BioPHYSICAL chemistry

CHAPTER 9 QUANTUM THEORY 183

The theoretical basis for relating the classical and wave parameters was
provided by Louis de Broglie (Nobel Prize winner in Physics in 1923),
who realized that the wavelength could be related to the momentum
according to:

(9.17)

As a particle moves faster the momentum increases and the wavelength
decreases. Because the proportionality constant is a very small constant,
h, the wavelength properties are normally not observed. For example, a
tennis ball with a mass of 50 g moving at 90 miles/h (40 m s−^1 ) has the
wavelength:

(9.18)

For small particles the wavelength is very different. For an electron orbit-
ing a nucleus with a velocity of 10^6 ms−^1 , the wavelength is:

(9.19)

This wavelength is comparable to the size of the orbital, and hence the
wave properties play an important role in describing the properties of
electrons in atoms.
The wavelength of an electron that is free, rather than orbiting in an
atom, is determined by the velocity. Since the electron is charged, the
velocity can be manipulated using electric fields. For an electron moving
through a potential difference, V, of 40 kV, as found in an electron micro-
scope, the potential energy gained by the electron is the product of the
potential difference and the charge of the electron, eV. This energy can
be equated to the kinetic energy, yielding:

(9.20)

Solving this equation for the velocity yields:

9 = (9.21)

2 eV

me

eV= me

1

2

92

λ

.

(. )

==

×

××

−

−

h

m 9

66 10

9 109 10

34

31

Js

kg (()

.

10

61 72 10^10

ms

− =×− m

90 90

5280 0 305

3600

miles

h

miles

h

ft

mile

m

ft

h

s

.

== 40

m

s

λ

.

(. ) ( )

==

×

×

−

−

h

m 9

66 10

0 050 40

34

1

Js

kg m s

=×33 10. −^34 m

λ=

h

p