Atoms, Electrons and Atomic/Electronic Energies

• The behavior of macroscopic objects/systems can be adequately described using Newtonian mechanics and thermodynamics. A train, a hockey puck and an iceberg are all examples of macroscopic objects. We might wish, for example, to consider how the kinetic energy of a hockey puck varies with velocity (goalies do!) or to calculate how much heat/energy is needed to melt an iceberg.

Pucks and Gas Molecules

• Hockey players who have studied physics know that the kinetic energy of a puck varies with velocity as described by the familiar equation.

• Ekinetic = ½ mv2

• During the course of a hockey game the puck will move with a range of velocities and, correspondingly, a range of kinetics energy values. We assume that the kinetic energy of the puck can be continuously varied. Why?

• In discussing gases one learns that the pressure exerted by a gas results from the fact that individual gas molecules have high velocities. The rapidly moving gas molecules have significant kinetic energy. At a given temperature not all gas molecules have the same velocity or kinetic energy. A range of velocity/kinetic energy values is seen for a gas at a particular temperature. As the gas temperature increases the average kinetic energy of gas molecules also increases.

• Atoms and molecules can possess energy in addition to translational kinetic energy. Two types of energy – rotational and vibrational – will be considered later (for molecules). Electronic energies will be considered first.

• The energies of individual atoms, electrons and molecules are best studied experimentally using spectroscopy (interaction of light with matter). Key spectroscopic experiments tell us that, for atoms and molecules, Newtonian mechanics does not work!

Atomic and Molecular Energies

• Spectroscopic experiments show us that atomic and molecular energies are not continuously variable.

• Experiments show us that atomic and molecular energies are quantized – only a small number of energy values are observed. The properties of light will be reviewed briefly. The wave/particle “dual” character of light (and electrons!) is particularly important.

Simple Frequency Wavelength Conversion

• In spectroscopic experiments both frequency and wavelength measurements are reported. Conversions using the velocity of light are needed. Example: What is the wavelength of the 2.450 GHz radiation (light) used in a typical microwave oven?

• Recognize Hz as equivalent to s-1.• Then ν = 2.450 GHz = 2.450 x 109 s-1

• Use c = λν to get λ = c/ν =(2.9979 x 108 m s∙ -1/2.450 x 109 s-1) =0.1224 m =12.24 cm (mention quarter-wave plates?)

• Higher frequency light is more energetic than lower frequency light. We know that infrared light makes us feel warm. Visible light and UV light on the other hand cause serious sunburn. The energy transported per photon of light is proportional to the frequency of light. The relationship between light frequency and energy was studied by Max Planck.

Photon Frequency (s-1) and Energy (J)

EPhoton

νPhoton (s-1)

Slope = Planck’s Constant = h = 6.626 x 10-34 J s ∙

• Example: How much energy is possessed by (a) 1 photon and (b) NA photons of 16.6 GHz electromagnetic radiation?

• EPhoton = hν = (6.626 x 10-34 J s)*(16.6 x 10∙ 9 s-1) = 1.10 x 10-23 J

• For one mole of photons• (1.10 x 10-23 J/photon)*(6.022 x 1023 photons/mol) =

6.62 J• This is a small amount of energy. It takes a lot of

microwave photons to heat up a cold cup of coffee! We could calculate the number of photons needed to heat a cup of coffee. How?

• Highly energetic/hot objects have a tendency to lose energy/heat to the surroundings. Atoms are no exception. Very hot atoms emit only a few frequencies of light. These so-called line spectra (together with E = hν and sometimes c = λν) indicate that atomic/electronic energies are quantized (not continuously variable). “Cold” atoms readily absorb light – as long as the energy of the photon corresponds to the difference in energy between two energy levels of the atom.

Simple Two Energy Level System

Energy perAtom (J)

EHIGH

ELOW

ΔEAbsorption ΔEEmission

• For the previous slide conservation of energy dictates that

• ΔEAbsorption + ΔEEmission = 0

• Careful measurements of light frequencies (or wavelengths) absorbed or emitted enables a “pattern” of atomic or molecular energy levels to be determined. In a few cases the energy level pattern and corresponding absorption/emission spectra can be described using simple equations.

Bohr Theory for the Hydrogen Atom• Bohr attempted to use the existing physics of

the early 20th century and a set of quantum numbers to model the observed absorption and emission spectra of the H atom.

• Tenets of the Bohr Theory:• (1) Energies of the H atom/electron are

quantized.• (2) The electron in a H atom moves around

the nucleus in a circular orbit.

• (3) The angular momentum of the electron in a H atom is quantized.

• (4) Energy and angular momentum values for the electron in a H atom are calculated using a quantum number, n. The quantum number (n) has integer values 1, 2, 3, 4, 5….. infinity.

• Angular momentum = nh/2π• Radius of orbit = n2ao = rn

• Electron energy = En = -RH/n2

• Notes: En is always negative. Why? The quantity ao is called the Bohr radius and specifies the radius of the electron orbit for the lowest energy state of the H atom.

Bohr Theory and the Ionization Energy of Hydrogen

Copyright © 2011 Pearson Canada Inc. General Chemistry: Chapter 8 Slide 15 of 50

ΔE = RH ( ni2

1

nf2

–1) = h

As nf goes to infinity for hydrogen starting in the ground state:

h = RH ( ni2

1) = RH

This also works for hydrogen-like species such as He+ and Li2+.

En = RH ( ni2

-Z2

) = -Z2RH

Bohr Picture/Theory• The Bohr theory has several shortcomings. It

implies that the electrons circle the nucleus in regular orbits. This picture, reminiscent of Newtonian mechanics, is incorrect. The Bohr picture does account for atomic energies – but only for one electron species such as H (and D and T), He+, Li2+, Be3+ and so on – a not insignificant problem.

The Rydberg Constant

• The equations presented on an earlier slide assume that the Rydberg constant has energy units (kJ mol∙ -1 or J atom∙ -1). Thus RH values of 1312 kJ mol∙ -1 or 2.179 x 10-18 J atom∙ -1 are seen. Historically, from optical spectroscopy, the Rydberg constant value was first expressed in “wavenumbers” (109679 cm-1 or 1.09678 x 107 m-1). (Text, Chapter 11, Page 512. Slightly different equations.)

Bohr Theory and Ionization Energies

• The equations presented earlier suggest that we can use ionization energies for one electron species to calculate values for the Rydberg constant (or vice versa). The ionization energy for a one electron species is the energy required to move the electron from the n=1 level to n=∞ (free electron). Calculations presented on the next slide take ionization energies from webelements.com.

Bohr Theory and Ionization Energies Species Ionization Energy

(kJ mol∙ -1)Atomic Number, Z

Z2 RH Estimated (kJ mol∙ -1)

H 1312.0 1 1 1312.0

He+ 5250.5 2 4 1312.6

Li2+ 11815.0 3 9 1312.78

Be3+ 21006.6 4 16 1312.91

B4+ 32826.7 5 25 1313.07

C5+ 47277.0 6 36 1313.25

Bohr Theory and Ionization Energies• The results of the previous slide suggest that

the Bohr picture, of an electron circling a “stationary” nucleus, has shortcomings. The small differences in calculated RH values in the table can be understood using a reduced mass calculation with the reduced mass defined as μ = memNucleus/(me + mNucleus)

• This effect will be considered in one of the Chemistry 2302 labs (H, D and T spectra).

Bohr Theory and Ionization Energies Species Ionization Energy

(kJ mol∙ -1)Atomic Number, Z

Z2 RH Estimated (kJ mol∙ -1)

H 1312.0 1 1 1312.0

He+ 5250.5 2 4 1312.6

Li2+ 11815.0 3 9 1312.78

Be3+ 21006.6 4 16 1312.91

B4+ 32826.7 5 25 1313.07

C5+ 47277.0 6 36 1313.25