piab - notes on the Ψ n ’ s

PIAB - Notes on the Ψn’s

• The PIAB wave functions exhibit nodes. As we move to higher energy (higher n) states the number of nodes increases. As well, as one moves to higher n values the characteristic wavelength decreases. (This is reminiscent of light where, again, the energy of a photon increases as the wavelength of the light decreases).The wave functions can have both positive and negative amplitude.

Amplitudes and Probabilities

• In quantum mechanics the probability of finding a particle at a particular point in space is proportional to the square of the amplitude of the wave function at that particular point. Probabilities must always be positive and the squares of the amplitudes for the first few one dimensional PIAB wave functions are shown on the next slide.

Copyright © 2011 Pearson Canada Inc. General Chemistry: Chapter 8 Slide 3 of 50

The probabilities of a particle in a one-dimensional boxFIGURE 8-21

Probabilities of Finding an e- ?• The graphs on the previous slide describe the

probability of finding a particle (electrons are especially of interest in chemistry) in different parts of the “box”. In all cases there is a 50% probability of finding a particle in the left half of the box. Calculus (or a graphing calculator) can tell us the probability of finding the particle in any arbitrarily chosen part of the box.

Probabilities – Class Examples• We will use Slide 12 to determine when

calculus must be used to calculate the probability of finding a particle in a given part of the box and when simpler “symmetry arguments” can be used. We’ll consider a number of cases where symmetry arguments can be used to specify exactly the probability of finding a particle in a certain part of the box.

PIAB Energies – Mass & Particle Size• The PIAB model allows us to calculate

energies for a particle in a particular quantum state as a function of quantum number (n), particle mass (m) and box length (L). As values of n increase energies increase. As particle mass increase and/or box size increases energy level spacings decrease rapidly. For the limiting case as m → and L → the energies vary continuously (familiar ground?).

PIAB – 3 Dimensions• The Schrodinger Eqtn. for the 3-dimensional

PIAB can also be solved.

• E = { + + }• Here a, b and c are the lengths of the 3 sides of

the box and n1, n2 and n3 are a set of three quantum numbers needed for this higher dimensionality case. We will need three quantum numbers when we discuss atoms!

One, Two and Three Dimensional PIAB Models

• The one dimensional PIAB model does account “qualitatively” for the electronic energy level patterns seen in some conjugated polyenes (nearly linear carbon chains). Most molecules are not linear. More complex PIAB models provide more realistic models for planar and three dimensional molecules. We will consider very briefly two PIAB models which serve to introduce the concept of energy degeneracy (degenerate energy levels).

Varying Dimension PIAB Models – Energies and Quantum Numbers

•

Square PIAB Model – Doubly Degenerate Energies (Energy Levels)

nx Value ny Value nx2 + ny

2

(Relative Energy)

1 1 2 2 1 5 1 2 5 2 2 8 3 1 10 1 3 10

Square PIAB Model – Doubly Degenerate Energy Levels

• In the previous slide relative energies are shown for clarity – this makes the degeneracies easier to spot. The precise energies can be calculated by multiplying the relative energies by h2/(8mL2). Similar arguments apply to the relative energies of the cubic PIAB box shown on the next slide (for the lowest energies!). Here triply degenerate energy levels are seen.

Cubic PIAB Model – Triply Degenerate Energies (Energy Levels)

nx Value ny Value ny Value nx2 + ny

2 +ny2

(Relative Energy)

1 1 1 3 2 1 1 6 1 2 1 6 1 1 2 6 2 2 2 12 3 1 1 11 1 3 1 11 1 1 3 11

Symmetry in Chemistry

• Symmetry is important throughout chemistry. The energy expression for the 2d square PIAB is simpler than for the 2d rectangular box. More importantly, due to molecular symmetry, the H2O molecule has only one bending vibration whereas the CO2 molecule has two degenerate bending vibrations. (Chemistry 2302 and 3500).

Spherical Polar Coordinates

• When we move from the one dimensional PIAB model to the wave functions for an electron in a H atom it’s simplest to use spherical polar coordinates when we construct the H atom wave functions. The correspondence between Cartesian and spherical polar coordinates is shown on the next slide.

Wave Functions of the Hydrogen Atom


FIGURE 8-22

•The relationship between spherical polar coordinates and Cartesian coordinates

Schrödinger, 1927 Eψ = H ψ

H (x,y,z) or H (r,θ,φ) ψ(r,θ,φ) = R(r) Y(θ,φ)

R(r) is the radial wave function.Y(θ,φ) is the angular wave function.

H Atom Wave Functions• The previous slide states that the H atom wave

functions, determined again by solving the Schrodinger equation, can be factored into an angular and a radial part if we employ spherical polar coordinates. The use of these coordinates makes it especially easy to locate nodes (regions of zero “electron density”) and to represent 3 dimensional probabilities (i.e. represent in 3 dimensions the probability of finding an electron in space/in an atom).

Subatomic Particles Can Move!

• Like larger objects subatomic particles can move. In this course the motion of electrons will be most important. When removed from an atom electrons can be focused and accelerated to high velocities using an electric field. Inside an atom, electrons move rapidly around the nucleus (electrons have orbital angular momentum!) and can also “spin”.

Subatomic Particles Can be Charged!

• Two common subatomic particles, the electron and the proton, are charged. Charged particles in motion can exhibit magnetic properties (act as small magnets). Such charged particles in motion will have their energies changed if they are placed in an “external” magnetic field. This can serve to remove or lift energy degeneracies by slightly changing, for example, the energies of electrons.

Energy Levels – H and More Complex Atoms

• The observed energies levels of a H atom can normally be described using the Bohr equation (which applies to other one electron species such as He+). For more complex atoms (extra electrons) more quantum numbers are required to describe energies. We’d expect to need three! In all atoms we will encounter quantization of energy and quantization of angular momentum.

8-7 Quantum Numbers and Electron Orbitals• Principle quantum number, n = 1, 2, 3…

• Angular momentum quantum number,l = 0, 1, 2…(n-1)

l = 0, sl = 1, pl = 2, dl = 3, f


Magnetic quantum number, ml= - l …-2, -1, 0, 1, 2…+l

H Atom Spectra and Energies

• The next two slides show the energy lelevl patterns seen for the H atom – a particularly simple case. Here degeneracies are very important. Subshells with the same n value but different l values have the same energy in the H atom. We say that these energy levels are degenerate. This result follows in the first instance from experiments.

Energy-level diagram for the hydrogen atomFIGURE 8-14


ΔE = Ef – Ei = -RH

nf2

-RH

ni2

–

= RH ( ni2

1nf

2–1

) = h = hc/λ

Principal Shells and Subshells


FIGURE 8-23

•Shells and subshells of a hydrogen atom

Atoms Beyond H (“Many” e-)

• For many e- atoms many energy level degeneracies seen for H disappear. Subshells with the same n but different l values are no longer degenerate (have the same energy). This follows 1st from experiment but we will gain some insight into this after we consider simple atomic wave functions and orbitals.


Orbital energy-level diagram for the first three electronic shells

FIGURE 8-36


Atomic Emission/Absorption Spectra

• The atomic absorption and emission spectra seen for H are very simple. Rather more complex spectra are seen for atoms with a number of electrons. How does this follow from the previous slide?

Test Example

• 3. The interstellar medium is comprised almost entirely of H atoms at a concentration of about 1.4 atom/cm3. (a) Find the gas pressure in outer space. (b) Find the volume in liters that contains 2.5 g of H atoms. Assume that the temperature of the interstellar medium is 3.5 K.

Test Example

• 7. Calculate the amount of pressure volume work done when 35.5 g of tin are reacted with excess acid at 25 OC and 1.02 atm.

piab - notes on the Ψ n ’ s

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