# GroupWk13Soln

advertisement ```Physics 208, Spring 08, Group Problem Week 13
Problem on the Bohr model of the Hydrogen atom
In the Bohr’s model the H-atom has a proton in the nucleus and an electron
orbiting in circular allowed orbits. Remember that:
Planck’s constant: h = 6.626 x 10-34 m2 kg/s
Reduced Planck constant: ℏ = h/2π= 1.055 x 10-34 J s.
Planck’s constant x velocity of light hc = 1240 eV x nm
electron mass: 9.11 x 10-31 kg, melectron c2 = 511,000 eV
A) Calculate the velocity and the radius of the electron in the first allowed orbit
from the fact that you know that there is an electric force between the nucleus
and the electron and the quantization condition on the angular momentum of the
electron postulated by Bohr.
For each of the results check the dimensions of your result.
Solution:
e2
v2
e2
F  k 2  m  a0  k
(1)
a0
a0
mv 2

On an orbit n the quantization condition of the angular momentum says:
L = mvr = pr= nℏ hence for the first orbit v= ℏ/(ma0) (2). Inserting (1) in (2):
v


ma 0

mv 2
ke2 9 10 9  (1.6 1019 ) 2

v


 2.18 10 6 m /s
2
34
mke
1.055 10
The radius of the first orbit (also called Bohr radius) is:
1.055 1034
a0 

 0.0.053nm
mv 9.11031 kg  2.18 10 6
B) Calculate the total energy of the electron in the first orbit.This is the energy
that needs to be provided to the electron in the first energy level to ionize the
atom.
Solution:
p2
e2
E
k
2m
a0
e2
e2m
2
2
F  k  mv  p  k
a0
a0
Ek

e 2m
e2
e2
9 10 9  (1.6 1019 ) 2
 k  k

 2.17 1018 J  13.6eV
9
a0 2m
a0
2a0
2  0.053 10
C) Calculate the wavelength of the light emitted as a hydrogen atom undergoes a
transition from state n = 4 to n = 2. To which series of the H-atom emission
spectrum does this line belong and in what range of the EM spectrum is it?
Solution:
The energy of the energy levels of the H-atoms are En = -13.6eV/n2. Hence
13.6 13.6
hc
hc
E4  E2  

 2.55eV 

1240nmeV /2.55eV  486nm
16
4

E
Balmer, Blue in the visible

```