Download I. Basic algebra, numbers, units.

I. Basic algebra, numbers, units. 1. Physical quantities In physical chemistry (and all chemistry) we deal with quantities, which describe physical
properties.
Physical quantity = numerical value + unit
E =
(energy)
 13.6
eV
(number) (unit: electronvolts)
Both the number and the unit are important: one without the other is nothing !
Numbers obey the laws of mathematics which is why math is an extremely important part of any
quantitative scientific discipline.
2. Numbers 


Natural numbers 1, 2, 3, …, 10, …
Integers 0, 1,  2, ….
Rational numbers (fractions)


Irrational numbers: cannot be expressed as rational numbers e, , 2
Real numbers = rational + irrational (of course they contain integers, which contain
naturals)
Complex numbers have real and imaginary part, e.g. 2 + 3i, where i = -1 is the
“imaginary unit”.
Real numbers are a subset of complex numbers (where imaginary part is zero).

numerator
denominator
2
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All physically observable quantities are real ! However, complex quantities can be extremely
useful (and in some cases necessary) for the mathematical formalism. More about complex
numbers later.
3. Rules of Algebra a+b=b+a
1.
2. a + (b + c) = (a + b) + c
ab = ba
3.
a(bc) = (ab)c
4.
a(b + c) = ab + ac
5.
(commutative law for addition)
(associative law for addition)
(commutative law for multiplication)
(associative law for multiplication)
(distributive law)
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rules of signs:
1.
2.
(a)(b) = ab
(a)( b) = ab
exponents
1.
2.
3.
am.an = am+n
am/an = amn (a  0)
(am)n = am.n
exponents add
exponents subtract
exponents multiply
fractions:
1. Equivalent fractions: multiples of numerator and denominator by the same number
(i.e. multiplication of both numerator and denominator by the same number does not change the
fraction)
2 4 10
 
3 6 15
2. addition, subtraction: common denominator !
2 6 7  2 3  6 14 18 14  18 32

 

 

3 7 7  3 3  7 21 21
21
21
3. multiplication
2 6 2  6 12

 
3 7 3  7 21
4. division
2
3  2  7  14  7
6 3  6 18 9
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5. raising to a power: both numerator and denominator are raised
n
2n
2
   n
3
3
6. fractional powers:
q
ap q  ap
this also means that roots can be expressed as powers:
a1 2  a
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4. Scientific Notation To simplify notation, powers of 10 are often used, e.g.
0.0001821 = 1.821 x10-4
1,248,000 = 1.248 x106
and particular powers of 10 have standardized prefixes, which go with particular units:
10-1
deci.. (decimeter)
d.. (dm)
101
deca (decaPascal)
D..(DPa)
10-2
centi.. (dentimeter)
c.. (cm)
102
10
-3
10
-6
milli.. (millimeter)
m..(mm)
hecto (hectoPascal)
h.. (hPa)
3
kilo (kiloPascal)
k.. (kPa)
6
mega (megaPascal)
M..(MPa)
10
micro..(micrometer)
.. (m)
10
10-9
nano .. (nanometer)
n.. (nm)
109
giga (gigaPascal)
G..(GPa)
10-12
pico..(picometer)
p.. (pm)
1012
tera (teraPascal)
T.. (TPa)
5. Units Units and their conversion are a great source of error in solving (physical) chemistry problems.
It is essential to use and keep track of the right units. Unfortunately, there are many different unit
systems that confuse things.
Most commonly used: SI (MKS) unit system:
Name Unit symbol
metre m
kilogram kg
second s
ampere A
kelvin K
candela cd
mole mol
SI base units
Quantity
Symbol
length
l (a lowercase L)
mass
m
time
t
electric current
I (a capital i)
thermodynamic temperature T
luminous intensity
Iv (a capital i with lowercase v subscript)
amount of substance
n
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Named units derived from SI base units
Expression in terms of other Expression in terms of
Quantity
units
SI base units
Name Symbol
hertz
Hz
radian rad
steradian sr
newton N
pascal Pa
joule
J
watt
W
coulomb C
volt
V
farad
F
ohm
Ω
siemens S
weber Wb
tesla
T
henry
H
Celsius °C
frequency
angle
solid angle
force, weight
pressure, stress
energy, work, heat
power, radiant flux
electric charge or quantity of
electricity
voltage, electrical potential
difference, electromotive force
electric capacitance
electric resistance, impedance,
reactance
electrical conductance
magnetic flux
magnetic field strength,
magnetic flux density
inductance
Celsius temperature
1/s
m·m-1
m2·m-2
kg·m/s2
N/m2
N·m = C·V = W·s
J/s = V·A
s-1
dimensionless
dimensionless
kg·m·s−2
m−1·kg·s−2
m2·kg·s−2
m2·kg·s−3
s·A
s·A
W/A = J/C
m2·kg·s−3·A−1
C/V
m−2·kg−1·s4·A2
V/A
m2·kg·s−3·A−2
1/Ω
J/A
m−2·kg−1·s3·A2
m2·kg·s−2·A−1
V·s/m2 = Wb/m2 = N/(A·m)
kg·s−2·A−1
V·s/A = Wb/A
K − 273.15
m2·kg·s−2·A−2
K − 273.15
In chemistry, often used are atomic units (more appropriate in the words of single atoms and
molecules):
Fundamental atomic units
Symbol
Dimension
Name
mass
charge
angular momentum
electric constant
electron rest mass
elementary charge
Reduced Planck's constant
Coulomb force constant
me
e
ħ = h/2
1 / (4πε0)
Value in SI units
9.1093826(16)×10−31 kg
1.60217653(14)×10−19 C
1.05457168(18)×10−34 J·s
8.9875517873681×109 kg·m3·s-2·C-2
Below are given a few derived units. Some of them have proper names and symbols assigned, as
indicated in the table. Most symbols are defined in this table or the table above, but also α is the
fine-structure constant, ε0 is the permittivity of vacuum, c is the vacuum speed of light, and kB is
Boltzmann constant.
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Derived atomic units
Dimension
Name
Expression
Value in SI units
Value in more
common units
a0
4πε0ħ2/(mee2)= ħ/(mec)
5.291772108(18)×10−11 m
0.052918 nm
=0.52918 Å
Eh
mee4/(4πε0ħ)2
4.35974417(75)×10−18 J
27.211 eV
time
velocity
ħ/Eh
a0Eh/ħ = c
2.418884326505(16)×10−17 s
2.1876912633(73)×106 m·s−1
force
Eh/a0
8.2387225(14)×10−8 N
temperature
pressure
Eh/kB
Eh/a03
3.1577464(55)×105 K
2.9421912(19)×1013 Pa
electric field
Eh/(ea0)
5.1421×1011 V·m−1
length
energy
Bohr
radius
Hartree
energy
Symbol
82.387 nN
=51.421 eV·Å−1
27.211 eV
514.21 V·nm−1
=51.421 V·Å−1
Also important units that are not SI, but recognized by SI:
Name
Liter
Molar
atomic mass unit
Quantity
volume
concentration
mass
Symbol
L
mol.L-1
amu
Value in SI
10-3 m3
1.660 538 782(83)×10−27 kg
Note that molar mass in grams/mol is equal to the atomic or molecular mass in amu.
And very often still used in chemistry
Name
calorie
wavenumber
(inverse centimeter)
Ångström
Quantity
energy
frequency
(inverse of wavelength)
Symbol
cal
Value in SI
4.184 J
cm-1
102 m-1
length
Å
10-10 m
To make things even worse, sometimes units are used for quantities that are related through some
fundamental physical constant or law. For example electronvolt (eV) - unit of energy, equals
kinetic energy of a charge of 1e accelerated by the potential diference (voltage) of 1 volt (V).
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Also, in quantum chemistry energy is commonly given in wavenumbers (cm-1) using the
relationship between energy and frequency, frequency and wavelength , and wavelength and
wavenumber:
E  h 
hc

 hc~
where h is Planck constant (in J.s-1),  is frequency (in Hz or s-1), c is speed of light (in cm.s-1)
and ~ is wavenumber (in cm-1). That gives energy in Joules, but
~ 
E
hc
is energy in cm-1.
Some of the fundamental universal constants (and their SI units)
Quantity
speed of light in vacuum
Planck constant
reduced Planck constant
Symbol
c
h
ħ = h/2
Value
299 792 458 m·s−1
6.626 068 96(33) × 10−34 J·s
1.054 571 628(53) × 10−34 J·s
atomic mass unit (unified atomic mass unit)
Avogadro's number
Boltzmann constant
Faraday constant
gas constant
m = 1u
NA, L
kB
F = NAe
R = NAkB
1.660 538 86(28) × 10−27 kg
6.022 141 5(10) × 1023 mol−1
1.380 650 4(24) × 10−23 J·K−1
96 485.338 3(83)C·mol−1
8.314 472(15) J·K−1·mol−1
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6. Unit conversions 1. Conversion factors are found in Tables.
2. The units must match (the rest cancels out).
Example: converting 13.6 eV to kcal.mol-1 (kilocalories per mol)
 1.602  10 -19 J  1cal  1kcal 
23
1

 13.6eV
 6.023  10 mol 

1eV

 4.184J  1000cal 
 13.6  1.602  6.023 10 1910 23
 13.6  1.602  6.023 10 4
1



kcal  mol 1
kcal
mol
4.184
4.184
10 3
10 3
 13.6  1.602  6.023

 10kcal  mol 1  31.363  10kcal  mol 1  3.1363  10 2 kcal  mol 1
4.184


Note: it’s good to eliminate the powers of ten first (without calculator ), then deal only with
small numbers on the calculator.
The safest strategy for dealing with (physical) chemistry problems is:
1.
convert everything to SI units (even though you may get crazy powers of 10)
2.
eliminate the crazy powers of 10
3.
do your calculation
4.
convert to other units (atomic, …) if desired.
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