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Difficulty Level: At Grade Created by: CK-12

## Lesson Objectives

• Explain how radioactivity involves a change in the nucleus of a radioisotope.
• Describe mass defect, and calculate the conversion of mass to energy according to Einstein’s equation.
• Describe the band of stability, the odd-even effect, and magic numbers in terms of their influence on the stability of nuclei.
• Describe and write equations for the primary types of radioactive decay.

## Lesson Vocabulary

• alpha particle
• band of stability
• beta particle
• gamma ray
• mass defect
• nuclear binding energy
• nuclear reaction
• nucleon
• nuclide
• positron
• transmutation

### Recalling Prior Knowledge

• What is the composition of an atomic nucleus?
• What quantities are represented by the terms atomic number and mass number?

Marie Curie (1867-1934) was a Polish scientist who pioneered research into nuclear radiation (Figure below). She was awarded the Nobel Prize in physics in 1903 along with her husband Pierre and Antoine Henri Becquerel for their work on radioactivity. She was awarded a second Nobel Prize in 1911, this time in chemistry, for her continued research on radioactive elements. In this lesson, you will learn about radioactivity, the reasons why certain elements and isotopes are radioactive, and the most common types of radioactive decay processes.

Marie Curie was one of the leading scientists in the field of radioactivity. She discovered two radioactive elements and was awarded two Nobel Prizes for her work.

Radioactivity was discovered quite by accident. In 1896, Henri Becquerel was studying the effect of certain uranium salts on photographic film plates. He believed that the salts had an effect on the film only when they had been exposed to sunlight. He accidentally found that uranium salts that had not been exposed to sunlight still had an effect on the photographic plates. The Curies, associates of Becquerel at the time, showed that the uranium was emitting a type of ray that interacted with the film. Marie Curie called this radioactivity. Radioactivity is the spontaneous breakdown of an atom’s nucleus by the emission of particles and/or radiation. Radiation is the emission of energy through space in the form of particles and/or waves.

Nuclear reactions are very different from chemical reactions. In chemical reactions, atoms become more stable by participating in a transfer of electrons or by sharing electrons with other atoms. In nuclear reactions, it is the nucleus of the atom that gains stability by undergoing a change of some kind. Some elements have no stable isotopes, which means that any atom of that element is radioactive. For some other elements, only certain isotopes are radioactive. A radioisotope is an isotope of an element that is unstable and undergoes radioactive decay. The energies that are released in nuclear reactions are many orders of magnitude greater than the energies involved in chemical reactions. Unlike chemical reactions, nuclear reactions are not noticeably affected by changes in environmental conditions, such as temperature or pressure.

The discovery of radioactivity and its effects on the nuclei of elements disproved Dalton’s assumption that atoms are indivisible. A nuclide is a term for an atom with a specific number of protons and neutrons in its nucleus. As we will see, when nuclides of one type emit radiation, they are changed into different nuclides. Radioactive decay is spontaneous and does not require an input of energy to occur. The stability of a particular nuclide depends on the composition of its nucleus, including the number of protons, the number of neutrons, and the proton-to-neutron ratio.

## Nuclear Stability

Atoms are made of protons, neutrons, and electrons. According to the well-established Law of Conservation of Mass, it would be reasonable to think that the mass of an atom should be exactly equal to the mass of all of its isolated particles. This turns out to be untrue. An atom of the most common isotope of helium consists of two protons, two neutrons, and two electrons. The combined mass of those six particles is 4.03298 amu. The mass of a helium atom, to the same number of decimal places, is 4.00260 amu. There has somehow been a loss of 0.03038 amu in going from the separated particles to the intact atom. The mass defect of an atom is the difference between the mass of an atom and the sum of the masses of its protons, neutrons, and electrons.

### Nuclear Binding Energy

If mass is lost as an atom forms from its particles, where does the mass go? Albert Einstein showed that mass and energy are interconvertible according to his famous equation, E = mc2. In other words, mass can be converted into energy and energy can be converted into mass. The mass represented by the mass defect undergoes a conversion to energy upon the formation of an atom from its particles. We can use Einstein’s equation to solve for the energy produced when a helium atom is formed. First, the mass defect must be converted into the SI units of kilograms.

$\mathrm{0.03038 \ amu \times \dfrac{1 \ g}{6.0223 \times 10^{23} \ amu} \times \dfrac{1 \ kg}{1000 \ g} = 5.0446 \times 10^{-29} \ kg}$

The energy is then calculated by multiplying the mass in kilograms by the speed of light in a vacuum (c) squared. The resulting unit of kg•m2/s2 is equal to the energy unit of a joule (J).

E = mc2 = (5.0446 × 10-29 kg)(3.00 × 108 m/s)2 = 4.54 × 10-12 J

This quantity is called the nuclear binding energy, which is the energy released when a nucleus is formed from its nucleons. A nucleon is a nuclear subatomic particle (either a proton or a neutron). The nuclear binding energy can also be thought of as the energy required to break apart a nucleus into its individual nucleons.

Overall, larger nuclei tend to have larger nuclear binding energies, because the inclusion of each additional nucleon generally results in an additional release of energy. When comparing the stabilities of different nuclei, a more relevant value than the absolute nuclear binding energy is the nuclear binding energy per nucleon. For example, the nuclear binding energy per nucleon for helium is 4.54 × 10−12 J / 4 nucleons = 1.14 × 10−12 J/nucleon. Nuclear binding energies per nucleon are shown as a function of mass number in the figure below (Figure below).

Nuclear binding energy as a function of the number of nucleons in an atom. A higher nuclear binding energy per nucleon corresponds to a more stable nucleus.

Notice that elements of intermediate mass, such as iron, have the highest nuclear binding energies per nucleon and are thus the most stable.

### The Band of Stability

Carbon-12, with six protons and six neutrons, is a stable nucleus, meaning that it does not spontaneously emit radioactivity. Carbon-14, with six protons and eight neutrons, is unstable and naturally radioactive. Among atoms with lower atomic numbers, the ideal ratio of neutrons to protons is approximately 1:1. As the atomic number increases, the stable neutron-proton ratio gradually increases to about 1.5:1 for the heaviest known elements. For example, lead-206 is a stable nucleus that contains 124 neutrons and 82 protons, a ratio of 1.51 to 1.

This observation is shown in the figure below (Figure below). The band of stability is the range of stable nuclei on a graph that plots the number of neutrons in a nuclide against the number of protons. Known stable nuclides are shown with individual black dots, while the 1:1 and 1.5:1 ratios are shown with a solid red line and a dotted red line, respectively.

A graph of the number of neutrons in a nucleus as a function of the number of protons. Each known stable nucleus is represented by a blue dot. The ideal neutron to proton ratio changes from 1:1 for light nuclei to 1.5:1 for the heaviest nuclei.

The band of stability can be explained by a consideration of the forces involved inside the nucleus. Protons within the nucleus repel one another due to electrostatic repulsion. However, another force called the strong nuclear force exists over very short distances within the nucleus. This strong nuclear force allows the nucleus to stay together despite the repulsive electrostatic force. However, as the number of protons increases, so does the electrostatic force. More and more neutrons are necessary so that the attractive nuclear force can overcome the electrostatic repulsion. Beyond the element lead (atomic number 82), the repulsive force is so great that no stable nuclides exist.

It should be noted that just because a nucleus is "unstable" (able to undergo spontaneous radioactive decay) does not mean that it will rapidly decompose. For example, uranium-238 is unstable because it spontaneously decays over time, but if a sample of uranium-238 is allowed to sit for 1000 years, only 0.0000155% of the sample will have decayed. However, other unstable nuclei, such as berkelium-243, will be almost completely gone (>99.9999% decayed) in less than a day.

There are several other characteristics common to stable nuclei. For example, stable nuclei with even numbers of both protons and neutrons are far more common than stable nuclei with odd numbers of one or both of those particles (Table below).

Protons Neutrons Number of Stable Isotopes
Odd Odd 4
Odd Even 50
Even Odd 53
Even Even 157

Nuclei that contain 2, 8, 20, 50, 82, or 126 protons or neutrons are unusually stable for a nucleus of that particular size. For example, there are ten stable isotopes of the element tin (atomic number 50), but only two stable isotopes of antimony (atomic number 51). These numbers are often referred to as magic numbers, and their existence lends evidence to the theory that nucleons, like electrons, exist in different energy levels within the nucleus. These numbers are thought to represent filled energy levels, and their stability is analogous to the stable electron configurations of noble gases.

Unstable nuclei spontaneously emit radiation in the form of particles and energy. This generally changes the number of protons and/or neutrons in the nucleus, resulting in a more stable nuclide. A nuclear reaction is a reaction that affects the nucleus of an atom. One type of a nuclear reaction is radioactive decay, a reaction in which a nucleus spontaneously disintegrates into a slightly lighter nucleus, accompanied by the emission of particles, energy, or both. An example is shown below, in which the nucleus of a polonium atom radioactively decays into a lead nucleus.

$\mathrm{ ^{210}_{84} Po \rightarrow ^{206}_{82} Pb + ^{4}_{2} He}$

Note that in a balanced nuclear equation, the sum of the atomic numbers and the sum of the mass numbers must be equal on both sides of the equation. Recall the notation system for isotopes, which shows both the atomic number and mass number along with the chemical symbol.

Because the number of protons changes as a result of this nuclear reaction, the identity of the element changes. Transmutation is a change in the identity of a nucleus as a result of a change in the number of protons. There are several different types of naturally occurring radioactive decay, and we will examine each separately.

### Alpha Decay

An alpha particle (α) is a helium nucleus with two protons and two neutrons. Alpha particles are emitted during some types of radioactive decay. The net charge of an alpha particle is 2+, and its mass is approximately 4 amu. The symbol for an alpha particle in a nuclear equation is usually $\mathrm{ ^{4}_{2} He}$ , though sometimes $\mathrm{ ^{4}_{2} \alpha}$ is used. Alpha decay typically occurs for very heavy nuclei in which the nuclei are unstable due to large numbers of nucleons. For nuclei that undergo alpha decay, their stability is increased by the subtraction of two protons and two neutrons. For example, uranium-238 decays into thorium-234 by the emission of an alpha particle (Figure below).

$\mathrm{ ^{238}_{92} U \rightarrow ^{234}_{90} Th + ^{4}_{2} He}$

The unstable uranium-238 nucleus spontaneously decays into a thorium-234 nucleus by emitting an alpha particle.

### Beta Decay

Nuclei above the band of stability are unstable because their neutron to proton ratio is too high. To decrease that ratio, a neutron in the nucleus is capable of turning into a proton and an electron. The electron is immediately ejected at a high speed from the nucleus. A beta particle (β) is a high-speed electron emitted from the nucleus of an atom during some kinds of radioactive decay (Figure below). The symbol for a beta particle in an equation is either $\mathrm{ ^{\ 0}_{-1} \beta}$ or $\mathrm{ ^{\ 0}_{-1} e}$. Carbon-14 undergoes beta decay, transmutating into a nitrogen-14 nucleus.

$\mathrm{ ^{14}_{6} C \rightarrow ^{14}_{7} N + ^{\ 0}_{-1} \beta}$

Note that beta decay increases the atomic number by one, but the mass number remains the same.

The beta decay of a carbon-14 nuclide involves the conversion of a neutron to a proton and an electron, with the electron being emitted from the nucleus.

### Positron Emission

Nuclei below the band of stability are unstable because their neutron to proton ratio is too low. One way to increase that ratio is for a proton in the nucleus to turn into a neutron and another particle called a positron. A positron is a particle with the same mass as an electron, but with a positive charge. Like the beta particle, a positron is immediately ejected from the nucleus upon its formation. The symbol for a positron in an equation is either $\mathrm{ ^{\ 0}_{+1} \beta}$ or $\mathrm{ ^{\ 0}_{+1} e}$. For example, potassium-38 emits a positron, becoming argon-38.

$\mathrm{ ^{38}_{19} K \rightarrow ^{38}_{18} Ar + ^{\ 0}_{+1} \beta}$

Positron emission decreases the atomic number by one, but the mass number remains the same.

### Electron Capture

An alternate way for a nuclide to increase its neutron to proton ratio is by a phenomenon called electron capture. In electron capture, an electron from an inner orbital is captured by the nucleus of the atom and combined with a proton to form a neutron. For example, silver-106 undergoes electron capture to become palladium-106.

$\mathrm{ ^{106}_{47} Ag + ^{\ 0}_{-1} e \rightarrow ^{106}_{46} Pd}$

Note that the overall result of electron capture is identical to positron emission. The atomic number decreases by one while the mass number remains the same.

### Gamma Ray Emission

Gamma rays (γ) are very high energy electromagnetic waves emitted from a nucleus. Gamma rays are emitted by a nucleus when nuclear particles undergo transitions between nuclear energy levels. This is analogous to the electromagnetic radiation emitted when excited electrons drop from higher to lower energy levels; the only difference is that nuclear transitions release much more energetic radiation. Gamma ray emission often accompanies the decay of a nuclide by other means.

$\mathrm{ ^{230}_{90} Th \rightarrow ^{226}_{88} Ra + ^{4}_{2} He} + \gamma$

The emission of gamma radiation has no effect on the atomic number or mass number of the products, but it reduces their energy.

The table below (Table below) summarizes the main types of nuclear radiation, including charge, mass, symbol, and penetrating power. Penetrating power refers to the relative ability of the radiation to pass through common materials. Radiation with high penetrating power is potentially more dangerous because it can pass through skin and do cellular damage.

Type Mass Charge Penetration Power Shielding
Alpha particle 4 amu 2+ Low Paper, skin
Beta particle ~0 1− Moderate Metal foil
Positron ~0 1+ Moderate Metal foil
Gamma ray 0 0 Very high Lead, concrete (incomplete)

## Lesson Summary

• Radioactive nuclides become more stable by emitting radiation in the form of small particles and energy.
• Mass defect is the difference between the mass of an atom and the sum of the masses of its particles. Upon the formation of a nucleus, that mass is converted to energy. The nuclear binding energy released during this process is an indicator of the stability of a nucleus.
• Nuclei tend to be unstable when they are either very large or have a proton to neutron ratio that does not fall within a band of stability. Greater stability comes from an even number of protons and/or neutrons, especially when one or both of these values is equal to certain magic numbers.
• Radioactive decay results in the formation of more stable nuclei. Types of radioactive decay include alpha decay, beta decay, positron emission, electron capture, and gamma ray emission.

## Lesson Review Questions

### Reviewing Concepts

1. How does an unstable nucleus release energy?
1. What is the relationship between mass defect and nuclear binding energy?
2. How does nuclear binding energy per nucleon relate to nuclear stability?
3. What changes in atomic number and mass number occur in each of the following types of radioactive decay?
1. positron emission
2. alpha decay
3. beta decay
4. electron capture
4. What kind of radiation is the most penetrating? The least penetrating?

### Problems

1. Given the following masses for isolated subatomic particles and for a neon-20 atom:
• proton = 1.00728 amu
• neutron = 1.00866 amu
• electron = 5.486 × 10−4 amu
• neon-20 = 19.99244 amu
1. Calculate the mass defect of a neon-20 atom in amu.
2. Calculate the nuclear binding energy in J. (Use the conversion 6.0223 × 1026 amu = 1 kg)
3. Calculate the nuclear binding energy per nucleon in J/nucleon.
2. Calculate the neutron-proton ratio for the following nuclides. Then, use the figure above (Figure above) to determine whether each nuclide is on, above, or below the band of stability.
1. $\mathrm{ ^{12}_{6} C}$
2. $\mathrm{ ^{13}_{7} N}$
3. $\mathrm{ ^{134}_{50} Sn}$
4. $\mathrm{ ^{59}_{27 } Co}$
3. For each pair of nuclides listed below, indicate which one you would expect to be radioactive. Explain.
1. $\mathrm{ ^{20}_{10} Ne}$ and $\mathrm{ ^{17}_{10} Ne}$
2. $\mathrm{ ^{96}_{42} Mo}$ and $\mathrm{ ^{96}_{43} Tc}$
3. $\mathrm{ ^{45}_{20} Ca}$ and $\mathrm{ ^{40}_{20} Ca}$
4. $\mathrm{ ^{195}_{80} Hg}$ and $\mathrm{ ^{196}_{80} Hg}$
4. For each pair of elements listed, predict which one has more stable isotopes. Explain.
1. Co or Ni
2. Hg or Pb
3. Ag or Am
4. Sr or Y
5. Fill in the blanks for each of the nuclear reactions below. State the type of decay in each case.
1. $\mathrm{ ^{198}_{79} Au \rightarrow }$ _______ $\mathrm{+ ^{\ 0}_{-1} e}$
2. $\mathrm{ ^{57}_{27} Co + ^{\ 0}_{-1}e \rightarrow}$ _______
3. $\mathrm{ ^{230}_{92} U \rightarrow }$ _______ $\mathrm{+ ^{4}_{2} He}$
4. $\mathrm{ ^{128}_{56} Ba \rightarrow }$ _______ $\mathrm{+ ^{\ 0}_{+1} e}$
5. $\mathrm{ ^{131}_{53} I \rightarrow ^{131}_{54} Xe +}$ _______
6. $\mathrm{ ^{239}_{94} Pu \rightarrow ^{235}_{92} U +}$ _______
6. Write balanced nuclear reactions for each of the following.
1. Francium-220 undergoes alpha decay.
2. Arsenic-76 undergoes beta decay.
3. Uranium-231 captures an electron.
4. Promethium-143 emits a positron.

## Points to Consider

The stability of a radioisotope is indicated by its half-life, which is the amount of time required for half of the nuclei in a given sample to undergo a decay process.

• What is the range of half-lives for known isotopes?
• How can half-life be used to determine the age of objects, such as fossils or certain rocks and minerals?

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Date Created:
Mar 29, 2013