Chemistry - A Molecular Science

The basic unit of charge is q = 1.602x10

-19
C, so the charge on an electron is q

= -q and e

the charge on a nucleus with Z protons is q

= +Zq. N

= 1 in a vacuum, and r is the radius ε

of the n

th orbit as given in Equatio

n 2.4. Substitution of these

known quantities into the

preceding energy expression yields the total energy of an electron in the n

th orbit in terms

of the n quantum number.

2

-18

n

Z^2

E = -2.180 10

J

⎛⎞n

×

⎜⎟⎜⎟⎝⎠

Eq 2.5a

The constant determined by Bohr, 2.180x10

-18

J, was the same as the experimental value

determined by Rydberg in Equation 2.3a, whic

h was strong support for the Bohr model.

Thus, the energy of the electron in the n

th orbit is also expressed as

⎛⎞⎜⎟⎜⎟⎝⎠

2

nH

Z^2

E = -hR

n

Eq.

2.5b

n=5n=4n=3 n=2 n=1

infrared

visible

ultraviolet

E=0¥ E =-hR/25^5 E =-hR/16^4 E =-hR/9^3 E =-hR/4^2 E =-hR/1^1

n=

free electron

Figure 2.6 Energy level diagram of the H atom Arrows represent the transitions that produce some of the observed lines in the emission

spectrum of a hydrogen atom.

The length of each line is proportional to the energy (photon frequency) of the transition. Re

versing the direction of the

arrows would result in absorpti

ons rather than emissions. Note

that it is the value of n

that dictates the series. lo

Equations 2.5 show that the energy of an electron in an atom is

quantized

; i.e.

, it can

have only discrete values that are dictated by

the value of the n quantum number. The

allowed energies are called

energy levels

, and Figure 2.6 shows an energy level diagram

for the hydrogen atom as determined with E

quation 2.5. Each horizontal line represents an

allowed energy level. Note how the ener

gy levels get closer with increasing n.

Example 2.3

What are the orbital radius and energy of an electron in the n = 1 level of a Li

2+ ion?

2+Li

has one electron, so we use Equations 2.

4 and 2.5 and the fact that Z = 3 for Li.

⎛⎞

⎛⎞

×××

⎜⎟

⎜⎟

⎝⎠

⎝⎠

22

-11

-11

-11

1

n1

r = 5.292 10

= 5.292 10

= 1.764 10 meters = 17.64 pm

Z3

22

-18

-18

-17

1

Z3^22

E = -2.180 10

= -2.180 10

= -1.962 10

J

⎛⎞n1

⎛⎞

×× ×

⎜⎟

⎜⎟

⎜⎟

⎜⎟

⎝⎠

⎝⎠

Note how the increased nuclear charge draws the electron closer to the nucleus and lowers its energy. We will use this trend in the next chapter to describe atomic properties. Atomic spectra can be understood in terms of the electron moving from one energy

level into another, which is referred to as an

electronic transition

. If n

is the higher hi

quantum number and n

is the lower quantum number, then the energy difference between lo

Chapter 2 Quantum Theory

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North

Carolina

State

University