Until now, we’ve taken it for granted that you know what protons, neutrons, and electrons are. Within the past century, these objects have gone from being part of vaguely conjectured theories by advanced physicists to common knowledge. Unfortunately, SAT II Physics is going to test you on matters that go far beyond common knowledge. That’s where we come in.

As you surely know, atoms are made up of a nucleus of protons and neutrons orbited by electrons. Protons have a positive electric charge, electrons have a negative electric charge, and neutrons have a neutral charge. An electrically stable atom will have as many electrons as protons.

Because objects on the atomic level are so tiny, it can be a bit unwieldy to talk about their mass in terms of kilograms. Rather, we will often use the atomic mass unit (amu, or sometimes just u), which is defined as one-twelfth of the mass of a carbon-12 atom. That means that 1 amu = kg. We can express the mass of the elementary particles either in kilograms or atomic mass units:

As you can see, the mass of electrons is pretty much negligible when calculating the mass of an atom.

You’re probably somewhat familiar with the periodic table and know that there are over 100 different chemical elements. An element is defined by the number of protons in the atomic nucleus. For instance, a nucleus with just one proton is hydrogen, a nucleus with two protons is helium, and a nucleus with 92 protons is uranium, the heaviest naturally occurring element. The number of protons in an atomic nucleus determines the atomic number, Z. In an electrically neutral atom of atomic number Z, there will be Z protons and Z electrons.

The number of neutrons in an atomic nucleus determines the neutron number, N. Different nuclei of the same atomic number—that is, atoms of the same element—may have different numbers of neutrons. For instance, the nuclei of most carbon atoms have six protons and six neutrons, but some have six protons and eight neutrons. Atoms of the same element but with different numbers of neutrons are called isotopes.

As we saw above, electrons weigh very little in comparison to protons and neutrons, which have almost identical masses. The sum of the atomic number and the neutron number, Z + N, gives us an atom’s mass number, A.

The standard form for writing the chemical symbol of an element, X, is:

The element’s mass number is written in superscript, and the atomic number is written in subscript. You can infer the neutron number by subtracting A – Z. For instance, we would write the chemical symbol for the two carbon isotopes, called carbon-12 and carbon-14, as follows:

The same sort of system can be used to represent protons, neutrons, and electrons individually. Because a proton is the same thing as a hydrogen atom without an electron, we can represent protons by writing:

where the + sign shows that the hydrogen ion has a positive charge due to the absence of the electron. Neutrons are represented by the letter “n” as follows:

Electrons and positrons, which are positively charged electrons, are represented, respectively, as follows:

The number in subscript gives the charge of the particle—0 in the case of the neutron and –1 in the case of the electron. The number in superscript gives the mass. Though electrons have mass, it is so negligible in comparison to that of protons and neutrons that it is given a mass number of 0.

On the SAT II, you will not need to apply your knowledge of any elementary particles aside from the proton, the neutron, and the electron. However, the names of some other particles may come up, and you will at least need to know what they are.

Quarks are the fundamental building blocks of the protons, neutrons, and mesons. They generally have positive or negative charges in units of one-third to two-thirds of the charge of the electron. Protons are neutrons composed of three quarks. Mesons are composed of a quark–antiquark pair.

Some configurations of protons and neutrons are more stable in a nucleus than others. For instance, the carbon-12 atom is more stable than the carbon-14 atom. While carbon-12 will remain stable, carbon-14 will spontaneously transform into a more stable isotope of nitrogen, releasing particles and energy in the process. Because these transformations take place at a very steady rate, archaeologists can date carbon-based artifacts by measuring how many of the carbon-14 atoms have decayed into nitrogen. These transformations are called radioactive decay, and isotopes and elements like carbon-14 that undergo such decay are called radioactive. There are three major kinds of radioactive decay.

When an atom undergoes alpha decay, it sheds an alpha particle, , which consists of two protons and two neutrons. Through alpha decay, an atom transforms into a smaller atom with a lower atomic number. For instance, uranium-238 undergoes a very slow process of alpha decay, transforming into thorium:

Notice that the combined mass number and atomic number of the two particles on the right adds up to the mass number and atomic number of the uranium atom on the left.

There are actually three different kinds of beta decay— decay, decay, and electron capture—but SAT II Physics will only deal with decay, the most common form of beta decay. In decay, one of the neutrons in the nucleus transforms into a proton, and an electron and a neutrino, , are ejected. A neutrino is a neutrally charged particle with very little mass. The ejected electron is called a beta particle, .

The decay of carbon-14 into nitrogen is an example of decay:

Note that the mass number of the carbon on the left is equal to the sum of the mass numbers of the nitrogen and the electron on the right: 14 = 14 + 0. Similarly, the atomic number of the carbon is equal to the sum of the atomic number of the nitrogen and the electron: 6 = 7 – 1. Because the neutrino has no charge and negligible mass, its presence has no effect on any aspect of beta decay that we will study. Still, it’s important that you know the neutrino’s there.

Gamma decay is the most straightforward kind of decay. An element in a high-energy state can return to a lower energy state by emitting a gamma ray, , which is an electromagnetic photon of very high frequency. No other particles are ejected and the nucleus doesn’t transform from one element to another. All we get is an ejected gamma ray, as in this example with technetium:

In the above reaction, a sodium nucleus transforms into some other element and gives off an electron. Electrons are only released in beta decay. A neutrino is also released but, because its effects are negligible, it is often left out of the equation.

We can calculate A and Z because the sum of the atomic numbers and the mass numbers on the right must add up to the atomic number and the mass number on the left. We can solve for A and Z with the following equations:

So A = 24 and Z = 12. The resulting element is determined by the atomic number, Z. Consult a periodic table, and you will find that the element with an atomic number of 12 is magnesium, so X stands in for the chemical symbol for magnesium, Mg.

Atomic nuclei undergo radioactive decay so as to go from a state of high energy to a state of low energy. Imagine standing on your hands while balancing a box on your feet. It takes a lot of energy, not to mention balance, to hold yourself in this position. Just as you may spontaneously decide to let the box drop to the floor and come out of your handstand, atomic nuclei in high-energy states may spontaneously rearrange themselves to arrive at more stable low-energy states.

So far, all the physical interactions we have looked at in this book result from either the gravitational force or the electromagnetic force. Even the collisions we studied in the chapters on mechanics are the result of electromagnetic repulsion between the atoms in the objects that collide with one another. However, neither of these forces explains why the protons in an atomic nucleus cling together. In fact, the electromagnetic force should act to make the protons push away from one another, not cling together. Explaining how things work on the atomic level requires two additional forces that don’t act beyond the atomic level: the strong and weak nuclear forces. The strong nuclear force binds the protons and neutrons together in the nucleus. The weak nuclear force governs beta decay. You don’t need to know any of the math associated with these forces, but you should know what they are.

As we have discussed, the mass of a proton is 1.0073 amu and the mass of a neutron is 1.0086 amu. Curiously, though, the mass of an alpha particle, which consists of two protons and two neutrons, is not 2(1.0073) + 2(1.0086) = 4.0318 amu, as one might expect, but rather 4.0015 amu. In general, neutrons and protons that are bound in a nucleus weigh less than the sum of their masses. We call this difference in mass the mass defect, , which in the case of the alpha particle is 4.0318 – 4.0015 = 0.0202 amu.

The reason for this mass defect is given by the most famous equation in the world:

As we discussed in the section on relativity, this equation shows us that mass and energy can be converted into one another.

The strong nuclear force binds the nucleus together with a certain amount of energy. A small amount of the matter pulled into the nucleus of an atom is converted into a tremendous amount of energy, the binding energy, which holds the nucleus together. In order to break the hold of the strong nuclear force, an amount of energy equal to or greater than the binding energy must be exerted on the nucleus. For instance, the binding energy of the alpha particle is:

Note that you have to convert the mass from atomic mass units to kilograms in order to get the value in joules. Often we express binding energy in terms of millions of electronvolts, MeV, per nucleon. In this case, J = 18.7 MeV. Because there are four nucleons in the alpha particle, the binding energy per nucleon is 18.7/4 = 4.7 MeV/nucleon.

Since there are two nucleons in a deuteron, the binding energy for the deuteron as a whole is MeV. That energy, converted into mass, is:

The mass of a free proton plus a free neutron is 1.0073 + 1.0086 = 2.0159 amu. The mass of the deuteron will be 0.0024 amu less than this amount, since that is the amount of mass converted into energy that binds the proton and the neutron together. So the deuteron will weigh 2.0159 – 0.0024 = 2.0135 amu.

On SAT II Physics, you probably won’t be expected to calculate how long it takes a radioactive nucleus to decay, but you will be expected to know how the rate of decay works. If we take a sample of a certain radioactive element, we say that its activity, A, is the number of nuclei that decay per second. Obviously, in a large sample, A will be greater than in a small sample. However, there is a constant, called the decay constant, , that holds for a given isotope regardless of the sample size. We can use the decay constant to calculate, at a given time, t, the number of disintegrations per second, A; the number of radioactive nuclei, N; or the mass of the radioactive sample, m:

, , and are the values at time t = 0. The mathematical constant e is approximately 2.718.

The decay constant for uranium-238 is about s^–1. After one million years, a 1.00 kg sample of uranium-238 (which has atoms) will contain

Uranium-238 is one of the slower decaying radioactive elements.

We generally measure the radioactivity of a certain element in terms of its half-life, , the amount of time it takes for half of a given sample to decay. The equation for half-life, which can be derived from the equations above, is:

You won’t need to calculate the natural logarithm of 2—remember, no calculators are allowed on the test. What you will need to know is that, at time t = , one-half of a given radioactive sample will have decayed. At time t = 2, one-half of the remaining half will have decayed, leaving only one-quarter of the original sample. You may encounter a graph that looks something like this:

The graph of decay vs. time will get steadily closer to the x-axis, but will never actually reach it. The fewer atoms that remain undecayed, the less activity there will be.

Nuclear reactions are effectively the same thing as radioactivity: new particles are formed out of old particles, and the binding energy released in these transitions can be determined by the equation E = mc². The difference is that nuclear reactions that are artificially induced by humans take place very rapidly and involve huge releases of energy in a very short time. There are two kinds of nuclear reaction with which you should be familiar for SAT II Physics.

Nuclear fission was used in the original atomic bomb, and is the kind of reaction harnessed in nuclear power plants. To produce nuclear fission, neutrons are made to bombard the nuclei of heavy elements—often uranium—and thus to split the heavy nucleus in two, releasing energy in the process. In the fission reactions used in power plants and atomic bombs, two or more neutrons are freed from the disintegrating nucleus. The free neutrons then collide with other atomic nuclei, starting what is called a chain reaction. By starting fission in just one atomic nucleus, it is possible to set off a chain reaction that will cause the fission of millions of other atomic nuclei, producing enough energy to power, or destroy, a city.

Nuclear fusion is ultimately the source of all energy on Earth: fusion reactions within the sun are the source of all the heat that reaches the Earth. These reactions fuse two or more light elements—often hydrogen—together to form a heavier element. As with fission, this fusion releases a tremendous amount of energy.

Fusion reactions can only occur under intense heat. Humans have only been able to produce a fusion reaction in the hydrogen bomb, or H-bomb, by first detonating an atomic bomb whose fission produced heat sufficient to trigger the fusion reaction. Scientists hope one day to produce a controllable fusion reaction, since the abundance of hydrogen found in this planet’s water supply would make nuclear fusion a very cheap and nonpolluting source of energy.