Chapter 7 - Atomic Structure + Periodicity

7/31/2019 Chapter 7 - Atomic Structure + Periodicity

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Atomic Structure & Periodicity

Electromagnetic Radiation and Radiation Energy

Photoelectric Effect and Frequency/Energy Dependence

Atomic Spectrum of Hydrogen Gas

The Bohrs Model of H-atom

Heisenberg Uncertainty Principle and Quantum MechanicalModel

Atomic Orbitals and Quantum Numbers

Electron Spin and Pauli Exclusion Principle

Electron Configurations of Polyelectronic Atoms

Periodic Trends and Atomic Properties


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Atomic Modeling in theEarly 20th Century: 1904-1913

Charles Baily

University of Colorado, Boulder

Oct 12, 2008


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Stabilityof the atom

Dynamics

of its parts

Chemical/spectral

properties

Key Themes to Atomic Modeling


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Various Depictions of the Plum Pudding Model


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Equal angular intervals

Thomsons Atomic Model* (1904)

negatively charged

corpuscle

*Joseph J. Thomson, On the Structure of the Atom

Philosophical Magazine and Journal of Science, Series 6, Vol. 7, No. 39, pp. 237-265

d ~ atomic dimensions

sphere of uniform

positive charge


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Electromagnetic Radiation

Light =

radiation energy that propagates through space in the form of

wave

In a given medium the speed of light is constant

Such that speed: c = ln is constant

Where, l = wavelength, n = frequency, and c ~ 3.00 x 108 m/s

Light with longer wavelength has lower frequency, and one with

higher frequency has shorter wavelength.

Quantum Theory: Radiation Energy depends on frequency:

En = hn; El = hc/l, where h = 6.626 x 10-34 J.s


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Electromagnetic Radiation


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Photoelectric Effect

Photoelectric current


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When light with energy greater than the minimum value

strikes a metal plate (the cathode), electrons are ejected

A potential gradient is created and electrons flow in the circuit

and photoelectric current is produced.

Different metals require different minimum energy, called

work function, to produce photoelectric effect.

If light with energy lower than the minimum value is used, no

photoelectric effect is observed.

The minimum energy needed to produce photoelectric effect

actually corresponds to the binding energy of electrons on the

metal surface.


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Light with minimum frequency needed to eject electrons


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Photoelectric Voltage & Current

The energy and speed of ejected electrons depends on frequency (n)

of incident light, which must be greater than the threshold

(minimum) value for the metal used.

Ee =EnEo (Eo = minimum energy)

Ee = h(nino) (no = minimum frequency)

= hc(1/li - 1/lo) (lo = longest wavelength)

Speed of electron: ve = (2Ee/me)

The photoelectric voltage is directly related to the energy of ejected

electrons, which depends on the frequency of light. The photoelectric current (or the Amps) depends on the intensity of

incident lighthigher light intensity produces more current.


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Einsteins Explanation of PhotoelectricEffect

Light is composed of energy particles calledphoton

Energy of each photon is dependent only on the frequency of light

emitting the photon: Ep = hn;

Total energy of electromagnetic radiation (light) =Nhn,

whereNbeing the number of photon.

When light strikes on the metal, the photon is absorbed by an

electrons on the metal surface, such that one electron absorbs only a

photon (a quantum of energy) and the electron becomes excited.

If the photon carries energy greater than the binding energy of themetal, that electron will be ejected from the metal surface. The

excess energy becomes the kinetic energy of electron.

Light is considered to have both wave and particle properties


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E= mc2 &E= hc/ll = h/mc

Portrait of Albert Einstein:


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Continuous Spectrum

White light produces a continuous spectrum


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Atomic Spectrum

Spectrum produced by hydrogen gas discharge contains

discrete lines:


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Hydrogen Spectrum

Balmers equation for hydrogen spectrum in the visible region:

1/l = 1.097 x 107 m-1(1/221/n2); (n > 2)

1/l = 1.097 x 10-2 nm-1(1/221/n2); (n > 2)

If n = 3, 1/l = 1.097 x 10-2 nm-1(1/221/32) = 1.524 x 10-3 nm-1

l = 656.3 nm

If n = 4, 1/l = 1.097 x 10-2 nm-1(1/221/42) = 2.057 x 10-3 nm-1

l = 486.2 nm

General equations for hydrogen spectrum:

1/l = 1.097 x 107 m-1(1/n121/n2

2); (n1 > 0, n2 > n1 )

n = 3.289 x 1015 s-1(1/n121/n2

2); (n1 > 0, n2 > n1 )


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Spectral Series of Hydrogen Spectrum

Recurring patterns of line spectra for hydrogen were observed in

different spectral regions, such as in ultraviolet region, visible

region, infrared region, etc.

Spectral lines in ultraviolet region are called theLyman series,

which are due to electronic transitions from higher energy levels tolevel n = 1;

Spectral lines observed in the visible region, called theBalmer

series, are due to electronic transitions from upper energy levels to

level n = 2;

Spectral lines that appear in the infrared region, called the Paschen

series, are due to electronic transitions from upper energy levels to

level n = 3.


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Balmers Equation: 1/l = RH(1/221/n2)

Portrait of Johann Balmer:


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Electronic Transitions in Hydrogen Discharge

Electronic transitions that produce different sets of line spectra


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Bohrs Model for Hydrogen

1. Electron orbits the nucleus in the manner Earth orbits the Sun2. Only a particular set of orbits is allowedeach orbit must satisfythe condition that the angular momentum:

mever = nh/2p (r = orbit radius)

3. While in a particular orbit, electron neither gains nor loses energyeach orbit is called stationary state

4. Electronic energy in a given orbit is given by the expression:

En = -2.18 x 10-18 J(1/n2) (Z = atomic number; n = 1, 2, 3,..)

5. Electron gains energy when it jumps from an inner orbit to theouter orbit, and loses energy when it jumps from an outer orbit toan inner one, such that,

DE= -2.18 x 10-18 J (1/nf2 - 1/ni2); (n = 1, 2, 3, )


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Energy in Hydrogen Atom:En = -B(Z2/n2)

Portrait of Niels Bohr:


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* Niels Bohr, On the Constitution of Atoms and Molecules


Electrons occupy discrete

orbits of constant

energy. These orbits aredescribed using the ordinary

mechanics, while thepassing of the system

between differentstationary statescannot be treated onthis basis


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* Niels Bohr, On the Constitution of Atoms and Molecules


In making a transition

between stationary states, a

single photon will beradiated


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Applying Bohrs Model to Hydrogen Atom

Consider an electron jumps from energy levels n = 3 to n = 2:

Ei =E3 = -2.178 x 10-18 J(1/32) = -2.420 x 10-19 J

Ef=E2 = -2.178 x 10-18 J(1/22) = -5.445 x 10-19 J

DE=E2E3 = -2.178 x 10-18 J(1/22 - 1/32) = -3.025 x x 10-19 J

Energy lost by electron is emitted as radiation energy,En

= hc/l

l = hc/En = (6.626 x 10-34 J.s)(2.998 x 108 m/s)/(3.025 x 10-19 J)

= 6.567 x 10-7 m = 656.7 nm

Calculated wavelength matches with observed wavelength of alpha(red) line in hydrogen spectrum.

Calculation of energy emitted when electron jumps from energylevels n = 4 to n = 2 yields a wavelength that matches with the beta(blue) line in H-spectrum, with l = 486.4 nm.


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Absorption and Emission Spectrum ofHydrogen

221

11

abR

abnBalmers Formula:

Eorbit 26.13

n

eV


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Limitation of Bohrs Model

Bohrs model only works for hydrogen atom and other one-electron(hydrogen-like) ionic species, such as He+, Li2+, etc.

For H-atom, electronic energy: En = -2.178 x 10-18J(1/n2)

For other one-electron particle: En = -2.178 x 10-18J(Z2/n2)

(Z = atomic number)

Bohrs model cannot explain atomic spectra of atoms having morethan one electron;

Bohrs model also cannot explain why each line in the hydrogen

spectrum appears as double-lines if the discharge tube is placed inmagnetic field.

Perhaps his treatment of electron as having only particulateproperties is insufficient.


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Traveling and Standing Waves

Light waves are traveling wavesvalues of wavelengths andfrequencies are infinite

Waves on plucked strings (guitar, violin, cello, etc.) are standingwavestheir motions limited within a boundary

Wavelengths of standing waves are limited by the length of thestringthe distance the wave has to travel, such that

L = n(l/2), whereL is the distance the wave has to travelwithin a boundary and n = 1, 2, 3,

Standing waves are quantizedthe distance the wave travels must

be an integer multiple of half-wavelength (nl/2), else destructiveinterference occurs.


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Traveling Waves


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Standing Wave


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Defined Wavelength for Standing Waves


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Particle-Wave Duality

According to Einstein light can be considered as having bothparticle and wave properties.

Louis de Broglie proposed that other particles too can have bothparticulate and wave properties.

According to de Broglie, a particle of mass m traveling at a speed vwill have a wave property, such that the wavelength is:

l = h/mv

For example, an electron (me = 9.11 x 10-31 kg) traveling at speed

v = 3.00 x 107 m/s will acquire a wave characteristic with

wavelength,l = (6.626 x 10-34 J.s)/{(9.11 x 10-31 kg)(3.00 x 108 m/s)}

= 2.42 x 10-11 m = 24.2 pm


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De Broglies Equation: l = h/mv

Portrait of Louis de Broglie:


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Heisenberg Uncertainty Principle

It is impossible to know simultaneously both the exactlocation and momentum of an electron in atom.

IfDx represents the uncertainty in the location and Dp representsthe uncertainty in the momentum, then

Dx.Dp > h/4p; Dx.D(mv) > h/4p;

(Such uncertainty is insignificant in macroscopic objects, but itbecomes very dominant when applied to a subatomic system.)

Thus, the more precisely the position of an electron is determined,the less precise the momentum will be known, and vice versa.

Therefore, it is not appropriate to assume that the electron ismoving around the nucleus in a well-defined orbit, as in the Bohrsmodel.


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Heisenberg Uncertainty:Dx.Dp > h/4p

Portrait of Werner Heisenberg:


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Quantum Mechanical Model

Also called wave mechanicstreating all motions of particles aswave-like;

Louis de Broglie originated the idea that, like light, all particulatemotions have wave characteristics;

a new mathematical formula that incorporates both particulate andwave characteristics was needed.

Heisenberg uncertainty principle implies that we cannot know theposition and energy of an electron in atom at the same time withsome degree of certainty.

If we determine precisely the energy of electrons in atoms, we can onlyapproximate their where about

Erwin Shrdinger derived a mathematical model for hydrogen thatassumed electron to behave a standing wave.


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Schrdingers Wave Function, y(x,y,z)

(h2/8p2me)[(d2y/dx2) + (d2y/dy2) + (d2y/dz2)](Zq1q2/r)y =Ey

The equation is a bit complicated and Schrdinger wasnt even sure

if it works

Well try to understand the meaning of this equation Actually the wave functiony(x,y,z) has no physical meaning

But [y(x,y,z)]2 impliesprobability

That is, the square of the wave function will yield the probability of

finding an electron with a particular energy at a particular locationin the atomonly the probability, not the definite location.

The sum of the squares of these wave functions yields a probability

space called orbital.


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Orbitals

Orbital

It is a probability space inside the atom where the chances of

finding an electron with particular energy value is greater than

90%

Each orbital is described by a set of three quantum numbers: n, l,and ml;

The number of orbitals in a subshell is equal to (2l + 1) and the

number or orbitals in a shell is equal to n2;

As a consequence of the Pauli exclusion principle, each orbitalcan accommodate a maximum of two electrons, which must

have opposite spins


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Quantum Numbers

A set of numbers that describe an orbital or an electron

Theprincipal quantum number (n)has the integral values: 1, 2,

3,, . It is related to the size and energy of the orbital

The angular momentum quantum number(l) has the integral values:

0, 1, 2,,(n1). It is related to the shape of atomic orbitals. Each

value ofl is designated a letter symbol, which is summarized below:

Values ofl: 0 1 2 3

Letter symbols: s p d f

The magnetic quantum number(ml) is related to the orientation ofthe orbital in the Cartesian coordinatesx, y, and z.

ml has values froml to +l (including 0)


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Other Meanings of The Quantum Numbers

Theprincipal quantum number(n) also describes theprimaryelectronic shell or main energy level

The angular momentum quantum number(l) also implies the sub-shell or energy sub-level

The number of sub-shell in a given energy shell is equal to n:

Shelln = 1 has one subshell - the 1s-subshell;

shell n = 2 has two subshells - the 2s- and 2p-subshells;

shell n = 3, has three subshells - the 3s-, 3p-, and 3d-subshells,and so on,

The number of orbitals in a given subshell is determined by thepossible values that ml can have, which is equal to (2l + 1):

subshell l = 0 has one orbital; l = 1 has three orbitals; l = 2 hasfive orbitals; l= 3 has seven orbitals, and so on


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Quantum Numbersand Orbital Designations

The combination ofquantum numbers: n, l, and ml, describes a

particular orbital in the atom.

n = 1, l = 0, and ml = 0, orbital 1s;



n = 2, l = 1, and ml = 0, orbital 2p;

n = 3, l = 1, and ml = 0, orbital 3p;

n = 3, l = 2, and ml = 0, orbital 3d;

All orbitals with l = 0 have spherical shape, but the size becomes

larger as the value ofn increases;

Each orbital-p has two lobes, like a dumb-bell, with a nodal plane


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Radial Probability Distribution for 1s in Hydrogen


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Radial Probability Distributions for 1s, 2s & 2p

in Hydrogen


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Radial Probability Distributions ofs andp


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Radial Probability Distributions of 3dand 4s


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Atomic Orbitals 1s, 2s, 2pz, 2py, and 2px


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Atomic Orbitals: 1s, 2p and 3d


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Experiment by Stern & Gerlac That Led to The

Electron Spins Concept


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The Spinning Electrons


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Quantum Numbers To Describe Electrons in Atoms and

The Limitation Set By Pauli Exclusion Principle

Sets ofthree quantum numbers: n, l, and ml, are needed to describeatomic orbitals;

As a consequence of Stern-Gerlac experiments, a fourth quantumnumber - the spin quantum number(ms), is also needed to describean electron in an atom.

The spin quantum number(ms) is assigned values + or -, whichdenote spin direction clockwise or counter-clockwise

Pauli Exclusion Principle states that two electrons in a given atomcannot have the same set of four quantum numbersat least one ofthe quantum numbers must be different.

If the first three quantum numbers (n, l, and ml) are the same, thefourth (ms) must be different. Therefore, an orbital canaccommodate only two electrons with opposite spins.


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Energy of Orbitals and Electrons in Hydrogen

and Multi-electrons Atoms

In hydrogen atom and other hydrogen-like ions, the energy of

orbitals are defined only by theprincipal quantum number(n).

In multi-electrons atoms and ions, the energy of orbitals are

primarily defined by theprincipal quantum number, n, but it is

also influenced (to some extent) by the angular momentumquantum number(l).

Energy trend in multi-electrons atoms:

1s < 2s < 2p < 3s < 3p < 4s < 3d< 4p < 5s < 4d< 5p < 6s < 4f

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Chapter 7 - Atomic Structure + Periodicity