Isotope Mass and Nuclear Energy – Peter Ashton  Purpose Nuclear energy is often a lightly-covered topic in many chemistry classes despite its increasing relevance to life in the modern world (e.g. fission/fusion reactors as an energy source, threats from nuclear weapons, etc.). Ultimately the source of nuclear energy lies at the intersection of atomic structure and the breakdown of the Law of Conservation of Mass, both topics with which beginning chemistry students will be familiar. This lesson presents a computational approach to considering nuclear energy while providing a segue to a variety of related topics.    Overview Students will compute the difference between the mass of a given isotope nucleus and the mass of its constituent subatomic particles (“nucleons”). Scaling this mass difference by the number of nucleons and plotting the result for a range of elements evinces a clear physical trend with implications for the production of elements in stars and energy in power plants.    Student Outcomes Students will be able to:

• Compute the number of protons and neutrons and calculate its mass • Explain why the provided mathematical formulation is a useful tool for evaluating the

Law of Conservation of Mass • Describe trends in binding energy/mass defect across the periodic table • Determine which nuclear reactions (fission/fusion) will be energy releasing/absorbing

and estimate relative yields.    Standards Addressed NGSS: HS-PS1-7: Use mathematical representations to support the claim that atoms, and therefore mass, are

conserved during a chemical reaction. HS-PS1-8: Develop models to illustrate the changes in the composition of the nucleus of the atom and

the energy released during the processes of fission, fusion, and radioactive decay. HS-ESS1-3: Communicate scientific ideas about the way stars, over their life cycle, produce elements.      Time Two 45 to 50 minute periods

         

Reach for the Stars is a GK-12 program supported by the National Science Foundation under grant DGE-0948017. However, any opinions, findings, conclusions, and/or recommendations are those of the investigators and do not necessarily reflect the views of the Foundation.

2          Level High school (nominally 10th grade) Regular or Honors Chemistry    Materials and Tools

Table of isotopes and their empirically determined masses Answer key for above Graph paper Calculators See below for alternatives

   Preparation None.    Prerequisites Some discussion of the Law of Conservation of Mass, e.g. vinegar + baking soda has same mass before and after reaction. Basic discussion of atomic structure, e.g. protons and neutrons in the nucleus, the number of protons determines the type of element.    Background Every atom has a nucleus made up of protons and neutrons. The number of protons determines what element that atom is. For example, all atoms of Hydrogen have 1 proton, all atoms of Helium have 2 protons, all atoms of Lithium have 3 protons, etc., all the way up the periodic table. The number of neutrons can be different, though. For example, you could have a Hydrogen atom with 1 proton and 0, 1, or 2 neutrons. These are called different “isotopes” of Hydrogen. The “mass number” of an isotope is the total number of particles in the nucleus (sometimes called “nucleons”). The mass number can be calculated by adding the number of protons and the number of neutrons. When we talk about isotopes, we name them by combining the name of the element and the mass number. For example, if we're talking about an atom of Carbon-14, we know that it has a mass number of 14, and that it has 6 protons (by consulting a periodic table). Thus we can conclude that the Carbon-14 nucleus has 8 neutrons.  The Law of Conservation of Mass says that in any process, the initial and final masses are equal. Another way to say this is that the mass of a whole is equal to the sum of the masses of its parts. If we want to check the Law of Conservation of Mass then, we'd want to find the difference between the mass of the whole and the mass of the pieces; if mass is conserved, the difference will always be zero.    Teaching Notes Begin by asking the class to recall the Law of Conservation of Mass as they've previously learned it. If they need refreshing on how to calculate proton and neutron numbers in a nucleus, ask them to work out a simple example (e.g. Lithium-7: 3 protons, 4 neutrons) and check them, and/or demonstrate on the board.  Organize students into groups of approximately 5, and distribute the attached handout, one to each student. Explain that this is a list of common isotopes of a variety of elements with their masses as

determined by experiment. Each group is assigned a set of approximately 5 isotopes to consider. For each, they should calculate the number of protons and neutrons and the total mass of that many protons and neutrons in amu. If students need support in constructing a mathematical expression, show them on the board that M = m_p * p + m_n * n, where m_p is the mass of a proton (1.007 amu), m_n is the mass of a neutron (1.009 amu), and p and n are the numbers of protons and neutrons, respectively. Emphasize that what they've just calculated is the mass of all the pieces of the nucleus when they're not assembled. We want to compare this to the mass of the isotope as observed by using a mathematical operation similar in form but conceptually different than percent error. The “mass defect” is the absolute value of the difference of the mass of the whole nucleus and the mass of all the pieces (M, above). We then want to scale this mass defect by dividing by the total number of particles in the nucleus (the mass number). Students should recognize that this process is similar to the error analysis they may be accustomed to (i.e. |observed – expected|/expected) but emphasize that this is not a difference due to random or systematic errors – this is a real physical effect that isn't explained by the Law of Conservation of Matter they've learned.  Each group is responsible for performing the above calculation for their assigned isotopes. After calculating the “mass defect per nucleon” for each, they should share their results with other groups, either by building a class table on the board or entering numbers in a spreadsheet that the teacher will project. Ultimately the entire class should have everyone's results copied into their tables. At this point each student should plot the class results on graph paper, with atomic number on the x-axis and mass defect per nucleon on the y-axis.  Once students have completed graphing, ask some questions to have them describe and interpret their graphs. For which elements is the line increasing? Flat? Decreasing? Where is the peak? Where is it steepest? Introduce that this effect comes as a result of Einstein's famous E = mc2 which says that energy (E) and mass (m) are related by a constant factor and that they are in some sense the same thing. In this case some of the mass of the nucleons becomes the “binding energy” that holds the nucleus together. Nuclei with more binding energy per nucleon/mass defect per nucleon are more tightly bound and more stable than those with less. The result then is that something like Hydrogen “knows” that if it could combine with more nucleons and become something like Helium or Lithium, it would be more stable and could release its excess energy to its environment. This process is known as “fusion” and is the primary source of energy in stars. In general, nuclei seek stability and will tend to undergo nuclear reactions like fusion and fission (the splitting of a nuclei into pieces) to climb up the curve students plotted. The amount of energy released in one of these reactions can be seen in the size of the vertical step between the “before” and “after” nuclei on the curve (e.g. large for H → He, smaller for U → Kr, zero for Fe: it's the most stable and doesn't react).    Assessment While students are calculating, teacher should stop at groups and check on progress. Final values of mass defect per nucleon should all be on the same order of magnitude, i.e. 0.00X, where X is somewhere between 6 and 9. If a student has made an error, it's easy to spot check and encourage to discuss with group members to find the error. While plotting, if a whole group has made an error, it will appear as a discontinuous section in the curve, calling attention to the need to correct. In addition to discussion questions listed above, the teacher can also ask students to estimate the mass defect per nucleon for another element without calculation, but by interpolating on their graphs. Then they can check how good an estimate it was by actually doing the calculation. For which elements is fusion/fission possible to extract energy? Which elements would make the best fuel (highest yields) for fusion/fission?  

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4            Additional Information This lesson plan was written assuming students wouldn't have access to computing resources like spreadsheets and plotting software. It could be adapted to use these resources, but in that case it would be better to spend more time letting students develop the mathematical expression on their own and gaining spreadsheet skills. Also all students (or all groups) could perform the operation for all elements, since more data points don't take significantly more time when working in a spreadsheet.