Nuclear Fuel Cycle

Q: Calculate the specific activity (Bq/g) of UO$_2$ fuel enriched to 4.5 wt% $^{235}$U. The U-234 weight fraction at this enrichment is approximately 0.035%. State which isotope dominates and by what factor.

Data: U-234 $T_{1/2}$ = 2.46 $\times$ 10$^5$ years; U-235 $T_{1/2}$ = 7.04 $\times$ 10$^8$ years; U-238 $T_{1/2}$ = 4.47 $\times$ 10$^9$ years; $N_A$ = 6.022 $\times$ 10$^{23}$; 1 year = 3.156 $\times$ 10$^7$ s; Mass of U in UO$_2$ = 238/270 = 0.8815.

A: Work through the same method as Tutorial Question 1 but with enriched compositions:

• Mass of U in 1 g UO$_2$ = 0.8815 g
• U-234: 0.8815 $\times$ 0.00035 = 3.09 $\times$ 10$^{-4}$ g $\rightarrow$ $\lambda$ = 0.693/(2.46 $\times$ 10$^5$ $\times$ 3.156 $\times$ 10$^7$) = 8.93 $\times$ 10$^{-14}$ s$^{-1}$ $\rightarrow$ $N$ = 3.09 $\times$ 10$^{-4}$ $\times$ 6.022 $\times$ 10$^{23}$/234 = 7.95 $\times$ 10$^{17}$ $\rightarrow$ $A_{234}$ = 7.10 $\times$ 10$^4$ Bq
• U-235: 0.8815 $\times$ 0.045 = 3.97 $\times$ 10$^{-2}$ g $\rightarrow$ $A_{235}$ = 3.17 $\times$ 10$^3$ Bq
• U-238: 0.8815 $\times$ 0.9547 = 8.415 $\times$ 10$^{-1}$ g $\rightarrow$ $A_{238}$ = 1.04 $\times$ 10$^4$ Bq
• Total $\approx$ 8.4 $\times$ 10$^4$ Bq/g. U-234 dominates (~85%), despite being only 0.035% by mass, because its half-life is ~18,000 times shorter than U-238.