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Chapter 2. Atomic Structure

Inorganic Chemistry1 CBNU T.-S.You

▪ 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.





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


▪ 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


© 2014 Pearson Education, Inc.

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.




▪ 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



▪ 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



▪ 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




▪ 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.



≥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


▪ 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



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)



▪ 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

particular orbital !!Inorganic Chemistry1 CBNU T.-S.You


▪ 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


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.



Ψ(x)


▪ 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)



▪ 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


*Ψ = Asin rx + Bcos sx

*r = s = √2mE(2π/h)


▪ ∴ 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



▪ 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



2.2.2 Quantum Number and Atomic Wave Functions

▪ Mathematically, atomic orbitals are discrete solutions of the three dimensional

Schrödinger equations.




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.




▪ 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



▪ 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.




▪ 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




▪ 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



▪ The different shading of the lobes represent diff. signs of the wave function Ψ.

→ but, the same probability..

→ bonding purpose..

Fig. 2.6




∴ 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




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



▪ ∴ 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)




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



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




▪ 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.




▪ Total pairing E, Π = Πc(+) + Πe(-)

∴




▪ 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



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



▪ “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




▪ 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



▪ 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




- 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



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




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..



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



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

Fig2.13Inorganic Chemistry1 CBNU T.-S.You


2.3.2 Electron Affinity

▪ electron affinity = E required to remove e- from a negative ion

▪ also described as the ‘zeroth ionization E’

▪ endothermic (+ ΔU value) except noble gases & alkaline earth metals

▪ smaller absolute # than IE - ∵ removal of e- from a negative ion is easier than removal

from a neutron atom

▪ noble gas has the lowest EA - ∵ removal of e- past noble gas is easy.

A-(g) → A(g) + e- electron affinity = ΔU (or EA)


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




▪ 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



▪ 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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