Learning Objectives
By the end of this section, you will be able to do the following:
- Define and discuss the nucleus in an atom
- Define atomic number
- Define and discuss isotopes
- Calculate the density of the nucleus
- Explain nuclear force
The information presented in this section supports the following AP® learning objectives and science practices:
- 3.G.3.1 The student is able to identify the strong force as the force that is responsible for holding the nucleus together. (S.P. 7.2)
What is inside the nucleus? Why are some nuclei stable while others decay (see Figure 14.6)? Why are there different types of decay and Why are nuclear decay energies so large? Pursuing natural questions like these has led to far more fundamental discoveries than you might imagine.
We have already identified protons as the particles that carry positive charge in the nuclei. However, there are actually two types of particles in the nuclei—the proton and the neutron, referred to collectively as nucleons, the constituents of nuclei. As its name implies, the neutron is a neutral particle that has nearly the same mass and intrinsic spin as the proton. Table 14.1 compares the masses of protons, neutrons, and electrons. Note how close the proton and neutron masses are, but the neutron is slightly more massive once you look past the third digit. Both nucleons are much more massive than an electron. In fact, (as noted in Medical Applications of Nuclear Physics and
Table 14.1 also gives masses in terms of mass units that are more convenient than kilograms on the atomic and nuclear scale. The first of these is the unified atomic mass unit (u), defined as
This unit is defined so that a neutral carbon atom has a mass of exactly 12 u. Masses are also expressed in units of These units are very convenient when considering the conversion of mass into energy—and vice versa—as is so prominent in nuclear processes. Using and units of in we find that cancels and comes out conveniently in MeV. For example, if the rest mass of a proton is converted entirely into energy, then
It is useful to note that 1 u of mass converted to energy produces 931.5 MeV, or
All properties of a nucleus are determined by the number of protons and neutrons it has. A specific combination of protons and neutrons is called a nuclide and is a unique nucleus. The following notation is used to represent a particular nuclide
where the symbols and are defined as follows: The number of protons in a nucleus is the atomic number as defined in Medical Applications of Nuclear Physics. X is the symbol for the element, such as Ca for calcium. However, once is known, the element is known; hence, and are redundant. For example, is always calcium, and calcium always has is the number of neutrons in a nucleus. In the notation for a nuclide, the subscript is usually omitted. The symbol is defined as the number of nucleons or the total number of protons and neutrons
where is also called the mass number. This name for is logical; the mass of an atom is nearly equal to the mass of its nucleus, since electrons have so little mass. The mass of the nucleus turns out to be nearly equal to the sum of the masses of the protons and neutrons in it, which is proportional to In this context, it is particularly convenient to express masses in units of u. Both protons and neutrons have masses close to 1 u, and so the mass of an atom is close to u. For example, in an oxygen nucleus with eight protons and eight neutrons, and its mass is 16 u. As noticed, the unified atomic mass unit is defined so that a neutral carbon atom—actually a atom—has a mass of exactly 12 Carbon was chosen as the standard, partly because of its importance in organic chemistry (see Exercise 17.31).
Particle | Symbol | kg | u | MeVc2 |
---|---|---|---|---|
Proton | p | 1.007276 | 938.27 | |
Neutron | n | 1.008665 | 939.57 | |
Electron | e | 0.00054858 | 0.511 |
Let us look at a few examples of nuclides expressed in the notation. The nucleus of the simplest atom, hydrogen, is a single proton, or —the zero for no neutrons is often omitted. To check this symbol, refer to the periodic table—you see that the atomic number of hydrogen is 1. Since you are given that there are no neutrons, the mass number is also 1. Suppose you are told that the helium nucleus or particle has two protons and two neutrons. You can then see that it is written There is a scarce form of hydrogen found in nature called deuterium; its nucleus has one proton and one neutron and, hence, twice the mass of common hydrogen. The symbol for deuterium is, thus, —sometimes is used, as for deuterated water An even rarer—and radioactive—form of hydrogen is called tritium, since it has a single proton and two neutrons, and it is written These three varieties of hydrogen have nearly identical chemistries, but the nuclei differ greatly in mass, stability, and other characteristics. Nuclei, such as those of hydrogen, having the same and different Ns are defined to be isotopes of the same element.
There is some redundancy in the symbols and If the element is known, then can be found in a periodic table and is always the same for a given element. If both and are known, then can also be determined—first find then, Thus the simpler notation for nuclides is
which is sufficient and is most commonly used. For example, in this simpler notation, the three isotopes of hydrogen are and while the particle is We read this backward, saying helium-4 for or uranium-238 for So for should we need to know, we can determine that for uranium from the periodic table, and, thus,
A variety of experiments indicate that a nucleus behaves something like a tightly packed ball of nucleons, as illustrated in Figure 14.7. These nucleons have large kinetic energies and, thus, move rapidly in very close contact. Nucleons can be separated by a large force, such as in a collision with another nucleus, but resist strongly being pushed closer together. The most compelling evidence that nucleons are closely packed in a nucleus is that the radius of a nucleus, is found to be given approximately by
where and is the mass number of the nucleus. Note that Since many nuclei are spherical, and the volume of a sphere is we see that is, the volume of a nucleus is proportional to the number of nucleons in it. This is what would happen if you pack nucleons so closely that there is no empty space between them.
Nucleons are held together by nuclear forces and resist both being pulled apart and pushed inside one another. The volume of the nucleus is the sum of the volumes of the nucleons in it, here shown in different colors to represent protons and neutrons.
Example 14.1 How Small and Dense Is a Nucleus?
(a) Find the radius of an iron-56 nucleus. (b) Find its approximate density in approximating the mass of to be 56 u.
Strategy and Concept
(a) Finding the radius of is a straightforward application of given (b) To find the approximate density, we assume the nucleus is spherical—this one actually is—calculate its volume using the radius found in part (a), and then find its density from Finally, we will need to convert density from units of to
Solution
(a) The radius of a nucleus is given by
Substituting the values for and yields
(b) Density is defined to be which for a sphere of radius is
Substituting known values gives
Converting to units of find
Discussion
(a) The radius of this medium-sized nucleus is found to be approximately 4.6 fm, and so its diameter is about 10 fm, or In our discussion of Rutherford’s discovery of the nucleus, we noticed that it is about in diameter—which is for lighter nuclei—consistent with this result to an order of magnitude. The nucleus is much smaller in diameter than the typical atom, which has a diameter of the order of
(b) The density found here is so large as to cause disbelief. It is consistent with earlier discussions we have had about the nucleus being very small and containing nearly all of the mass of the atom. Nuclear densities, such as found here, are about times greater than that of water, which has a density of only cubic meter of nuclear matter, such as found in a neutron star, has the same mass as a cube of water 61 km on a side.