Chemistry - A Molecular Science

ν = R

H

(^12) nlo

• 
  

(^12) nhi
⎛ ⎜⎜⎝
(^) ⎞ ⎟⎟⎠
Eq. 2.3a
nhi

n
are lo
integers
and R
=H
3.290x10
15 s
-1
is the Rydberg constant for the hydrogen
atom. The Rydberg constant had no theoretical basis at the time; it was simply the number that made the equation work! Multiplication of
Equation 2.3a by Planck's constant yields
the energy of the
emitted
photon (E = h
) as shown in Equation 2.3b, ν
-18
photon
H
22
22
lo
hi
lo
hi
11
11
E
= h = hR

• 
  

= 2.180 10

• 
  

J

nn

nn

ν

⎛⎞

⎛⎞

×

⎜⎟

⎜⎟

⎜⎟

⎜⎟

⎝⎠

⎝⎠

Eq.

2.3b

Example 2.2

What are the wavelength and the frequency of the n

= 3 hi

→

n

= 2 emission lo

transition in hydrogen? We use the values n

= 2 and nlo

= 3 in the Rydberg Equation (Eq 2.3a) to obtain the hi

emitted wavelength. ν = 3.290

×^10

15

(^122)

• 
  

(^123)
⎛ ⎜⎝
⎞ = 4.569⎟⎠
×^10
14
-1 s
λ =
c = ν
2.998
×^10
8 m
-1⋅s
4.569
×^10
14
-1 s
= 6.562
×^10
-7 m = 656.2 nm
This is the red line in the visible spectrum, the lowest energy (frequency) line in the visible region of the hydrogen spectrum. According to Rydberg, the frequency
of each emitted photon obeyed a relatively
simple mathematical expression; it is proportio
nal to the difference of the reciprocals of
two squared integers! Scientis
ts had another simple relati
onship, and understanding its
origin would unlock yet anothe
r of nature’s secrets. It was time for a new model of the
atom, one that would explain 40 different lin
es in the hydrogen spectrum and account for
the relatively simple relationship of their fre
quencies. Rutherford’s model of electrons
orbiting the nucleus like planets around the sun would serve as the starting point for the new model, but the new model had to
incorporate the ideas of quantization.
Chapter 2 Quantum Theory
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Carolina
State
University