Electron crashes into the nucleus!?

Physics had reached a turning point in 1900 with Planck’s hypothesis of the quantum behavior of radiation.

Slide 15

The Bohr Model of the Hydrogen Atom

Bohr’s general assumptions:

1. Stationary states, in which orbiting electrons do not radiate energy, exist in atoms and have well-defined energies, En. Transitions can occur between them, yielding light of energy:

E = En − En’ = hn

2. Classical laws of physics do not apply to transitions between stationary states, but they do apply elsewhere.

3. The angular momentum of the nth state is: where n is called the Principal Quantum Number.

Angular momentum is quantized!

Slide 16

4.4:

Bohr’s general assumptions:

“Stationary states” (orbiting electrons do not radiate energy) exist in atoms.

E = E1 − E2 = hf

Classical laws of physics do not apply to transitions between stationary states.

The mean kinetic energy of the electron-nucleus system is K = nhforb/2, where forb is the frequency of rotation. This is equivalent to ask that the angular momentum L=nh/(2p)

Slide 17

Consequences of the Bohr Model

The angular momentum is:

But:

So:

Solving for rn:

So the velocity is:

where:

a0 is called the Bohr radius. It’s the diameter of the Hydrogen atom (in its lowest-energy, or “ground,” state).

Slide 18

The diameter of the hydrogen atom for stationary states is

Where the Bohr radius is given by

The smallest diameter of the hydrogen atom is

n = 1 gives its lowest energy state (called the “ground” state)

Slide 19

Emission of light occurs when the atom is in an excited state and decays to a lower energy state (nu → nℓ).

where n is the frequency of a photon.

R∞ is the Rydberg constant.

The Hydrogen Atom

The energies of the stationary states where E0 = 13.6 eV.

Slide 20

Transitions in the Hydrogen Atom

The atom will remain in the excited state for a short time before emitting a photon and returning to a lower stationary state. All hydrogen atoms exist in n = 1 (invisible).

Slide 21

Fine Structure Constant

The electron’s velocity in the Bohr model:

On the ground state,

v1 = 2.2 × 106 m/s ~ less than 1% of the speed of light.

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