Physical Chemistry Third Edition

14.4 The Old Quantum Theory 649

where we assign the value ofV to approach zero asr→∞. (Remember that we

can always pick the state at which we assignVto equal zero.) The kinetic energy is

given by

K 

1

2

meν^2 

1

2

e^2

4 πε 0 r



1

2

|V| (14.4-18)

where Eq. (14.4-13) has been used to replacev^2. The kinetic energy is equal to half

of the magnitude of the potential energy. This is one of the consequences of thevirial

theoremof mechanics and holds for any system acted upon only by electrostatic forces.^4

The energy of the hydrogen atom is given by the formula:

EEnK +V−

1

2

e^2

4 πε 0 r

−

2 π^2 mee^4

(4πε 0 )^2 h^2 n^2

(14.4-19a)

where we have used Eq. (14.4-15) for the value ofr. The energy is quantized and is

determined by the value of the quantum number,n. Figure 14.14 depicts the first few

energy levels. With the accepted values of the physical constants, we can write

En−

2. 1797 × 10 −^18 J

n^2

−

13 .605 eV

n^2

(14.4-19b)

where the electron-volt (eV) is the energy required to move an electron through a

potential difference of 1 volt:

1eV 1. 60218 × 10 −^19 J

Bohr postulated that a photon is emitted or absorbed only when the electron makes

a transition from one energy level to another. The energy of an emitted or absorbed

photon is equal to the difference between two quantized energies of the atom:

E(photon)−∆Eatom−(En 2 −En 1 )

2 π^2 mee^4

(4πε 0 )^2 h^2

(

1

n^22

−

1

n^21

)

(14.4-20)

wheren 2 is the quantum number of the final state andn 1 is the quantum number of the

initial state of the atom. Figure 14.15 depicts the first few transitions corresponding to

emission of photons. Using the Planck–Einstein relation for the energy of the photon,

Eq. (14.4-8), we obtain

1

λ



En 1 −En 2

hc



2 π^2 mee^4

(4πε 0 )^2 h^3 c

(

1

n^22

−

1

n^21

)

(14.4-21)

This is the formula of Rydberg, Eq. (14.4-10), with the constant given by

R∞

2 π^2 mee^4

(4πε 0 )^2 h^3 c

 1. 097373 × 107 m−^1 (14.4-22)

The subscript∞is used to indicate that this value corresponds to an infinitely heavy

nucleus.

n 55 n^56

n 54

n 53

n 51

215

210

25

0

Energy/eV

n 52

Figure 14.14 The Quantized Elec-

tron Energies According to the Bohr

Theory.The energy values are all

negative, since an energy value of zero

corresponds to enough barely energy to

remove the electron from the atom.

(^4) I. N. Levine,Quantum Chemistry, 4th ed., Prentice-Hall, Englewood Cliffs, NJ, 1991, p. 434ff.