Electronic Structure of Atoms

1. Atoms and molecules can only exist in certain states characterized by definite amounts of energy. When an atom or molecule changes its state, it absorbs or emits an amount of energy (electromagnetic radiation) just sufficient to bring it to another state.

Electronic energy

---form of energy that arises from the motion of e^- about the nucleus, and from the interactions among the e^- and between the e^- and the nucleus.

--only certain values of electronic energy are allowed for an atom

--said to be quantized

--a change of electronic energy level (state) of an atom involves the absorption or emission of a definite amount, quantum, of energy

--lowest electronic energy state is called the ground state

--any state with energy greater than that of the ground state is an excited state

2. When atoms or molecules absorb or emit light in moving from one energy state to another, the wavelength of the light is related to the energies of the two states by the equation

E_final - E_initial = hc/λ

h = Planck's constant
c = speed of light

--ray of light can be considered to consist of photons

--each photon of wavelength has an energy of hc/λ

--an atom or molecule can move from one electronic energy state to another by absorbing or emitting a photon of energy

--if it absorbs a photon of energy, hc/λ , it moves from a lower to a higher energy state, and its energy increases by hc/λ

therefore: ΔE = E_final - E_initial = E_hi - E_lo = hc/λ

3. The allowed energy states of atoms and molecules can be described by sets of numbers called quantum numbers

--quantum numbers are associated with individual electrons in an atom.

Relation between Energy difference, ΔE, and wavelength.

--wavelength is inversely related to the energy difference, ΔE
a large ΔE---short wavelength........<400 nm--ultraviolet
a small ΔE---long wavelength.........>700 nm--infrared

--e^- moves from one energy state to another by absorbing or emitting radiation of a particular wavelength.

--by measuring the wavelength associated with e^- transition, we can find ΔE of the two energy states involved

h = 6.626 x 10^-34 J ^. s/ particle
c = 2.998 x 10⁸ m/s
ΔE is in J/particle when wavelength is in meters
Ordinarily: ΔE is in kJ/mol, and wavelength is in nm
1 mole = 6.02 x 10²³ particles
1 kJ = 10³ J
1 nm = 10^-9 m

Relation between wavelength and frequency:

--frequency, represented by the Greek letter ν (nu), is the number of wavecycles that pass a given point in a unit time ( 1 cycle per second is equal to 1 Hertz (Hz) )

ex. 10⁸ cycles per second = 10⁸ /s = 10⁸ Hertz

--the velocity at which a wave moves can be found by multiplying the length of a wave, λ, by the number of cycles passing a point in unit time, ν

c = λν

ν = c /λ
frequency is inversely related to wavelength

For example:
Blue light, at a wavelength of 450 nm, has a higher frequency than red light with a wavelength of 650 nm.

blue.....frequency = c/450 nm = 6.66 x 10¹⁴/s

red......frequency = c/650 nm = 4.61 x 10¹⁴/s

--ΔE is directly proportional to the frequency of the light absorbed or emitted

--if ΔE is large, the light will have a high frequency

--if ΔE is small, the light will have a low frequency

The Atomic Spectrum of Hydrogen and the Bohr Model

1911--Niels Bohr, Danish physicist

-developed mathematical model for the behavior of an e^- in the hydrogen atom
-based: on the Rutherford atom & quantum theory of Planck

spectroscope--instrument that breaks up light into its component colors

--continuous--contains all the colors (such as in white light)

--sodium vapor lamps, neon signs: the light given off looks different
-spectrum not continuous
-the light consists of several discrete colors which appear as lines of definite wavelength when seen in a spectroscope
-each element has its own characteristic spectrum

Hydrogen most studied because it is the simplest with only 1 e^-

--several series of lines representing electronic transitions
-Lyman series (ultraviolet)
-Balmer series (visible)
-Paschen series (infrared)

--Bohr's ideas about H atom
-central proton
-single e^- moves around the p⁺ in a circular orbit
-force of attraction of the proton for the e^- related to the circular motion of the e^-
-expressed energy of e^- in terms of the radius of the orbit
-combined classical Newtonian Laws of Motion and Coulomb's Law of Electrostatic Attraction with Quantum Theory

angular momentum = mvr
m = mass
v = velocity (speed)
r = radius of its orbit around the nucleus
mvr = nh/2π
h = Planck's constant
n = a quantum number that can have any positive integral value (1,2,3...)

According to Bohr

-angular momentum is quantized (i.e. cannot have just any value)
-restricted to values for which "n" is a positive interger (2, h, and ψ are constants)
-thus angular momentum can change only by discrete amounts (i.e. integral multiples of h/2π)

Also, Bohr found the allowed energies are restricted
The energy of an energy level is given by the equation:

E = -B/n² ( where n = 1, 2, 3...) and B = 2.179 x 10^-18 J/particle
E = -2.179 x 10^-18J/particle/n² = -1312kJ/mol/n²

Example
ionization energy of the H atom can be calculated from the Bohr model
H_(g) -------> H⁺_(g) + e^-
ΔE = ionization energy
ΔE = 0 - (-1312 kJ/mol) = 1312 kJ.mol

Quantum Mechanical Atom

--Bohr's model was great for the H atom (error of .1%); however, even with He error increased to 5%

Wave Nature of the Electron:

--deBroglie: suggested that particles might exhibit wave properties
wavelength = h/mv
m = mass
v = speed
2πr = νλ
mvr = hν/2π

--big question arises--how does one specify the position of a wave at a particular instant?

--can determine wavelength, energy, amplitude--but no way to tell just where the electron is because a wave extends over space

--Quantum mechanics was developed to describe the motion of small particles confined to very small portions of space.

--deals with the probability of finding a particle within a given region of space

--Schrodinger derived an equation from which one could calculate the amplitude (height) ψ of an electron wave at various points in space.

--Several expressions for ψ that will satisfy the equation. Each is assoicated with a set of quantum numbers.

--Becomes possible to determine the probability of finding a particle in a given region of space.

--ψ² proportional to the probability of finding the particle at that point

--ψ² proportional to the electron charge density at that point

Back to Notes Menu