Ch 9pages 444-445; 469-470

Lecture 19 – The Hydrogen atom

Planck’s theory of black body radiation proposed that energy is emitted by oscillators in discrete packets E=h

These packets, called photons, are treated as energy particles

As we shall see later, at least two additional experiments further support the particle nature of radiation.

Summary of lecture 18

Photon Emission

• Relaxation from one energy level to another by emitting a photon.

• WithE = hc/

• If = 440 nm,

= 4.5 x 10-19 J

Em

issi

on

Emission spectrum of H

“Continuous” spectrum “Quantized” spectrum

Any E ispossible

Only certain E areallowed

E E

Emission spectrum of H

Light Bulb

Hydrogen Lamp

Quantized, not continuous

Emission spectrum of H

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

22

1

n2n = 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:

Ry1

n12

1

n22

n1 = 1, 2, 3, …..

• This suggests that the energy levels of the H atom are proportional to 1/n2

n2 = n1+1, n1+2, …

Ry = 3.29 x 1015 1/s

Bohr’s Atom

The stability of the nuclear atom cannot be explained by classical mechanics. Consider the hydrogen-like nuclear atom: an electron of mass m and charge –e is a distance R from the much-heavier nucleus of mass M and charge +Ze. Because the proton and electron are charged, they have a Coulombic interaction

U rZe

R( )

1

4 0

2

Where 0 is the vacuum permittivity (8.85x10-12 C2/Nt-m2)

As a result of the Coulombic Force, the electron will be compelled to move toward the nucleus, eventually colliding with it

Bohr’s Atom

To mechanically stabilize the atom, the electron might be assumed to orbit the nucleus, just as a planet orbits the sun. In a planetary system, it is the motion of the planets and resulting centrifugal force that prevents them from colliding with the sun under the influence of gravitation

We can by analogy propose that the electron orbits the nucleus such that the Coulomb Force, which is directed inward toward the nucleus, effects a centripedal acceleration of the electron

U rZe

R( )

1

4 0

2

V

R

2

Where V is the velocity of the electron in its orbit

Bohr’s Atom

Let us use F=ma where F is the Coulomb force and a is the acceleration to balance the two forces:

U rZe

R( )

1

4 0

2

V

R

2

One would think that the analogy would be sufficient to explain the stability of atoms

FZe

R

V

R

mV

R

1

4 0

2

2

2 2

mM

m M

(the reduced mass is used because we are discussing the motion of the electron relative to the center-of-mass of the atom, which is located at the nucleus to good approximation)

Bohr’s Atom

However, according to classical electrodynamics, an orbiting electron charge (any charge) must similarly emit radiation at the frequency of its orbital motion. As a consequence, atoms should be constantly radiating energy as a result of the motion of their electrons. As the electrons radiate, energy is lost, their orbits would decay and eventually they would collide with the nucleus, thus annihilating the atom. Such an atom should self-annihilate in about 10-12 sec

U rZe

R( )

1

4 0

2

V

R

2

FZe

R

V

R

mV

R

1

4 0

2

2

2 2

Bohr’s Atom

Furthermore, atoms do not spontaneously emit radiation, but only radiate when they absorb energy and then they give off radiation only at certain discrete frequencies

Niels Bohr used a quantum hypothesis to explain the emission spectrum of hydrogen. He reasoned that the angular momentum mVR of an orbiting electron is quantized

The orbit of an electron is assumed constant when the integral of the momentum along the circular orbit equals nh, where n=1,2,3. More explicitly:

2mVR nh

FZe

R

V

R

mV

R

1

4 0

2

2

2 2

Bohr’s Atom

When this condition holds, the electron does not radiate and the orbit is mechanically stable. Bohr’s momentum quantization equation:

2mVR nh

mVnh

R

2

FZe

R

V

R

mV

R

1

4 0

2

2

2 2

and the force equation requirement are combined to yield:

2

2

0

22

4

1

2 R

Ze

mR

nh

R

m

R

mV

This equation can be solved to yield the allowed orbital radii:

Z

an

mZe

nhRn

02

20

24

2

Bohr’s Atom

The Bohr orbit is:

2mVR nh FZe

R

V

R

mV

R

1

4 0

2

2

2 2

The force balance requirement gives the expression for the total quantized energy of Bohr’s atom:

Z

an

mZe

nhRn

02

20

24

2

cmxme

ha 8

20

2

0 1053.04

2

2 2 2 2 42

2 2 20 0 0 0

1

2 8 4 8 8n n n

Ze Ze Ze Z e mE K U mV U

R R R h n

Bohr’s Atom

The energy equation for the hydrogenic atom predicts that, as in the case of the black body radiator, energy can only be emitted in discrete quantities. If an atom absorbs energy, its electron will be promoted from the nth orbit to, say, the kth orbit. To return to the nth orbit, the atom must emit energy.

The frequency of the energy particle or photon emitted by the atom is given by:

2 2 2 2 42

2 2 20 0 0 0

1

2 8 4 8 8n n n

Ze Ze Ze Z e mE K U mV U

R R R h n

2 4

2 2 2 20

1 1

8k n

Z e mE E E hv

h n k