# 18.3.1 Nuclear Reactions

Nuclear transmutation is the conversion of one chemical element or isotope into another, either by nuclear reactions (which requires an external trigger) or by radioactive decay (which are random and spontaneous).

While chemical reactions involve the rearrangement of atoms (to form new compounds), nuclear reactions involve the rearrangement of nucleons (to form new nuclei). While chemical reactions are represented symbolically by chemical equations, nuclear reactions are represented by nuclear equations.

Take a look at this one,

$\displaystyle \displaystyle {}_{2}^{4}He+{}_{7}^{{14}}N\to {}_{8}^{{17}}O+{}_{1}^{1}H$

where alpha-particles (very energetic He-4 nuclei) are smashed against nitrogen nuclei (N-14) to produce oxygen (O-17) and hydrogen nuclei (He-1, or protons)[1].

Let’s do some calculations using the following given data:

Mass of $\displaystyle {}_{2}^{4}He=4.002\text{ }602u$                        Mass of $\displaystyle \displaystyle {}_{8}^{{17}}O=16.999\text{ }133\text{ }u$

Mass of $\displaystyle {}_{7}^{{14}}N=14.003\text{ }074u$                        Mass of $\displaystyle {}_{1}^{1}H=1.007\text{ }825\text{ }u$

Mass of reactant nuclei before the reaction            $\displaystyle \sum {{m}_{i}}=4.002602u+14.003074u=18.005676u$

Mass of product nuclei after the reaction            $\displaystyle \sum {{m}_{f}}=16.999133u+1.007825u=18.006958u$

Apparently, there is an increase in mass after the reaction.

\displaystyle \begin{aligned}\Delta m&=\sum {{m}_{f}}-\sum {{m}_{i}}\\&=18.006958u-18.005676u\\&=0.001282u\end{aligned}

By the mass-energy equivalence principle, the increase in mass implies that energy is absorbed by the system during the reaction. And the amount of energy absorbed is[2]

\displaystyle \begin{aligned}\Delta E&=\Delta m.{{c}^{2}}\\&=(0.001282)(1.66\times {{10}^{{-27}}}).{{(3.00\times {{10}^{8}})}^{2}}\\&=1.915\times {{10}^{{-13}}}\text{ J}\\&=1.20\text{ MeV}\end{aligned}

Where does this energy come from? The KE of the alpha particles! This reaction actually requires the alpha particles to have at least 1.20 MeV of KE, (since the nitrogen atoms have practically zero KE). This 1.20 MeV is required to re-arrange the reactant nuclei’s nucleons to form the product nuclei, since the product nuclei have more rest-mass energy than the reactant nuclei (analogous to how energy is required to reassemble short Jenga blocks into taller ones). This explains why this reaction does not occur spontaneously in our atmosphere. When ordinary helium atoms encounter nitrogen atoms in the atmosphere, there is simply not enough KE to “fund” the nuclear reaction.

If you’re very vigilant, you will realize that 1.20 MeV is actually not enough. Why?

Notice that the momentum of the system before the reaction is non-zero (due to the momentum of the alpha particles). By PCOM, the momentum should remain non-zero after the reaction. This means that the oxygen and hydrogen nuclei must be “born” with some KE, so to speak. The alpha particles must therefore provide for the KE of the reactants in addition to the 1.20 MeV required to re-arrange the nucleons. In other words, the alpha-particle must have KE of at least 1.20 MeV + X, where X represents the KE of the reactants. Get it?

Concept Test

[1] Historically, this was the nuclear reaction that led to the discovery of the proton.

[2] Note that Dm here does not denote the mass defect (of any specific nuclide). It merely denotes the change in mass brought about by the nuclear reaction.