Spontaneous fission rate for sample of U-235 (1mg/cm^2)

Goal: Calculate the fission rate of the bomb grade [math]\displaystyle{ U^{235} }[/math] that has a mass of [math]\displaystyle{ .91*10^{-3} }[/math] Procedure:

 Givens- [math]\displaystyle{  T_{1/2} = 7.038 \cdot 10^{8} year \times 3.1536 \cdot 10^{7} sec = 2.2195 \cdot10^{16} sec }[/math]

         [math]\displaystyle{  m = .91 \cdot 10^{-3} g }[/math]

 Calculations- [math]\displaystyle{ \lambda = \frac{\ln{2}}{T_{1/2}} = \frac{\ln{2}}{2.2195 \cdot 10^{16} sec} = 3.12 \cdot 10^{-17} sec^{-1} }[/math]

               [math]\displaystyle{ N = \frac{N_a}{A} m = \frac{6.022 \cdot 10^{23} \frac{nuclei}{mol}}{235 \frac{g}{mol}} .91 \cdot 10^{-3} g = 2.33 \cdot 10^{18} nuclei }[/math]

               [math]\displaystyle{ R = \lambda N = 3.12 \cdot 10^{-17} sec^{-1} \times 2.33 \cdot 10^{18} nuclei = 72.70  fissions/sec }[/math]

 Probability of fission per decay = [math]\displaystyle{ 2.0 \cdot 10^{-7} % => 2.0 \cdot 10^{-9} \times 72.70 fissions/sec }[/math]

                [math]\displaystyle{ = 1.5 \cdot 10^{-5}% }[/math] chance there will be a fission each second [math]\displaystyle{ => \textbf{not probable} }[/math]