Topic 13 Quantum and Nuclear physics Atomic spectra and atomic energy states

Topic 13

Quantum and Nuclear physics

Atomic spectra and atomic energy states

How do you excite an atom?

1. Heating to a high temperature

2. Bombarding with electrons

3. Having photons fall on the atom

I’m excited!

Atomic spectra

When a gas is heated to a high temperature, or if an electric current is passed through the gas, it begins to glow.

cathode anode

electric current

Light emitted

Low pressure gas

Emission spectrum

If we look at the light emitted (using a spectroscope) we see a series of sharp lines of different colours. This is called an emission spectrum.

Absorption Spectrum

Similarly, if light is shone through a cold gas, there are sharp dark lines in exactly the same place the bright lines appeared in the emission spectrum.

Some wavelengths missing!Light source gas

Why?

Scientists had known about these lines since the 19th century, and they had been used to identify elements (including helium in the sun), but scientists could not explain them.

Rutherford

At the start of the 20th century, Rutherford viewed the atom much like a solar system, with electrons orbiting the nucleus.

Rutherford

However, under classical physics, the accelerating electrons (centripetal acceleration) should constantly have been losing energy by radiation (this obviously doesn’t happen).

Radiating energy

Niels Bohr

In 1913, a Danish physicist called Niels Bohr realised that the secret of atomic structure lay in its discreteness, that energy could only be absorbed or emitted at certain values.

At school they called me “Bohr the

Bore”!

The Bohr Model

Bohr realised that the electrons’ angular momentum is an integral (whole number) multiple of the unit h/2π. This meant that the electron could only be at specific energy levels (or states) around the atom.

The Bohr Model

We say that the energy of the electron (and thus the atom) can exist in a number of states n=1, n=2, n=3 etc. (Similar to the “shells” or electron orbitals that chemists talk about!)

n = 1

n = 3

n = 2

The Bohr Model

The energy level diagram of the hydrogen atom according to the Bohr model

n = 1 (the ground state)

n = 2

n = 3

n = 4n = 5

High energy n levels are very close to each other

Energy eV

-13.6

0

Electron can’t have less energy than this

The Bohr ModelAn electron in a higher state than the ground state is called an excited electron. It can lose energy and end up in a lower state.

High energy n levels are very close to each other

n = 1 (the ground state)

n = 2

n = 3

n = 4n = 5

-13.6

Energy eV

0

Wheeee!

Atomic transitions

If a hydrogen atom is in an excited state, it can make a transition to a lower state. Thus an atom in state n = 2 can go to n = 1 (an electron jumps from orbit n = 2 to n = 1)

n = 1 (the ground state)

n = 2

n = 3

n = 4n = 5

-13.6

Energy eV

0

electronWheeee!

Atomic transitions

Every time an atom (electron in the atom) makes a transition, a single photon of light is emitted.

n = 1 (the ground state)

n = 2

n = 3

n = 4n = 5

-13.6

Energy eV

0

electron

Atomic transitions

The energy of the photon is equal to the difference in energy (ΔE) between the two states. It is equal to hf. ΔE = hf

n = 1 (the ground state)

n = 2

n = 3

n = 4n = 5

-13.6

Energy eV

0

electron

ΔE = hf

The Lyman Series

Transitions down to the n = 1 state give a series of spectral lines in the UV region called the Lyman series.

n = 1 (the ground state)

n = 2

n = 3

n = 4n = 5

-13.6

Energy eV

0

Lyman series of spectral lines (UV)

The Balmer Series

Transitions down to the n = 2 state give a series of spectral lines in the visible region called the Balmer series.

n = 1 (the ground state)

n = 2

n = 3

n = 4n = 5

-13.6

Energy eV

0

UV

Balmer series of spectral lines (visible)

The Pashen Series

Transitions down to the n = 3 state give a series of spectral lines in the infra-red region called the Pashen series.

n = 1 (the ground state)

n = 2

n = 3

n = 4n = 5

-13.6

Energy eV

0

UV

visible

Pashen series (IR)

Emission Spectrum of Hydrogen

Which is the emission spectrum and which is the absorption spectrum?

The emission and absorption spectrum of hydrogen is thus predicted to contain a line spectrum at very specific wavelengths, a fact verified by experiment.

Pattern of lines

Since the higher states are closer to one another, the wavelengths of the photons emitted tend to be close too. There is a “crowding” of wavelengths at the low wavelength part of the spectrum

n = 1 (the ground state)

n = 2

n = 3

n = 4n = 5

-13.6

Energy eV

0

Spectrum produced

Limitations of the Bohr Model

1. Can only treat atoms or ions with one electron

2. Does not predict the intensities of the spectral lines

3. Inconsistent with the uncertainty principle (see later!)

4. Does not predict the observed splitting of the spectral lines

The “electron in a box” model!

☺

Hi! I’m Erica the electron

The “electron in a box” model!

• Imagine an electron is confined within a linear box length L.

☺

L

The “electron in a box” model!

• According to de Broglie, it has an associated wavelength λ = h/p

☺

L

L

The “electron in a box” model!

• Imagine then the electron wave forming a stationary wave in the box.

L

The “electron in a box” model!

• Therefore we have a stationary wave with nodes at x = 0 and at x = L (boundary conditions)

The “electron in a box” model!• The wavelength therefore of any stationary

wave must be λ = 2L/n where n is an integer.

L

The “electron in a box” model!

• The momentum of the electron is thus

• P = h/λ = h/2L/n = nh/2L

The “electron in a box” model!

• The kinetic energy is thus = p2/2m = (nh/2L)2/2m = n2h2/8mL2

The “electron in a box” model!

• Ek = n2h2/8mL2

☺

L

Energy states

This can be thought of like the allowed frequencies of a standing wave on a string (but this is a crude analogy).

Erwin Schrödinger

The many problems with the Bohr model were corrected by Erwin Schrödinger, an Austrian physicist.

I like cats!

d2Ψ/dx2 = -8π2m(E – V)Ψ/h2

The Schrödinger equation

Erwin Schrödinger

Schrödinger introduced the wave function, a function of position and time whose absolute value squared is related to the probability of finding an electron near a specific point in space and time.

I don’t believe that God

plays dice!

Erwin Schrödinger

In this theory, the electron can be thought of as being spread out over a large volume and there are places where it is more likely to be found than others! This can be thought of as an electron cloud.

Rubbish!

Wave function

Ψ = (2/L)½(πnx/L) where n is the state, x is the probability of finding the electron and L is the “length” of the orbital.

From this we also get the energy to be

EK = h2n2/8meL2

Beware!This wave function is only a mathematical model that fits very well. It also links well with the idea of wave particle duality (electron as wave and particle).

But it is only one mathematical model of the atom. Other more elegant mathematical models exist that don’t refer to waves, but physicists like using the wave model because they are familiar with waves and their equations. We stick with what we are familiar!

.

I’m used to the idea of waves, so

I like using Schrödinger’s

model

Heisenberg Uncertainty Principle

Heisenberg Uncertainty Principle

• It is not possible to measure simultaneously the position AND momentum of a particle with absolute precision.

ΔxΔp ≥ h/4π

Also ΔEΔt ≥ h/4π

That’s it for Quantum physics!

Next week we’ll be looking at nuclear physics!

Let’s try some questions.