# Chemistry - What happens to a radioactive carbon dioxide molecule when its carbon-14 atom decays?

## Solution 1:

An article by Snell and Pleasanton, 'The Atomic and Molecular Consequenses of Radioactive Decay', (J. Phys. Chem., 62 (11), pp 1377–1382, $$1958$$) supports Ben Norris's comment.

It is clear ... that $$\ce{^{14}CO2}$$ remains predominantly bound as $$\ce{NO2+}$$, a result that is perhaps not surprising. [This occurs in] $$81$$% of the decays. In $$\ce{^{14}CO2 -> NO2^+}$$ dissociation yielding $$\ce{NO+}$$, $$\ce{O+}$$ and $$\ce{N+}$$ follows [in], respectively, $$8.4$$, $$5.9$$, and $$3.6$$% of the decays.

A table summarising the results is given.

$$\begin{array}{|c|c|} \hline \mathbf{Ion} & \mathbf{\%\ abundance} \\ \hline \ce{NO2+} & 81.4(16) \\ \ce{NO+} & 8.4(4) \\ \ce{O+} & 5.9(6) \\ \ce{N+} & 3.6(4) \\ \ce{NO2^{2+}} & 0.40(06)\\ \hline \end{array}$$

## Solution 2:

The antineutrino is not completely irrelevant. The decay energy is shared between the beta particle and the neutrino. Therefore, the beta particles appear with an energy distribution that ranges from zero to the maximum beta energy. The maximum beta energy for the decay of $\ce{^14C}$ is $E_\text{max}=0.1565\ \mathrm{MeV}$; the average beta energy is $E_\text{avg}=0.0495\ \mathrm{MeV}$.

Nevertheless, even beta particles with a relatively low beta energy have a much larger kinetic energy than any chemical bond dissociation energy or ionization energy. Thus, the beta particle cannot be captured in the electron shell of the affected atom. The beta particle leaves the atom with almost relativistic speed before the rest of the molecule understands what went wrong. This process is too fast for immediate chemical reactions. The daughter nuclide remains in the same chemical structure. Thus, the beta decay of a $\ce{^14C}$ atom in $\ce{CO2}$ simply leaves a $\ce{NO2+}$ ion in an excited state. This primary product is unstable and can dissociate slightly later. The resulting secondary products can be radicals or ions, which tend to react with almost any other molecule or atom.

However, the energy released in beta decay is not only distributed between the neutrino and the beta particle. A small amount is also allocated to the recoil of the nucleus. Since the mass of the nucleus is much larger than the mass of the beta particle, the recoil energy is much smaller than the energy of the neutrino and the beta particle and can usually be neglected in a first approximation.

For the beta decay of $\ce{^14C}$, the maximum recoil energy of the $\ce{^14N}$ nucleus can be calculated as $E_\text{recoil, max}=7.08\ \mathrm{eV}$. This value is much smaller than the total decay energy of $0.1565\ \mathrm{MeV}$ and could thus be neglected, but it is larger than the energy of a chemical bond. By way of comparison, the bond dissociation energy in $\ce{NO2}$ is only about $306\ \mathrm{kJ/mol}$ or $3.17\ \mathrm{eV}$. Therefore, the recoil of the $\ce{^14N}$ nucleus can be sufficient to break a bond immediately during the beta decay of $\ce{^14C}$.

## Solution 3:

There seem to be three processes to consider here:

1. As the high-energy electron exits, it creates a rapidly varying electrical current, which produces intense electromagnetic fields for a short time. This may act on the other charged particles that are present, possibly creating further ionization and/or breaking the chemical bond.
2. The nitrogen nucleus recoils, and this recoil could break the bond and/or create further ionization.
3. The charge of the nucleus changes, so even if effects 1 and 2 didn't exist, the state of the electrons is no longer the ground state. It is now a linear combination of different states of the new element, which means there is some probability of excitation or ionization.

A paper by Oksyuk and Gerasimenko says that effect 2 is normally the important one.

From Linear Christmas's answer, we know that in the case of $\ce{^14C}$ decay, which has an unusually low energy, there is a fairly small but still appreciable probability of breaking the bond, and this is presumably caused by one or more of the above processes.

For process 1, it certainly isn't safe to assume that it's negligible because the electron leaves the molecule so quickly. If this were true, then beta particles wouldn't produce ionization when they came along from the outside and hit atoms. Although the time-scale for the electron to exit is short, its electromagnetic fields are intense. By the way, the motion of this electron is not insanely fast compared to the velocities of the other electrons. It has a typical velocity in this decay of about $0.5c$, which could be compared to an estimate of $Zc/137\approx 0.04c$ for an inner-shell electron in carbon.

To estimate process 1, let's use the mean rate of energy loss for beta particles in a solid. For $\pu{0.1 MeV}$ betas in a solid made of fairly light elements, this is about $(dE/dx)/\rho\approx 0.3$ MeV.m2/kg. Taking $\rho$ to be the density of water, and $\Delta x=0.2$ nm, we find $\Delta E=0.06$ eV, which seems to be a couple of orders of magnitude too low to break a bond. However, energy loss of betas is a process that has a lot of random variation about the mean, so it doesn't seem unreasonable to me to imagine that there is something like a 1% chance that it deposits 100 times this energy in the parent atom on its way out. This puts us in the right ballpark for this mechanism to contribute significantly to the observed probability of breaking a bond in NO.

So now let's consider process 2. Let $Q$ be the energy released in the decay and $M$ the mass of the recoiling nitrogen nucleus. We'll see that for kinematical reasons, almost all of the energy goes into the electron and antineutrino, not the nucleus. Suppose we want to find the maximum energy of the recoiling nucleus. This is attained in the case where the electron gets almost 100% of the energy, because for a fixed energy, a more massive particle carries more momentum. If the electron and neutrino were to share the energy, then their momentum vectors could also partially cancel, further reducing the recoil.

The decay energy of $\ce{^14C}$ is unusually low, but in most beta decays the beta is much more relativistic. Let's do the ultrarelativistic case first, both because the math is simpler and because it's a better guide to our intuition about what happens in general.

In the approximation that the beta is ultrarelativistic, its momentum (in the case where it carries all the energy in this example) is $p\approx Q/c$, and by conservation of momentum, this is also the momentum of the recoiling nucleus. Since the nucleus is nonrelativistic, its kinetic energy is $K=p^2/2M\approx Q^2/2Mc^2$.

As a typical example, let's take $\ce{^40K}$, which is the strongest source of naturally occurring beta radioactivity in our environment. It has an 89% probability of decaying to $\ce{^40Ca}$ plus an electron and an antineutrino. The energy for this decay mode is $\pu{1.33 MeV}$, which is almost $10$ times that of $\ce{^14C}$. The ultrarelativistic approximation for the electron is not too ridiculous; in the case where it gets almost all the energy (none to the neutrino), its velocity is about $0.96c$. The maximum kinetic energy of the recoiling calcium nucleus in this approximation is about $\pu{24 eV}$, which is clearly plenty of energy to break a chemical bond.

Without the ultrarelativistic approximation, the momentum of the beta, in the maximum-recoil case, isn't $Q/c$ but rather $\sqrt{(x+Q)^2-x^2}/c$, where $x=mc^2$, and $m$ is the mass of the electron. Even in the case of $\ce{^40K}$, it turns out that the ultrarelativistic approximation is not so great. The actual maximum recoil energy for the calcium is $\pu{41.7 eV}$, so although the approximation gives the right order of magnitude, it's off by almost a factor of $2$.

In the example of $\ce{^14C}$, the result for the maximum energy of the recoiling nitrogen is $\pu{7.0 eV}$, in agreement with Loong's answer. This is plenty of energy to break the bond. Another interesting case with a very low energy is $\ce{^3H}$ , in which the recoiling $\ce{^3He}$ has a maximum energy of only about $\pu{3 eV}$. People have used this to try to study the chemistry of helium.

So the take-away here is that in almost all cases, beta decay is very likely to break up the molecule in which it occurs.