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Basic Chem Math
Gialih Lin
Numbers, variables, and units
• 1.1 Concepts
• Chemistry, in common with the other
physical sciences, comprises (1)
experiment and (2) theory.
• 1. ideal gas
• 2. Bragg’s Law
• 3. the Arrhenius equation
• 4. the Nernst equation
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Function f(x)
Algebra 代數
Constant and variable變數
Independent variable (x of the following
equation)
• y= 3x10-3 x /(1x10-6 + x)
• Dependent variable (y of the above
equation)
1.2 Real numbers
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Nature number
Integers 整數
Addition (+)
Subtration (–)
multiplication (x)
Division (÷)
m/n: m over n or m divided by n
Rational numbers 有理數 such as 2/3
Irrational number 無理數 such as (2)1/2, p
Surds such as root of 2, √2, (2)1/2
The Cubice root of 3, 3√2, (2)1/3
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Fractions 分數; One half, ½; One quarter, ¼
Numerator 分子
Denominator 分母
Algebraic equations
a0+a1x+a2x2+a3x3+… +anxn=0
Algebraic numbers, a0 a1 a2 a3 an
There exist also other numbers that are not algebraic;
they are not obtained as solutions of any finite 有限
algebraic equation. These numbers are irrational
numbers called transcendental 超越numbers.
• Euler number e=2.71828
• Archimedean (阿基米德, Greek mathematician) number
p =3.141592653589793
Transcendental numbers 超越數
• In mathematics, a transcendental number
is a number (possibly a complex number)
that is not algebraic that is it is not a root
of a non-constant polynomial equation with
rational coefficients.
Real number
• The rational and irrational numbers from
the continuum 連續of numbers; together
they are called the real numbers.
1.3 Factorization, factors, and
factorials
• Factorization, factorize 因數分解
• Prime number 質數
• Prime number factorization 因數分解
examples1.6
• Common factor 公因子
• The fundamental theorem of arithmetic算術 is
that every natural number can be factorized as a
product of prime numbers in only one way.
• 30=2x3x5
• Simplification of fractions 分數簡化 examples 1.7
Factorials 階乘
• n!=1x2x3x…xn
• Read as n factorial
• The Euler number e = 1+ 1/1! + 1/2! +
1/3!+ 1/4!+ …
• =1+1+0.5+0.16667+0.041667+
• =2.71828
1.4 Decimal reprsentation of
numbers十進位
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372=300+70+2=3x102+7x10+2
Three hundred and seventy-two
Three seventy two
3816
Thirty-eight sixteen
Significant figures floating number
representation : fixed number of significant
figures with zeros on the left
• 3210=0.3210x104, 3.210=0.3210x101,
0.003210=0.3210x10-2 all have 4 significant
figures.
rounding
四捨五入
• 五入 If the first digit dropped is greater
than or equal to 5, the proceeding digit is
increased by 1; the number is rounded up.
• 四捨 If the first digit dropped is less than 5,
the proceeding digit is left unchangede;
the number is rounded down.
1.5 Variables 變數
• A quantity that can take as its value any
value chosen from a given set of values is
called a variable.
• The set forms the domain 域 of variable
• Continuous variable, discrete 不連續
variable
• Constant variable or constant 常數
Example 1.11. The spectrum of
hydrogen atom氫原子光譜
• Two energy levels of hydrogen atom:
• 2. Continuous energy levels, which all positive
energies, E>0. The corresponding states of the
atom are those of free (unbound) electron
moving in the presence of the electrostatic field
of the nuclear charge. Transitions between these
energy levels and those of the bound states give
rise to continuous ranges of spectra frequencies.
Early Models of the Atom
Therefore, Rutherford’s
model of the atom is
mostly empty space:
The Nature of Energy
• Einstein used this
assumption to explain the
photoelectric effect.
• He concluded that energy is
proportional to frequency:
E = h
where h is Planck’s
constant, 6.626  10−34 J-s.
Photon Theory of Light and the
Photoelectric Effect
The particle theory assumes that an electron
absorbs a single photon.
Plotting the kinetic energy vs. frequency:
This shows clear
agreement with the
photon theory, and
not with wave
theory.
The Uncertainty Principle
Heisenberg showed
that the more
precisely the
momentum of a
particle is known, the
less precisely is its
position is known:
(x) (mv) 
h
4p
The Uncertainty Principle
In many cases, our
uncertainty of the
whereabouts of an
electron is greater
than the size of the
atom itself!
39.6 X-Ray Spectra and Atomic Number
The continuous part of the X-ray spectrum
comes from electrons that are decelerated by
interactions within the material, and therefore
emit photons. This radiation is called
bremsstrahlung (“braking radiation”).
Example 1.11. The spectrum of
hydrogen atom氫原子光譜
• 1. Discrete不連續(quantized量子化 階梯化)
• En = -1/2n2, n = 1, 2, 3, …
• The corresponding states of the atom are
‘bound states’, in which the motion of the
electron is confined 限制to the vicinity of
the nucleus. Transitions between the
energy levels give rise to discrete lines in
the spectrum of the atom.
The Nature of Energy
• For atoms and
molecules, one does
not observe a
continuous spectrum,
as one gets from a
white light source.
• Only a line spectrum
of discrete
wavelengths is
observed.
The Nature of Energy
The energy absorbed or emitted
from the process of electron
promotion or demotion can be
calculated by the equation:
E = −hcRH (
1
1
n f2
ni2
)
where RH is the Rydberg
constant, 1.097  107 m−1, and ni
and nf are the initial and final
energy levels of the electron.
39.2 Hydrogen Atom: Schrödinger Equation and
Quantum Numbers
The time-independent Schrödinger
equation in three dimensions is then:
Equation 39-1 goes here.
where
Equation 39-2 goes here.
1.6 The algebra of real numbers
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Commutative law of addition
Commutative law of multiplication
Associative law of addition
Associative law of multiplication
Distribution law
ab= axb =a．b
Product of a and b. ．(dot), x (cross)
Modulas of a real number a, read as ‘mod a’
∣a∣=+√a2
The index rule
• am
• Read as a to the power m, a to the m, or the mth
power of a.
• a is called the base
• m is the index or exponent
• a1/m =m√a
• Read as the mth root of a
• 21/3 is a cube root of 2
• If x=am then m=logax is the logarithm of x to
base a (see Section 3.7)
Rules of precedence 居先 for
arithmetic operation
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先乘除後加減
Parentheses 括號
小括號 ( ) innermost brackets
中括號 [ ] square brackets
大括號 { } braces (curly brackets)
1.7 Complex numbers 複數
• x2 = -1 are x = ± √-1
• The square root of -1 as a new number which is
usually represented by symbol I (sometimes j)
with the property
• i2 = 1
• I, 4i, and -4i are numbers called imaginary to
distinguish them from real number.
• z= x + iy
• Such numbers are called a complex number
(Discuss in Chapter 8)
1.8 Units 單位
• A physical quantity has two essential
attributes, magnitude and dimensions.
• 2 meters has the dimension of length and
has magnitude equal to 2.
SI Units
• Système International d’Unités
• A different base unit is used for each quantity.
1-4 Units, Standards, and the SI System
Quantity Unit
Length
Time
Mass
Meter
Standard
Length of the path traveled
by light in 1/299,792,458
second
Second
Time required for
9,192,631,770 periods of
radiation emitted by cesium
atoms
Kilogram Platinum cylinder in
International Bureau of
Weights and Measures, Paris
Dimensional Analysis
• We use dimensional analysis to convert
one quantity to another.
• Most commonly, dimensional analysis
utilizes conversion factors (e.g., 1 in. =
1 in.
2.54 cm
2.54 cm)
or
2.54 cm
1 in.
Dimensional Analysis
Use the form of the conversion factor
that puts the sought-for unit in the
numerator:
Given unit 
Conversion factor
desired unit
given unit
= desired unit
Dimensional Analysis
• For example, to convert 8.00 m to inches,
– convert m to cm
– convert cm to in.
100 cm
1 in.
8.00 m 

= 315 in.
1m
2.54 cm
1-5 Converting Units
Unit conversions always involve a conversion
factor.
Example:
1 in. = 2.54 cm.
Written another way: 1 = 2.54 cm/in.
So if we have measured a length of 21.5
inches, and wish to convert it to centimeters,
we use the conversion factor:
1-5 Converting Units
Example 1-2: The 8000-m peaks.
The fourteen tallest peaks in the world are
referred to as “eight-thousanders,” meaning
their summits are over 8000 m above sea level.
What is the elevation, in feet, of an elevation of
8000 m?
1-7 Dimensions and Dimensional Analysis
Dimensions of a quantity are the base units
that make it up; they are generally written
using square brackets.
Example: Speed = distance/time
Dimensions of speed: [L/T]
Quantities that are being added or subtracted
must have the same dimensions. In addition, a
quantity calculated as the solution to a
problem should have the correct dimensions.
1-7 Dimensions and Dimensional Analysis
Dimensional analysis is the checking of
dimensions of all quantities in an equation to
ensure that those which are added, subtracted,
or equated have the same dimensions.
Example: Is this the correct equation for
velocity?
Check the dimensions:
Wrong!
Examples 1.16 Dimentions and
units
• 1. velocity is the rate of change of position
with time, and has dimensions of
length/time: LT-1
• 2. Acceleration is the rate of change of
velocity with time, and has dimensions of
velocity/time: LT-2
• g=9.80665 ms-2=980.665 Gal
• Gal = 10-2 m s-2 (cm s-2) is called the
galileo
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3. force has dimensions of mass x acceleration: [M] x [LT-2] = MLT-2
With SI unit the newton, N=kgms-2
4. Pressure has dimensions of force per unit pascal,
Pa=Nm=kgm2s-2
Standard pressure: bar=105 Pa
Atmosphere: atm= 101325 Pa
Torr: torr=(101325/760)Pa=133.322Pa
5. Work, energy and heat are quantities of the same kind, with the
same dimensions and unit.
Thus, work has dimensions of force x distance:
[MLT-2] x [L] = ML2T-2, with SI unit the joule, J = Nm=kgm2s-2
And kinetic energy ½ mv2 has dimensions of mass x (velocity)2:
[M]x[LT-1] = ML2T-2, with SI unit J
Large and small units
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Table 1.3 SI prefixes
Avogadro’s constant,
NA=6.02214x1023 mol-1
The mass of a mole of 12C is 12 g.
The mass of an atom of 12C atom is therefore
12 g mol-1/NA = 2x10-26 kg = 12 u
• u =(1/NA) g = 1.66054 x10-27 kg
• Is called the unified atomic mass unit
(sometimes called a Dalton, with symbol Da or
amu)
Table 1.3 SI prefixes
Prefixes convert the base units into units that
are appropriate for the item being measured.
fs 飛秒
Examples 1.18 Molecular properties: mass,
length and moment of inertia
• 1. mass, Ar for an atomic mass, Mr for
molecular mass
• Mr(1H216O) = 2x Ar(1H)+ Ar(16O) =
2x1.0078+ 15.9948 =18.0105
• M(1H216O)= 18.0105 g mol-1
• m(1H216O)= Mr(1H216O) x u = 2.9907x10-26
kg
41.1 Structure and Properties of
the Nucleus
A and Z are sufficient to specify a nuclide.
Nuclides are symbolized as follows:
X is the chemical symbol for the element; it
contains the same information as Z but in a
more easily recognizable form.
41.1 Structure and Properties of
the Nucleus
Masses of atoms are measured with
reference to the carbon-12 atom, which is
assigned a mass of exactly 12 u. A u is a
unified atomic mass unit.
Example 1.18
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2. length
mm=10-6 m
micrometer
微米
nm = 10-9m
nanometer 奈米
pm =10-12 m
picometer
微微米或漠米
Å = 10-10 m = 0.1 nm
埃
1 nm =10 Å
Bohr radius a0 = 0.529177 x 10-10 m = 0.529177
Å
• Thus for O2, the bond length of the oxygen
molecule, Re = 1.2075 Å =120.75 pm
Examples 1.18-3. reduced mass 折算
質量and moment of inertia貫性力距
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3. reduced mass and moment of inertia
m = mAmB/(mA+mB)
Thus for CO, Ar(12C)=12 and Ar(16O)=15.9948
reduced mass of 12C16O is
m(12C16O)=
=(12x15.9948/27.9948)(u2/u)=6.8562 u
• =6.8562x1.66054x10-27 kg
• = 1.1385x10-26 kg
the moment of inertia of CO
• The bond length of CO is 112.81 pm =
1.1281x10-10 m, so that the moment of
inertia of the molecule is
• I = mR2 =
• (1.138x10-26 kg)x (1.1281x10-10m)2
• =1.4489x10-46 kg m2
Physics Chapter 10
Rotational Motion
Physcis Chapter 11
Angular Momentum;
General Rotation
11-9 The Coriolis Effect
The Coriolis effect is
responsible for the
rotation of air around
low-pressure areas—
counterclockwise in
the Northern
Hemisphere and
clockwise in the
Southern. The Coriolis
acceleration is:
10-5 Rotational Dynamics; Torque and
Rotational Inertia
The quantity
is called the
rotational inertia of an object.
The distribution of mass matters here—these
two objects have the same mass, but the one on
the left has a greater rotational inertia, as so
much of its mass is far from the axis of rotation.
10-5 Rotational
Dynamics; Torque
and Rotational
Inertia
The rotational inertia of
an object depends not
only on its mass
distribution but also the
location of the axis of
rotation—compare (f)
and (g), for example.
10-8 Rotational Kinetic Energy
The kinetic energy of a rotating object is given
by
By substituting the rotational quantities, we find
that the rotational kinetic energy can be written:
A object that both translational and rotational
motion also has both translational and rotational
kinetic energy:
40.4 Molecular Spectra
The overlap of orbits alters energy levels in
molecules. Also, more types of energy levels
are possible, due to rotations and vibrations.
The result is a band of closely spaced energy
levels.
40.4 Molecular Spectra
Figure 40-16 goes
here.
A diatomic molecule
can rotate around a
vertical axis. The
rotational energy is
quantized.
40.4 Molecular Spectra
These are some
rotational energy
levels and allowed
transitions for a
diatomic molecule.
40.4 Molecular Spectra
Example 40-4: Reduced mass.
Show that the moment of inertia of a
diatomic molecule rotating about its center
of mass can be written
I = m r 2,
where
m1m2
m=
m1  m2
.
Examples 1.19 Molecular properties:
wavelength, frequency, and energy
• The wavelength l and frequency  of electromagnetic radiation are
related to the speed of light by
• c = l
• E = h
• h = 6.62608x10-34 Js
 E= E1-E2
• The wavelength of one of yellow D lines in the electronic spectrum
of the sodium atom is l=589.76 nm.
 =c/l=3x108 ms-1/ 5.8975x10-7 m = 5.0833x1014 s-1
 E =h
• =(6.62608x10-34 Js)x(5.0833x1014 s-1)
• =3.368x10-19 J=2.102 eV=202.8 kJ mol-1
• eV = 1.60218x10-19 J
• For sodium
• eV x NA=(1.60218x10-19 J)x(6.02214x1023 mol-1)=96.486 kJ mol-1
Wavenumber
• Wavenumber ύ
• =1/l=/c=E/hc
• The wavelength of sodium
• 1/ 5.8975x10-5 cm =16956 cm-1
• The second line of sodium doublet lines at
16973 cm-1, and the fine structure splitting
due to spin-orbit coupling in the atom is 17
cm-1.
Approximation calculations
• Powers of 10 are often used as a description of
order of magnitude; for example, if a length is
two orders of magnitude larger than length B
then it is about 102=100 times larger.
• I = mR2 =
• (1.138x10-26 kg)x (1.1281x10-10m)2
• =1.4489x10-46 kg m2
• approximation
• (10-26 kg)x (10-10m)2
• =10-46 kg m2
Atomic units
• Schrödinger Equation for the motion of the stationary nucleus in the
hydrogen atom
• m the rest mass of the electron
• e the charge on the proton
• h Plank’s constant
• ħ = h/ 2p
 eo the permittivity of a vacuum
• ▽ : gradient see p467
• grad f = ▽ = (f/x)I + (f/y) j + (f/ z) k
• Unit of x-axis I = (1,0,0); unit of y-axis j=(0,1,0);unit of z-axis k=(0,0,1)
• The Laplacian operator
• ▽2 = (2/x2) + (2/y2) + (2/z2)
Chapter 38
Quantum Mechanics
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
The time-independent Schrödinger
equation in three dimensions is then:
Equation 39-1 goes here.
where
Equation 39-2 goes here.
Atomic units (Table 1.4)
• bohr, a0 =4pe0ħ2/mee2=5.29177x10-11 m =0.53 Å
• The unit of energy, Eh (sometimes au), is called
the hartree, and is equal to twice the ionization
energy of the hydrogen atom.
• Eh = mee4/16p2e02ħ2 = 4.35074x10-18 J=2620
kJ/mol
• For one mole of an atom or molecular,
• NAx Eh = (6.02x1023)x(4.35074x10-18 J)=2620 kJ
Example 1.21 The atomic unit of
energy
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Coulomb’s law
V=q1q2/4pe0r
q1=Z1e and r=Ra0
V=(Z1Z2/R)(e2/4pe0a0)
1. To show that the unit is the hartree unit Eh in
Table 1.4, use a0= 4pe0ħ2/mee2;
• e2/4pe0a0 =(e2/4pe0)/(4pe0ħ2/mee2)=
• (e2/4pe0)/(mee2/4pe0ħ2)= mee4/16p2e02ħ2
• =Eh
• 2. To calculate the value of Eh in SI units,
use the values of e and a0 given in Table
1.4.
• e2/4pe0a0 =
(1.602182/4x3.14159x8.85419x5.29177)
x(10-19x10-19/10-12x10-11)x(C2/Fm-1 m) =
• (4.35975x10-3)x(10-15)x(C2F-1)
• From Table 1.2, J = C2F-1
• e2/4pe0a0 = 4.35975x10-18 J = Eh
理論計算(Gaussian 03) 聯苯(biphenyl)轉動中心C-C鍵能量變化曲線圖
旋轉聯苯中心C-C鍵而成之三構形(conformation)能階示意圖
理論計算(Gaussian 03) 4-n-butylcarbamyloxy-4’-acetyloxy-biphenyl 轉動中心C-C鍵
能量變化曲線圖
旋轉4-n-butylcarbamyloxy-4’-acetyloxy-biphenyl中心C-C鍵而成之
三構形(conformation)能階示意圖