atomic physics chapter iii : bohrs model of hydrogen atom ch. # 42 1 1431 - 1432

Atomic Physics Chapter III : Bohr’s Model of Hydrogen AtomCh. # 42

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1431 - 1432

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1- Atomic Spectra of GasesAll objects emit thermal radiation characterized by a continuous distribution of wavelengths.

A line spectra observed when a low-pressure gas is subject to an electric discharge, this produce lines called EMISSION lines.

Lecture # 1Ch# 42

When white gas from a continuous source passes through a gas or dilute solution, ABSPORPTION lines produces.

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Balmer series of hydrogen

• The empirical equation by Johannes Rydberg (1854-1919):

– RH: Rydberg constant = 1.0973732 x 107

m-1.

– The measured spectral lines agree with the empirical equation to within 0.1%

,...5,4,3 1

2

1122

n

nRH

The Balmer series of spectral lines for atomic hydrogen.

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Johann Jocob Balmer (1825-1898)

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Balmer series

Where RH is Rydberg constant =109673732 m-1

Pashen

Bracket

Lyman

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2- Early Models of the Atom

Thomson’s model of the atom: negatively charged electrons in a volume of continuous positive charge.

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2- Early Models of the Atom

Rutherford’s planetary model of the atom.

The classical model of the nuclear atom. Because the accelerating electron radiates energy, the orbit decays until the electron falls into the nucleus.

Difficulties with Rutherford’s planetary model

• Cannot explain the phenomenon that an atom emits (and absorbs) certain characteristic frequencies of electromagnetic radiation and no others.

• Predication of the ultimate collapse of the atom as the electron plunges into the nucleus.

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3-The Bohr Model of the Atom

Bohr proposed that the possible energy states for atomic electrons were quantized – only certain values were possible. Then the spectrum could be explained as transitions from one level to another.

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Bohr’s Quantum Model of the Atom

1. The electron moves in circular orbits around the proton under the influence of the attractive Coulomb force.

Q: What are Bohr’s postulates?

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2. Only certain special orbits are stable (STATIONARY STATES). While in one of these orbits, the electron does not radiate (emit) energy. Its total energy is constant. (electron will not spiraling into the nucleus).

3. Radiation (e.g., light) is emitted by the atom when the electron transits “jumps” from a higher energy orbit or “state” to a lower energy orbit or “state”.

the frequency of the emitted radiation is found to be

E E h fi f

whereEi = energy of the initial stateEf = energy of the final stateEi > Ef

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4. The size of the stable or “allowed” electron orbits is determined by a “quantum condition”

(angular orbital momentum equal to an integral multiple of )

Using the Bohr’s four assumptions enable us to calculate the allowed energy level and emission wavelength of the

hydrogen atom

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,3,2,12

h

n

nnh

mvr

2

h

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The electric potential energy

Where Ke is the Coulomb constant and the negative sign arises from the charge -e.

The kinetic energy is given by

Thus the total energy of the atom is as follow

Form Newton’s 2nd low, the electric force must equal to its mass and its centripetal acceleration

then

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So the total energy can be given as

,3,2,1 here w2

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nemk

nr

en

Q: Prove that the radii of the allowed orbits given as

nmvr

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The following constant is called Bohr’s radius

nmnrn )(0529.0 2

And hence the quantization of orbit radii is given as

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The quantization of orbit radii leads to energy quantization which is the allowed values of energy of H atom (i.e. allowed energy levels of H atom), is given as follow

The minimum energy required to ionize the atom in its ground state is called the IONIZATION ENERGRY

to completely remove an electron from the proton’s influence = 13.6 eV for

hydrogen.

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Q: Show that the frequency and the wavelength of an emitted photon if transits from higher (outer orbit) state f to lower (inner orbit) state i is given by the following expressions

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Z, is the atomic number of the element (the number of protons in the nucleus)

Bohr extended his model for hydrogen to other elements in which all but one electron had been removed

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Q: If the electron in the hydrogen atom was 207 times heavier (a muon), the Bohr radius would be

1. 207 Times Larger

2. Same Size

3. 207 Times Smaller

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emk

nr

en

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Q: A hydrogen atom is in its ground state. Many photons are incident on the atom, each having an energy of 10.5 eV. The result is that:

(a) the atom is excited to a higher allowed state

(b) the atom is ionized

(c) the photons pass by the atom without interaction

to completely remove an electron from the proton’s influence = 13.6 eV for hydrogen.

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Electron A falls from energy level n=2 to energy level n=1 (ground state), causing a photon to be emitted.

Electron B falls from energy level n=3 to energy level n=1 (ground state), causing a photon to be emitted.

Which photon has more energy? n=2

n=3

n=1

• Photon A

• Photon BA B

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Q: Calculate the wavelength of photon emitted when an electron in the hydrogen atom drops from the n=2 state to the ground state (n=1).

12 EEhf

eVhc2.10

n=2

n=3

n=1

V2.10)V6.13(V4.3 eee

nm1242.10

1240

hc

E photon

E1= -13.6 eV

E2= -3.4 eV

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Q: Compare the wavelength of a photon produced from a transition from n=3 to n=2 with that of a photon produced from a transition from n=2 to n=1.

n=2

n=3

n=1E32 < E21 so 32 > 21

32 < 21

32 = 21

32 > 21

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. If the energy difference between the electronic states of hydrogen atom is 214.68 kJ mol-1, what will be the frequency of light emitted when the electron jumps from the higher to the lower energy state? (Planck's constant = 39.79 x 10-14 kJ mol-1)

SolutionThe frequency (v) of emitted light is related to the energy difference of two levels (ΔE) as

E = 214.68 kJ mol-1, h =39.79 x 10-14 kJ mol-1

= 5.39 x 1014 s

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According to quantum mechanics, each electron is described by four quantum numbers:

1. Principal quantum number (n)2. Angular momentum quantum number (l)

3. Magnetic quantum number (ml)

4. Spin quantum number (ms)

•The first three define the wave function for a particular electron. The fourth quantum number refers to the magnetic property of electrons.

Lecture # 2Ch. 42

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A wave function for an electron in an atom is called an atomic orbital (described by three quantum numbers—n, l, ml). It describes a region of space with a definite shape where there is a high probability of finding the electron.

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•Principal Quantum Number, n

•This quantum number, which refers to energy level or shell, is the one on which the energy of an electron in an atom primarily depends. The smaller the value of n, the lower the energy and the smaller the orbital.

•The principal quantum number can have any positive value: 1, 2, 3, . . .

•Orbitals with the same value for n are said to be in the same shell.

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•Shells are sometimes designated by uppercase letters:

Lettern

K1

L2

M3

N4

. . .

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•Orbital Quantum Number, l•Sometimes called the azimuthal quantum number, this quantum number distinguishes orbitals within a given n (shell) having different shapes.

•It can have values from 0, 1, 2, 3, . . . to a maximum of (n – 1).

•For a given n, there will be n different values of l, or n types of subshells.

•Orbitals with the same values for n and l are said to be in the same shell and subshell.

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•Subshells are sometimes designated by lowercase letters:

l ≤Letter

0s

1p

2d

3f

. . .

Not every subshell type exists in every shell. The minimum value of n for each type of subshell is shown above.

n = 1 2 3 4

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•Magnetic Quantum Number, ml

•This quantum number distinguishes orbitals of a given n and l , that is, of a given energy and shape but having different orientations.

•The magnetic quantum number depends on the value of l and can have any integer value from –l to 0 to +l. Each different value represents a different orbital. For a given subshell, there will be (2l + 1) values and therefore (2l + 1) orbitals.

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•Let’s summarize:

•When n = 1, l has only one value, 0.

When l = 0, ml has only one value, 0.

So the first shell (n = 1) has one subshell, an s-subshell, 1s. That subshell, in turn, has one orbital.

•When n = 2, l has two values, 0 and 1.

When l = 0, ml has only one value, 0. So there is a 2s subshell with one orbital.

When l = 1, ml has only three values, -1, 0, 1. So there is a 2p subshell with three orbitals.

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•When n = 3, l has three values, 0, 1, and 2.

When l = 0, ml has only one value, 0. So there is a 3s subshell with one orbital.

When l = 1, ml has only three values, -1, 0, 1. So there is a 3p subshell with three orbitals.

When l = 2, ml has only five values, -2, -1, 0, 1, 2. So there is a 3d subshell with five orbitals.

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•Spin Quantum Number, ms

This quantum number refers to the two possible orientations of the spin axis of an electron.

It may have a value of either +1/2 or -1/2.

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Q: For a hydrogen atom, determine the number of allowed states corresponding to the principal quantum number n=2, and calculate the energies of these states.

When n = 2l can be 0 or 1

When l = 0 the only value that ml can have is 0, for l = 1 , ml = can be -1,0,1States:one state designated as the 2s state, that is associated with the quantum numbers

n=2, l=0 and ml=0

three states, designated as the 2p state, that is associated with the quantum numbers

n=2, l=1 , ml=-1n=2, l=1 , ml=0n=2, l=1 , ml=1

Because all four of these states have the same principal quantum number n=2, they have the same energy

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The Wave functions for Hydrogen

The 1s state of the hydrogen atom, ψ1s(r):

• Ψ1s is spherically symmetric.

• This symmetry exists for all s states.

• The probability density for the 1s state:

0/

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1

1)( ar

s ea

r

0/230

2

1

1 ars e

a

2

12

22

1

2

1

4)(

4)(

s

ss

rrP

drrdVdrrP

Radial probability density function P(r):

The probability per unit radial length of finding the electron in a spherical shell at radius r:

Lecture # 3Ch. 42

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Radial probability density function for the hydrogen atom in the 1s state:

0/230

22

12 4

4)( ars e

a

rrrP

• The peak indicates the most probable location of the electron.

• The peak occurs at the Bohr radius.• The average value of r for the 1s state of hydrogen is

3/2 a0.

Electron cloud:The charge of the electron is extended throughout a diffuse region of space, commonly called an electron cloud.

• This figure shows the probability density as a function of position in the xy plane.

• The darkest area corresponds to the most probable region.

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The 2s state of the hydrogen atom:

02/

0

2

3

02 2

1

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1)( ar

s ea

r

ar

• ψ2s depends only on r and is spherically symmetric.

• The radial probability density for the 2s state has two peaks.

• The highest value of P corresponds to the most probable value (r ≈5a0).

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Quantum numbers of the hydrogen atom

42.6 Physical interpretation of the quantum numbers

I. Principle quantum number n :Determines the energy of an atomic state.

II. Orbital quantum number l :1) In the Bohr model, the angular momentum of the electron is restricted to L=mevr = n ħ.

2) According to quantum mechanics, an atom in a state with principle quantum number n can take on the following discrete orbital angular momentum:

L can equal zero, which causes great difficulty when attempting to apply classical mechanics to this system.

1,,3,2,1,0 )1( n lllL

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III. Orbital magnetic quantum number ml:• The atom possesses an orbital angular momentum L. A magnetic moment

exists due to this angular momentum. (IA=(-e/2m)L).• There are distinct directions allowed for the magnetic moment vector with

respect to the magnetic field vector B.• Because the magnetic moment of the atom is related to the angular

momentum vector L, the discrete direction of translates into the fact that the direction of L is quantized.

• Therefore, Lz, the projection of L along the z axis, can have only discrete values. The orbital magnetic quantum number ml specifies the allowed values of the z component of orbital angular momentum:

lll mmL llz ,,1,

Space quantization: The quantization of the possible orientations of L with respect to an external magnetic field.

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Vector model of space quantization:• L can never be parallel or antiparallel to B. (Lz < L).• L lies anywhere on the surface of a cone that makes an angle θ with the z axis. The angle θ is also quantized:

)1(cos

ll

mL lz

L

Zeeman effect: the splitting of spectral lines in a strong magnetic field.

Figure: The upper level has l = 1 and splits into three different levels corresponding to the three different directions of .

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IV. Spin magnetic quantum number ms:The fourth quantum number, spin magnetic quantum number ms, does not come from the Schrödinger equation.

Electron spin:• Only two directions exist for electron spins. The

electron can have spin up (a) or spin down (b).• In the presence of a magnetic field, the energy of the

electron is slightly different for the two spin directions. This produces doublets in the spectra of some gases.

• The electron cannot be considered to be actually spinning. The experimental evidence supports that the electron has some intrinsic angular momentum that can be described by ms.

• Dirac showed the electron spin from the relativistic properties of the electron.• Stern-Gerlach experiment: A beam of silver atoms is split in two by a

nonuniform magnetic field.

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Spin angular momentum:• Electron spin can be described by a single quantum number s, whose value

can only be s = 1/2. The magnitude of the spin angular momentum S is

2

3)1( ssS

• The spin angular momentum can have two orientations relative to the z axis, specified by the spin quantum number ms = ± 1/2:

ms = + 1/2 corresponds to the spin up case; ms = - 1/2 corresponds to the spin down case.

The z component of spin angular momentum is

2

1 sz mS

The spin magnetic moment: .spin Sμem

e

The z component of the spin magnetic moment: .2spin,

ez m

eμ

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Exclusion principle and the periodic table

42.7 The exclusion principle and the periodic table The four quantum numbers, n, l, ml, ms can be used to describe all the electronic states of an atom regardless of the number of electrons in its structure.Question: How many electrons can be in a particular quantum states?

Pauli’s exclusion principle: No two electrons can ever be in the same quantum state; Therefore, no two electrons in the same atom can have the same set of quantum numbers.

Sequence of filling subshells: Once a subshell is filled, the next electron goes into the lowest-energy vacant state.

Orbital: The atomic state characterized by the quantum numbers n, l and ml.From the exclusion principle, only two electrons can be present in any orbital. One electron will have spin up and one spin down.

Lecture # 4Ch. 42

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Q: Which of the following are permissible sets of quantum numbers?

A. n = 4, l = 4, ml = 0, ms = ½

B. n = 3, l = 2, ml = 1, ms = -½

C. n = 2, l = 0, ml = 0, ms = ³/²

D. n = 5, l = 3, ml = -3, ms = ½

(A) Not permitted. When n = 4, the maximum value of l is 3.

(B) Permitted.

(C) Not permitted; ms can only be +½ or –½.

(B) Permitted.

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Question: How are the electrons aligned in an orbital?Hund’s rule: When an atom has orbitals of equal energy, the order in which they are filled by electrons is such that a maximum number of electrons have unpaired spins. (Exceptions exist).

Electronic configuration:The filling of the electronic states must obey both Pauli’s exclusion principle and Hund’s rule.

The periodic table:

An arrangement of the atomic elements according to their atomic masses and chemical similarities. The chemical behavior of an element depends on the outermost shell that contains electrons.