Hydrogen Spectra explained

1. In 1885 Johann Balmer, a Swiss physicist,
discovered, by trial and error, that the
energies in the emission spectrum of
hydrogen were given by the formula:
Explanation of spectra
Johann Balmer
(1825-1898)

DE=−R
z
gZ
21
n
f
2
−
1
n
i
2
⎛
⎝
⎜
⎞
⎠
⎟
where n may take integer values 3, 4, 5, … and R is a
constant number

2. Since the emitted light from a gas carries energy, it is reasonable
to assume that the emitted energy is equal to the difference
between the total energy of the atom before and after the
emission.
Since the emitted light consists of photons of a specific
wavelength, it follows that the emitted energy is also of a specific
amount since the energy of a photon is given by:
Explanation of spectra
l
hc
hE==f
This means that the energy of the atom is discrete, that is, not
continuous.

3. If the energy of the atom were continuous the emission of light
wouldn't always be a set of specific amounts.
The first attempt to explain these observations came with the
“electron in a box” model.
Imagine that an electron is confined in a box of linear size L.
If the electron is treated as a wave, it will have a wavelength
given by:
The “electron in a box” model
p
h
=l
x=0 x=L
the electron can only be found somewhere
along this line

4. If the electron behaves as a wave, then:
The wave is zero at the edges of the box
The wave is a standing wave as the electron does not
lose energy
This means that the wave will have nodes at x=0 and x=L.
This implies that the wavelength must be related to the size
of the box through:
The “electron in a box” model
n
2L
=l
Where n is an integer

5. Therefore, the momentum of the electron is:
The “electron in a box” model
2L
mh
n
2L
hh
p ===
l
The kinetic energy is then:
2
22
2
2
k
8mL
hn
2m
2L
mh
2m
p
E =
÷
ø
ö
ç
è
æ
==

6. This result shows that, because the electron was treated as
a standing wave in a “box”, it was deduced that the
electron’s energy is quantized or discrete:
The “electron in a box” model
3n
8mL
h
9
2n
8mL
h
4
1n
8mL
h
1
2
2
2
2
2
2
=´
=´
=´
=
k
E
However, this model is not correct but because it shows
that energy can be discrete it points the way to the correct
answer.

7. In 1926, the Austrian physicist Erwin Schrödinger
provided a realistic quantum model for the
behaviour of electrons in atoms.
The Schrödinger theory assumes that there is a
wave associated to the electron (just like de
Bröglie had assumed)
This wave is called wavefunction and
represented by:
The Schrödinger theory
),(txy
Erwin
Schrödinger
(1887-1961)
This wave is a function of position x and time t. Through
differentiation, it can be solved to find the Schrödinger function:
),r(ψ )(),r(ψ
2
),r(
2
2
trVt
m
t
t
i +Ñ-=
¶
¶ 
y

8. The Schrödinger theory
The German physicist Max Born interpreted Schrödinger's
equation and suggested that:
2
),(txy
can be used to find the probability of finding an electron
near position x at time t.
This means that the equation cannot tell exactly where to
find the electron.
This notion represented a radical change from classical
physics, where objects had well-defined positions.

9. The Schrödinger theory
Solving for Hydrogen, it is found that:
In other words, this theory predicts that the electron in the
hydrogen atom has quantized energy.
The model also predicts that if the electron is at a high
energy level, it can make a transition to a lower level.
In that process it emits a photon of energy equal to the
difference in energy between the levels of the transition.
eV
2
n
13.6
E-=

10. The Schrödinger theory
Because the energy of the photon is given by E = hf, knowing the
energy level difference, we can calculate the frequency and
wavelength of the emitted photon.
Furthermore, the theory also predicts the probability that a
particular transition will occur.
energy
This is essential to
understand why
some spectral lines
are brighter than
others.
Thus, the
Schrödinger theory
explains atomic
spectra.
0 eV
-13.6 eV
n=1
n=2
n=3
n=4
n=5
high n
energy
levels very
close to
each other