Thermonuclear Fusion: The Basic Process — Question 8

Question 8

Fuel depletion makes the cumulative fusion yield sublinear in time

In a fixed volume with constant temperature, equal reactant densities start at $n_D=n_T=n_0$. Their common density obeys $dn/dt=-Kn^2$, with no replenishment. Let $Kn_0=.200\,\mathrm{s^{-1}}$.

1. Solve for n(t) and the fraction of either initial species consumed.

2. Find that fraction after 5.00 s and compare the cumulative event count with the constant-initial-rate estimate.

Question 8 - Solution

Fuel depletion makes the cumulative fusion yield sublinear in time

(a) Method and calculation

$$\int_{n_0}^{n}\frac{dn\prime}{n\prime{}^2}=-Kt\Rightarrow\boxed{n(t)=\frac{n_0}{1+Kn_0t}}.$$$$\boxed{f_{\mathrm{consumed}}=1-n/n_0=\frac{Kn_0t}{1+Kn_0t}}.$$

(b) Interpretation and further calculation

$$\boxed{f(5)=1/2,\quad N_{\mathrm{events}}/V=n_0-n=n_0/2}.$$$$R_0t=Kn_0^2t=n_0\Rightarrow\boxed{\text{initial-rate estimate is twice the actual yield}}.$$

Physical check

One atom of each species is consumed per event. Constant reactivity is imposed; temperature evolution is outside this depletion model.