Derivation of theRydberg Constant

An introduction to the Bohr Modelof the Atom

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