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Chapter 2. Atomic Structure
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▪ The theory of atomic and molecular structure depend on quantum mechanics
to describe atoms and molecules in mathematical terms. Fortunately, it is
possible to gain a practical understanding of the principles of atomic and
molecular structure with only a moderate amount of mathematics rather than
the mathematical sophistication involved in quantum mechanics.
▪ This chapter presents the fundamentals needed to explain atomic and
molecular structures in qualitative or semiquantitative terms.
Chapter 2. Atomic Structure
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Chapter 2. Atomic Structure
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Finding of the subatomicparticle:
Balmer’s Equation
원자의 subatomic particle인전자가 E를방출하거나
흡수할수있음.
Generalized by Niels Bohr하지만이러한이론은전자의 wave nature때문에 H외
에는잘맞지않음.
Heisenberg’s Uncertaintyprinciple
그래서 wave property를잘나타내는 equation사용하기
로함:
Schroedinger equation
Schroedinger equation의realistic solution을위한조건
들을적용한예:
Particle in a box (1-D)
3-D Schrodinger equation의solution이바로
atomic orbitals
Quantum number로
AO표현함.
Ψ표현하는두가지방법:
1) Cartesian,
2) Spherical
QN의제한으로인해 aufbauprinciple필요:
1) Pauli’s
2) Hund’s
If more than 1 e-,
shielding effect: Z
Ionization energy, electron
affinity, covalent/ ionic radii.
▪ Many Chemists had considered the idea of arranging the elements into a periodic
table.
▪ But, due to either insufficient data or incomplete classification scheme, it was not
done until Mendeleev and Meyer’s time.
▪ Using similarities in chemical behavior and atomic weight, Mendeleev arranged those
families in rows and columns,,,
▪ and, he predicted the properties of unknown elements, such as Ga, Sc, Ge, Po.
2.1.1 The Periodic Table
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▪ In the modern periodic table :
: periods (horizontal row of elements)
: group/family (vertical column)
▪ 3 different ways of designations of groups:
: IUPAC, American, European
1) American: main group→ ІA – ⅧA;
TMs→ ⅢB – ⅧB –ⅡB
2) IUPAC: numbering from 1 through 18
for all group
2.1.1 The Periodic Table
Fig2.1
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2.1.2 Discovery of Subatomic Particles and the Bohr Atoms
▪ During the 50 years after the Mendeleev’s periodic table was proposed, there had been
experimental advances and discoveries as shown in Table 2.1.
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2.1.2 Discovery of Subatomic Particles and the Bohr Atoms
▪ The discovery of atomic spectra showed that each elements emits light of specific
energy when excited.
: Balmer’s equation (1885) → energy of visible light emitted by H atom
2 2
1 1= -
2H
h
E Rn
nh: integer, with nh>2
RH: Rydberg constant for hydrogen = 1.097 X 107 m-1 = 2.179 X 10-18J
------------------------------------------------------------------------------------
* E is related to the wavelength, frequency, and wave number of the light!!
hcE = hυ = = hcυ
λh = Planck constant = 6.626 X 10-34 Js
υ = frequency of the light, in s-1
c = speed of light = 2.998 x 108 ms-1
λ = wavelength of the light, frequently in nm
υ = wavenumber of the light, usually in cm-1
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2.1.2 Discovery of Subatomic Particles and the Bohr Atoms
▪ Balmer’s equation becomes more general by replacing 22 by n2l. (nl < nh)
▪ Niels Bohr’s quantum theory of the atom : - negative e- in atoms move in circular
orbitals around positive nucleus.
- e- may absorb or emit light of specific E
ln
2 2
1 1= -H
h
E Rn
μ = reduced mass of the electron-nucleus combination
* me = mass of the electron
mnucleus = mass of the nucleus
Z = charge of the nucleus
e = electronic charge
h = Planck constant
nh = quantum number of describing the higher energy state
nl = quantum number of describing the lower energy state
4πε0 = permittivity of a vacuum
2 2 4
2 20(4 )
Z e
h
2R =
π µ
πε
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2.1.2 Discovery of Subatomic Particles and the Bohr Atoms
▪ Electron transition among E levels for the hydrogen atom (Fig. 2.2)
▪ E release: as e- drops from nh to nl
▪ If correct E is absorbed: e- is raised from nl to nh
▪ According to Bohr’s model and equation, E is
inverse-squarely proportional to n.
▪ Thus, at small n → large E gap
at large n → small E gap
▪ Exercise 2.1
Fig.2.2
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2.1.2 Discovery of Subatomic Particles and the Bohr Atoms
▪ However, Bohr’s theory works only for H and fails to atoms w/ more e- because of
the wave nature of e-.
▪ de Broglie equation: all moving particles have wave properties, which can be expressed
as shown below
hλ =
mυ
λ = wavelength of the particle
h = Planck constant = 6.626 X 10-34 Js
m = mass of the particle
υ = velocity of the particle
▪ e-’s wave property is observable due to the very small mass.(1/1836 of the H atom)
▪ But, we can not describe the motion of e- w/ the wave property precisely because of
Heisenberg’s uncertainty principle.
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2.1.2 Discovery of Subatomic Particles and the Bohr Atoms
≥x
hΔx Δp
4π
Δx = uncertainty in the position of the electron
Δpx = uncertainty in the momentum of the electron
▪ Thus, there is the inherent uncertainty in the location and momentum of e-
(Δx is large) (Δpx is small)
▪ e- should be treated as wave (due to its uncertainty in location), not simple particles.
▪ We can’t describe orbits of e-, but can describe orbitals !!!
region that describe the probable location of e-
electron density
Heisenberg’s Uncertainty Principle
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▪ Therefore, one should use an equation which describes wave property well !!!
2.2 The Schrödinger Equation
▪ The equation describes the wave properties of e- in terms of its position, mass, total E
and potential E.
H = the Hamiltonian operator
E = energy of the electron
Ψ = the e-s wave function
▪ Hamiltonian operator (H) includes derivatives that operate on the wave function.
▪ The result is a constant (E) times Ψ.
▪ Different orbitals have 1) different wave functions, and 2) different E values
HΨ = EΨ
The Schrödinger equation
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2.2 The Schrödinger Equation
h = Planck constant
m = mass of the particle (e-)
e = charge of the e-
√(x2+y2+z2) = r =distance from the nucleus
Z = charge of the nucleus
4πε0 = permittivity of a vacuum
Kinetic E of the e-
2 2 2 2 2
2 2 2 2 2 2 20
-h δ δ δ ZeH = + + -
8π m δx δy δz 4πε x + y + z
potential E of the e-
(electrostatic attraction b/w
the e- and the nucleus)
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2.2 The Schrödinger Equation
▪ If applied to a wave function Ψ,,,
▪ V (potential E): result of electrostatic attractions b/w e- and nucleus→ negative value
1) e- near nucleus (small r) → large attraction
2) e- far from nucleus (large r)→ small attraction
3) e- at infinite distance (r = ∞) → no attraction
( , , )V x y z
+ ψ ψ
2 2 2 2
2 2 2 2
-h δ δ δ+ + x,y,z = E x,y,z
8π m δx δy δz( ) ( )
=2 2
2 2 20 0
-Ze -ZeV =
4πε r 4πε x + y + z
▪ every atomic orbital has a unique Ψ→ no limit to the number of solution of the Schrödinger
equation
→ describes the wave properties of a given e- in a
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2.2 The Schrödinger Equation
▪ Ψ2 : the probability of finding an e- at a given point in space
▪ conditions for a physically realistic solution for Ψ
1) The wave function Ψ must be single-valued.
2) The wave function Ψ and its first derivatives must be continuous.
3) The wave function Ψ must approach zero as r approaches infinity.
4) The integral
5) The integral
∫*ψ ψ =1A Aall spacedr
∫*ψ ψ = 0A Ball spacedr
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2.2.1 Particle in a Box
▪ the 1-D particle in a box shows how these conditions are used.
Fig.2.3 Potential E well for the particle in a box
ψ
2 2
2 2
-h δ= E x
8π m δx( )
▪ Thus, wave equation within a box is,,
,,,since V(x) = 0
1-D box
▪ potential E V(x) = 0, inside the box
V(x) = ∞, outside a box
- A particle is completely trapped in the box.
- It would require an infinite amount of E to leave the box.
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Ψ(x)
2.2 The Schrödinger Equation
▪ The wave characteristics of our particles can be described by a combination of sine and
cosine functions
since they have properties associated w/ waves → a well-defined wavelength and
amplitude.
▪ a general solution to describe a possible wave in the box,,,
Ψ = Asin rx + Bcos sx
* A, B, r, s are constants.
* solution for r and s,, r = s = √2mE(2π/h)
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2.2 The Schrödinger Equation
▪ Ψ must be → continuous and equals 0… at x < 0 and x > a,
∴ Ψ must go to 0… at x = 0 and x = a
1) because for x = 0, cos sx = 1 → ∴ Ψ = 0, only if B = 0
↓
∴ Ψ = Asin rx
2) for x = a, Ψ must be 0 → ∴ sin ra must be 0
it occurs,, only if ra = integral multiple of π
(ra = ±nπ or r = ±nπ/a, n = any integer ≠ 0)
↓
positive & negative values give the same result.
so, substitute positive r for the solution r,,,
↓
r = nπ/a = √2mE(2π/h)
∴ E = n2h2/8ma2
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*Ψ = Asin rx + Bcos sx
*r = s = √2mE(2π/h)
2.2 The Schrödinger Equation
▪ ∴ E = n2h2/8ma2 → E level predicted by the particle in-a-box model for any particle in
a one-dimensional box of length a.
→ E levels are quantized: n = 1, 2, 3,,,
▪ For the wave function: r = nπ/a, Ψ = Asin rx
Ψ = Asin nπx/a
if applying normalizing requirement
A = √(2/a)
Then, total solution is,
∴ Ψ = √(2/a) sin nπx/a
∫*ψ ψ =1dr
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2.2 The Schrödinger Equation
▪ The resulting wave function(Ψ) & their squares(Ψ2) for the first three states → Fig. 2.4
▪ Ψ2 is a probability densities → shows difference b/w classical and quantum mechanical
behavior
1) classical: equal probability of being at any point in a
box
2) quantum: high and low probability at different
location in a box
Fig. 2.4
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2.2.2 Quantum Number and Atomic Wave Functions
▪ Mathematically, atomic orbitals are discrete solutions of the three dimensional
Schrödinger equations.
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2.2.2 Quantum Number and Atomic Wave Functions
explains two experimental observations (1) doubled emission spectra of alkali-metal
(2) split beams of alkali-metal in magnetic field
∵ magnetic moment of the electron
(spin of the electron, tiny bar magnet)
▪ n, l, ml define an atomic orbital, ms describes the electron spin within the orbital.
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2.2.2 Quantum Number and Atomic Wave Functions
▪ There is i (=√-1) in the p and d orbital wave equation.
→ need to convert to real function rather than complex functions
→ use linear combination (sum or differences) of functions
ex) for p orbitals having ml = +1 and -1
1) sum + normalizing (by multiplying the const. 1/√2)
Ψ2px = 1/√2 (Ψ+1 + Ψ-1) = 1/2√(3/π)[R(r)]sinθcosΦ
2) difference + normalizing (by multiplying the const. i/√2)
Ψ2py = i/√2 (Ψ+1 - Ψ-1) = 1/2√(3/π)[R(r)]sinθsinΦ
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2.2.2 Quantum Number and Atomic Wave Functions
▪ The same procedure can be used on the d orbitals, and the results are shown in the column
headed θΦ (θ, φ) in Table 2.3.
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2.2.2 Quantum Number and Atomic Wave Functions
▪ Ψ can be expressed in terms of Cartesian coordinates (x, y, z) or in terms of spherical
coordinates (r, θ, Φ ).
x = rsinθcosΦ
y = rsinθsinΦ
z = rcosθ
▪ This is especially useful because r represents the distance from the nucleus. (Fig 2.5)
▪ volume elements: rdθ, rsinθdΦ, dr
→ products: r2sinθdθdΦdr (= dxdydz)
▪ volume of the thin shell b/w r and (r + dr)
: 4πr2dr (= integral over Φ (from 0 → π),
over θ (from 0 → 2π))
Describing the electron density as a function
of distance from the nucleusFig. 2.5
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2.2.2 Quantum Number and Atomic Wave Functions
▪ Ψ can be factored into a radial component & two angular components.
R θ, Φ
∴ Ψ (r, θ, Φ) = R(r)Θ(θ)Φ(φ) = R(r) Y(Θ, Φ)
e- density at diff. distancefrom the nucleus
shape of orbitals (s, p, d)
orientations of orbitals
Y
1) Angular functions: θ, Φ
shapeorientations
determined by l
determined by ml
(1) + (2) 3-D shape in the far-right column in Table 2.3
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2.2.2 Quantum Number and Atomic Wave Functions
▪ The different shading of the lobes represent diff. signs of the wave function Ψ.
→ but, the same probability..
→ bonding purpose..
Fig. 2.6
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2.2.2 Quantum Number and Atomic Wave Functions
∴ Ψ (r, θ, Φ) = R(r)Θ(θ)Φ(φ) = R(r) Y(Θ, Φ)
2) Radial functions(R(r)): determined by n and l
: probability of finding e- at a given distance from nucleus
Radial probability function: 4πr2R2
→ at the center of nucleus, 4πr2R2 = 0 (∵r = 0)
→ 4πr2 X R2
→ ∴show which orbitals are most likely to be involved in reactions
r↑ → 4πr2R2↑r↑ → R2 ↓ (generally)
Fig. 2.7
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2.2.2 Quantum Number and Atomic Wave Functions
3) Nodal surface: - a surface w/ zero e- density
ex) 2s at r = 2a0
- appears as a result of the wave equation for the e-
→ result from solving the wave function is 0 as it changes sign
Ψ = 0, probability of finding e- is 0.
Ψ2 = 0∴ Ψ = 0
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2.2.2 Quantum Number and Atomic Wave Functions
▪ ∴ Ψ (r, θ, Φ) = R(r) Y(Θ, Φ): for Ψ = 0 → either R(r) = 0 or Y(Θ, Φ) = 0
▪ Thus, we can determined nodal surfaces by detecting under what conditions R(r) = 0 or Y = 0.
1) Angular Nodes (Y(Θ, Φ) = 0): planar or conical→ l angular nodes (conical nodes counts two
nodes)
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2.2.2 Quantum Number and Atomic Wave Functions
2) Radial Nodes [R(r) = 0] → n - l - 1
→ the lowest E orbitals of each classification (1s, 2p, 3d, 4f,,,) have
no radial nodes.
Fig. 2.8
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2.2.3 The Aufbau Principle
▪ in German: aufbau – “building up”
▪ when there is more than one e-s, interactions b/w e-s require that the order of filling orbitals
be specified
1) to give the lowest total E to the atom
→ lowest n, l are filled first
→ ml & ms’s values don’t matter
2) Pauli exclusion principle: each e- in an atom have a unique set of quantum number
3) Hund’s rule of maximum multiplicity: e- should be placed in orbitals so as to give the max.
total spin possible (max. number of parallel spins)
→ to avoid electrostatic repulsion
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2.2.3 The Aufbau Principle
▪ Coulombic energy of repulsion, Πc: causes a repulsive force b/w e-s in the same orbitals
▪ exchange energy, Πe: depends on the number of possible exchanges b/w two e-s w/ the same
E & the same spin
ex) C: 1s2 2s2 2p2 → three arrangements for 2p2 are possible
Πc (e- - e- repulsion)so, higher E than other two
two possible ways to arrange e-s
(one exchange of e-)Only one way to arrange the e-
to give the same diagram
The higher the # possible exchange,the lower the E.
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2.2.3 The Aufbau Principle
▪ Total pairing E, Π = Πc(+) + Πe(-)
∴
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2.2.3 The Aufbau Principle
▪ The order of filling of atomic orbitals
1) Klechkowsky’s rule → lowest available value for the
sum n + l (e.g. 4s: n + l = 4 + 0; 3d: n + l = 3 + 2)
→ if same, the smallest value of n
2) the blocked out periodic table
▪ Group 1-2: filling s orbitals, l = 0
▪ Group 13-18: filling p orbitals, l = 1
▪ Group 3-12: filling d orbitals, l = 2
▪ lanthanides & actinides: f orbitals, l = 3
Fig2.9
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2.2.4 Shielding Effect
▪ If there is more than one e-, prediction of E of specific levels are difficult.
use the concept of shielding
each e- acts as a shield for e- farther from the nucleus
reduce the attraction b/w nucleus and distant e-
the E changes are somewhat irregular due to the shielding of outer e- by inner e-
*see Table 2.7
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2.2.4 Shielding Effect
▪ “the higher E w/ a higher quantum #”
→ this is true only for orbitals w/ the lowest n
(Fig.2.10)
▪ for higher value of n,,,
→ l is also important
ex) 4s orbital is lower than 3d orbital
→ thus, 3s, 3p, 4s, 3d, 4p
→ also, 5s before 4d
6s before 5d
Fig2.10
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2.2.4 Shielding Effect
▪ Slater’s effective nuclear charge Z* as a measure of the attraction of the nucleus for a
particular e-.
Z* = Z – S
▪ Rules for determining S
1) write order of e- str. and group them
(1s) (2s, 2p) (3s, 3p) (3d) (4s, 4p) (4d) (4f) (5s, 5p) (5d),,,,
2) right side e-s do not shield left side e-s
3) for ns & np valence e-
a. same group e-→ 0.35
(exception: 1s e-→ 0.30)
b. n-1 group e-→ 0.85
c. n-2 or the lower group e-→ 1.00
4) for nd and nf
a. same group e-→ 0.35
b. left side e- → 1.00
nuclear charge shielding constant
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2.2.4 Shielding Effect
▪ Examples) Oxygen’s 2p,
Nickel’s 3d and 4s
▪ Justification for the Slater’s rule comes from e- probability curves for the orbitals. (Fig. 2.7)
- for the same n, s & p orbitals have higher probability near the nucleus than d orbitals
ex) (3s, 3p) shield 3d w/ 100% effectiveness→ 1.00
- shielding 3s or 3p by (2s, 2p) shows 85 % effectiveness→ 0.85
∵ 3s & 3p orbitals have regions of significant probability close to the nucleus
Fig2.7
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2.2.4 Shielding Effect
- Cr : [Ar]4s13d5 ← [Ar]4s23d4
Cu : [Ar]4s13d10 ← [Ar]4s23d9
∵ “special stability of half-filled subshell” (Fig. 2.11)
- Rich’s explanation for e- - e- interactions
consider E difference b/w 1 e- in 1 orbital
2 e-s in 1 orbital → electron pairing E is added due to the
electrostatic repulsion
place one extra e- in the 3d &remove one e- from the 4s
▪ Complicate cases
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2.2.4 Shielding Effect
Fig2.12
▪ Fig. 2.12: Schematic E levels for TM
- as nuclear change↑ → e- more strongly attracted → E level decrease → more stable
- the filling order: from bottom to top
* d orbitals changes more rapidly than s, p
→ due to the lack of shielding
ex) V: [Ar] 4s23d3, Cr: [Ar] 4s13d5, Cu: [Ar]4s13d10
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2.2.4 Shielding Effect
Fig2.12
▪ Fig. 2.12(b) – formation of positive ions by removing e-
→ among TM, reduce the overall e- repulsion & E of the d orbitals more
e- w/ highest n are removed first, as ions are formed
no s, occupied d
TM cations have no s electrons, only d electrons in their outer levels
▪ lanthanide & actinide series show similar patterns..
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2.3 Periodic Properties of Atoms
= ionization potential (E required to remove an e- from a gaseous atom or ion)
2.3.1 Ionization Energy
▪ in general, as nuclear charge↑ → ionization E ↑
▪ experimentally observed breaks in trend → at Boron and Oxygen
- B: new p e- is farther away from the nucleus than others
→ easier to remove than Be
- O: the fourth p e- → pairing of 2p e- at the same orbital
→ easier to remove
Fig2.13
A+(g) → A(n+1)+(g) + e- ionization E = ΔU
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2.3.1 Ionization Energy
▪ TM - smaller diff. in IE
- usually lower value for heavier atoms in the same group due to the shielding effect
▪ each new period: much larger decrease in IE
∵ change the next major quantum # → e- w/ much higher E
▪ noble gas: maximum of IE decreases w/ increasing Z
∵ e- farther from nucleus in the heavier elements
▪ overall, in periodic table,, higher IE from left to rightlower IE from top to bottom
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2.3.3 Covalent and Ionic Radii
▪ two effects exist as Z ↑: 1) e- are pulled toward nucleus → orbital size ↓
2) e- # ↑ → mutual repulsion → orbital size ↑
Gradual decrease in atomic size across each periods
▪ Table 2.8: Nonpolar covalent radii
→ sufficient for general comparison
▪ other measure of size of elements → difficult to obtain consistent data
sum
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2.3.3 Covalent and Ionic Radii
▪ ion size: the stable ions of the diff. elements have different charges, e- #, crystal str.
∴ difficult to determine the size of ion
▪ old method: Pauling’s approach → isoelectronic ion’s ratio of radii was assumed to be equal to
the ratio of their effective nuclear charge
▪ recent method: Shanon’s approach → consider many things (e.g. e- density map from X-ray)
(Table 2.9)
▪ factors influencing ionic size: 1) coordination of ion
2) covalent character of bonding
3) distortions of regular crystal geometries
4) delocalization of e- (metallic or semi-metallic)
5) the size & charge if the cation
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2.3.3 Covalent and Ionic Radii
▪ for the ions w/ the same # e-, as Z ↑ → size ↓
▪ within a group, as Z ↑ → size ↑
▪ same element, as charge on the cation ↑
→ size ↓
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