Empirical formula discoveredby Balmer to describe thehydrogen spectra
1λ= R
1
22−
1
n2
Lyman, Paschen, BrackettPfund Series
1λ= R
1
12−
1
n2
1λ= R
1
22−
1
n2
1λ= R
1
32−
1
n2
1λ= R
1
42−
1
n2
1λ= R
1
52−
1
n2
Energy of a photon
E
i− E
f= hf =
hcλ
Bohr Model
Ei− E
f= hf =
hcλ
E = KE + EPE
Energy of an atom
Ei− E
f= hf =
hcλ
E = KE + EPE
E =12
mv2 −kZe2
r
Newtonian mechanics
Fc= F
e
Fnet equals Coulombicattraction
Fc= F
e
mv2
r=
kZe2
r 2
Convenient expression
Fc= F
e
mv2
r=
kZe2
r 2
mv2 =kZe2
r
Energy of an atom
E =
12
mv2 −kZe2
r
After simplification
E =12
mv2 −kZe2
r
E =12
kZe2
r
−
kZe2
r= −
kZe2
2r
Recall rotational mechanics
L = Iω
I = mr 2
ω =vr
L = mvr
Bohr Postulate
Electrons in a stable orbit do not radiate.That is, atoms are stable and do notcollapse.
Electrons in transition from a higher toa lower energy state radiate a photon ofenergy hf.
Bohr Postulate
L
n= mvr
n= n
h2π
;n = 1,2,3...
Angular Momentum
Ln= mvr
n= n
h2π
;n = 1,2,3...
v =n
mrn
h2π
Combining expressions
v =n
mrn
h2π
mv2 =kZe2
r
General formula for the radius
r
n=
h2
4π 2mke2
n2
Z;n = 1,2,3...
Bohr radius
r1= 5.29 ⋅10−11( )m
Energy levels of the atom
E
n=−kZe2
2r⇒−
2π 2mk 2e4
h2
Z 2
n2
Bohr Energy Levels
En= −
2π 2mk 2e4
h2
Z 2
n2
= −2.18 ⋅10−18 J( ) Z 2
n2
En= −13.6eV( ) Z 2
n2
Energy of a photon
hcλ= ΔE = E
i− E
f
hcλ= ΔE = E
i− E
f
1λ=
2π 2mk 2e4
h3c
Z 2( ) 1
nf2−
1
ni2
Rydberg Constant!
1λ= R Z 2( ) 1
nf2−
1
ni2
R = 1.097 ×107 m−1
Limitations of the Bohr Model
Works well for hydrogen, and for othersingle electron atoms, but not as wellfor helium and even less well for otheratomsBased upon postulates that angularmomentum is quantized