Materials Included In Kit

Safety Precautions

This laboratory is considered nonhazardous. Follow all normal laboratory safety procedures.

Lab Hints

• This laboratory activity can reasonably be completed in one 50-minute class period. All materials are reusable.

Teacher Tips

• Quantum mechanics is perhaps the most difficult topic to teach accurately at the high school level. Think about the challenge—the principles of quantum mechanics were developed based on the work of at least eight Nobel Prize-winning physicists. Teachers should strive for a conceptual understanding of three main ideas: the quantization of energy levels, the wave properties of the electron, and the relationship of probability to electron structure.
• This is a wonderful activity to introduce the use of computer spreadsheet and graphing programs. One way to do this is by carrying out the exercise as a collaborative classroom activity. Have all student groups enter their data on one spreadsheet program. Graph the class results to illustrate the effects of probability. If the activity is carried out using classroom data, then each group may reduce the number of dice rolls to 30.
• Emphasize to the students that the rolling of dice is only an analogy for the probability calculations for the electron location. Each roll of the dice represents the relative distance of the 1s electron away from the nucleus. The probability of finding the electron at a given distance from the nucleus will be modeled by plotting the number of times each “dice value” is obtained. The graph—a histogram—will be examined to determine the most frequent distances and the least frequent.
• The radial graph is another way to represent the dice rolls. By shading the various rings, students get a different visual representation of the 1s orbital electron distribution.

Answers to Prelab Questions

1. What did Rutherford add to the Atomic Model?

   The atom has a positively charged nucleus with electron particles circling the nucleus.

2. What problem with Rutherford’s model did Niels Bohr solve?

   In Rutherford’s model, the electron would fall into the nucleus from electrostatic attraction, which doesn’t occur. Niels Bohr proposed that electrons can only have fixed energies and occupy discrete orbits or distances. Therefore, they cannot fall into the nucleus.

3. As with any model, Bohr’s model was modified as new data was discovered. What piece of data led to the model’s modification?

   Matter can have both wave-like and particle-like properties. These properties tell us that if the energy of an electron is known, its position can only be estimated.

4. What is the fundamental difference between an electron orbit and an electron orbital?

   The electron orbit has each electron in fixed distances from the nucleus. An electron orbital pictures the locations where there is a positive probability of finding an electron with a given energy.

Sample Data

Energy Level Worksheet

Referring back to your original data, place a circle in the Energy Level Worksheet for your first 32 rolls of the dice. For example: If roll 1 gave a 9, then place a circle (•) in the 9 box for roll 1.

With a pencil, shade in the rings for each number. Shade the rings containing the fewest circles lightly. Darken the shading of the rings as the number of circles in the rings increases.

Answers to Questions

Plot the numbers obtained for each roll of the dice as a histogram for energy level 1 (1s orbital). The x-axis represents the number rolled and the y-axis represents the frequency each number was rolled. Put an X in the box corresponding to the dice roll. If the dice roll value has been previously entered, place the X in the box above the last X entry. Note: The shaded area represents the Bohr radius for the electron in the lowest energy level.

1. What is the most frequent value (electron distance from nucleus) for two dice?

   It should be seven.

2. How does the distribution of an electron in the wave mechanical model compare with the distribution for an electron using the Bohr model?

   The Bohr model puts the electron at a given distance from the nucleus at all times, while the wave mechanical model has the electron distributed over time at various distances from the nucleus. The frequency that the electron is at a specific distance at any given time is determined by the probability calculations for that electron at a given energy.

3. Explain where the electron in the lowest energy level (1s orbital) has the greatest probability of being found based on 50 “snapshots” or rolls of the dice.

   At the same position for the Bohr Model.

4. Explain where the electron in the lowest energy level (1s orbital) has the lowest probability of being found based on 50 “snapshots” or rolls of the dice.

   It should be two and twelve.

5. Referring to the graph again, are there any regions where the probability of finding the 1s electron is zero? At zero, one, and beyond twelve.
6. Explain how the wave mechanical model and the Bohr model are similar.

   The greatest probability for the location of the electron is the same as the Bohr distance.

7. Explain how the wave mechanical model and the Bohr model are different.

   The electron has a “probability” of being found at other distances from the nucleus where Bohr’s model predicts only one distance.

8. Describe the pattern observed when all the rings are shaded. If the graph was rotated about an axis through the center, what 3-D pattern would be observed?

   Rings go from lightest shading increasing to darkest in the middle, then fading to lightest again at the edge. In 3-D, a sphere would be observed with lightly shaded inner and out cores, sandwiching densely shaded spheres.

9. Explain why rolling the dice is a good model for describing where an electron may be found at any given instant for the wave mechanical model of the atom.

   It shows there are regions of high and low probabilities of finding the electron. Would rolling one die be a better or worse model? Why? Since the probability for rolling each number from 1 to 6 is the same, all distances for the electron would be equally probable, not the Bohr radius.

References

Special thanks to Gary Schiltz, Glenbard West High School, Glen Ellyn, IL, for providing the idea and the instructions for this activity to Flinn Scientific.

Introduction

Model the difficult concept of wave mechanics by actually “rolling the dice” to determine where the electron may be located.

Concepts

• 1s orbital
• Wave mechanics
• Electron structure

Background

The investigation and development of an accurate Atomic Model has been ongoing for many years. John Dalton (1766–1844) gave us the first model based on experimental evidence—atoms are the smallest discrete parts of a substance that cannot be divided. He also determined that elements differ due to the mass of their atoms. When new data became available, the Atomic Model was modified to fit the new evidence.

Ernest Rutherford (1871–1962) performed experiments which led him to postulate that the atom was made up of a small positively charged core, or nucleus, with negatively charged electrons moving around the nucleus at relatively large distances. Rutherford described the electron as a particle circling the nucleus, but could not explain why the electron did not “fall” toward to the nucleus. If the electron “fell,” then matter would collapse and radiate energy, an event that obviously does not occur.

Niels Bohr (1885–1962) solved this dilemma by proposing that electrons exist in quantized energy levels or orbits. The electrons in these levels would be at a fixed distance from the nucleus and couldn’t fall into the nucleus.

For example: Imagine taking 10 photographic snapshots of the electron with an atomic camera in the first energy level for the hydrogen atom. The result would be a series of pictures with the electron at a defined distance from the nucleus each time (see Figure 1).

If data are collected from the “Bohr Atomic Camera,” then a graph or histogram can be created. Figure 2 shows a histogram plotting the distance the electron is from the nucleus for each snapshot. Bohr based the theory of electron structure on experimental evidence showing that atoms, when energetically excited, generate well defined spectral lines. Bohr proposed that these spectral lines resulted from the release of energy when an electron that has been excited through the absorption of energy “falls” back down to a lower or more stable quantized energy level or its ground state.

For a model to be useful, it must allow for accurate predictions. Bohr’s model correctly described the electron energy levels for the hydrogen atom, which has only one electron, but failed to explain the data for atoms containing two or more electrons. To explain the observed data for multi-electron atoms, Bohr’s atomic model needed to be modified.

New experiments suggested that matter, like light, has both wave-like and particle-like characteristics. From this wave-like nature of matter, Werner Heisenberg (1901–1976) concluded that knowing both the position and energy of an electron at the same time would be impossible. If the energy of an electron is known, then the position of the electron can only be estimated. Likewise, if the precise location of the electron is known, then its energy can only be estimated.

The probability of finding an electron with a given energy within a given space is calculated using a complex mathematical equation called a wave function, Ψ. When this wave function is squared, the result is the probability of finding an electron with a specific energy in a given region of space. The electron orbital can be thought of as the region of space where the probability of finding an electron with a given energy is greater than zero. At large distances from the nucleus, the probability of finding the electron is very small, but never reaches zero. Scientists therefore defined the relative 1s orbital size as the radius of a sphere where 90% of the time the electron is in that sphere.

The modified wave–mechanical model still treats the electron as a particle, but adds a new property. This new property is that the electron movement is best described as a standing wave pattern around the nucleus.

How will the wave nature of the electron affect the position of the electron if we were to again take snapshots of the electron now exhibiting this wave property around the nucleus?

Experiment Overview

The purpose of this experiment is to simulate the probability of finding an electron by “rolling the dice.” Each roll of the dice will represent the electron location results for the wave mechanical probability calculations. The total value of the dice generated for each of fifty rolls of the dice will be recorded and the results will be plotted to create a histogram. The histogram will show the frequency of occurance for each value, that is, it will model the probability of finding the 1s electron at a particular location from the nucleus.

Materials

Prelab Questions

1. What did Rutherford add to the Atomic Model?
2. What problem with Rutherford’s model did Niels Bohr solve?
3. As with any model, Bohr’s model was modified as new data was discovered. What piece of data led to the model’s modification?
4. What is the fundamental difference between an electron orbit and an electron orbital?

Safety Precautions

This laboratory is considered nonhazardous. Follow all normal laboratory safety procedures.

Procedure

Each roll of the dice represents taking a snapshot of the distance of the electron from the nucleus for the lowest energy level (1s orbital) at that instant.

1. Obtain a pair of dice.
2. Roll the pair of dice.
3. Record the sum of the dice in the data table.
4. Repeat steps 1–3 for 49 more rolls.

Student Worksheet PDF

13951_Student1.pdf