**Electronic Structure of Atoms**

The dual nature of the electron

The Dual Nature of the Electron

Depending on the experimental circumstances, EM radiation appears to have either a *wavelike or a particlelike (photon) character*.

Louis de Broglie (1892-1987) who was working on his Ph.D. degree at the time, made a daring hypothesis:

if radiant energy could, under appropriate circumstances behave as though it were a stream of particles, then could matter, under appropriate circumstances, exhibit wave-like properties?

For example,** the electron in orbit around a nucleus**. DeBroglie suggested that the electron could be thought of *as a wave with a characteristic wavelength.*

He proposed that the wavelength of the electron was a function of its mass (m) and its velocity (u):

i.e. the wavelength for "matter waves", where *h* is Planck's constant and is u velocity (not, the frequency). The quantity **mu** for any object is its ** momentum** (mass * velocity).

What is the characteristic wavelength of an electron with a velocity of 5.97 x 10^{6} m/s? (the mass of the electron is 9.11 x 10^{-28} g)

Planck's constant (h) is 6.63 x 10^{-34} J s (also, recall that 1J = 1 kg m^{2}/s^{2})

converting g to kg:

Converting from kg m^{2}/s^{2} to Joules:

- The relationship between energy (
*E*) and frequency (n ) for electromagnetic radiation (Planck's quantum of energy)

- The relationship between wavelength (l ) and frequency (n ) for electromagnetic radiation

From these relationships, we can determine the relationship between energy and wavelength:

or, rearranging:

- The relationship between wavelength (l ) and momentum (m*v) for DeBroglie's "particle wave"

From the above relationships, we can calculate the relationship between energy (*E*) and momentum (m*v)

Simplify, and solve for *E*:

The highest velocity (v) attainable by matter is the speed of light (*c*), therefore, the maximum energy would seem to be:

or

Why do nuclear bombs make such a loud "pop"?

- The Fuel value of hydrogen is 142 x 10
^{3}J g^{-1} - If all matter were converted into energy

E = 1x10^{-3}kg * (3x10^{8}m/s)^{2}

E = 9x10^{13 }kg m^{2} s^{-2}

E = 9x10^{13} J

In other words, we can get out about 9 orders of magnitude greater energy if the hydrogen is converted directly into energy, rather than combusting it.

Nuclear fission and fusion reactions convert a fraction of their matter into energy. The bomb that was dropped on Hiroshima contained about 15 kg of the ^{235}Uranium isotope, a fissionable material. The actual amount of mass that was converted into energy is estimated at about 1 kg (releasing around 1 x 10^{17} J of energy in a split second). The estimated temperature at the moment of detonation is estimated to have been around 5 million degrees. In addition to the sun-like heat, much of the damage was due to the pressure wave that was produced.

**The Uncertainty Principle**

For a relatively large solid object, like a bowling ball, we can determine its position and velocity at any given moment with a high degree of accuracy.

However, if an object (like an electron) has wave-like properties then how can we accurately define its' position?

Werner Heisenberg (1901-1976) concluded that due to the dual nature of matter (both particle and wavelike properties) it is impossible to simultaneously know both the position and momentum of an object as small as an electron.

Thus, it is not appropriate to imagine the electrons as moving in well-defined circular orbits about the nucleus.

*1996 Michael Blaber*