ   I  I 1 J s rad–1
z-component of spin angular momentum
I z  mz  J s rad–1
Transition energy
Chemical shift
Magnetisation
E   B0 joule
Frequency
v
 v  vref 
  106 
ppm
 vref 


n  n
M0  N
 z evaluates to
n  n
Magnetisation on z-axis (longitudinal component)
dM z
  M 0  M z  / T1
dt
 B0 –1
s or    B0 rad s–1
2
M 0  Nmz
 22
B0
2 k BT
M z  M 0   M z 0  M 0  e  t /T 1

M z  M 0 1  2et /T1

if
M z 0  M z (t  0)   M 0
Chemical Exchange

 T 
G  19.134 103 Tc 10.32  log10  c   kJ mol–1 where k  v / 2 s–1
 k 

then
Chemical Kinetics
First order reaction
k
A 
P
k
Second order reaction A  A 
P
Half–life of a first order reaction
 At   A0 exp  kt 
1
1

= kt
[ A]t [ A]0
t1/2  ln  2 / k sec
Half–life of a second order reaction (A + A) t1/2 
1
sec
k [A]0
 E 
k  A exp   a 
 RT 
 E 
k   d 2 v exp   a  m3s–1
Collision Theory
 RT 
 S ‡ 
 H ‡  –1
k BT
Thermodynamic formulation of TST
k=
exp 
 exp  
 s
h
 R 
 RT 
k  4000 rD dm3 s–1 where D  kBT /  4r  m2 s–1
Diffusion limited rate coefficient
Pressure dependence of activation controlled reaction in solution:
  ln  k  / P T  V ‡ / RT
Arrhenius Equation
14
Intermolecular Forces
Charge–Charge (Coulomb) energy
1 q1q2
 4 0  r
Charge–Dipole energy (at angle )

Dipole–Dipole energy
 μ1r  μ1 r  
1  μ1  μ2
 3 3
 (1, 2 and r are vectors)
4 0  r
r5

Freely rotating dipoles (Keesom Energy)


Dipole–non–polar molecule
1 q cos  
4 0
r2
12  22
 4 0 2 k BT 3r 6
1

 2 1  3cos 2  
1
 4 0 
2
2r
6

(dipole at angle )
  12   6 
 4      
 r  
 r 
Lennard–Jones 6–12 potential
VLJ
Van der Waals equation of state

an 2 
P


2 
 V  nb   nRT
V


Virial equation of state
B T  C T 
PVm
 1


RT
Vm
Vm2
X-ray Crystallography
Structure factor
Fhkl   f j e
i hkl  j 

where hkl  j   2 hx j  ky j  lz j

j
Fourier synthesis of electron density   r  
1
2 i hx  ky lz 
Fhkl e 

V hkl
1
2 2 i hx  ky lz 
Fhkl e 

V hkl
Patterson synthesis
Pr  
Bragg’s law
n  2dhkl sin  
Orthorhombic lattice
1
h2 k 2 l 2
 2 2 2
2
d hkl
a
b
c
15
16
17
Thermodynamics and Statistical Mechanics
ni gi  ( i  0 )/ kBT
ni gi ei / kBT

e

also
n0 g 0
N
q
Boltzmann equation
Molecular partition function is Sum over States of energy levels j
q   g je
(energy is j, degeneracy gj)
  j / k BT
j
Factorisation
qtot  qtrans qrot qvib qelec  qext qint
Distinguishable particles
Q  qN
Indistinguishable particles
qN
Q
N!
First Law of Thermodynamics
dU  dq  dw
Enthalpy
H  U  PV
Heat Capacity
dq
C
dT
For pure materials
 dU 
CV  

 dT V
 dH 
CP  

 dT  P
For reactions
 d U 
CV  

 dT V
 d H 
C P  

 dT  P
Second Law of Thermodynamics
dS 
U  q  w
_

dq
T
S  kB ln W 
Combined First and Second Laws
dU  TdS  PdV
dH  TdS  VdP
closed systems
Helmholtz energy
A  U  TS
dA  PdV  SdT
closed systems
Gibbs energy
G  H  TS
dG  VdP  SdT
closed systems
 G T , P  nFErev
Chemical potential
 dG 
i  

 dni T , P ,n j
For pure substances

G
.
n
For pure solids and liquids
  O
18
Pure ideal gas
 P 
   O  RT ln  O  , P O  105 Pa (1 bar)
P 
Gas mixtures
 p 
i  i O  RT ln  Oi 
P 
 i xi
, x O 1
O
x
Liquid mixtures and solvents i  iO  RT ln  ai  , ai 
Solutes
Equilibrium
For the reaction
i  i O  RT ln  ai  , ai 
 i mi
, m O  1 mol kg -1
O
m
 I   IO  RT ln  ai  , ai 
 i ci
, c O  1 mol dm-3
O
c
 i dni  0
 vi i  0
A g   B g   C g 
K
p
p
C
A
/ po
/ po
 p
B

/ po

G O   RT ln  K 
  ln  K  
r H O

 
RT 2
 T  P
ln  K 2   ln  K1   
Raoult’s law
pi  P O xi
Henry's law
pi  ki xi
r H O
R
1 1
  
 T2 T1 
Solution Equilibria


Equilibrium and free energy
K  exp G O / RT
Definition of pH
pH   log10  H  
Acid dissociation constant
Ka 
pH and pKa
 [A  ] 
pK a  pH  log10 

 [HA] 
Basicity constant
[BH  ][OH  ]
Kb 
[B]
pH and pKb
pK b  14  pK a


[H  ][A  ]
[HA]
19
 [B] 
pKa  pH  log10 
 
 [BH ] 
K a1 K a2 F  K a1 KW
Isoionic point
[H  ] 
Isoelectric point
pH  pK a1  pK a2 / 2
K a1  F


Electrochemistry
Nernst equation for a half-cell
Solution redox equilibria
mOx  ne  rRed
E  Eo 
RT  [Red] r
ln 
nF  [Ox] m

 ,

E  Eo 
 [Red] r
0.05916
log10 
m
n
 [Ox]

 at 25 C

E  E  E 
0.05916
log10  Q 
n
G  nF E
E  E o 
at 25 C
0
0
G  Edonor
 Eacceptor
Electrochemical cells
Replace E with Ecell in redox equilibrium formulae.
Phase Equilibria
Adsorption equilibria
Kads,A 
Vaporisation equilibria
K vap,A 
nb,A
[A]M  nempty
Cation–exchange equilibria
Kexch 
Partition coefficient
Kp,A 
Distribution ratio
DA 
pA
 S,A  A
or 
nb,A
pM,A  nempty
 pA
[M  ]res [H ]sol
[M  ]sol [H  ]res
[A]org
[A]aq
[A in all of its forms]org
[A in all of its forms]aq
20
Solvent extraction
nex
D

ni  Vaq 

  D
V
org


Chromatography and Electrophoresis
tr  tm
V
D s
tm
Vm
Chromatographic capacity factor
k 
Retention volume
Vr  VM  DVs
Chromatographic resolution
R
2 t R1  t R2
N    1  k  
or
R




4    1  k  
wb1  wb2
Gas chromatography
k A 
RT  S Vs
 S , A pA M S VM
Electrophoretic mobility
e 
q
6 r
Retention factor
Rf 
Separation factor


Migration in capillary electrophoresis u  (e  eo )

or
R
tm
tr
k2
k1
t R
wb2
E
L
Mass Spectrometry
Relative intensity of M+1 isotope peak
 I M 1 

 100%  nH  0.012  nC 1.07  ...
 IM 
Exact Masses of Isotopes and their Natural Abundance
Element
H
C
N
O
Cl
Fe
Mass number
Mass / Da
Abundance / %
1
2
12
13
14
15
16
17
18
35
37
54
56
57
1.00783
2.01410
12 (exact)
13.00335
14.00307
15.00011
15.99491
16.99913
17.99916
34.96885
36.96590
53.93961
55.93494
56.93540
99.988
0.012
98.93
1.07
99.632
0.368
99.757
0.038
0.205
75.78
24.22
5.845
91.754
2.119
21
58
57.93328
0.282
Understanding Character Tables
Symmetry operations.
In C3v there are a total
of h = 6 operations
Class C3
Point Group
name
Number of operations in class v
C3v
E
2C3
3v
A1
1
1
1
z
A2
1
1
-1
Rz
E
2
-1
0
(x, y), (Rx, Ry)
Mulliken labels are
shorthand for
Character
representations of
molecular, orbital &
vibrational
Irreducible
symmetry species.
representation
A2 in point group C3v.
(+1 if operation
symmetric, –1 if not)
x2+y2, z2
(x2-y2, xy), (xz, yz)
Product transformations.
x2+y2 represent s orbitals, the
others, d orbitals.
All are operators for Raman
spectroscopy selection rules
Linear transformations.
x, y, z operators used in transition
dipole selection rules.
Rotation operators Rx, Ry, Rz used
in spin orbit coupling and whole
body rotation about axis shown.
Brackets indicate degeneracy
Mulliken Labels (Principal Axis is labelled as Cn)
A
B
E
T
subscript 1
subscript 2
g
u
superscript '
superscript ''
singly degenerate, symmetric about Cn axis (+1 in table)
singly degenerate, anti-symmetric about Cn axis (–1 in table)
double degenerate
triply degenerate
symmetric about C2 axis  to Cn axis or v if no C2 present, e.g. A1
anti-symmetric about C2 axis  to Cn axis or v if no C2 present, e.g. A2
‘gerade’, symmetric to inversion i , e.g. E2g
‘ungerade’, anti- symmetric to inversion i, e.g. B2u
symmetric to h, e.g. A'
anti-symmetric to h, e.g. E''
22
Point Groups and their Symmetry Operations (excluding the identity E).
Cn  360/n fold rotation; n = 2  180; n = 3  120; n = 4  90 rotation.
i = inversion;
h = horizontal mirror plane, v = vertical mirror, d = dihedral mirror plane.
Sn = rotation-reflection. Angle is 360/n
Cn2 means Cn applied twice over and Sn5 means Sn applied 5 times etc.
( note Cnn = E, S2 = i, S2nn = Cn )
Point
group
C1
Cs
Ci
C2
C3
C2
C3
Symmetry Operations
C4,5,6
i
v
h
d
Sn
all the rest
E identity only
h
i
C2
C32
C3
C2v
C3v
C4v
C2
C2h
C3h
D2
D3
C2
3C2
3C2
D2h
D3h
D4h
D5h
D6h
3C2
3C2
C2
5C2
C2
v, v'
3v
2v
2C3
C2
2C4
2d
h
h
i
C3
S3
C32, S35
C2  C2(x),C2(y)C2(z)
2C3
i
2C3
2C3
2C4
5C5
2C6
i
i
2v
3v
2v
5v
3v
h
h
h
h
h
  (xy),(xz),(yz)
2d
3d
2S3
2S4
2S5
2S6
2C2' , 2C2''
2C52
2S3, 3C2', 3C2''
C2
2S4
C2 '
D2d
2d
3C2
2C3
i
2S6
D3d
3d
C2
2C4
2S8
4C2' , 2S83
D4d
4d
5C2
2C5
i
2S10
2C52, 2S103
D5d
5d
C2
S4
S43
S4
3C2
8C3
6S4
Td
6d
6
6C2
8C3
6C4
i
6S4
8S6 , 3C2
Oh
3h
d
Identify Dh (e.g. homonuclear diatomic) and Cv (e.g. heteronuclear diatomic) directly by their
shape.
23
‘Road Map’ for Systematically Determining a Point Group
Special groups?
No h
No i
C1
i
Ci
No Cn axis
No
h
i
Cs
C∞v
D∞h
Td
Oh
Ih
Yes
Cn axis
n-C2’s perp to Cn axis
No n-C2’s perp
to Cn axis
Cnh
Cnv
S2n
h
Dnh
h
No h
n-v
No h
n-d
No v
S2 × n
No S2 × n
Cn
Dnd
Nod
Dn
24
Statistics
Mean
x
Sample standard deviation
s
Gaussian probability distribution
Confidence interval
Student's t–test, case 1
1 n
 xi
n i 1
1 n
 xi  x 2

n  1 i 1
   x   2 

p( x) 
exp 
2
2


2

2


ts
x
n
x k
tcalc 
n  tcrit
s
1
s 2  n  1  s2 2  n2  1
n1n2
with s pooled  1 1
n1  n2
n1  n2  2
x1  x2
t–test, case 2a
tcalc 
t–test, case 2b
tcalc 
with

s 
 2 

 n1 n2 
s pooled
x1  x2
s12 n1  s22 n2
s12
2
d
sd
2

 

 s2 n 2
s2 2 n2
 1 1
 n 1  n 1
2
 1

t–test, case 3
tcalc 
Calibration by standard addition
I S  X [ X ] f  [S ] f

IX
[X ] i
2
sd 
n with
1


  2 degrees of freedom


  di  d 
2
i
n 1
V 
V 
I
I S  X  total   I X  X [S ]i  S 
[X ] i
 V0 
 V0 
sX 
sy
m D2
Calibration with an internal standard [X ]  F
n
n
i 1
i 1
X 2 n   xi 2  2 X  xi
IX
[S ]
IS
25
Student’s t–distribution table
Two tailed confidence
90%
0.05
/2 
v = n –1
1
2
3
4
5
6
7
8
9
10
15
20
30
40
50
95% 99%
0.025 0.005
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.753
1.725
1.697
1.684
1.676
 (Normal distribution) 1.645
12.71
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.131
2.086
2.042
2.021
2.009
1.960
63.66
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
2.947
2.845
2.750
2.704
2.678
2.576
Propagation of Errors
2
If y  f  u, v  , the variance squared is
 y2
2
 y 
 y 
    u2     v2
 u v
 v u
y2
f(u, v)
u  v,
u v
u2  v2
uv
v2 u2  u 2 v2
u/v
v 2 u2  u 2 v2
v4
1 1
 ,
u v
1 1

u v
v 4 u2  u 4 v2
u 4v 4
a 2 v2e2av
eav
uev
u ln  v 

2
u

 u 2 v2 e2v
v 2 u2  u 2 v2
v
2
ln  v 
2
26
Useful Mathematical Formulae
Equation of straight line y  mx  c with gradient m intercept c
y y 
Given points (x1, y1) and (x2, y2) then y   2 1   x  x1   y2
 x2  x1 
 a  b2  a2  2ab  b2
1
 x  b
a
  x  b
 a  b  a  b   a 2  b2
a
xa
 x a b
b
x
xa  xb  xab
ax2  bx  c  0 with solution (roots) r 
x!  x  x 1 x  21
n
x  x1/n
b  b 2  4ac
2a
0!  1
 x n  1  x 2  x3  x 4  
n 0
Absolute value x
x
log    log  x   log  y 
 y
log  xy   log  x   log  y 
 
 
log x a  a log  x 
log  x  means take log to power a: log xa  log  x 
logk  x   logk  m logm  x  ,
loge  x   loge 10 log10  x   2.3026  log10  x 
a
a
ln  x   loge  x 
sin 
opposite
hypotenuse
cos 
Arithmetic mean  a  b / 2
adjacent
hypotenuse
tan 
opposite
adjacent
Geometric mean
ab
sin  all 
tan  cos
Harmonic mean
Probability = number of desired outcomes /total number of possible outcomes
2ab
a b
p n/k
Number of Combinations: order does not matter.
select k items out of n;
Number of Permutations: order does matter
Cnk 
n
n!
 
k ! n  k  !  k 
Pnk 
n!
 n  k !
Pnk  Cnk
27
Differentials and Integrals (c is an arbitrary constant)
ax n1
c
n 1
d n
ax  nax n1
dx
n
 ax dx 
d
1
ln  x  
dx
x

d ax
e  aeax
dx
ax
ax
 ae dx  e  c
dx
 ln  x   c
x
d
sin  x   cos  x 
dx
 sin  x  dx   cos  x   c
d
cos  x    sin x
dx
 cos  x  dx  sin  x   c
d
tan  x   1  tan 2  x 
dx
 tan  x  dx   ln  cos  x    c
d
dv
du
(uv)  u  v
dx
dx
dx
 udv  v   vdu (by parts)
dy dy dz
 
(chain rule)
dx dz dx
 dx   1dx  x  c
Substitution. Let u be a function of x, e.g. u  x2
 dx 
then  f  x  dx   f  u    du
 du 
dy
dx
 1/
dx
dy
Trigonometric Functions
In a right angled triangle, x horizontal, y vertical, hypotenuse r then,
sin    y r
r  x2  y 2
cos    x r
tan   
sin  
cos  
y x
sin 2    cos 2    1
ei  cos    i sin  
sin  2   2sin   cos  
cos  2   cos2    sin 2    2cos 2    1  1  2sin 2  
sin(   )  sin   cos    cos   sin  
cos(  )  cos   cos    sin   sin  
  
   
sin    sin    2sin 
 cos 

 2 
 2 
     
sin    sin    2 cos 
 sin 

 2   2 
  
   
cos    cos    2 cos 
 cos 

 2 
 2 
       
cos    cos    2sin 
 sin 

 2   2 
Cosine rule
a 2  b 2  c 2  2bc cos  A 
Sine rule
a
b
c


sin  A  sin  B  sin  C 
28
Series Expansion
x 2 x3
e  1  x    ... ,
2! 3!
e0  1,
x
e
x
x 2 x3
 1  x    ... ,
2! 3!
ln(1  x)  x 
sin  x   x 
x 2 x3
  ...
2 3
x 3 x5
  ...
3! 5!
sinh  x   x 
x3 x 5
 
3! 5!
e  
e  0
ln 1  0, ln  0  , ln    
for |x| < 1
cos  x   1 
x 2 x4
  ...
2! 4!
cosh  x   1 
1
 1  x  x 2  x3  x 4  
1 x
x x 2 x3
5 4
1 x  1   
x 
2 8 16 128
tan  x   x 
x3 2 5
 x 
3 15
x2 x4
x3 2
  tanh  x   x   x5 
2! 4 !
3 15
1
 1  2 x  3x 2  4 x3  5 x 4  6 x5 
2
1  x 
1
x 3
5
25 4
 1   x 2  x3 
x 
2 8
16
128
1 x
Binomial Series: if x  1
(1  x) n  1  nx  n(n  1)
n n
 
x2
x3
 n(n  1)(n  2)  ... , equivalently (1  x)n    x k
2!
3!
k 0  k 
Binomial Expansion
 n  n1  n  n 2 2
 n  n 1  n  n
n
n!
 x y    x y   
 xy    y where   
1 
2
 n  1
n
 k  k ! n  k  !
 x  y n  x n  
Maclaurin Series Expansion of a Function f(x)
x2  d 2 f 
x3  d 3 f 
xn  d n f 
 df 
f  x   f  0   x     2    3      n   
n !  dx 0
 dx 0 2!  dx 0 3!  dx 0
The derivative’s subscript means evaluate at x = 0
Taylor Series Expansion of a Function f(x)
 x  x0 
 df 
f  x   f  x0    x  x0    
2!
 dx  x0
2
 d2 f
 2
 dx

 x  x0 
   
n!
 x0
n
 dn f
 n
 dx

  
 x0
The derivative’s subscript means evaluate at x = x0
29
i  1, i 2  1, i  1/ i
Complex Numbers
Cartesian representation z  x  iy
‘real’ part Re( z )  x ‘imaginary’ part Im( z )  y
Re  z   r cos  
Polar representation
z  rei
Complex conjugate
z  x  iy  rei
Modulus (Absolute value) squared
Im( z)  r sin  
z  z z   x  iy  x  iy   x2  y 2
2
Relationship between representations
    arctan  y / x  if x  0
x  r cos  
y  r sin  
  / 2 if x  0 and y  0
  / 2 if x  0 and y  0
r  x2  y 2
 arctan  y / x  if x  0
Euler’s Identity
ei  cos    i sin  
eix  eix
sin  x  
2i
eix  eix
cos  x  
2
sinh  x   i sin  ix  cosh  x   i cos  x 
e x  e x
sinh  x  
2
De Moivre’s Formula
e x  e x
cosh  x  
2
z n  r n einx  r n cos  x   i sin  x    r n cos  nx   i sin  nx  
n
Spherical Polar to Cartesian Coordinate Conversion
A point [ x, y, z] in Cartesian coordinates is represented by a polar angle , an azimuthal (or equatorial)
angle  and a radius r, i.e. [ r, , ]. These are related as
which means that
x  r sin   cos  
y  r sin   sin  
z  r cos  
r  x2  y 2  z 2
cos    z / r
tan    y / x
The volume element is dV   r sin   d   dr  rd   r 2 sin   drd d
30
Glossary of selected Mathematical Symbols
Symbols
Meaning
ab
Equality, with numbers   3.14159 three dots are added.
ab
a is not equal to b
ab
Identity; a is identical to b  a  b   a 2  2ab  b 2 . Rarely used.
ab
a is less than b,
ab
a is greater than b.
≤
less than or equal to
≥
greater than or equal to

mean much less than
2

much greater than
ab
a is approximately equal to b
ab
a is of the order of b, or a changes at the same rate as b
373 K  100°C
Indicates change of units to equivalent value
a
values are plus and minus a
a
values are minus and plus a
a^b
a
, a / b, a  b
b

a raised to power b. Used only in computer languages
a  b, a  b
a is perpendicular and parallel to b respectively

Infinity

tends to, or approaches and used as in a   or a  0
a is divided by b
Angle
n

Summation,
 xi  x0  x1  x2    xn
i 0
n

Product,
 xi  x0 x1x2  xn
i 0
n
O(x )
Big ‘O’ notation. In series expansion to indicate that the next, unwritten terms
do not grow faster than xn.
  x   1 if x  0 else is zero
(x)
delta function
nm
Kronecker delta  nm  1 if m  n else is zero. m and n are integers
df  x  / dx
f   x  , f   x 
f , 
f
[A, B]
Derivative of function f with respect to x.
Alternative notation. First and second derivatives of function f with respect to x.
Alternative notation. First and second derivatives, usually with respect to time.
Commutator brackets, with operators A and B
31
Using MAPLE
Maple can be used in different modes, the simplest is to use the worksheet mode which means that the
text is in red preceded by the red symbol >. To force the programme to use this mode, click in the [>
icon on the toolbar at the top of the screen, then click the text icon in the lower tool bar on the left part
of the screen. Next, using the tools menu, locate and click on options, then interface and select default
for new worksheets and using the drop down menu select worksheet. Complete the setup by closing the
options box by clicking on apply globally. Maple should start in the worksheet mode next time it is
used.
When manipulating algebraic expressions, the syntax always has the form
>
Your_name_for_something:= expression;
Curved brackets always follow functions such as sin, cos, log or your own functions, e. g.
sin(3*x)*ln(x^2) or pre-defined terms such as plot(x^2, x=-2..2);
> restart:
# always start this way. The symbol # starts a comment
> z:= sin( x /(1+x) );
# use brackets when dividing
> a_Gaussn:= exp( -x^2/20 );
> f1:= simplify( (x^2)^(1/2), symbolic );
> y:= diff( x^2, x);
# differentiate by x
> eV:= convert(1,units, kJ, electronvolt);
# convert kJ to eV
> plot(3+x-x^4,x=-3..3,view=[-1..2,-3..4],color=blue);
> ans:= solve(a*x^2+b*x+c=0,x);
# solve for x
> # Your own functions are defined as
> func:= ( x, y)-> exp(-x^2) + sin(y^2) + 2;
# user defined function
> # and used as, for example,
> sin(x)* func(3.0);
> y:= func(x^2);
> plot(func(s),s=0..3*Pi,colour=[blue],view=[0..10, 0..4],numpoints =1000);
> # convenient way to show integration (or differentiation) and result
> Int(sin(x)^2,x = a..b): % = value(%);
Further Maple instructions can be found on the Phys. Chem. (Porter) lab computers
32