Basic Chem Math

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Basic Chem Math

• Gialih Lin

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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. Braggs Law
• 3. the Arrhenius equation
• 4. the Nernst equation

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• Function f(x)
• f(x) is a function of x, x is an independent
  variable then f(x) is called as dependent
  variable.
• Algebra ??
• Constant and variable??
• Independent variable (x of the following
  equation)
• y 3x10-3 x /(1x10-6 x)
• Dependent variable (y of the above equation)

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Example

• In the following equation,
• v 3.1x10-3 S / (1.2x10-6 S),
• how many constants in this equation?
• Ans 2

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1.2 Real numbers

• Nature number
• Integers ??
• Addition ()
• Subtration ()
• multiplication (x)
• Division ()
• m/n m over n or m divided by n
• Rational numbers ??? such as 2/3
• Irrational number ??? such as (2)1/2, p
• Surds such as root of 2, v2, (2)1/2
• The Cubic root of 3, 3v2, (2)1/3

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Example

• rational numbers ??
• 2.71828 ,
• 3.14159,
• 1 1/1! 1/2! 1/3! 1/4!,
• (1)1/3

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Example

• irrational numbers?
• e,
• p,
• (3)1/2

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• Fractions ?? One half, ½ One quarter, ¼
• Numerator ??
• Denominator ??
• Algebraic equations
• a0a1xa2x2a3x3 anxn0
• 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 ??algebraic equation. These numbers
  are irrational numbers called transcendental
  ??numbers.
• Euler number e2.71828
• Archimedean (????, Greek mathematician) number p
  3.141592653589793

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Transcendental numbers ???

• 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.

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Real number ??

• The rational and irrational numbers from the
  continuum ??of numbers together they are called
  the real numbers.

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1.3 Factorization, factors, and factorials

• Factorization, factorize ????
• Prime number ??
• Prime number factorization ???? examples1.6
• Common factor ???
• The fundamental theorem of arithmetic?? is that
  every natural number can be factorized as a
  product of prime numbers in only one way.
• 302x3x5
• Simplification of fractions ???? examples 1.7

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• The fundamental theorem of arithmetic is that
  every natural number can be factorized as a
  product of prime numbers (??) in only one way.

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Factorials ??

• n!1x2x3xxn
• Read as n factorial
• The Euler number e 1 1/1! 1/2! 1/3! 1/4!
  
• 110.50.166670.041667
• 2.71828

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Example

• The Euler number
• e 1 1/1! 1/2! 1/3! 1/4! 1/n!
• Use a calculator to calculate the Euler number
  to n7 in 4 significant numbers.
• Ans 2.718

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1.4 Decimal reprsentation of numbers???

• 3723007023x1027x102
• 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
• 32100.3210x104, 3.2100.3210x101,
  0.0032100.3210x10-2 all have 4 significant
  figures.

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rounding????

• ?? If the first digit dropped is greater than or
  equal to 5, the proceeding digit is increased by
  1 the number is rounded up.
• ?? If the first digit dropped is less than 5, the
  proceeding digit is left unchangede the number
  is rounded down.

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Example

• After rounded, 1.6 becomes 2. This process is
  called as rounding up.
• How many significant numbers for 4.23100?
• Ans6

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1.5 Variables ??

• 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 ? of variable
• Continuous variable, discrete ???variable
• Constant variable or constant ??

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Example 1.11. The spectrum of hydrogen atom?????

• Two energy levels of hydrogen atom
• 2. Continuous energy levels, which all positive
  energies, Egt0. 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.

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Early Models of the Atom
Therefore, Rutherfords model of the atom is
mostly empty space
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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 Plancks constant, 6.626 ? 10-34 J-s.

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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.
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The Uncertainty Principle

• Heisenberg showed that the more precisely the
  momentum of a particle is known, the less
  precisely is its position is known

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The Uncertainty Principle

• In many cases, our uncertainty of the
  whereabouts of an electron is greater than the
  size of the atom itself!

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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).
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Example 1.11. The spectrum of hydrogen atom?????

• 1. Discrete???(quantized??? ???)
• 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 (??)to the vicinity of the nucleus.
• Transitions between the energy levels give rise
  to discrete lines in the spectrum of the atom.

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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.

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The Nature of Energy

• The energy absorbed or emitted from the process
  of electron promotion or demotion can be
  calculated by the equation

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.
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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.
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1.6 The algebra of real numbers

• 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
• ava2

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Example

• - 0.201 ? Ans 0.201

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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 mva (Read as the mth root of a)
• 21/3 is a cubic root of 2
• If xam then mlogax is the logarithm of x to
  base a (see Section 3.7)

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Example

• What is the square root of -1 ? Ansi
• What is the cubic root of -1? Ans -1

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Rules of precedence ?? for arithmetic operation

• ??????
• Parentheses ??
• ??? ( ) innermost brackets
• ??? square brackets
• ??? braces (curly brackets)

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1.7 Complex numbers ??

• x2 -1 are x v-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)

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1.8 Units ??

• A physical quantity has two essential attributes,
  magnitude and dimensions.
• 2 meters has the dimension of length and has
  magnitude equal to 2.

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SI Units

• Système International dUnités
• A different base unit is used for each quantity.

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1-4 Units, Standards, and the SI System
Quantity Unit Standard
Length Meter Length of the path traveled by light in 1/299,792,458 second
Time Second Time required for 9,192,631,770 periods of radiation emitted by cesium atoms
Mass Kilogram Platinum cylinder in International Bureau of Weights and Measures, Paris
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Dimensional Analysis

• We use dimensional analysis to convert one
  quantity to another.
• Most commonly, dimensional analysis utilizes
  conversion factors (e.g., 1 in. 2.54 cm)

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Dimensional Analysis

• Use the form of the conversion factor that puts
  the sought-for unit in the numerator

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Dimensional Analysis

• For example, to convert 8.00 m to inches,
• convert m to cm
• convert cm to in.

?
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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
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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?
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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.
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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!
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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
• g9.80665 ms-2980.665 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, Nkgms-2
• 4. Pressure has dimensions of force per unit
  pascal,
• PaNmkgm2s-2
• Standard pressure bar105 Pa
• Atmosphere atm 101325 Pa
• Torr torr(101325/760)Pa133.322Pa
• 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
  Nmkgm2s-2
• And kinetic energy ½ mv2 has dimensions of mass x
  (velocity)2
• MxLT-1 ML2T-2, with SI unit J

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Large and small units

• Table 1.3 SI prefixes
• Avogadros constant,
• NA6.02214x1023 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)

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Example

• The mass of an atom of 12C atom is therefore 12 g
  mol-1/NA a kg b u, where u is (1/ NA) g c
  kg is called the unified atomic unit (amu).
• What are a, b, and c?
• Ans a2x10-26 kg, b12, and c1.66054x10-27 kg

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Table 1.3 SI prefixes

• Prefixes convert the base units into units that
  are appropriate for the item being measured.

fs ??
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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

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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.
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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.
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Example 1.18

• 2. length
• mm10-6 m micrometer ??
• nm 10-9m nanometer ??
• pm 10-12 m picometer ??????
• Å 10-10 m 0.1 nm ?
• 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

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Examples 1.18-3. reduced mass ????moment of
inertia????

• 3. reduced mass and moment of inertia
• m mAmB/(mAmB)
• 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

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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

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Physics Chapter 10 Rotational Motion
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Angular Momentum General Rotation
Physcis Chapter 11
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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
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10-5 Rotational Dynamics Torque and Rotational
Inertia
The quantity is called the rotational inertia
of an object. The distribution of mass matters
herethese 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.
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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 rotationcompare (f) and
(g), for example.
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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
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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.
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40.4 Molecular Spectra
A diatomic molecule can rotate around a vertical
axis. The rotational energy is quantized.
Figure 40-16 goes here.
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40.4 Molecular Spectra
These are some rotational energy levels and
allowed transitions for a diatomic molecule.
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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
where
.
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Examples 1.19 Molecular properties wavelength,
frequency, and energy

• The wavelength l and frequency n of
  electromagnetic radiation are related to the
  speed of light by
• c ln
• E hn
• h 6.62608x10-34 Js
• DE E1-E2
• The wavelength of one of yellow D lines in the
  electronic spectrum of the sodium atom is
  l589.76 nm.
• nc/l3x108 ms-1/ 5.8975x10-7 m 5.0833x1014 s-1
• DE hn
• (6.62608x10-34 Js)x(5.0833x1014 s-1)
• 3.368x10-19 J2.102 eV202.8 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

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Wavenumber

• Wavenumber ?
• 1/ln/cDE/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.

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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 102100 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)210-46 kg m2

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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 Planks constant
• h 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)

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Chapter 38Quantum Mechanics
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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.
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Atomic units (Table 1.4)

• bohr, a0 4pe0h2/mee25.29177x10-11 m 0.53 Å
• The unit of energy, Eh (sometimes au), is called
  the hartree Eh, and is equal to twice the
  ionization energy of the hydrogen atom.
• Eh mee4/16p2e02h2 4.35074x10-18 J2620 kJ/mol
• For one mole of an atom or molecular,
• NAx Eh (6.02x1023)x(4.35074x10-18 J)2620 kJ

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Example 1.21 The atomic unit of energy

• Coulombs law
• Vq1q2/4pe0r
• q1Z1e and rRa0
• V(Z1Z2/R)(e2/4pe0a0)
• 1. To show that the unit is the hartree unit Eh
  in Table 1.4, use a0 4pe0h2/mee2
• e2/4pe0a0 (e2/4pe0)/(4pe0h2/mee2)
• (e2/4pe0)/(mee2/4pe0h2) mee4/16p2e02h2
• Eh

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• 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

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????(Gaussian 03) ??(biphenyl)????C-C????????
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??????C-C???????(conformation)?????
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????(Gaussian 03) 4-n-butylcarbamyloxy-4-acetylox
y-biphenyl ????C-C? ???????
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??4-n-butylcarbamyloxy-4-acetyloxy-biphenyl??C-C?
??? ???(conformation)?????