Transcript Lecture 17: Bohr Model of the Atom
Bohr Model of the Atom Outline – Emission spectrum of atomic hydrogen.
– The Bohr model.
– Extension to higher atomic number.
Photon Emission • Relaxation from one energy level to another by emitting a photon.
• With D E = hc/ l • If l = 440 nm, DE = 4.5 x 10 -19 J
Emission spectrum of H “Continuous” spectrum “Quantized” spectrum Any D E is possible Only certain D E are allowed
Emission spectrum of H (cont.) Light Bulb Hydrogen Lamp Quantized, not continuous
Emission spectrum of H (cont.) We can use the emission spectrum to determine the energy levels for the hydrogen atom.
Balmer Model • Joseph Balmer (1885) first noticed that the frequency of visible lines in the H atom spectrum could be reproduced by: 1 2 2 1
n
2 n = 3, 4, 5, …..
• The above equation predicts that as n increases, the frequencies become more closely spaced.
Rydberg Model • Johann Rydberg extends the Balmer model by finding more emission lines outside the visible region of the spectrum:
R y
1
n
1 2 1
n
2 2 n 1 = 1, 2, 3, …..
n 2 = n 1 +1, n 1 +2, … R y = 3.29 x 10 15 1/s • This suggests that the energy levels of the H atom are proportional to 1/n 2
The Bohr Model • Niels Bohr uses the emission spectrum of hydrogen to develop a quantum model for H.
• Central idea: electron circles the “nucleus” in only certain allowed circular orbits.
• Bohr postulates that there is Coulomb attraction between e- and nucleus. However, classical physics is unable to explain why an H atom doesn’t simply collapse.
The Bohr Model (cont.) • Bohr model for the H atom is capable of reproducing the energy levels given by the empirical formulas of Balmer and Rydberg.
E
2.178
x
10 18
J
Z n
2 2 Z = atomic number (1 for H) n = integer (1, 2, ….) • R y x h = -2.178 x 10 -18 J (!)
The Bohr Model (cont.)
E
2.178
x
10 18
J
Z n
2 2 • Energy levels get closer together as n increases • at n = infinity, E = 0
The Bohr Model (cont.) • We can use the Bohr model to predict what D E is for any two energy levels D
E
E final
E initial
D
E
2.178
x
10 18
J
1
n
2
final
D
E
( 2.178
x
10 18
J
) 1
n
2
initial
2.178
x
10 18
J
n
2 1
final
1
n
2
initial
The Bohr Model (cont.) • Example: At what wavelength will emission from n = 4 to n = 1 for the H atom be observed?
D
E
D
E
2.178
x
10 18
J
n
1 2
final
1
n
2
initial
1 2.178
x
10 18
J
1 1 16 4 2.04
x
10 18
J
D
E
2.04
x
10 18
J
hc
l l 9.74
x
10 8
m
97.4
nm
The Bohr Model (cont.) • Example: What is the longest wavelength of light that will result in removal of the e from H?
D
E
2.178
x
10 18
J
n
1 2
final
1
n
2
initial
1 D
E
2.178
x
10 18
J
2.178
x
10 18
J
D
E
2.178
x
10 18
J
hc
l l 9.13
x
10 8
m
91.3
nm
Extension to Higher Z • The Bohr model can be extended to any single electron system….must keep track of Z (atomic number).
E
2.178
x
10 18
J
Z n
2 2 Z = atomic number n = integer (1, 2, ….) • Examples: He + (Z = 2), Li +2 (Z = 3), etc.
Extension to Higher Z (cont.) • Example: At what wavelength will emission from n = 4 to n = 1 for the He + atom be observed?
D D
E E
2.178
x
10 2.178
x
10 18
J
18 2
n
1 1 1 16 2 1
final
n
2 1
initial
4 8.16
x
10 18
J
D
E
8.16
x
10 18
J
hc
l l 2.43
x
10 8
m
24.3
nm
l
H
l
He
Where does this go wrong?
• The Bohr model’s successes are limited: • Doesn’t work for multi-electron atoms.
• The “electron racetrack” picture is incorrect.
• That said, the Bohr model was a pioneering, “quantized” picture of atomic energy levels.