### Table of Contents

# Nuclear chemistry

In ordinary chemical reactions matter and energy are conserved separately and the identities of the atoms don‘t change as the bonds of the reactants are broken and those of the products are formed. However, elements can be transformed into one another by radioactive decay. To account for the conservation of matter and energy, a new „set“ of conservation laws needed to be formulated. The solution to this problem was provided by Einstein‘s special theory of relativity which introduced the concept of mass-energy equivalence and is summarized by the (probably the world‘s most famous equation) equation:

what the equation states is that mass and energy are fundamentally equivalent, they can be interconverted and it‘s only their sum that is conserved.

## Nuclear structure and nuclear decay

We shall begin by reviewing the essential features of nuclear structure, which will help you visualize nuclear decay processes. Nuclei are built up from two kinds of nucleons, protons and neutrons, with the number of protons being given by the **atomic number Z** and the number of neutrons given by the **neutron number N**. The mass number A is the sum of the number of protons and neutrons, that is:

Nuclides (definition: a distinct kind of atom or nucleus characterized by a specific number of protons and neutrons) are distinct atomic species characterized by their atomic number Z, mass number A, and nuclear energy state (discussed later). Each nuclide is identified by the symbol , where Z is the chemical symbol for the element and the subscript z is the atomic number. **Isotopes** are elements with the same number of protons but different numbers of neutrons - that is, they differ only in their mass.

The symbols for and masses of a few selected elementary particles and atoms are listed in **Table 1**. The atomic number is given by the left subscript and the mass number by the left superscript for nucleons. The symbol for the electron is and that of its antimatter counterpart the **positron** is , with the charge also being specified by the left subscript and the mass number (essentially zero, relative to the nucleons) being specified by the left superscript. There is some redundancy in this notation, as the chemical symbol of the element implies its atomic number and the charges of the electron and positron are written in two places. This redundancy is useful when balancing nuclear reaction. Masses of the elementary particles are listed both in atomic mass units (u) as well as in kilograms, but the masses of the atoms are listed only in atomic mass units. An **atomic mass unit** is defined as exactly 1/12 the mass of a single atom of , which can be calculated by dividing the atomic mass of by *Avogadro's number* (value: and converting the result to kilograms.

Atomic mass units are very convenient units to use when carrying out calculations involving individual elementary particles and atoms. They are numerically equal to atomic masses expressed in grams per mole, and they allow us to do calculations without carrying along large negative powers of 10. The latter advantage is particularly apparent when calculating mass changes associated with nuclear reactions as we discuss shortly.

## Nuclear decay processes

There are thousands of known isotopes of the elements, only about 275 of which are considered to be “stable”, that is, they show no evidence of radioactive decay whatsoever. A general rule to get some idea about the stability of nuclides is that: *if the neutron number N divided by the atomic number Z is equal to (or close to) 1, then the nuclide is stable*. That is, if (or close to 1), then we can assume that the nuclide is stable.

### The decay of the radioactive isotopes

The radioactive isotopes decay by three basic processes, so called decays, where the beta decay has three distinct, but related modes.

Alpha decay is the spontaneous emission of particles ( nuclei) from heavy nuclei (Z < 83) that reduces the Coulomb repulsion between protons; the parent nucleus loses two protons and two neutrons. There are three different modes of decay: (electron) emission, (positron) emission, and EC (electron capture).

These processes convert neutrons into protons (and vice versa) to bring the N/Z ratio closer to 1, that is to gain more stability. Neutron-rich nuclei decay by electron emission, whereas proton-rich nuclei decay either by positron emission or by electron capture. The decay energy is shared among all particles in electron and positron emission (becomes the kinetic energy of the released particles), whereas almost all of the energy is carried away by the neutrino in electron capture.

### Balancing nuclear reactions

Nuclear reactions are written and balanced much like chemical reactions, with both the mass number A and the electric charge being conserved. We represent the elementary particles by the symbols: for the electron, for the proton and for the neutron to help remind us to include their mass number and charge when balancing nuclear reaction. Examples include:

### Alpha decay

Proton-rich nuclei can decay into more stable isotopes by emitting particles ( nuclei), which reduces the atomic number Z and the neutron number N by 2, resulting in a decrease of the mass number A of 4. Most of the energy is carried away by the lighter particles, with a small fraction appearing as recoil energy of the heavier daughter nuclei. An example of alpha-decay is:

### Beta decay

Proton-deficient nuclei can decay by transforming a neutron into a proton, which results in the emission of a particle and a *antineutrino* (). Note that we have to remember to use the symbol in the equation to denote the emitted electron - that is, to help us balance the nuclear reactions. The daughter nuclide produced by beta decay (electron emission) has the same mass number A as the parent nuclide, but its atomic number Z has been increased by 1 because a neutron has been transformed into a proton. The energy liberated is carried off in the form of kinetic energy by the beta particle (electron) and the anti-neutrino, because the daughter nucleus produced is heavy enough that its recoil energy is small and can be neglected. A few examples of beta decay are as follows:

### Positron emission

Proton-rich nuclei may decay by emitting particles as an alternative to decay. A proton is converted to a neutron, leading to the emission of a high-energy positron () and a neutrino (v). The mass number A of the daughter nuclide is unchanged, but the atomic number Z has *decreased* by 1. The kinetic energy () is distributed between the positron and the neutrino. Examples of *positron emission are:*

### Electron capture

Electron capture is another process by which proton-rich nuclei decay, converting a proton to a neutron; it is an important alternative when energetic considerations don’t allow positron emission. The nucleus captures an orbital electron, thereby converting a proton into a neutron. The mass number is unchanged and the atomic number decreases by 1, as in positron emission, but the only particle emitted is a neutrino. An example is:

The three beta decay processes may be summarized by the flowing equations in which P represents the parent nucleus, D represents the daughter nucleus, and the rest of the symbols have their usual meanings:

# Mass-energy relationships

Let's now consider what the driving force for spontaneous nuclear decay is. For regular chemical reactions, their spontaneity depends on the change in Gibbs free energy where IF and only IF the change in Gibbs free energy is lower than zero (is negative: ) - then they are spontaneous. In nuclear chemistry, it turns out that the change in the Gibbs free energy for nuclear reactions is dominated by the enormous energy released, so our criterion for spontaneous nuclear decay becomes (they are spontaneous WHEN):

Now let's see if we can identify some characteristics of parent and daughter nuclei that allow us to predict which decay processes are spontaneous. Einstein showed the equivalence of mass and energy in his special theory of relativity as expressed by the famous relation:

There are two important consequences of this result. First, it predicts that matter and energy can be converted into one another and that the conversion of very small quantities of mass can produce very large quantities of energy. Second, the laws of conservation of mass and conservation of energy must be modified; it is only their sum that must be conserved. Equation 1 implies that there is a change in mass associated with the change in energy for any reaction, which we can calculate using:

Equation 2 implies that all exothermic reactions must be accompanied by a loss in mass, so our thermodynamic criterion for spontaneity can be rewritten in terms of mass as:

Let’s examine the mass changes associated with some simple nuclear reactions to see if this conclusion is true and to calculate the associated energy changes. Example of such nuclear reaction is the decay of a free neutron into a proton and an electron according to the following reaction:

Neutrons are stable inside the nucleus, but they decay with a half-life of about 10 minutes in free space. The mass of the neutron is 1.0086649 u, the mass of the proton is 1.007276 u, and the mass of the electron is 0.000548 u. We calculate the mass change associated with the decay of the neutron as:

Thus we get a confirmation that mass is indeed lost in this spontaneous nuclear transformation. The energy associated with the decay of the neutron is (we use equation 2 as shown above):

It is convenient to calculate changes in energy directly from
changes in mass. **We define an energy equivalent to 1 u as 931.494 MeV**, which allows
us to calculate the energy released from the decay of a neutron using:

# Energy changes in nuclear reactions

Now we shall calculate the energy changes associated with various nuclear decay processes discussed earlier, beginning with decay as represented by the following reaction:

And we know that nuclear reactions are spontaneous when , which can be expressed as:

because the mass of the anti-neutrino is almost zero. The masses in this inequality are those of the parent and daughter nuclei and the emitted electron, the daughter nucleus being positively charged when initially created. We can rewrite this equation in terms of atomic masses by adding Z electrons to both sides of the expression, the right-hand side now representing a positive ion and the emitted electron, which can combine to form a neutral daughter atom, **resulting in the following criterion for spontaneous decay**:

Thus, by comparing the atomic masses (see some of them in Table 1 - these masses are always given in some tables and on tests as well), then we can determine immediately whether a particular transformation can occur via or not. **The energy change associated with a particular reaction is calculated using:**

with the liberated energy being carried off in the form of kinetic energy of the lighter particles, the electron and the anti-neutrino. There are no restrictions on how the available energy is distributed between these two particles, so the kinetic energy of emitted electrons falls in a continuous range between . Thus, the maximum kinetic energy of a electron released in a decay of this kind is the maximum energy released.

Positron () emission is represented by the reaction: with the criterion for spontaneous decay being

in which the masses given are those of the bare nuclei. We convert this inequality to one expressed in terms of atomic masses, as before, by adding Z electrons to both sides of the expression. We now have Z+1 electrons on the right-hand side, in contrast to the expression for decay, however, so we get:

Mother nature has found an alternate way to convert protons into neutrons without paying the cost associated with positron emission. Electron capture is spontaneous when

a much less restrictive requirement than that for positron emission. This decay mode is an important alternative to positron emission for the heavier neutron-deficient nuclei for which the mass changes are not large enough to permit positron emission, for example, in the reaction:

# Kinetics of radioactive decay

The decay of any given unstable nucleus is a random event and is independent of the number of surrounding nuclei that have decayed. When the number of nuclei is large, we can be confident that during any given period a definite fraction of the original number of nuclei will have undergone a transformation into another nuclear species. In other words, the rate of decay of a collection of nuclei is proportional to the number of nuclei present, showing that nuclear decay follows a first-order rate equation.

## The integrated rate law

Is described by the equation:

where N is the number of nuclides „now“, N_{i} is the number of nuclides initially present at the time t = 0. e is a constant and the decay constant k is related to a half-life t_{1/2} through the equation:

Note that **half-life** is defined as the time required for the nuclei in a sample to decay to one-half their initial number. Table 2 lists the half-lives and decay modes of some unstable nuclides.

## The decay rate of radioactive nuclides

This decay rate — the average disintegration rate in numbers of nuclei per unit time is called the **activity A**, given by:

The S.I. unit of activity is the becquerel (Bq), defined as 1 radioactive disintegration per. second and thus has the unit s^{-1}.

Because the activity is proportional to the number of nuclei N, it also decays exponentially with time:

By rewriting the equations given above, we can solve them and arrange so that when we know the values of the activity (A) and the decay constant (k) – then we can calculate the number of nuclei (N) at that time, see:

# Summary

The identities of the elements are not preserved in nuclear reactions — elements decay into lighter daughter elements in fission reactions, and heavier elements are synthesized from lighter elements in fusion reactions. Mass changes in nuclear reactions are relatively small, but the accompanying energy changes are enormous; they are related by Einstein’s famous formula E = mc^{2}. The isotopes of the lighter elements (Z < 40 or so) are stable when the ratio of the number of neutrons to the number of protons (N/Z) is approximately equal to 1. Isotopes with N/Z > 1 will decay via positron emission or electron capture to increase the number of protons in the nucleus, whereas those with N/Z > 1 will decay via beta emission to decrease the number of protons in the nucleus. A fourth decay channel, alpha particle emission, becomes important for heavier nuclei. Radioactive decay follows first order kinetics, and the half-life t_{1/2} is a convenient measure of the timescale of the reaction.