KAHIBARO
Table of Contents
Energy Balance in Nuclear Reactions
Reaction energetics studies how energy changes during a nuclear reaction. A nuclear reaction can absorb energy or release energy, depending on the masses of the particles before and after the reaction. The key idea is that mass and energy are related. A small change in mass can correspond to a large change in energy.
A general nuclear reaction may be written as
$$
a + A \rightarrow b + B
$$
where $a$ is the projectile, $A$ is the target nucleus, and $b$ and $B$ are the reaction products. To understand the energetics of this reaction, we compare the total mass before and after the reaction.
Mass and Energy
Einstein’s relation connects mass and energy,
$$
E = mc^2
$$
In nuclear physics, this means that if the total rest mass of the initial particles is greater than the total rest mass of the final particles, the reaction can release energy. If the final mass is greater, energy must be supplied.
The energy available from the mass difference is called the reaction $Q$ value. It is defined by
$$
Q = \left(m_{\text{initial}} - m_{\text{final}}\right)c^2
$$
For the reaction
$$
a + A \rightarrow b + B
$$
the formula becomes
$$
Q = \left(m_a + m_A - m_b - m_B\right)c^2
$$
Here, all masses must be expressed consistently, either as nuclear masses or as atomic masses, with proper care about electrons.
The central rule of reaction energetics is
$$
Q = \left(m_{\text{initial}} - m_{\text{final}}\right)c^2
$$
If $Q > 0$, energy is released.
If $Q < 0$, energy must be supplied.
Exothermic and Endothermic Reactions
If $Q$ is positive, the reaction is called exothermic. The products have less rest mass than the reactants, and the missing mass appears as kinetic energy or radiation.
If $Q$ is negative, the reaction is called endothermic. In this case, the reaction cannot happen unless enough initial kinetic energy is provided.
The sign of $Q$ gives immediate physical meaning:
| Reaction type | Condition | Meaning |
|---|---|---|
| Exothermic | $Q > 0$ | Energy released |
| Endothermic | $Q < 0$ | Energy required |
| Neutral | $Q = 0$ | No net rest-mass energy change |
Where the Energy Goes
The reaction energy does not disappear. It is shared among the products in forms such as kinetic energy and electromagnetic radiation. In many reactions, the outgoing particles move away with some kinetic energy. Sometimes a product nucleus is formed in an excited state, and part of the energy goes into nuclear excitation rather than motion.
If the reaction produces gamma radiation, some of the released energy may be carried by photons.
So, in a simple energy balance,
$$
\text{initial rest energy} + \text{initial kinetic energy}
=
\text{final rest energy} + \text{final kinetic energy}
$$
Using the $Q$ value, this can be written as
$$
K_{\text{final}} = K_{\text{initial}} + Q
$$
for a reaction with no other stored internal excitation energy. If excitation energy is present, that must also be included in the balance.
Energy conservation in a nuclear reaction requires that the total energy before and after the reaction be equal.
A useful form is
$$
K_{\text{final}} = K_{\text{initial}} + Q
$$
A positive $Q$ increases the final kinetic energy available.
Using Atomic Masses
In many practical calculations, atomic masses are used instead of bare nuclear masses. This works well when the number of electrons is the same on both sides of the reaction, because the electron masses cancel.
For example, in
$$
^{14}\mathrm{N} + \alpha \rightarrow ^{17}\mathrm{O} + p
$$
the total electronic contribution can be handled consistently if atomic masses are used carefully.
In nuclear physics, masses are often given in atomic mass units, abbreviated as u. The conversion between mass and energy is
$$
1\,\mathrm{u}\,c^2 \approx 931.5\,\mathrm{MeV}
$$
So if the mass difference is $\Delta m$ in atomic mass units, then
$$
Q = \Delta m \times 931.5\,\mathrm{MeV}
$$
Example of a Q-Value Calculation
Consider the reaction
$$
^{6}\mathrm{Li} + ^{2}\mathrm{H} \rightarrow 2\,^{4}\mathrm{He}
$$
Suppose the total initial mass is slightly greater than the total final mass by $\Delta m$. Then
$$
Q = \Delta m \, c^2
$$
If, for example,
$$
\Delta m = 0.024\,\mathrm{u}
$$
then
$$
Q = 0.024 \times 931.5 \,\mathrm{MeV} \approx 22.4\,\mathrm{MeV}
$$
This positive value means the reaction releases about $22.4\,\mathrm{MeV}$ of energy.
Excited States and Effective Reaction Energy
Sometimes the final nucleus is not produced in its lowest energy state. Instead, it is left excited. Then some of the reaction energy goes into internal nuclear energy.
If the excitation energy is $E^*$, then the effective kinetic energy balance becomes
$$
K_{\text{final}} = K_{\text{initial}} + Q - E^*
$$
This means that even if $Q$ is positive, less kinetic energy may appear in the products because part of the energy is stored inside the nucleus.
Threshold for Endothermic Reactions
For an endothermic reaction, the projectile must have a minimum kinetic energy for the reaction to occur. This minimum is called the threshold energy. A full treatment of threshold energy belongs to a separate topic, but the basic idea follows directly from energetics. Since $Q < 0$, the initial kinetic energy must at least compensate for the negative $Q$ and satisfy momentum conservation as well.
So, negative $Q$ means that simply having enough energy in principle is not quite the whole story. The motion of all particles must also be consistent with conservation laws.
Energy Diagram
A simple way to picture reaction energetics is to compare the total energy levels of reactants and products.
In the upper case, the final state lies lower in energy, so energy is released. In the lower case, the final state lies higher, so energy must be supplied.
Practical Importance
Reaction energetics tells us whether a nuclear reaction is energetically allowed and how much energy can be released or must be provided. This is important in nuclear power, fusion, particle production, and laboratory experiments. Before studying how likely a reaction is, one must first know whether the reaction is energetically possible.
Reaction energetics answers two basic questions.
First, is the reaction energetically allowed?
Second, how much energy is released or required?
These are determined by the mass difference through
$$
Q = \left(m_{\text{initial}} - m_{\text{final}}\right)c^2
$$
Summary
The energetics of a nuclear reaction comes from comparing the total rest mass before and after the reaction. The difference defines the $Q$ value. Positive $Q$ means the reaction releases energy, and negative $Q$ means energy must be supplied. The released or required energy affects the kinetic energy of the products and may also appear as excitation energy or radiation. This simple energy balance is one of the first and most important checks for any nuclear reaction.