Chapter 38 Atoms
The solution to the Schrodinger Equation predicts that the electron in hydrogen is a standing wave. The solution also predicts the energy levels of the electron. This standing wave is three dimensional in the space around the nucleus. The standing wave can be used to predict the probability that an electron is in a certain volume of the space.
page 2 Solutions of Schrodinger’s Equation for Hydrogen
Schrodinger developed a wave equation that is a fundamental equation in Quantum Mechanics. In practice, one specifies the potential energy in a situation and then solves the equation. For atoms the solution produces a set of "quantum numbers" and a "wave function." The quantum numbers are 1) "n", the principle quantum number, "l", the total angular momentum quantum number, and "ml", the magnetic quantum number. All three of these numbers are integers. The wave function can be used to compute the probability of finding the electron in a certain volume of space. Figure 1 essentially shows the probabilities, with a brighter region more likely to have an electron present during some time interval.
There is a relation between the quantum numbers.
n is a positive integer: 1,2,3,4,….
l is a positive integer that can have a value of 0 up to the value of n-1. e.g. if n =4 l can be 0,1,2, or 3.
ml can be a negative integer, zero, or a positive integer with a magnitude of at most, l e.g. if l = 1 ml would be –1,0,1
It turns out than when n=1, l=0 and ml = 0, the solution to Schrodinger’s Equation predicts that the most likely place for the electron is at a location matching the first Bohr radius. Also, the energy value for this set of quantum numbers is –13.6 eV. The energy values of the higher n values agree for the Quantum Mechanical solution and the Bohr model.
page 4 The l = 0 Patterns
All of the l = 0 patterns are spherically symmetric.
*Define spherically symmetric.
The l = 0 patterns only vary based on the distance from the nucleus of the atom. They do not depend on any angle measured with the nucleus at the vertex of the angle. The l=0 wave pattern has no net angular momentum. This is in disagreement with the Bohr model of the atom. Which model is correct?
page 5 The l not equal 0 Patterns
These patterns do have a net angular momentum.
Intensity at the Origin
The l=0 patterns have a maximum in the radial part of the wave function at the origin (the nucleus). However, the maximum probability of the location of the electron is not at the nucleus. For l > 0 the value of the wave function at r=1 is zero.
Quantized Projections of Angular Momentum
The solution to the Schrodinger Equation predicts that the electron is moving in different directions for the case of ml < 0 compared to ml >0. We can say that in one situation the electron is traveling clockwise and the other situation the electron is traveling counterclockwise. The ml value is used to calculate the amount of angular momentum in a certain direction. The angular momentum is ml times h-bar. h-bar is (h/(2 pi))
*What does it mean to say that the projections of the angular momentum vector are quantized? (state this in your own words)
page 7 The Angular Momentum Quantum Number
The angular momentum of the electron can be calculated using L = sqrt( l ( l+1) ) h-bar
Calculate L for l=0 for l=1 for l=100
Comment on the number of ml values for each case above.
Calculate the angular momentum for a bicycle wheel that has a mass of 2kg, a radius of 60 cm and an angular speed of 0.5 radians/second. You may have to look at your notes from first semester … or, use L = I w , with I = mr2 .
It is likely that you have used a different notation for the l quantum number. In Chemistry the l=0 state is called a "s" shell. The l=1 state is called a "p" shell. The l=2 state is called a "d" shell. Recall that the fundamental data for the development of our knowledge of electron structure in the atom comes from the measurements of spectral lines. In the 1800’s the spectral lines were grouped into categories such as sharp (s), principal (p), diffuse (d), etc.
page 8 An Expanded Energy Level Diagram
All of the quantum numbers can have an effect on the energy value of the electron. i.e. The energy of an electron in the n=3, l=2, ml = 1 state is different than the energy of an electron in the n=3, l=2, ml = 0 state. However, the value of n has the most influence on the energy of the electron.
*Why aren’t the Bohr orbits labeled with l and ml values?
Examine Figure 4. Why does the n=1 level only have one line to the right while the n=3 level has three lines to the right?
The diagram in Figure 4 is helpful in describing the transitions that electrons make from one state to another state. The most probable transitions involve the value of l changing by 1. These are called "allowed" transitions. Other changes in the value of l are called "forbidden" transitions. Is the change in state of an electron from the n=4, l=2, ml = 1 state to the n=3, l=2, ml = 1 state "allowed" or "forbidden?" "Forbidden" actually means that there is a very low probability that this transition will occur. The spectral lines from some gas clouds in our galaxy were a puzzle until astronomers realized that they were recording light emitted from "forbidden" transitions.
page 9 Multi Electron Atoms
Why would it be more difficult to specify the potential energy function for electrons in helium than for the electron in hydrogen?
The experimental values of the energy required to ionize different elements (see page 11) provided physicists a clue in the 1920’s that the electron in an atom has a fourth quantum number, ms . Evidence for this fourth quantum number came from the study of the deflection of electrons that were sent through a nonuniform magnetic field. Half of the electrons were deflected up and half of the electrons were deflected down. The interpretation is that the electron has an inherent property of "spin" angular momentum that can either be directed up or down (relative to a magnetic field). This is NOT to say that the electron should be pictured as an object that spins like a basketball on the finger of a basketball player, or the spin of a curve ball in baseball or as any other object you know that spins on its axis. This "spin" effect is a quantum mechanically property of the electron. The size of an electron is unknown.
With a fourth quantum number the Pauli Exclusion Principle can provide a prediction that explains the broad features of the chart on page 11. The Pauli Exclusion Principle states that no two electrons in one atom can have the same four quantum numbers.
How many electrons can be accommodated in the n=1 state?
…in the n=2 state? in the n=3 state?
page 10 The Periodic Table
How many electrons are in Lithium, Sodium? Potassium?
The Pauli Exclusion principle allows us to understand why the elements on the left column of the periodic table have such relatively low ionization energies. For these atoms the last electron to be accounted for is outside of closed shells. It is farther from the nucleus and more easily removed from the atom.
It also allows us to understand the chemical behavior of the two right columns in the periodic table. An element that has filled shells is very stable, it does not seek to gain or lose an electron. An element that is lacking one electron to create a filled shell is very reactive and will tend to remove an electron from another atom.
These are the broad outlines that underlie the periodic table. Of course, for the details, study Chemistry.
It is useful to be able to write how many electrons are in each n,l state in an atom. The notation uses a number to identify n and a letter (s,p,d,f, etc.) to identify l. This notation is called the electron configuration of the atom. We will write the electron configurations for some elements. I will put a chart on the board that will help you fill up the subshells (l values) in the proper order.
Carbon Neon Aluminum Calcium*
Electron Screening
The electrons in a multi-electron atom do have an effect on each other. They repel each other and reduce the net attractive force in the atom. Imagine that you are riding on the outermost electron in Lithium. What is the approximate net charge that you observe as you look towards the nucleus?
This is a simplified view of the situation. The electrons do not reside in definite orbits as given in the Bohr model. The electrons have probabilities of being located in any volume of space near the atom. These probabilities have to be included in detailed studies of the effect called electron screening. Hopefully you can get the general idea of why the ionization energies are low for elements on the left side of the periodic table from the discussion above.
pages 12 - 14 Effective Nuclear Charge We will skip this section. It is worthwhile reading if you want more background on the periodic table.
page 15 Ionic Bonding We will skip this section.
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