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## Group Activity: Isotope Notation and the Table of Isotopes

### Introduction

This work should be preceded by two things. First, students should have an opportunity to explore electric forces on a macroscopic scale by using pieces of plastic that obtain an electric charge when rubbed. Second, students should be introduced to the main concepts in the text below in a lecture-style or class discussion format. Most importantly, they should be shown a full table of the isotopes: one is available from Wikipedia.
and Positron Emission and Electron Capture

In this lesson you will learn about the four fundamental forces of nature and how two of them determine whether an isotope is stable or unstable. Stable isotopes do not decay or change over time. Unstable isotopes produce radiation and are transmuted into different isotopes and different elements. You will learn more about radiation in another lesson.

### Four Fundamental Forces

In modern physics there are four forces which, taken together, explain all possible interactions between pieces of matter. The forces are called Gravity, Electro-magnetism, the Strong Nuclear Force, and the Weak Nuclear Force.

You are probably already familiar with Gravity since it is the force that keeps you from floating up out of your chair. Gravity is an attractive force between two objects with mass. The strength of gravity depends on the size of the two masses and the distance between them. Gravity acts over all distances but its strength drops off with the square of the distance between two objects. That is, if something is twice as far from the Earth then it will feel a force only 1/4 as great. Gravity never has a repulsive effect. Gravity is responsible for keeping the Moon in orbit around the Earth and the Earth in orbit around the Sun. Gravity is only really important for objects with large masses: its effects are unmeasurable on the scale of atoms and molecules.

Electro-magnetism is more complicated than gravity. We will concentrate on just the electric part of this force. Electric forces are much stronger and more versatile force than gravity. For example, the force of gravity between two protons in the nucleus of an atom is about 1 × 1036 times weaker than the electric force between the same two protons.

 Electric Forces + + + >–< – – –

Electric forces can be attractive or repulsive (see table at right). As you have seen in the lab, two objects with the same charge repel each other: they push one another away. On the other hand, two objects with opposite charges are attracted to one another. This is where the expression that ‘opposites attract’ comes from. Scientific investigations of electric forces led to calling one kind of charge positive and the other kind negative. Objects with all the positive and negative charges balanced out are called neutral and have a charge of zero. If two charges are positive (or if they are both negative) they will be repelled from one another. But if one charge is positive and the other is negative they will be attracted to each other. Incidentally, it is because of electric forces that you do not fall through the floor and that you cannot push objects right through each other. The electric forces that bind atoms together are very strong and cannot be broken easily.

The Strong Nuclear Force is what holds atomic nuclei together. It is a force that is always attractive like gravity but it can only work over extremely short distances: from 1.3 × 10-15 m to 2.5 × 10-15 m. For purposes of this activity it is enough to note that the strong nuclear force acts to make two nucleons attract one another inside an atomic nucleus. The word nucleon stands for either a proton or a neutron, both of which are found in atomic nuclei. This means that the strong nuclear force causes protons to be attracted to protons and neutrons and causes neutrons to be attracted to protons and neutrons.

The Weak Nuclear Force is not really important for the purposes of this lesson. It governs certain kinds of radioactive processes and is important for describing how radioisotopes (radioactive isotopes) decay. Radioisotopes are isotopes of an element that are unstable because of unbalanced forces in the nucleus of that isotope.

### Electric Force and Atomic Nuclei

There are two kinds of subatomic particles in atomic nuclei. There are positively charged protons and neutrally charged neutrons. The protons are repelled from one another due to electric forces but neutrons are unaffected. Since they are so close together in the extremely tiny atomic nucleus the repulsive electric force is very, very high. If no other forces were acting then atomic nuclei could not hold themselves together. All the protons, since they are all positively charged, would push each other as far away as possible.

### Strong Nuclear Force and Atomic Nuclei

The strong nuclear force makes protons and neutrons stick together. It is not quite strong enough to make two protons stick together. The electric repulsion is stronger. But if there is also at least one neutron then the strong nuclear force can balance out the electric repulsion. This works because neutrons are not repelled or attracted by electric forces. They do respond to the strong nuclear force and can help hold atomic nuclei together. For example, take a nucleus with two protons and one neutron. The protons are electrically repelled but the neutron is neither repelled nor attracted. The strong nuclear force acts on all three particles to cause them to be attracted to one another. The neutron contributes just enough extra strong nuclear force to enable the nucleus to hold together. The protons are attracted to each other, too, but without the contribution of the neutron that attraction is not enough to keep the nucleus from being unstable. By the way, what element was used in this example?

### Isotope Stability

Isotopes can be stable or unstable. Their stability depends on the balance of forces. If either the electric repulsion or the strong nuclear force is stronger than the other then the isotope will be an unstable radioisotope. A simple way to think about nuclear stability is to consider the ratio of the number of neutrons to the number of protons. Every isotope has some number of protons and some number of neutrons (sometimes the number is zero). The protons alone will not stick together to make an atomic nucleus. Some number of neutrons is necessary to provide enough strong nuclear force to balance the electric repulsion. All of the known isotopes of all of the elements can be shown in a table of isotopes, as your teacher has already shown you. In the graph shown at left all of the isotopes are depicted as small squares placed to show the number of protons (x-axis) and neutrons (y-axis) in each one. There is a line on the graph showing the places on the graph where the number of neutrons equals the number of protons (n0 = p+ or N = Z).

Notice that the elements from hydrogen (Z = 1) to calcium (Z = 20) have their stable isotopes (the filled-in squares) on the line of n0 = p+. These isotopes have a n0/p+ ratio of 1 (or very nearly 1). Isotopes to the left and right of the line do not have a 1/1 ratio of n0/p+ and are unstable (radioactive). As you look at elements with a higher and higher Z (more and more protons in the nucleus) you should see that the n0/p+ ratio for the stable isotopes gets larger. Eventually, near the top of the graph, there are no more stable isotopes. Essentially, this means that no matter how many neutrons you add to atoms of these elements you cannot find a completly stable ratio of n0/p+.

Unstable isotopes decay by breaking up or by emitting particles or energy. Most of the unstable isotopes on this graph only exist when they are manufactured in a lab (the empty squares). These radioisotopes decay so rapidly that if they are made by natural processes they are not observed because they have already transmuted into other isotopes by the time scientists come across them. Even though the n0/p+ ratio of some isotopes is not ideal they occur in nature (the squares with dots). These naturally occurring radioisotopes are unstable but do not decay too rapidly and so can be mined from minerals on Earth. Examples include 40K, 48Ca, 235U, and 238U.

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##### Comprehension Questions

Answer questions with complete sentences, when appropriate. Fill in the following table with the required information about the isotopes in the picture at left.
 Name Symbol p+ n0 Mass hydrogen-1 1 1H 1 0 1 hydrogen-2 2 1H
1. What is the meaning of the letter ‘Z’ when speaking about atomic nuclei?
2. What is the definition of the atomic mass number (A)?
1. What is the meaning of the number that often follows the name of an element?
2. When the name of an element is followed by a number (as in calcium-40), what does it refer to, an isotope or an element? Why?.

1. Name and define the four fundamental forces.
2. Describe a stable nucleus. What forces are in balance and how is that balance achieved? Use an example.
3. Describe an unstable nucleus. What forces are unbalanced? There are two ways in which a nucleus can be unstable. Describe both and use examples.

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##### More Questions 1. What does the x-axis of this graph show? The y-axis?
2. What do the little dots on this graph represent?
3. What does A - Z equal? What does it mean on this graph?
4. What does Z = A - Z mean? What does it mean on this graph?
5. What do the numbers next to horizontal lines mean (2, 8, 20, 28, 50, 82, 126)?
6. What do the numbers above vertical lines mean?
1. Why are there only dots on part of the graph of Z vs. A – Z? In other words, why are the possible isotopes limited to that narrow curved patch on the graph?
2. Up to a certain point the path of the possible isotopes on the graph follows the line of Z = A – Z. Then it curves upward showing more and more neutrons per proton. Explain why this happens in terms of what you have learned about the fundamental forces important in the stability of an atomic nucleus.
1. Look at the graph of Z vs. A – Z. Write down how many neutrons (approximately) are needed to make the following elements stable:
element 4, Z = 25, Zr, element 79, and lead.
2. No elements above Z = 83 have any stable isotopes. Why do you think that is?
3. Look at the numbers and lines that are written on the graph (2, 8, 20, etc.). Up to 20 the the lines meet in the same place. After that they are increasingly far apart. In terms of the stability of atomic nuclei describe and explain the pattern that you see.

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Stable isotopes have a dark grey background. 1. The table above is a close-up of part of the table of isotopes. How many isotopes of hydrogen are there? Which ones are stable? Write both the name and the symbol for each isotope of hydrogen.
2. How many isotopes of carbon are there? Which ones are stable? Write both the name and the symbol for each isotope of carbon.
3. Make a chart on a separate piece of paper with the headings shown below. Include in your chart all of the isotopes shown in the table at the top of this page. When you calculate the n0/p+ ratio divide the no. of n0 by the no. of p+ and record the result as a decimal to three places. When you finish this task make a small version of the Z vs. A - Z graph on graph paper: record the number of protons and neutrons for each isotope as a dot. Draw a line on this graph for the places where Z = A - Z.
 Name Symbol Mass No. No. of p+ No. of n0 Ratio n0/p+ Stable (Y/N)
4. What pattern do you see?
5. What ratio corresponds to the largest number of stable isotopes?
6. Explain what makes nuclei unstable when they have a n0/p+ ratio greater than the stable ratio.
7. Explain what makes nuclei unstable when they have a n0/p+ ratio less than the stable ratio.
8. Using the information in the table at the top of the next page make another chart on a separate piece of paper or add to your existing chart. Use the same headings as before. Record the relevant data for the isotopes inside the space defined by corners at Pt-196 and Tl-210 at the top and Rn-204 and Rn-215 at the bottom. Make a new graph of Z vs. A – Z, being sure to leave room to draw in a line for where the number of protons equals the number of neutrons.

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Ā 1. What is different about these isotopes as compared with the isotopes you charted from the beginning of the periodic table?
2. What is the most common (or average) stable ratio of n0/p+ here? How does it compare with the stable ratio for isotopes near the beginning of the periodic table? That is, why is it different?
3. Why are there fewer stable isotopes in this part of the periodic table (Z = 76 to Z = 92)?
4. Do you think that there will be any new element with Z > 83 discovered that have stable or long-lived isotopes? Why or why not?

Last updated: Dec 31, 2007        Home