Nuclear Reactor Project: When a uranium atom inside the reactor of a nuclear power plant

Chapter 9, Problem C.1

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Nuclear Reactor Project: When a uranium atom inside the reactor of a nuclear power plant is hit by a neutron, it splits (fissions), releasing energy and some new neutrons (Figure 9-9b). The mathematically expected number of new neutrons per fission is 2.3. Figure 9-9b a. Suppose there are 100 neutrons in the reactor initially. If all of these neutrons cause fissions, how many neutrons would you expect there to be after this first generation of fissions? b. If all of the neutrons from the first generation cause fissions, how many would you expect after two generations? After three generations? After four generations? What kind of sequence do these numbers form? c. If each generation takes 0.001 sec, how many neutrons would you expect there to be after 1 sec? Does this answer surprise you? This is what makes atomic bombs explode! d. Not all of the neutrons from one generation actually do cause fissions in the next generation. Some leak out of the reactor, some are captured by atoms other than uranium, and some that are captured by uranium atoms do not cause fission. Assume that P(leaking) = 0.36 P(capture by other atom) = 0.2 P(nonfission capture) = 0.15 Calculate the probability that none of these things happens and thus that the neutron does cause a fission in the next generation. e. Use the probability in part d and the fact that there are 2.3 new neutrons per fission to calculate k, the expected number of new neutrons in the second generation caused by one neutron in the first generation. f. How many neutrons would you expect there to be after 1 sec under the conditions in part e if each generation still takes 0.001 sec as in part c? Would the reactor explode like a bomb? g. Why can you say that the number of neutrons is increasing exponentially with time? h. The constant k in part e is called the multiplication factor. The chain reaction in a nuclear reactor is controlled by moving control rods out or in to absorb fewer or more neutrons. If k is slightly more than 1, the power level increases. If k is less than 1, the power level decreases. If k = 1, the power level remains constant and the reactor is said to be critical. What would P(capture by other atom), mentioned in part d, have to equal to make the reactor critical?