Question

a.) A neutron can cause ²³⁵U to fission, producing two daughter nuclei (tin and molybdenum) and three more neutrons. These three neutrons can then, in turn, cause three more ²³⁵U nuclei to split. One typical reaction is summarized as follows:

n + ²³⁵U ® ¹³¹Sn + ¹⁰²Mo + 3 ¹n

Find the energy released in this process (in MeV), given that M(²³⁵U)=235.0439 u; M(¹³¹Sn)=130.9169 u; M(¹⁰²Mo)=101.9103 u.

b.)Lise Meitner’s work in 1938 predicted that the fission of ²³⁵U would produce 200 MeV of energy per fission event. How did she do? Find a %-error.

c.)¹³¹Sn is radioactive, and typical undergoes three successive b^– decays. What is the final product?

d.)How many nuclei are in 1 kg of ²³⁵U? This is, incidentally, how many fission events occur if 1 kg of ²³⁵U were to fission.

e.)In each “generation” of a chain reaction of ²³⁵U, the number of events is three times the number in the previous generation. Hence events go as 1® 3® 9® 27® 81® 243 etc. If n tells you which generation, find the limit of the ratio of (events in n^th generation)/(total # of events) as n ® large. (You don’t need to use calculus, just use your calculator and notice the pattern. The series starts 1, 3/4, 9/13, etc.)

f.)Given these results, estimate the number n of chain reaction “generations” that would occur if 1 kg of ²³⁵U were to fission.

g.)If each “generation” lasts on the order of 10^–7 seconds, estimate the time for the entire fission chain reaction to occur.

Answer 1

a) Energy released E = (M(235U) - M(131Sn) - M(102Mo) - 2 x M(n))

=>E = (235.0439 - 130.9169 - 101.9103 - 2 x 1.008664) 931.5 MeV = 185.72 MeV

b) %age error = (Exact value - approximate value) x 100 / Exact value = (185.72 - 200) x 100 / 185.72 = 7.69 %

The information about the accurate mass of U, Sn and Mo lead to the error in the estimation of released energy.

c) ¹³¹Sn - -> ¹³¹Sb - -> ¹³¹Te - -> ¹³¹I

Hence the final product will be ¹³¹I.

d) One nuclei of ²³⁵U weighs 235.0439 u = 235.0439 x 1.66 x 10^-27 kg =

Number of nuclei in 1 kg = 1 / (235.0439 x 1.66 x 10^-27 ) = 2.56 x 10²⁴ nuclei.