Nuclear Physics/Change of isotope composition of natural uranium

Natural uranium consists of three isotopes. These are uranium234, uranium235 and Uranium238. Hereby is uranium234 a decay product of uranium238, whose concentration is in a fix relation to uranium238. The concentration of uranium235 is independent of the content of uranium238. As uranium238 and uranium235 decay with different speed, the isotope composition of natural uranium changes by time.

This has drastical consequences:

2 billion years ago in Oklo in today's Gabun, there was a natural reactor moderated by normal water. At those days natural uranium consisted of 3% uranium 235, this is the same concentration as in nuclear fuel for boiling water reactors in power plants.

Today natural uranium has only a uranium 235 concentration of 0.72 %. A chain reaction in normal water is impossible. Usage of heavy water or enrichment of uranium 235 is required.

Should there exist in a few billion years, before the sun gets a red giant, intelligent life in our solar system, it will have great problems to exploit nuclear fission, as concentration of uranium235 will be very low.

The development of isotope composition can be calculated as described below:

Lot of uranium235 in dependence of time t

$p(Uranium235)*0.5^{\frac {t}{HalfLifeTimeU235}}$

Lot of uranium238 in dependence of time t

$p(Uranium238)*0.5^{\frac {t}{HalfLifeTimeU238}}$

Lot of uranium234 in dependence of time t

$p(Uranium238)*{\frac {HalfLifeTimeU234}{HalfLifeTimeU238}}*(0.5^{\frac {t}{HalfLifeTimeU238}})$

The following table contains the values for frequency and half-life time of natural uranium isotopes

Isotope	Frequency	Half Life Time
Uranium238 (U238)	99.27390 %	4510000000 years
Uranium235 (U235)	0.7205 %	710000000 years
Uranium234 (U234)	0.0056 %	254000 years

Concentration of uranium235 at time t

$p(Uranium235)(t)={\frac {p(Uranium235)*0.5^{\frac {t}{HalfLifeTimeU235}}}{p(Uranium235)*0.5^{\frac {t}{HalfLifeTimeU235}}+p(Uranium238)*0.5^{\frac {t}{HalfLifeTimeU238}}+p(Uranium238)*{\frac {HalfLifeTimeU234}{HalfLifeTimeU238}}*(0.5^{\frac {t}{HalfLifeTimeU238}})}}$

Concentration of uranium238 at time t

$p(Uranium238)(t)={\frac {p(Uranium238)*0.5^{\frac {t}{HalfLifeTimeU238}}}{p(Uranium235)*0.5^{\frac {t}{HalfLifeTimeU235}}+p(Uranium238)*0.5^{\frac {t}{HalfLifeTimeU238}}+p(Uranium238)*{\frac {HalfLifeTimeU234}{HalfLifeTimeU238}}*(0.5^{\frac {t}{HalfLifeTimeU238}})}}$

Concentration of uranium234 at time t

$p(Uranium234)(t)={\frac {p(Uranium238)*{\frac {HalfLifeTimeU234}{HalfLifeTimeU238}}*(0.5^{\frac {t}{HalfLifeTimeU238}})}{p(Uranium235)*0.5^{\frac {t}{HalfLifeTimeU235}}+p(Uranium238)*0.5^{\frac {t}{HalfLifeTimeU238}}+p(Uranium238)*{\frac {HalfLifeTimeU234}{HalfLifeTimeU238}}*(0.5^{\frac {t}{HalfLifeTimeU238}})}}$

Composition of natural uranium as function of time

Time (Million years)	Uranium 238	Uranium 235	Uranium 234
-5000	69.26852 %	30.72757 %	0.00391 %
-4500	77.27535 %	22.72029 %	0.00436 %
-4000	83.68689 %	16.30839 %	0.00472 %
-3500	88.55684 %	11.43817 %	0.00500 %
-3000	92.10948 %	7.88532 %	0.00520 %
-2500	94.62553 %	5.36913 %	0.00534 %
-2000	96.37030 %	3.62426 %	0.00544 %
-1500	97.56263 %	2.43187 %	0.00550 %
-1000	98.36929 %	1.62516 %	0.00555 %
-500	98.91133 %	1.08309 %	0.00558 %
today	99.27390 %	0.72050 %	0.00560 %
500	99.51568 %	0.47871 %	0.00561 %
1000	99.67658 %	0.31780 %	0.00562 %
1500	99.78351 %	0.21086 %	0.00563 %
2000	99.85451 %	0.13986 %	0.00563 %
2500	99.90162 %	0.09274 %	0.00564 %
3000	99.93287 %	0.06149 %	0.00564 %
3500	99.95360 %	0.04076 %	0.00564 %
4000	99.96734 %	0.02702 %	0.00564 %
4500	99.97645 %	0.01791 %	0.00564 %
5000	99.98249 %	0.01187 %	0.00564 %