Download Maths for Chemistry Facts and Formulae

43799 D&PS Jan11
The sample unbiased estimate �
of standard deviation, s, is the
�n
2
i=1 (xi − x̄)
. The standard
square root of the variance: s =
n−1
deviation of the sample mean is called the standard error of the
mean and is equal to √σn , and is often estimated by √sn .
Population values, or parameters, are denoted by Greek letters.
Population mean = µ. Population variance = σ 2 . Population
standard deviation = σ. Sample values, or estimates, are denoted
by roman letters.
The mean of a sample of n observations x1 , x2 , . . . xn is
�n
x1 + x2 + · · · + xn
i=1 xi
=
x̄ =
n
n
The sample mean x̄ is an unbiased estimate of the population
mean µ. The unbiased
of the variance of these n sample
�estimate
n
2
i=1 (xi − x̄)
which can be written as
observations is s2 =
n−1
n
1 � 2
nx̄2
s2 =
xi −
n − 1 i=1
n−1
c
ln x + c or ln c� x
ex + c
ekx
k +c
− k1 cos kx + c
1
k sin kx + c
�
k, constant
x
kx + c
x2
2 +c
x2
x3
3 +c
xn+1
n+1 +
For more copies contact the Network at
[email protected]
This leaflet has been produced in conjunction with the
Higher Education Academy Maths, Stats & OR Network,
the UK Physical Sciences Centre, and sigma
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f (x) dx
Integration
b2 − 4ac > 0
b2 − 4ac = 0
b2 − 4ac < 0
(1)
x
−b/2a
nt
adie
e gr
ativ
neg
(x2 ,y2 )
(x1 ,y1 )
d
df
(k × f (x)) = k ×
dx
dx
for k constant.
1
x
y = f (x)
dy
dx
k, constant
x
x2
xn , constant n
sin(kx)
cos(kx)
ex
ekx
ln kx = loge kx
0
1
2x
nxn−1
k cos(kx)
−k sin(kx)
ex
kekx
x
minimum point
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Facts & Formulae
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dy
>0
dx
dy
<0
dx
y
= f � (x)
Arithmetic
Second priority
Second priority
Second priority
as a percentage is
5
8
1
3
× 100% = 33 13 %
Algebra
Trigonometry
Degrees and radians:
2π
π
360◦ = 2π radians, 1◦ =
=
radians
360
180
180
degrees ≈ 57.3◦
1 radian =
π
Trig ratios for an acute angle θ:
e
b
side opposite to θ
s
u
=
ten
po c
hypotenuse
c
b
hy
sin θ =
Removing brackets:
a(b + c) = ab + ac,
a(b − c) = ab − ac
a
ac
(a + b)(c + d) = ac + ad + bc + bd
c=
b
b
Formula for solving a quadratic equation: √
−b ± b2 − 4ac
if ax2 + bx + c = 0 then x =
2a
Laws of Indices:
am
= am−n (am )n = amn (ab)n = an bn
am an = am+n
an
√
√
m
1
a0 = 1
a−m = m
a1/n = n a
a n = ( n a)m
a
Logarithms: for any positive base b (with b = 1)
logb A = c means A = bc .
Logarithms to base e, denoted loge or alternatively ln are called
natural logarithms. The letter e stands for the exponential constant which is approximately 2.718.
loge A or ln A = c means A = ec .
opposite
When multiplying or dividing positive and negative numbers the
sign of the result is given by:
positive × positive = positive positive × negative = negative
negative × positive = negative negative × negative = positive
positive
positive
= positive
= negative
positive
negative
negative
negative
= negative
= positive
positive
negative
The BODMAS rule reminds us of the order in which operations
are carried out. BODMAS stands for:
Brackets ()
First priority
Of
Division ÷
Multiplication ×
Addition +
Third priority
Subtraction −
Third priority
numerator
Fractions:
fraction =
denominator
Adding and subtracting fractions: to add or subtract two fractions first rewrite each fraction so that they have the same denominator. Then, the numerators are added or subtracted as
appropriate and the result is divided by the common denomina3
16
15
31
4
+
=
.
tor: e.g. + =
5
4
20
20
20
Multiplying fractions: to multiply two fractions, multiply their
numerators and then multiply their denominators: e.g.
3
5
3×5
15
×
=
=
7
11
7 × 11
77
Dividing fractions: to divide two fractions, invert the second
3
2
3
3
9
and then multiply: e.g. ÷ = × =
.
5
3
5
2
10
Proportion and Percentage:
To convert a fraction to a percentage multiply by 100 and label
the result as a percentage.
as a percentage is × 100% = 62.5%
5
8
1
3
a
side adjacent to θ
θ
=
cos θ =
a
hypotenuse
c
adjacent
side opposite to θ
b
Pythagoras’ theorem:
=
side adjacent to θ
a
a2 + b2 = c2
tan θ =
Standard triangles:
√
2
2
60◦ 1
1
45◦
1
30◦
√
1
sin 45◦ = √ ,
2
cos 60◦ =
c, the natural logarithm of a number A, is the power to which
e would have to be raised to equal A. Note:
eln A = A if A > 0;
ln(eA ) = A
Logarithms to base 10: log10 A = c means A = 10c .
pH: of a solution measures its acidity or basicity.
3
1
cos 45◦ = √ ,
2
tan 45◦ = 1
√
3
cos 30◦ =
,
2
1
sin 30◦ = ,
2
√
3
,
2
sin 60◦ =
1
tan 30◦ = √
3
1
,
2
y
Some common conversions are:
1
1
3
1
= 10%,
= 25%,
= 50%,
= 75%
10
4
2
4
Ratios are an alternative way of expressing fractions. Consider
dividing £200 between two people in the ratio of 3:2. For every
£3 the first person gets, the second person gets £2. So the
first gets 35 of the total, and the second gets 52 of the total;
that is £120 and £80.
(1)
(2)
(3)
(2)
(1)
Quadratic functions: y = ax2 + bx + c
x
x
(x2, y2 )
y2 − y1
m=
x2 − x1
c
Differentiation
Graphs of common functions
Generally, to split a quantity in the ratio m : n, the quantity is
m
n
of the total and m+n
of the total.
split into m+n
b2 − 4ac < 0
b2 − 4ac = 0
b2 − 4ac > 0
(2)
(3)
x
if a is negative
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© mathcentre 2010
The
linearity rule:
�
�
�
(af (x) + bg(x)) dx = a f (x) dx + b g(x) dx, (a, b constant)
� b
� b
dv
du
u dx = [uv]ba −
Integration by parts:
v dx.
dx
dx
a
a
xn , (n �= −1)
x−1 = x1
ex
ekx
sin kx
cos kx
f (x)
Statistics
(1)
(2)
(3)
−b/2a
d2 f
df
, means differentiate
dx2 � �
dx
df
d2 f
d
.
with respect to x. That is,
=
dx2
dx dx
Partial derivatives: If f = f (x, y) is a function of two (or more)
∂f
independent variables, ∂x
means differentiate f with respect to
means differentiate f
x treating y as if it were a constant. ∂f
∂y
with respect to y treating x as if it were a constant.
Higher derivatives: f �� (x), or
(3)
if a is positive
The product and quotient rules:
v du − u dv
dv
du
d �u�
d
(uv) = u
+v
= dx 2 dx .
dx
dx
dx
dx v
v
The chain rule:
dy
dy
If y = y(u) where u = u(x) then dx
= du
× du
.
dx
Graph of y = ex and y = e−x Graph of y = ln x and y = log x
10
showing exponential growth/ decay
1
-1
(x1 ,y1 )
pos
a
The linearity rules:
d
du dv
(u(x)±v(x)) =
±
dx
dx dx
x
1
1
y=ln x
y = log1 0 x
Exponential and log functions:
e ≈ 2.718 is the exponential constant.
y
y = ex
y= e-x
x
c
gr
itive
t
dien
y
dy
Differentiating a function, y = f (x), we obtain its derivative dx
.
This new function tells us the gradient (slope) of the original
dy
= 0 the gradient is zero.
function at any point. When dx
The straight line: y = mx + c.
m=gradient (slope), c = vertical intercept.
tan 60◦ =
√
◦
pH = − log10 ([H+ ]/c−
)
◦
[H+ ] = 10−pH c−
so
where [H+ ] = hydrogen ion concentration in mol dm−3 and
◦
c−
= 1 mol dm−3 . Equivalently, pH = − log10 aH3 O+ where
aH O+ = hydronium ion activity.
3
Laws of Logarithms: for any positive base b, with b = 1,
A
logb A − logb B = logb ,
logb A + logb B = logb AB,
B
3
Bragg’s Law: a fundamental law of x-ray crystallography.
nλ = 2d sin θ
λ = wavelength of the x-rays incident on a crystal lattice with
planes a distance d apart, n is an integer (whole number), and
θ, the Bragg angle, is the angle between the incident beam and
the reflecting planes.
n
logb
A=
An ,
logb
logb
Formula for change of base:
ln x
log10 x =
.
ln 10
Inequalities:
Matrices and Determinants
�
�
a b
The 2 × 2 matrix A =
has determinant
c� d �
�a b�
�
= ad − bc
|A| = ��
c d�


a11 a12 a13
The 3 × 3 matrix A =  a21 a22 a23  has determinant
a31 a32 a33
�
�
�
�
�
�a
�a
a23 ��
a23 ��
a22 ��
− a12 �� 21
+ a13 �� 21
a33 �
a31 a33 �
a31 a32 �
1 = 0,
loga x =
logb
logb x
.
logb a
b = 1.
Specifically,
a > b means a is greater than b
a < b means a is less than b
a  b means a is greater than or equal to b
a  b means a is less than or equal to b
Sigma notation: The Greek letter sigma, Σ, is used to abbrevix , . . . xn , and add them,
ate addition. If we have n values,
2
n

xi .
x1 ,
�
�a
|A| = a11 �� 22
a32
the sum x1 + x2 + . . . xn is written
i=1
Note that i takes all whole number values from 1 to n. So, for
3

xi means x1 + x2 + x3 .
The inverse
of�a 2 × 2 matrix:
�
�
�
1
a b
d −b
If A =
then A−1 =
c d
ad − bc −c a
provided that ad − bc �= 0.
Matrix multiplication: for 2 × 2 matrices
�
��
� �
�
a b
α γ
aα + bβ aγ + bδ
=
c d
β δ
cα + dβ cγ + dδ
instance
i=1
Example:

5
2
2
2
2
2
2
i=1 i means 1 + 2 + 3 + 4 + 5 .
n

xi = x1 · x2 · . . . · xn .
Product notation:
i=1
Produced in conjunction with the Higher Education Academy Maths,
Stats & OR Network, the UK Physical Sciences Centre, sigma, and
the HE STEM Programme. Thanks to the many individuals who
c 2010. - version draft 1, August 2010.
offered detailed advice. 
Remember that AB �= BA except in special cases.
www.mathcentre.ac.uk
© mathcentre 2010
2∗
2
1
0
www.mathcentre.ac.uk
© mathcentre 2010
(* A + B → P reaction.)
= kt
d[A]
dt
= −k[A][B]
d[A]
dt
d[A]
dt
1
[B]0 −[A]0
ln
[B][A]0
[A][B]0
Rate Law
Integrated form
Half-life
Common
unit of k
= −k
[A]0 − [A] = kt
[A]0
2k
mol dm−3 s−1
= −k[A]
[A] = [A]0 e−kt
ln 2
k
s−1
1
k[A]0
mol−1 dm3 s−1
-
mol−1 dm3 s−1
= −k[A]2
d[A]
dt
Rate Law
Differential form
a×b
=
1
[A]
−
1
[A]0
= kt
(a2 b3 − a3 b2 )i + (a3 b1 − a1 b3 )j + (a1 b2 − a2 b1 )k
�
�
� i
j
k ��
�
� a1 a2 a3 �
�
�
� b1 b2 b3 �
b
a · b = a 1 b1 + a 2 b2 + a 3 b3
If a = a1 i + a2 j + a3 k and b = b1 i + b2 j + b3 k then
a
θ
a · b = |a| |b| cos θ
initial volumes.
where Vfinal and Vinitial are the
 and 
 final
∆H
∆G
∂
=− 2 .
Gibbs-Helmholtz equation:
∂T
T
T
p
First Law: For a closed system, ∆U = q + w. Here ∆U is the
change in internal energy of a system, w is the work done on
the system, and q is the heat energy transferred to the system.
Enthalpy: H = U +pV where U = internal energy, p=pressure
and V = volume.
� 
Heat capacity at constant volume:
CV = ∂U
�∂T V
Heat capacity at constant pressure:
Cp = ∂H
∂T p
In general Cp depends upon T . Values of Cp at temperatures
not much different from room temperature can be estimated
c
from
Cp = a + bT + 2
T
where a, b and c are experimentally determined constants.
Second Law of thermodynamics:
During a spontaneous change, the total entropy of an isolated
system and its surroundings increases: ∆S > 0. For a reversible
process, at constant temperature, T , change in entropy
qrev
∆S =
T
where qrev = energy reversibly transferred as heat.
Boltzmann formula: S = kB ln W where W = ‘weight’ of
the most probable configuration of the system and kB is the
Boltzmann constant.
Helmholtz energy: A = U − T S.
Gibbs energy: G = H − T S.
Change in Gibbs energy:
∆G = ∆H − T ∆S (at constant
temperature).
Entropy change for isothermal expansion of an ideal gas:


Vfinal
∆S = nR ln
Vinitial
General Thermodynamics
A
B
Γ
∆
E
Z
H
Θ
b
Scalar product:
+
y2
+
z2 .
α
β
γ
δ

ζ
η
θ
alpha
beta
gamma
delta
epsilon
zeta
eta
theta
I
K
Λ
M
N
Ξ
O
Π
ι
κ
λ
µ
ν
ξ
o
π
NA = 6.022 × 1023 mol−1
kB = 1.381 × 10−23 J K−1
h = 6.626 × 10−34 J s
e = 1.602 × 10−19 C
R = 8.314 J K−1 mol−1
0 = 8.854 × 10−12 J−1 C2 m−1
c = 2.998 × 108 m s−1
F = e NA = 96.485 kC mol−1
iota
kappa
lambda
mu
nu
xi
omicron
pi
P
Σ
T
Υ
Φ
X
Ψ
Ω
ρ
σ
τ
υ
φ
χ
ψ
ω
rho
sigma
tau
upsilon
phi
chi
psi
omega
The Greek alphabet
Units & Conversions
If r = xi + yj + zk then |r| =
x2
Avogadro constant
Boltzmann constant
Planck constant
Elementary charge
Ideal gas constant
Vacuum permittivity
Speed of light (vacuum)
Faraday constant
Physical constants
�
Dalton’s Law of partial pressures: states that the pressure
exerted by a mix of ideal gases is the sum of the partial pressures
each would exert if they were alone in the same volume:
n

pi
Amount of substance etc.
Scientific notation: is used to express large or small numbers
concisely. Each number is written in the form a × 10n where a
is usually a number between 1 and 10. We make use of
. . . 0.01 = 10−2 , 0.1 = 10−1 , . . . , 100 = 102 , 1000 = 103 , . . .
Then, for example, 6859 = 6.859 × 1000 = 6.859 × 103 and
0.0932 = 9.32 × 0.01 = 9.32 × 10−2 .
SI base units: for most quantities it is necessary to specify the
units in which they are measured.
Quantity
SI unit
Symbol
length
metre
m
mass
kilogram kg
time
second
s
temperature
kelvin
K
amount of substance mole
mol
current
ampere
A
luminous intensity
candela
cd
Derived units are formed from the base units. For example, the
unit of force is found by combining units of mass, length and
time in the combination kg m s−2 . This combination is more
usually known as the newton, N.
Property
unit name
unit symbols
frequency
hertz
Hz = s−1
force
newton
N = kg m s−2
pressure
pascal
Pa = N m−2
energy
joule
J = N m = kg m2 s−2
charge
coulomb
As
potential difference
volt
V = J C−1
power
watt
W = J s−1
Celsius temperature degree Celsius ◦ C
Capacitance
farad
F = C V−1
Resistance
ohm
Ω
Common prefixes: a prefix is a method of multiplying the SI
unit by an appropriate power of 10 to make it larger or smaller.
J
Order
J
where k = rate constant, Ea = the activation energy for the
reaction, R = ideal gas constant, T = absolute temperature,
and A is a constant.
By taking logarithms this can be expressed as
A
Ea
k
= ln −
−
ln −
k◦
k◦
RT
◦
where k−
is a chosen standard rate constant. Together, A and
Ea are called the Arrhenius parameters.
Reaction Thermodynamics
i
Kinetics
=
i
a
θ
a × b = |a| |b| sin θ ê
eˆ
Vector product:
Vectors
Multiple Prefix Symbol Multiple Prefix Symbol
1012
tera
T
10−1
deci
d
109
giga
G
10−2
centi
c
106
mega
M
10−3
milli
m
103
kilo
k
10−6
micro µ
102
hecto h
10−9
nano
n
101
deca
da
10−12
pico
p
Interconversion of units:
Sometimes alternative sets of units are used and conversion
between these is needed.
Pressure is often quoted in units corresponding to the
Standard atmosphere (atm).
Then 1 atm = 1.01325×105 Pa = 1.01325 bar = 760mm Hg =
760 Torr. (1 bar = 1 × 105 Pa).
Length (of bonds) is sometimes quoted in Ångströms, Å where
1 Å = 10−10 m. 1 nm = 10−9 m. 0.1nm=100pm.
Energy is often measured in calories (cal): 1 cal = 4.184 J.
d
∆r H
.
ln K =
dT
RT 2




◦
K2
1
∆r H −
1
ln
−
=−
K1
R
T2
T1
van’t Hoff equation:
−
◦
◦
= RT ln K
At equilibrium Q = K, ∆r G = 0 and −∆r G−
where
 v
K=
(aJJ )equilibrium
where aJ is the activity of species J and νJ is its stoichiometric
number.
Rate Laws
In the table, [A] = molar concentration of reactant A at time
t. [A]0 = concentration of reactant A at time t = 0.
Standard state: The standard state of a substance is the pure
substance at a pressure of 1 bar. The standard state value is
◦
denoted by the superscript symbol −
◦ , as in G−
.
∆G
Reaction Gibbs energy:
∆r G =
is the slope of the
∆ξ
graph of Gibbs energy against the progress of the reaction.
Here, ∆ξ = ∆nJ /νJ for all species J in the reaction. Reaction
Gibbs energy at any composition of the reaction mixture can
be written
 v
◦
∆r G = ∆r G−
+ RT ln Q
where Q =
aJ J
Arrhenius equation: The rate at which most chemical reactions proceed depends upon the temperature. The amount of
energy necessary for the reaction to take place at all is called
the activation energy. These quantities are related by the
Arrhenius equation:
k = Ae−Ea /(RT )
Vm,i is the partial molar volume of the ith substance.
Total Gibbs energy for the mixture is G = nA GA + nB GB
where GA and GB are the partial molar Gibbs energies of substances A and B respectively. The partial molar Gibbs energies
are also denoted µA and µB so that G = nA µA +nB
µB . More
n i µi .
generally, with mixtures of several substances G =
where Vm,A = partial molar volume of A and
Vm,B = partial
ni Vm,i , where
molar volume of B. More generally, V =
ê is a unit vector perpendicular to the plane containing a and
b in a sense defined by the right hand screw rule.
If a = a1 i + a2 j + a3 k and b = b1 i + b2 j + b3 k then
where µ∗A = chemical potential of pure A and xA is the mole
fraction.
Properties of mixtures: suppose an amount nA of substance
A is mixed with nB of substance B. The total volume of the
mixture is
V = nA Vm,A + nB Vm,B
Mixtures
Raoult’s law: states that the partial vapour pressure, pA , in a
liquid mixture, A, is proportional to its mole fraction, xA , and
its vapour pressure when pure, p∗A :
pA = xA p∗A .
Henry’s law: states that the vapour pressure, pB , of a volatile
solute, B, is proportional to its mole fraction, xB , in a solution:
pB = xB KB . Here KB is Henry’s law constant.
Chemical potential of a solvent:
µA = µ∗A + RT ln xA
The mole is the amount of substance that contains 6.0221415×
1023 (Avogadro constant/ mol−1 ) atoms or molecules of the
pure substance being measured. For example 1 mole (mol) of
potassium will contain NA atoms. 1 mole of water contains NA
water molecules. A mole of any substance contains as many
atoms or molecules (as specified) as there are atoms in 12g of
the carbon isotope 612 C. (To avoid confusion, always specify the
type of particle.)
The molar mass, M , of a substance is the mass divided by
chemical amount. For pure materials, molar mass, M = molecular weight (or relative atomic or molecular mass) × g mol−1 .
The concentration or molarity of a solute is c = Vn where
n = amount of solute, V = volume of solution. Molarity is
usually quoted in mol dm−3 . The unit 1 mol dm−3 is commonly
denoted 1 M and read as ‘molar’.
The molality, b, of a solute is the amount of solute divided by
n
the mass of solvent: b = m
. So, molality represents 1 mole of
a solute in 1 kg of solvent.
The mole fraction, x, of a solute is the amount of solute divided
by the total amount in the solution:
n
x=
.
ntotal
The mass density (or just density), ρ, of a substance is its
mass divided by its volume, that is ρ = m
.
V
The mass percentage is the mass of a substance in a mixture
as a percentage of the total mass of the mixture.
The mass-volume percentage is the number of grams of solute
in 100 millilitres of solution.
The volume-volume percentage is the number of millilitres of
liquid solute in 100 millilitres of solution.
Parts per million (ppm):
mg/kg.
Parts per billion (ppb):
µg/kg.
Mass obtained
Yield:
% yield =
× 100
Theoretical yield
ptotal = p1 + p2 + . . . + pn
or
ptotal =
i=1
The partial pressure, pi , of one of the gases can be calculated
by multiplying the gas mole fraction, xi , by the total pressure
of all the gases, ptotal .
Perfect Gas Law: pV = nRT where R = ideal gas constant.
p 2 V2
p 1 V1
=
.
Combined Gas Laws:
T1
T2
Virial equation of state: This improves the perfect gas law
because it takes into account intermolecular forces.
B
C
pVm = RT (1 +
+ 2 + . . .)
Vm
Vm
Vm = Vn = molar volume, B, C etc are the virial coefficients.
Observe that when the molar volume is very large, the terms
B
and VC2 become increasingly less important, and in the limit
Vm
m
we obtain the ideal gas law.
Van der Waals equation takes into account the finite distance
between molecules and interparticle attractions:


an2
RT
a
p + 2 (V − nb) = nRT
p=
− 2 or
Vm − b
Vm
V
a is a measure of attraction between particles, b is the volume
excluded by a mole of particles.
Phases
Gibbs’ phase rule: F = C − P + 2, where F is the number of degrees of freedom, C = number of independent components, P = the number of phases in equilibrium with each
other. This is a relationship used to determine the number of
state variables, F , chosen from amongst temperature, pressure
and species compositions in each phase, which must be specified
to fix the thermodynamic state of a system in equilibrium.
Clapeyron equation relates change in pressure to change in
temperature at a phase boundary. The slope of the phase
dp
boundary is dT
.
dp
∆H
=
dT
T ∆V
Here ∆H = molar enthalpy of transition, ∆V = change in
molar volume during transition.
Gases
The general form of an equation of state is p = f (T, V, n)
where p = pressure, T is temperature, V = volume, n = amount
of substance.
Boyle’s Law relates the pressure and volume of a gas when the
temperature is fixed: p1 V1 = p2 V2 .
1
An equivalent form is: pV = nM c2 , where n = number of
3
molecules, m = mass of a molecule, M (= mNA ) is the molar
mass of the molecules, c2 = mean square speed.
Charles’s Law relates the volume and temperature of a gas at
V1
T1
=
.
V2
T2
Clausius-Clapeyron equation is an approximation of the
Clapeyron equation for a liquid-vapour phase boundary. Plotting
vapour pressure for various temperatures produces a curve. For
1
pure liquids, plotting ln pp−
◦ against T produces a straight line
◦
with gradient − ∆H
. (Here, p−
= any standard pressure).
R


p1
∆H
1
1
=−
−
p2
R
T1
T2
constant pressure:
ln
Here ∆H = molar enthalpy of vaporisation. This equation relates the natural logarithm of the vapour pressure to the temperature at a phase boundary.
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The sample unbiased estimate �
of standard deviation, s, is the
�
n
2
i=1 (xi − x̄)
. The standard
square root of the variance: s =
n−1
deviation of the sample mean is called the standard error of the
mean and is equal to √σn , and is often estimated by √sn .
Population values, or parameters, are denoted by Greek letters.
Population mean = µ. Population variance = σ 2 . Population
standard deviation = σ. Sample values, or estimates, are denoted
by roman letters.
The mean of a sample of n observations x1 , x2 , . . . xn is
�n
x1 + x2 + · · · + xn
i=1 xi
=
x̄ =
n
n
The sample mean x̄ is an unbiased estimate of the population
mean µ. The unbiased estimate of the variance of these n sample
�
n
2
i=1 (xi − x̄)
which can be written as
observations is s2 =
n−1
n
1 � 2
nx̄2
xi −
n − 1 i=1
n−1
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© mathcentre 2010
43799 D&PS Jan11
a
(af (x) + bg(x)) dx = a
�
Integration by parts:
b
The linearity rule:
�
s2 =
1
k
x
n+1 + c
ln x + c or ln c� x
ex + c
ekx
k +c
− k1 cos kx + c
sin kx + c
xn , (n �= −1)
x−1 = x1
ex
ekx
sin kx
cos kx
x2
2
x3
3
x2
kx + c
+c
�
f (x)
(2)
(3)
x
if a is negative
(1)
(3)
(2)
x
−b/2a
Quadratic functions: y = ax2 + bx + c
Graph of y = ex and y = e−x Graph of y = ln x and y = log x
10
showing exponential growth/ decay
x
x
1
y=ex
Exponential and log functions:
e ≈ 2.718 is the exponential constant.
y
x
c
t
posi
ive
i
grad
ent
x
c
(x , y )
2
2
y2 − y1
m=
x2 − x1
y
y
ive
neg
at
grad
ient
(x2 ,y2 )
(x1 ,y1 )
1
x
0
1
2x
nxn−1
k cos(kx)
−k sin(kx)
ex
kekx
k, constant
x
x2
xn , constant n
sin(kx)
cos(kx)
ex
ekx
ln kx = loge kx
dy
dx
y = f (x)
x
minimum point
dy
>0
dx
Mathematics for
Chemistry
Facts & Formulae
For the help you need to
support your course
dy
<0
dx
y
= f � (x)
dy
Differentiating a function, y = f (x), we obtain its derivative dx
.
This new function tells us the gradient (slope) of the original
dy
= 0 the gradient is zero.
function at any point. When dx
The straight line: y = mx + c.
m=gradient (slope), c = vertical intercept.
Differentiation
Graphs of common functions
3
,
sin 60 =
2
◦
opposite
3
1
cos 45◦ = √ ,
2
√
3
cos 30◦ =
,
2
1
,
2
√
(x1 ,y1 )
The linearity rules:
d
du dv
d
df
(u(x)±v(x)) =
±
(k
× f (x)) = k ×
dx
dx dx
dx
dx
for k constant.
The product and quotient rules:
v du − u dv
dv
du
d �u�
d
(uv) = u
+v
= dx 2 dx .
dx
dx
dx
dx v
v
The chain rule:
dy
dy
If y = y(u) where u = u(x) then dx
= du
× du
.
dx
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students and staff free resources to support the transition from school mathematics to university
mathematics in a range of disciplines.
Generally, to split a quantity in the ratio m : n, the quantity is
m
n
of the total and m+n
of the total.
split into m+n
sin 30◦ =
y=e-x
y=ln x
y = log1 0 x
d2 f
df
Higher derivatives: f �� (x), or
, means differentiate
dx2 � �
dx
df
d2 f
d
.
with respect to x. That is,
=
dx2
dx dx
Partial derivatives: If f = f (x, y) is a function of two (or more)
∂f
independent variables, ∂x
means
differentiate
f with respect to
means differentiate f
x treating y as if it were a constant. ∂f
∂y
with respect to y treating x as if it were a constant.
Some common conversions are:
1
1
3
1
= 10%,
= 25%,
= 50%,
= 75%
10
4
2
4
Ratios are an alternative way of expressing fractions. Consider
dividing £200 between two people in the ratio of 3:2. For every
£3 the first person gets, the second person gets £2. So the
first gets 35 of the total, and the second gets 52 of the total;
that is £120 and £80.
1
sin 45◦ = √ ,
2
1
f (x) dx
Integration
b2 − 4ac > 0
b2 − 4ac = 0
b2 − 4ac < 0
60◦ 1
◦
30
√
1
www.mathcentre.ac.uk
× 100% =
33 13 %
45
1
-1
This leaflet has been produced in conjunction with the
Higher Education Academy Maths, Stats & OR Network,
the UK Physical Sciences Centre, and sigma
+c
2
(1)
For more copies contact the Network at
[email protected]
n+1
k, constant
x
Statistics
(a, b constant)
a
side adjacent to θ
θ
=
a
hypotenuse
c
adjacent
side opposite to θ
b
Pythagoras’ theorem:
tan θ =
=
side adjacent to θ
a
a2 + b2 = c2
Standard triangles:
2
if a is positive
�
as a percentage is
1
3
b
cos θ =
1
Third priority
Third priority
numerator
fraction =
denominator
Adding and subtracting fractions: to add or subtract two fractions first rewrite each fraction so that they have the same denominator. Then, the numerators are added or subtracted as
appropriate and the result is divided by the common denomina3
16
15
31
4
+
=
.
tor: e.g. + =
5
4
20
20
20
Multiplying fractions: to multiply two fractions, multiply their
numerators and then multiply their denominators: e.g.
3
5
3×5
15
×
=
=
7
11
7 × 11
77
Dividing fractions: to divide two fractions, invert the second
3
2
3
3
9
and then multiply: e.g. ÷ = × =
.
5
3
5
2
10
Proportion and Percentage:
To convert a fraction to a percentage multiply by 100 and label
the result as a percentage.
5
as a percentage is 58 × 100% = 62.5%
8
1
3
2π
π
=
radians
360
180
180
degrees ≈ 57.3◦
π
Trig ratios for an acute angle θ:
e
b
side opposite to θ
us
=
sin θ =
ten
po c
hypotenuse
c
y
h
◦
Addition +
Subtraction −
Fractions:
Removing brackets:
a(b + c) = ab + ac,
1 radian =
√
−b/2a
g(x) dx,
�
du
v dx.
dx
Second priority
Second priority
Second priority
1◦ =
(1)
(2)
(3)
f (x) dx + b
b
Of
Division ÷
Multiplication ×
Algebra
Trigonometry
360◦ = 2π radians,
b2 − 4ac < 0
b2 − 4ac = 0
b2 − 4ac > 0
�
a
dv
dx = [uv]ab −
dx
The BODMAS rule reminds us of the order in which operations
are carried out. BODMAS stands for:
Brackets ()
First priority
Degrees and radians:
(1)
(2)
(3)
u
Arithmetic
When multiplying or dividing positive and negative numbers the
sign of the result is given by:
positive × positive = positive positive × negative = negative
negative × positive = negative negative × negative = positive
positive
positive
= positive
= negative
positive
negative
negative
negative
= negative
= positive
positive
negative
1
cos 60 = ,
2
◦
tan 45◦ = 1
1
tan 30◦ = √
3
√
◦
tan 60 = 3
Bragg’s Law: a fundamental law of x-ray crystallography.
nλ = 2d sin θ
λ = wavelength of the x-rays incident on a crystal lattice with
planes a distance d apart, n is an integer (whole number), and
θ, the Bragg angle, is the angle between the incident beam and
the reflecting planes.
Matrices� and�Determinants
a b
has determinant
c� d �
�a b�
� = ad − bc
|A| = ��
c d�


a11 a12 a13
The 3 × 3 matrix A =  a21 a22 a23  has determinant
a31 a32 a33
�
�
�
�
�
�
� a22 a23 �
�
�
�
�
� − a12 � a21 a23 � + a13 � a21 a22 �
|A| = a11 ��
� a31 a33 �
� a31 a32 �
a32 a33 �
The 2 × 2 matrix A =
The inverse
of�a 2 × 2 matrix:
�
�
�
1
a b
d −b
If A =
then A−1 =
c d
ad − bc −c a
provided that ad − bc �= 0.
Matrix multiplication: for 2 × 2 matrices
�
��
� �
�
a b
α γ
aα + bβ aγ + bδ
=
c d
β δ
cα + dβ cγ + dδ
Remember that AB �= BA except in special cases.
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© mathcentre 2010
a(b − c) = ab − ac
a
ac
(a + b)(c + d) = ac + ad + bc + bd
c=
b
b
Formula for solving a quadratic equation: √
2
−b ± b − 4ac
if ax2 + bx + c = 0 then x =
2a
Laws of Indices:
m
a
= am−n (am )n = amn (ab)n = an bn
am an = am+n
an
√
√
m
1
a0 = 1
a−m = m
a1/n = n a
a n = ( n a)m
a
Logarithms: for any positive base b (with b = 1)
logb A = c means A = bc .
Logarithms to base e, denoted loge or alternatively ln are called
natural logarithms. The letter e stands for the exponential constant which is approximately 2.718.
loge A or ln A = c means A = ec .
c, the natural logarithm of a number A, is the power to which
e would have to be raised to equal A. Note:
eln A = A if A > 0;
ln(eA ) = A
Logarithms to base 10: log10 A = c means A = 10c .
pH: of a solution measures its acidity or basicity.
◦
pH = − log10 ([H+ ]/c−
)
so
◦
[H+ ] = 10−pH c−
where [H+ ] = hydrogen ion concentration in mol dm−3 and
◦
c−
= 1 mol dm−3 . Equivalently, pH = − log10 aH3 O+ where
aH3 O+ = hydronium ion activity.
Laws of Logarithms: for any positive base b, with b = 1,
A
logb A − logb B = logb ,
logb A + logb B = logb AB,
B
logb 1 = 0,
logb b = 1.
n logb A = logb An ,
bx
. Specifically,
Formula for change of base:
loga x = log
logb a
ln x
log10 x =
.
ln 10
Inequalities: a > b means a is greater than b
a < b means a is less than b
a  b means a is greater than or equal to b
a  b means a is less than or equal to b
Sigma notation: The Greek letter sigma, Σ, is used to abbreviate addition. If we have n values, x1 , x2 , . . . xn , and add them,
n

the sum x1 + x2 + . . . xn is written
xi .
i=1
Note that i takes all whole number values from 1 to n. So, for
3

instance
xi means x1 + x2 + x3 .
i=1
5
i2 means 12 + 22 + 32 + 42 + 52 .
n

xi = x1 · x2 · . . . · xn .
Product notation:
Example:
i=1
i=1
Produced in conjunction with the Higher Education Academy Maths,
Stats & OR Network, the UK Physical Sciences Centre, sigma, and
the HE STEM Programme. Thanks to the many individuals who
c 2010. - version draft 1, August 2010.
offered detailed advice. 
www.mathcentre.ac.uk
© mathcentre 2010
(* A + B → P reaction.)
= kt
ln

K2
K1

van’t Hoff equation:
◦
d
∆r H −
.
ln K =
dT
RT 2


◦
∆r H −
=−
R
2∗
1
1
−
T2
T1
◦
= RT ln K
At equilibrium Q = K, ∆r G = 0 and −∆r G−
where

(aJvJ )equilibrium
where aJ is the activity of species J and νJ is its stoichiometric
number.
Standard state: The standard state of a substance is the pure
substance at a pressure of 1 bar. The standard state value is
◦
denoted by the superscript symbol −
◦ , as in G−
.
∆G
Reaction Gibbs energy:
∆r G =
is the slope of the
∆ξ
graph of Gibbs energy against the progress of the reaction.
Here, ∆ξ = ∆nJ /νJ for all species J in the reaction. Reaction
Gibbs energy at any composition of the reaction mixture can
be written
 v
◦
∆r G = ∆r G−
+ RT ln Q
where Q =
aJ J
J
Reaction Thermodynamics
Vm,i is the partial molar volume of the ith substance.
Total Gibbs energy for the mixture is G = nA GA + nB GB
where GA and GB are the partial molar Gibbs energies of substances A and B respectively. The partial molar Gibbs energies
are also denoted µ and µ so that G = n µ +n µ . More
A
B
A
A
B
B

n i µi .
generally, with mixtures of several substances G =
i
where Vm,A = partial molar volume of A and
Vm,B = partial
ni Vm,i , where
molar volume of B. More generally, V =
Order
∗
where µA
= chemical potential of pure A and xA is the mole
fraction.
Properties of mixtures: suppose an amount nA of substance
A is mixed with nB of substance B. The total volume of the
mixture is
mol dm−3 s−1
[A]0
2k
Rate Law
Integrated form
Common
unit of k
Half-life
Rate Laws
In the table, [A] = molar concentration of reactant A at time
t. [A]0 = concentration of reactant A at time t = 0.
Arrhenius equation:
k = Ae−Ea /(RT )
where k = rate constant, Ea = the activation energy for the
reaction, R = ideal gas constant, T = absolute temperature,
and A is a constant.
By taking logarithms this can be expressed as
A
Ea
k
= ln −
−
ln −
k◦
k◦
RT
◦
where k−
is a chosen standard rate constant. Together, A and
Ea are called the Arrhenius parameters.
Arrhenius equation: The rate at which most chemical reactions proceed depends upon the temperature. The amount of
energy necessary for the reaction to take place at all is called
the activation energy. These quantities are related by the
Kinetics
=
a×b
=
(a2 b3 − a3 b2 )i + (a3 b1 − a1 b3 )j + (a1 b2 − a2 b1 )k
�
�
�
j
k ��
� i
�
�
� a1 a2 a3 �
� b1 b2 b3 �
ê is a unit vector perpendicular to the plane containing a and
b in a sense defined by the right hand screw rule.
If a = a1 i + a2 j + a3 k and b = b1 i + b2 j + b3 k then
a
θ
a × b = |a| |b| sin θ ê
eˆ
Vector product:
ln xA
b
a · b = a 1 b1 + a 2 b2 + a 3 b3
If a = a1 i + a2 j + a3 k and b = b1 i + b2 j + b3 k then
a
θ
a · b = |a| |b| cos θ
b
Scalar product:
If r = xi + yj + zk then |r| =
�
General Thermodynamics
NA = 6.022 × 1023 mol−1
kB = 1.381 × 10−23 J K−1
h = 6.626 × 10−34 J s
e = 1.602 × 10−19 C
R = 8.314 J K−1 mol−1
0 = 8.854 × 10−12 J−1 C2 m−1
c = 2.998 × 108 m s−1
F = e NA = 96.485 kC mol−1
Avogadro constant
Boltzmann constant
Planck constant
Elementary charge
Ideal gas constant
Vacuum permittivity
Speed of light (vacuum)
Faraday constant
Physical constants
x2 + y 2 + z 2 .
Vectors
The general form of an equation of state is p = f (T, V, n)
where p = pressure, T is temperature, V = volume, n = amount
of substance.
Boyle’s Law relates the pressure and volume of a gas when the
temperature is fixed: p1 V1 = p2 V2 .
1
An equivalent form is: pV = nM c2 , where n = number of
3
molecules, m = mass of a molecule, M (= mNA ) is the molar
mass of the molecules, c2 = mean square speed.
Charles’s Law relates the volume and temperature of a gas at
V1
T1
constant pressure:
=
.
V2
T2
Mixtures
s−1
ln 2
k
= kt
First Law: For a closed system, ∆U = q + w. Here ∆U is the
change in internal energy of a system, w is the work done on
the system, and q is the heat energy transferred to the system.
Enthalpy: H = U +pV where U = internal energy, p=pressure
and V = volume.
� 
Heat capacity at constant volume:
C = ∂U
V
�∂T V
Heat capacity at constant pressure:
Cp = ∂H
∂T p
In general Cp depends upon T . Values of Cp at temperatures
not much different from room temperature can be estimated
c
from
Cp = a + bT + 2
T
where a, b and c are experimentally determined constants.
Second Law of thermodynamics:
During a spontaneous change, the total entropy of an isolated
system and its surroundings increases: ∆S > 0. For a reversible
process, at constant temperature, T , change in entropy
qrev
∆S =
T
where qrev = energy reversibly transferred as heat.
Boltzmann formula: S = kB ln W where W = ‘weight’ of
the most probable configuration of the system and kB is the
Boltzmann constant.
Helmholtz energy: A = U − T S.
Gibbs energy: G = H − T S.
Change in Gibbs energy:
∆G = ∆H − T ∆S (at constant
temperature).
Entropy change for isothermal expansion of an ideal gas:


Vfinal
Vinitial
∆S = nR ln
A
B
Γ
∆
E
Z
H
Θ
α
β
γ
δ

ζ
η
θ
alpha
beta
gamma
delta
epsilon
zeta
eta
theta
I
K
Λ
M
N
Ξ
O
Π
ι
κ
λ
µ
ν
ξ
o
π
iota
kappa
lambda
mu
nu
xi
omicron
pi
P
Σ
T
Υ
Φ
X
Ψ
Ω
ρ
σ
τ
υ
φ
χ
ψ
ω
rho
sigma
tau
upsilon
phi
chi
psi
omega
The Greek alphabet
Gases
Raoult’s law: states that the partial vapour pressure, pA , in a
liquid mixture, A, is proportional to its mole fraction, xA , and
∗
∗
its vapour pressure when pure, pA
:
p A = xA p A
.
Henry’s law: states that the vapour pressure, pB , of a volatile
solute, B, is proportional to its mole fraction, xB , in a solution:
pB = xB KB . Here KB is Henry’s law constant.
Chemical potential of a solvent:
=
+ RT
mol−1 dm3 s−1
1
k[A]0
Multiple Prefix Symbol Multiple Prefix Symbol
1012
tera
T
10−1
deci
d
109
giga
G
10−2
centi
c
106
mega
M
10−3
milli
m
103
kilo
k
10−6
micro µ
2
10
hecto h
10−9
nano
n
101
deca
da
10−12
pico
p
Interconversion of units:
Sometimes alternative sets of units are used and conversion
between these is needed.
Pressure is often quoted in units corresponding to the
Standard atmosphere (atm).
Then 1 atm = 1.01325×105 Pa = 1.01325 bar = 760mm Hg =
760 Torr. (1 bar = 1 × 105 Pa).
Length (of bonds) is sometimes quoted in Ångströms, Å where
1 Å = 10−10 m. 1 nm = 10−9 m. 0.1nm=100pm.
Energy is often measured in calories (cal): 1 cal = 4.184 J.
∗
µA
mol−1 dm3 s−1
-
Amount of substance etc.
The mole is the amount of substance that contains 6.0221415×
1023 (Avogadro constant/ mol−1 ) atoms or molecules of the
pure substance being measured. For example 1 mole (mol) of
potassium will contain NA atoms. 1 mole of water contains NA
water molecules. A mole of any substance contains as many
atoms or molecules (as specified) as there are atoms in 12g of
the carbon isotope 12
6 C. (To avoid confusion, always specify the
type of particle.)
The molar mass, M , of a substance is the mass divided by
chemical amount. For pure materials, molar mass, M = molecular weight (or relative atomic or molecular mass) × g mol−1 .
The concentration or molarity of a solute is c = Vn where
n = amount of solute, V = volume of solution. Molarity is
usually quoted in mol dm−3 . The unit 1 mol dm−3 is commonly
denoted 1 M and read as ‘molar’.
The molality, b, of a solute is the amount of solute divided by
n
the mass of solvent: b = m
. So, molality represents 1 mole of
a solute in 1 kg of solvent.
The mole fraction, x, of a solute is the amount of solute divided
by the total amount in the solution:
n
x=
.
ntotal
The mass density (or just density), ρ, of a substance is its
mass divided by its volume, that is ρ = m
.
V
The mass percentage is the mass of a substance in a mixture
as a percentage of the total mass of the mixture.
The mass-volume percentage is the number of grams of solute
in 100 millilitres of solution.
The volume-volume percentage is the number of millilitres of
liquid solute in 100 millilitres of solution.
Parts per million (ppm):
mg/kg.
Parts per billion (ppb):
µg/kg.
Mass obtained
Yield:
% yield =
× 100
Theoretical yield
µA
1
[A]0
where Vfinal and Vinitial
[B][A]0
[A][B]0
[A]0 − [A] = kt
= −k
Rate Law
Differential form
i
V = nA Vm,A + nB Vm,B
−
ln
[A] = [A]0 e−kt
= −k[A]
d[A]
dt
0
1
[A]
= −k[A]2
d[A]
dt
1
1
[B]0 −[A]0
= −k[A][B]
d[A]
dt
2
J
K=
d[A]
dt
are the
initial volumes.
 and 
 final
∆H
∆G
∂
=− 2 .
∂T
T
T
p
Gibbs-Helmholtz equation:
Units & Conversions
Scientific notation: is used to express large or small numbers
concisely. Each number is written in the form a × 10n where a
is usually a number between 1 and 10. We make use of
. . . 0.01 = 10−2 , 0.1 = 10−1 , . . . , 100 = 102 , 1000 = 103 , . . .
Then, for example, 6859 = 6.859 × 1000 = 6.859 × 103 and
0.0932 = 9.32 × 0.01 = 9.32 × 10−2 .
SI base units: for most quantities it is necessary to specify the
units in which they are measured.
Quantity
SI unit
Symbol
length
metre
m
mass
kilogram kg
time
second
s
temperature
kelvin
K
amount of substance mole
mol
current
ampere
A
luminous intensity
candela
cd
Derived units are formed from the base units. For example, the
unit of force is found by combining units of mass, length and
time in the combination kg m s−2 . This combination is more
usually known as the newton, N.
Property
unit name
unit symbols
frequency
hertz
Hz = s−1
force
newton
N = kg m s−2
pressure
pascal
Pa = N m−2
energy
joule
J = N m = kg m2 s−2
charge
coulomb
As
potential difference
volt
V = J C−1
power
watt
W = J s−1
Celsius temperature degree Celsius ◦ C
Capacitance
farad
F = C V−1
Resistance
ohm
Ω
Common prefixes: a prefix is a method of multiplying the SI
unit by an appropriate power of 10 to make it larger or smaller.
Dalton’s Law of partial pressures: states that the pressure
exerted by a mix of ideal gases is the sum of the partial pressures
each would exert if they were alone in the same volume:
n

or
ptotal =
pi
ptotal = p1 + p2 + . . . + pn
i=1
The partial pressure, pi , of one of the gases can be calculated
by multiplying the gas mole fraction, xi , by the total pressure
of all the gases, ptotal .
Perfect Gas Law: pV = nRT where R = ideal gas constant.
p 2 V2
p 1 V1
=
.
Combined Gas Laws:
T1
T2
Virial equation of state: This improves the perfect gas law
because it takes into account intermolecular forces.
B
C
pVm = RT (1 +
+ 2 + . . .)
Vm
Vm
Vm = Vn = molar volume, B, C etc are the virial coefficients.
Observe that when the molar volume is very large, the terms
B
and VC2 become increasingly less important, and in the limit
Vm
m
we obtain the ideal gas law.
Van der Waals equation takes into account the finite distance
between molecules and interparticle attractions:


an2
RT
a
p + 2 (V − nb) = nRT
p=
− 2 or
Vm − b
Vm
V
a is a measure of attraction between particles, b is the volume
excluded by a mole of particles.
Phases
Gibbs’ phase rule: F = C − P + 2, where F is the number of degrees of freedom, C = number of independent components, P = the number of phases in equilibrium with each
other. This is a relationship used to determine the number of
state variables, F , chosen from amongst temperature, pressure
and species compositions in each phase, which must be specified
to fix the thermodynamic state of a system in equilibrium.
Clapeyron equation relates change in pressure to change in
temperature at a phase boundary. The slope of the phase
dp
boundary is dT
.
dp
∆H
=
dT
T ∆V
Here ∆H = molar enthalpy of transition, ∆V = change in
molar volume during transition.
Clausius-Clapeyron equation is an approximation of the
Clapeyron equation for a liquid-vapour phase boundary. Plotting
vapour pressure for various temperatures produces a curve. For
1
pure liquids, plotting ln pp−
◦ against T produces a straight line
−
◦
∆H
with gradient − R . (Here, p = any standard pressure).


p1
∆H
1
1
ln
=−
−
p2
R
T1
T2
Here ∆H = molar enthalpy of vaporisation. This equation relates the natural logarithm of the vapour pressure to the temperature at a phase boundary.
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