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Directions

This simulation models a fission-type chain reaction in which a neutron strikes a nucleus of isotope A, splitting it into isotopes B and C and releasing X additional neutrons:

A + ¹n → B + C + X·¹n

Click Example (top right) to switch to a specific, proton + neutron-balanced reaction, ²³⁵U + ¹n → ¹⁴⁴Xe + ⁸⁹Sr + 3·¹n. Note that X = 3 is the only value of X that balances the reaction. Click Example again to return to the general A/B/C reaction, where X is adjustable.

Drag the X slider (0–3) to set how many neutrons each fission event releases, and the k slider (1–10) to set the fission rate constant. The plot recalculates immediately, starting from 1000 nuclei of the parent isotope and 100 free neutrons.

Watch how X controls whether the reaction dies out, holds steady, or accelerates:

• X = 0: no replacement neutrons — the starting neutrons are consumed quickly and the reaction stops (sub-critical).
• X = 1: each fission replaces the neutron it consumes — the neutron population stays constant and the parent isotope decays smoothly (critical).
• X = 2 or 3: each fission produces a net surplus of neutrons, so the neutron population grows and the reaction accelerates until the parent isotope is depleted (super-critical).

Hover over the plot to read exact values at any time. Click Details for the governing equations.

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Details

Let N_A, N_B, N_C, and N_n be the numbers of parent nuclei (A), the two product nuclei (B and C), and free neutrons. Each fission event consumes one A and one neutron, and produces one B, one C, and X neutrons, at a rate r = k·N_A·N_n:

dN_A/dt = −k·N_A·N_n

dN_B/dt = k·N_A·N_n

dN_C/dt = k·N_A·N_n

dN_n/dt = −k·N_A·N_n + X·k·N_A·N_n = (X−1)·k·N_A·N_n

Because dN_B/dt and dN_C/dt are both equal to −dN_A/dt, and N_B(0) = N_C(0) = 0, it follows that N_B(t) = N_C(t) = N_A(0) − N_A(t) at every instant.

The equations are solved numerically (4th-order Runge–Kutta) from t = 0, with N_A(0) = 1000, N_n(0) = 100, N_B(0) = 0, and N_C(0) = 0, using the current slider values of X and k.

Click Example to see these same equations applied with A = ²³⁵U, B = ¹⁴⁴Xe, and C = ⁸⁹Sr. In that view X is fixed at 3 (the only mass-balanced choice), and ⁸⁹Sr (= N_C) is also plotted alongside ¹⁴⁴Xe (= N_B) for direct comparison with the reaction given in the original problem statement.

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About

This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.