Announcements
Early Models of the Atom
The Bohr Model
Final Questions
Early Quantum Theory and Models of the Atom
Sections 27.10 - 27.13
Early Quantum Theory and Models of the Atom
Announcements
Early Models of the Atom
The Bohr Model
Reading Assignment
Read section 28.1 - 28.4
Homework Assignment 12
Homework for Chapter 27 (due at the beginning of class on Friday, November 19)
Q: 2, 7, 14, 20, 28
P: 12, 24, 38, 44, 48, 56
Early Quantum Theory and Models of the Atom
Final Questions
Announcements
Early Models of the Atom
The Bohr Model
Final Questions
A brief history
450 BC: The Greek philosopher Democritus proposed that all matter is made up of small, indivisible
particles called atomos (“indivisible”)
1896: J. J. Thomson (and colleagues) “discovered” the electron
1904: J. J. Thomson proposed the plum pudding model of the atom
1909: Hans Geiger and Ernest Marsden, under the direction of Ernest Rutherford, performed the Gold foil
experiment which demonstrated for the first time the existence of the atomic nucleus
1911: Rutherford proposed a new model (also called the planetary model) of the atom
Early Quantum Theory and Models of the Atom
Announcements
Early Models of the Atom
The Bohr Model
Final Questions
A brief history
450 BC: The Greek philosopher Democritus proposed that all matter is made up of small, indivisible
particles called atomos (“indivisible”)
1896: J. J. Thomson (and colleagues) “discovered” the electron
1904: J. J. Thomson proposed the plum pudding model of the atom
1909: Hans Geiger and Ernest Marsden, under the direction of Ernest Rutherford, performed the Gold foil
experiment which demonstrated for the first time the existence of the atomic nucleus
1911: Rutherford proposed a new model (also called the planetary model) of the atom
The Rutherford model
The Rutherford model described the atom as having a tiny, positively charged center (the nucleus) about
which negatively charged electrons orbited
While it was an important step in our understanding of the atom, it had its drawbacks
The Rutherford model was unable to explain the spectral lines of atoms (such as the Rydberg formula for
hydrogen)
It predicted that the electrons would undergo a death spiral into the nucleus; as a result, it predicted that
the frequency of the radiation emitted by the electrons would increase, and that the atom was not stable
It did not take into account the work of Planck and Einstein (quantum theory)
In 1913, Niels Bohr introduced a new model of the hydrogen atom
Early Quantum Theory and Models of the Atom
Announcements
Early Models of the Atom
The Bohr Model
Final Questions
The Bohr model of the hydrogen atom
The electrons are only allowed to travel in special circular orbits, which are certain discrete distances from
the nucleus and have certain energies associated with them
While in one of these orbits (called stationary states), the electron does not radiate energy
An electron can only gain (or lose) energy by jumping from one allowed orbit (with energy E1 ) to another
allowed orbit (with energy E2 ); in doing so, they absorb (or emit) electromagnetic radiation with a
frequency f which is determined by the energy difference ∆E between the two orbits
∆E = E2 − E1 = hf
The angular momentum L of the electron is quantized such that
L = mvrn = n
h
= n~,
2π
n = 1, 2, 3, . . .
where ~ = h/2π and n is called the principal quantum number of the orbit
Early Quantum Theory and Models of the Atom
Announcements
Early Models of the Atom
The Bohr Model
Final Questions
The Bohr radius
Using these assumptions, Bohr found that the radius of the smallest orbit in hydrogen (called the Bohr
radius) is
h2
−10
r1 =
= 0.529 × 10
m
4π 2 me ke e 2
The radii rn of the other allowed orbits for the electron are
2
rn = n r1 ,
n = 1, 2, 3, . . .
Notice that the radii of the allowed orbits go as n2 :
r1 = 0.529 × 10−10 m
r2 = 4r1 = 2.12 × 10−10 m
r3 = 9r1 = 4.76 × 10−10 m
In the Bohr model, rn = n2 r1 are the only allowed orbits; there are no orbits in between
Early Quantum Theory and Models of the Atom
Announcements
Early Models of the Atom
The Bohr Model
Final Questions
Binding energy
For each allowable orbit, the electron has a total energy
En =
1
2
2
me v −
ke e 2
rn
Using our previous result for rn , as well as the angular momentum condition L = me vrn = nh/2π, this
expression for the total energy becomes
En = −
2π 2 e 4 me ke2 1
h2
n2
=−
13.6 eV
n2
,
n = 1, 2, 3, . . .
The lowest energy level (n = 1) for hydrogen is
E1 = −13.6 eV
The energy state that has energy E1 is called the ground state; the higher energy states, which have
energies E2 , E3 , etc. are called the excited states
The minimum energy required to remove an electron from the ground state is called the binding energy
For hydrogen, the binding energy has experimentally been found to be 13.6 eV, which is in agreement with
Bohr’s model
Early Quantum Theory and Models of the Atom
Announcements
Early Models of the Atom
The Bohr Model
Final Questions
The Rydberg formula
In 1888 (25 years before Bohr’s model), the Swedish physicist Johannes Rydberg invented an empirical
formula for the spectral lines of hydrogen
Rydberg found that the wavelengths of the spectral lines satisfied the equation
1
λ
= R∞
1
n12
1
−
n22
!
where n1 < n2 are integers and R∞ = 1.097 × 107 m−1 is called the Rydberg constant
From his assumption that ∆E = hf , Bohr was able to derive this expression and, in the process, found that
R∞ =
2π 2 e 4 me ke2
h3 c
The great success of Bohr’s model of the hydrogen atom was in its ability to explain the emission and
absorption spectrum for hydrogen
Hydrogen spectral series
Lyman series: n1 = 1 (all wavelengths in this series lie in the ultraviolet band)
Balmer series: n1 = 2 (the four visible spectral lines belong to the Balmer series)
Paschen series: n3 = 3 (all wavelengths in this series lie in the infrared band)
Early Quantum Theory and Models of the Atom
Announcements
Early Models of the Atom
Early Quantum Theory and Models of the Atom
The Bohr Model
Final Questions
Announcements
Early Models of the Atom
The Bohr Model
Question #1
What is the wavelength of the first Balmer line (the transition from n = 3 to n = 2)?
Early Quantum Theory and Models of the Atom
Final Questions
Announcements
Early Models of the Atom
The Bohr Model
Question #1
What is the wavelength of the first Balmer line (the transition from n = 3 to n = 2)?
Answer
Using the Rydberg formula
1
λ
= R∞
1
1
n12
−
n2
!
where n1 = 2 and n2 = 3, we find that λ = 656 nm
This wavelength corresponds to the red spectral line of hydrogen
Early Quantum Theory and Models of the Atom
Final Questions
Announcements
Early Models of the Atom
The Bohr Model
Question #1
What is the wavelength of the first Balmer line (the transition from n = 3 to n = 2)?
Answer
Using the Rydberg formula
1
λ
= R∞
1
1
n12
−
n2
!
where n1 = 2 and n2 = 3, we find that λ = 656 nm
This wavelength corresponds to the red spectral line of hydrogen
Question #2
What is the shortest wavelength in the Balmer series?
Early Quantum Theory and Models of the Atom
Final Questions
Announcements
Early Models of the Atom
The Bohr Model
Final Questions
Question #1
What is the wavelength of the first Balmer line (the transition from n = 3 to n = 2)?
Answer
Using the Rydberg formula
1
λ
= R∞
1
1
n12
−
n2
!
where n1 = 2 and n2 = 3, we find that λ = 656 nm
This wavelength corresponds to the red spectral line of hydrogen
Question #2
What is the shortest wavelength in the Balmer series?
Answer
For the Balmer series, n1 = 2; the shortest wavelength corresponds to the electron moving from the
n2 = ∞ energy level
4
= 365 nm
λ=
R∞
This wavelength lies in the ultraviolet band
Early Quantum Theory and Models of the Atom
Announcements
Early Models of the Atom
The Bohr Model
Final Questions
Successes of the Bohr model
Unlike Rutherford’s model, Bohr’s model predicts that the atom is stable (which it is!)
Bohr’s model also accurately predicted the ionization energy of 13.6 eV for hydrogen
It also gave an explanation for the mysterious absorption and emission spectra of hydrogen
Failures of the Bohr model
Aside from hydrogen, the model was not so successful for other atoms
Though Bohr could not have known it at the time, his model violates the uncertainty principle
Conclusion
Though it has now been replaced by a fully quantum mechanical model, Bohr’s model is still often used as an
introductory model of the atom, primarily due to its simplicity and elegance
Early Quantum Theory and Models of the Atom
Announcements
Early Models of the Atom
The Bohr Model
Reading Assignment
Read section 28.1 - 28.4
Homework Assignment 12
Homework for Chapter 27 (due at the beginning of class on Friday, November 19)
Q: 2, 7, 14, 20, 28
P: 12, 24, 38, 44, 48, 56
Early Quantum Theory and Models of the Atom
Final Questions