mass defect - total mass loss of one pellet

Question

I have a task where there is a uranium pellet (mass=10g) with 3.5% U-235 and 96.5% U-238. Now I should calculate how much mass the pellet loses. I shall only consider the decay of U-235 and consider that 0.27g of U-235 decayed.

U-235 gets split according to following formula: U-235 + neutron -> 100Zr + 133Te + 3*neutrons

I now calculate the number of uranium atoms in those 0.27g N=0.00027kg/m(U-235). Since the free neutrons decay on their own and should not contribute to the mass of the pellet I assume that the mass loss per split is Δm = m(U-235) - m(Zr-100) - m(Te-133). Now I multiply that with the number of decays I calculated earlier and I should get the mass loss of the pellet. However, it seems as if the proposed solution of the tasks wants me to simply use the normal mass defect equation (using the neutrons). My teacher explained it confusingly, something about "since I also subtract the neutrons from the Uranium mass they are considered to be gone (what I wanted to achieve with my approach)". So it seems as if the correct mass loss would be Δm = m(U-235) - m(Zr-100) - m(Te-133) - 2*m(neutron).

I don't understand that, since that mass difference Δm would be tinier as the one I calculated earlier. And shouldn't it be bigger since neutrons are missing? Or am I overthinking this and ignore that neutrons decay - since they decay into a proton, which effectively has almost the same mass as a neutron, so their mass still contributes to the mass of the pellet?

Answer 1

It sounds like you’re doing a calculation like this one, where you are computing the mass lost due to the difference in binding energy between the uranium nucleus and its daughter products. However, you have made the clever and complicating observation that the neutrons don’t hang around in the fuel pellet: the neutrons’ mean free path is longer than the pellet. In a working reactor the fuel pellets are embedded in some “moderator” (often water) whose purpose is to exchange heat with the neutrons, slowing them down so that they are more likely to engage in capture reactions, and this matrix is shot through with “control rods” which contain some neutron absorber or “neutron poison” (often boron) so that the reaction does not run away.

A fission event releases three-ish very fast neutrons, each of which instantly leaves the fuel pellet. Some of the neutrons will scatter back into the pellet and trigger another fission; some of them will scatter back into the pellet and participate in a non-fission capture reaction; some of them will scatter into the control rods and capture there. (The half-life for free neutron decay is about fifteen minutes; nearly every neutron in the reactor will capture on some other nucleus instead of decaying to a free proton.) The number of neutrons per fission which trigger a new fission is called $k_\text{effective}$. Reactor operators modulate the amount of neutron poison to maintain $k_\text{eff} \approx 1$, so that the rate of fission is stable rather than growing or shrinking. A device with $k_\text{eff} \approx 2$ is a bomb; a device with $k_\text{eff} \approx \frac12$ is shutting down.

So you’ve identified two related issues here, both of which have to be resolved to know how much the pellet’s mass changes:

1. The binding energy issue: each fission releases something like 200 MeV of energy, which is equivalent to a mass change thanks to $E=mc^2$.

2. An actual issue of mass flow: neutrons leave the fuel pellet and end up in the control rods. (If I cut a wooden plank in half, the mass of the halves doesn’t add up to the mass of the whole; I have to weigh the sawdust.)

I’m a little murky on which approach is yours and which is your teacher’s. The discovery of point #1 is one of the major events in 20th-century physics, and is enormously important, so most homework assignments stop there for pedagogical purposes. But if your job were actually to weigh a fuel pellet, #2 is a bigger effect by a factor of ten. (The neutron’s mass is about $1000\,\mathrm{MeV}/c^2$.)

If you were my student, I’d tell you to just do the binding-energy calculation; we might call it the mass loss, per fuel pellet, of the entire reactor assembly.