Derivation of the
Rydberg Constant
An introduction to the Bohr Model
of the Atom
Empirical formula discovered
by Balmer to describe the
hydrogen spectra
 1
1
1
= R 2 − 2 
λ
2
n 
Lyman, Paschen, Brackett
Pfund Series
1 1
1
= R 2 − 2 
λ
1 n 
 1
1
1
= R 2 − 2 
λ
2
n 
 1
1
1
= R 2 − 2 
λ
3 n 
 1
1
1
= R 2 − 2 
λ
4
n 
 1
1
1
= R 2 − 2 
λ
5
n 
Energy of a photon
hc
Ei − E f = hf =
λ
Bohr Model
hc
Ei − E f = hf =
λ
E = KE + EPE
Energy of an atom
hc
Ei − E f = hf =
λ
E = KE + EPE
1 2 kZe2
E = mv −
2
r
Newtonian mechanics
Fc = Fe
Fnet equals Coulombic
attraction
Fc = Fe
2
mv
kZe
= 2
r
r
2
Convenient expression
Fc = Fe
mv 2 kZe2
= 2
r
r
2
kZe
2
mv =
r
Energy of an atom
1 2 kZe
E = mv −
2
r
2
After simplification
2
1 2 kZe
E = mv −
2
r
2
2
2


1 kZe
kZe
kZe
E= 
−
=−

2 r 
r
2r
Recall rotational mechanics
L = Iω
2
I = mr
v
ω=
r
L = mvr
Bohr Postulate
Electrons in a stable orbit do not radiate.
That is, atoms are stable and do not
collapse.
Electrons in transition from a higher to
a lower energy state radiate a photon of
energy hf.
Bohr Postulate
 h 
Ln = mvrn = n   ;n = 1,2,3...
 2π 
Angular Momentum
 h 
Ln = mvrn = n   ;n = 1,2,3...
 2π 
n  h 
v=
mrn  2π 
Combining expressions
n  h 
v=


mrn  2π 
kZe
mv =
r
2
2
General formula for the radius
2
2


h
n
rn =  2
;n
=
1,2,3...
2
 4π mke  Z
Bohr radius
(
r1 = 5.29 ⋅10
−11
)m
Energy levels of the atom
2
2
2 4
2


−kZe
2π mk e Z
En =
⇒ −

2
2
2r
h

n
Bohr Energy Levels
 2π 2 mk 2 e4  Z 2
En = − 

2
2
h

n
(
= −2.18 ⋅10
−18
Z
J 2
n
Z2
En = −13.6eV 2
n
(
)
2
)
Energy of a photon
hc
= ΔE = Ei − E f
λ
hc
= ΔE = Ei − E f
λ
2
2 4

1
2π mk e  2  1
1
=
Z  2 − 2

3
λ 
hc 
 n f ni 
( )
Rydberg Constant!
 1

1
1
2
= R Z  2 − 2
λ
 n f ni 
( )
7
R = 1.097 × 10 m
−1
Limitations of the Bohr Model
Works well for hydrogen, and for other
single electron atoms, but not as well
for helium and even less well for other
atoms
Based upon postulates that angular
momentum is quantized