Lecture 24 Quantum mechanics: Atoms
Lecture 24 Quantum mechanics: Atoms
The Hydrogen atom One proton + one electron Bohr’s model: En = -13.6 eV/ n n: quantum number Actually there are 4 quantum numbers Each set of q.n. specifies the “state” of the electron.
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The quantum numbers: •Principal quantum number: quantifies the energy. n= 1,2,3…. •Orbital quantum number: quantifies the magnitude of the angular momentum l = 0,1,2,…n-1 Angular momentum L = ◊ l ( l+1) h •Orbital magnetic quantum number: quantifies the components of the angular momentum ml = -l,…,-2,-1,0,1,2,….l Lz = ml h •Spin quantum number: doesn't have a classical analogy. ms = +1/2 , -1/2 2
For Hydrogen, the energy only depends on n (in absence of a magnetic field) For atoms with more than one electron E depends on n and l. •Ground state (g.s.): n=1, E = -13.6 eV 2 states: n=1, l=0, ml=0, ms = ±1/2 •First excited state: n=2, E = -3.4 eV 8 states: n=2, l=0, ml=0, ms = ±1/2 ml= -1, ms = ±1/2 n=2, l=1,
ml=0, ms = ±1/2 ml= 1, ms = ±1/2
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Complex atoms Pauli’s Exclusion Principle: No two electrons in an atom can have the same state (same set of quantum numbers n, l, ml, ms) •Principal q.n.: All states of same n form a shell n: Symbol:
1 2 3 4 5 ... K L M N P ...
•Orbital q.n.: States with same n and l form a subshell l: Symbol:
0 1 2 3 4 ... s p d f g
1s1
Example: H g.s.: n
one electron
l 4
Number of electrons in subshell l: 2 x (2l+1)
2 possible values of ms
l
Symbol
0 1 2 3 4
s p d f g
# of possible values of ml
Population 2 6 10 14 18
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Examples, ground states: Both are equally probable H: 1 e-
He: 2 e-
Li: 3 e-
1s1
1s2
2s1 1s2
Be: 4 e-
2s2 1s2
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2p6 Ne: 10 e-
2s2 1s2
3 s1 2p6 Na: 11 e-
2s2 1s2
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The periodic Table Atoms are grouped according to increasing mass Columns: similar properties. Examples: •The noble gases, in the last column, all have completely filled shells. Are spherically symmetric, electrons are not easily lost or attracted. Nobel gases are “nonreactive”. •Halogens, column seven. Lack one electron from filled shell: quite reactive. •Alkali metals (column 1) There is a single electron in a outer s subshell. Very reactive. •Periods have same number of occupied shells.
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Z: atomic number 1
(number of protons = number of electrons)
H 1.008
Alkali
Atomic mass, in units a.m.u = 1.66 x 10-27 kg
Halogens
noble gases
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Question: There are 18 states with n=3. How many are there with n=4? We have to add those states with l=3, that are 2x(2x3+1) = 14 The total is 18 + 14 = 32
Selection rule for transition of one electron between two states: Transitions can only occur between states with ∆l = ± 1 (a phonon is absorbed or emitted)
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Example: Which of the following transitions are permitted? a) 3p b) 4p c) 4p
2p 2s 5d
Ans: b and c, a) is forbidden by the selection rule because ∆l =0 Example: Which of the following configurations are possible? a) 1s2 2s2 2p3 b) 1s2 2s2 2p7 3s1 Ans: a, in b) 2p7 violates the exclusion principle. 11
Example: What is the angular momentum of an electron in the state 3d? d
l=2, L = ◊ l ( l+1) h
= ◊6 h
When the principal number is n=5, how many different values of l are possible? a)2 b)3 c)4 d)5 Which of the following electron transitions in a Hydrogen atom results in the emission of the highest energy photon? a)n = 6 to n = 2 b)n = 2 to n = 6 c)n = 5 to n = 1 d)n = 1 to n = 5
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The Hydrogen atom One proton + one electron Bohr’s model: En = -13.6 eV/ n n: quantum number Actually there are 4 quantum numbers Each set of q.n. specifies the “state” of the electron.
1
The quantum numbers: •Principal quantum number: quantifies the energy. n= 1,2,3…. •Orbital quantum number: quantifies the magnitude of the angular momentum l = 0,1,2,…n-1 Angular momentum L = ◊ l ( l+1) h •Orbital magnetic quantum number: quantifies the components of the angular momentum ml = -l,…,-2,-1,0,1,2,….l Lz = ml h •Spin quantum number: doesn't have a classical analogy. ms = +1/2 , -1/2 2
For Hydrogen, the energy only depends on n (in absence of a magnetic field) For atoms with more than one electron E depends on n and l. •Ground state (g.s.): n=1, E = -13.6 eV 2 states: n=1, l=0, ml=0, ms = ±1/2 •First excited state: n=2, E = -3.4 eV 8 states: n=2, l=0, ml=0, ms = ±1/2 ml= -1, ms = ±1/2 n=2, l=1,
ml=0, ms = ±1/2 ml= 1, ms = ±1/2
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Complex atoms Pauli’s Exclusion Principle: No two electrons in an atom can have the same state (same set of quantum numbers n, l, ml, ms) •Principal q.n.: All states of same n form a shell n: Symbol:
1 2 3 4 5 ... K L M N P ...
•Orbital q.n.: States with same n and l form a subshell l: Symbol:
0 1 2 3 4 ... s p d f g
1s1
Example: H g.s.: n
one electron
l 4
Number of electrons in subshell l: 2 x (2l+1)
2 possible values of ms
l
Symbol
0 1 2 3 4
s p d f g
# of possible values of ml
Population 2 6 10 14 18
5
Examples, ground states: Both are equally probable H: 1 e-
He: 2 e-
Li: 3 e-
1s1
1s2
2s1 1s2
Be: 4 e-
2s2 1s2
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2p6 Ne: 10 e-
2s2 1s2
3 s1 2p6 Na: 11 e-
2s2 1s2
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The periodic Table Atoms are grouped according to increasing mass Columns: similar properties. Examples: •The noble gases, in the last column, all have completely filled shells. Are spherically symmetric, electrons are not easily lost or attracted. Nobel gases are “nonreactive”. •Halogens, column seven. Lack one electron from filled shell: quite reactive. •Alkali metals (column 1) There is a single electron in a outer s subshell. Very reactive. •Periods have same number of occupied shells.
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Z: atomic number 1
(number of protons = number of electrons)
H 1.008
Alkali
Atomic mass, in units a.m.u = 1.66 x 10-27 kg
Halogens
noble gases
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Question: There are 18 states with n=3. How many are there with n=4? We have to add those states with l=3, that are 2x(2x3+1) = 14 The total is 18 + 14 = 32
Selection rule for transition of one electron between two states: Transitions can only occur between states with ∆l = ± 1 (a phonon is absorbed or emitted)
10
Example: Which of the following transitions are permitted? a) 3p b) 4p c) 4p
2p 2s 5d
Ans: b and c, a) is forbidden by the selection rule because ∆l =0 Example: Which of the following configurations are possible? a) 1s2 2s2 2p3 b) 1s2 2s2 2p7 3s1 Ans: a, in b) 2p7 violates the exclusion principle. 11
Example: What is the angular momentum of an electron in the state 3d? d
l=2, L = ◊ l ( l+1) h
= ◊6 h
When the principal number is n=5, how many different values of l are possible? a)2 b)3 c)4 d)5 Which of the following electron transitions in a Hydrogen atom results in the emission of the highest energy photon? a)n = 6 to n = 2 b)n = 2 to n = 6 c)n = 5 to n = 1 d)n = 1 to n = 5
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