Physical Chemistry Third Edition

23.2 The Spectra of Atoms 959

23.7 a.Calculate the wavelength of the light absorbed when an
electron in a box of length 1.000 nm makes the
transition fromn1ton2. In what region of the
electromagnetic spectrum (X-ray, ultraviolet, visible,
infrared, microwave) does this light lie?
b.The contribution of this electron to the dipole moment is
−e(x−a/2), where−eis the charge on the electron,
xis its coordinate, andais the length of the box.
Argue that the transition dipole moment for this
transition is nonzero for all transitions.Hint:Consider
graphs of the factors in the integrand function in the
integral used to calculate the transition dipole
moment.

23.8 a.A solution of a certain dye in ethanol has a

concentration of 0.0001000 mol L−^1 and gives an

absorbance of 1.034 at a wavelength of 480 nm in a

cell of length 1.000 cm. Find the molar absorptivity of

this dye in this solvent at this wavelength. If this is the

wavelength of maximum absorbance, what is the color

of the solution?

b.A solution of a different dye in methanol gives an

absorbance of 0.987 at a wavelength of 680 nm. If the

molar absorptivity of this dye in this solvent at this

wavelength is equal to 1. 80 × 104 L mol−^1 cm−^1 ,

what is its molar concentration?

23.2 The Spectra of Atoms

The Hydrogen Atom

The spectra of atoms are due to electronic transitions. The following selection rules are

derived when the hydrogen atom orbitals are substituted in the integral of Eq. (23.1-8):^5

Hydrogen Atom Selection Rules

∆mmfinal−minitial0,± 1 (23.2-1a)

∆llfinal−linitial± 1 (23.2-1b)

∆n:no restrictions (23.2-1c)

These selection rules correspond to conservation of angular momentum in the atom–

radiation system, because the angular momentum component of a photon is±h ̄.

Because∆l±1, a hydrogen atom in anssubshell can make a transition only to

apsubshell, whereas an atom in apsubshell state can make a transition to anssub-

shell or to adsubshell, and so on. All states in the same shell have the same energy

in hydrogen atoms, and a simple spectrum is obtained as was shown in Figure 23.2.

Figure 23.6 shows schematically some of the transitions that take place, with line seg-

ments connecting each pair of states between which transitions can occur. This type of

diagram is called aGrotrian diagram.

More states not shown

3 s

2 s

1 s

25

210

215

2 p

3 p 3 d

El

eV

Figure 23.6 The Energy Levels
of the Hydrogen Atom and the
Allowed Transitions between
Them. This type of diagram is
known as a Grotrian diagram.

EXAMPLE23.3

From Rydberg’s formula in Eq. (14.4-10), find the wavelength and frequency of the photons

emitted by a hydrogen atom undergoing then 2 →n1 transition.

Solution

1

λ

R

(

1

n 1

−

1

n 2

)

(109678 cm)−^1

(

1

n^21

−

1

n^22

)

(^5) J. C. Davis,op. cit., pp. 256–257 (note 2).