A particular nuclear chain reaction is initiated when a neutron colliding with a

Question

A particular nuclear chain reaction is initiated when a neutron colliding with a particular nucleus causes the nucles to fission into fragments along with the release of two neutrons. These two neutrons go on to trigger two more fission events each of which releases two neutrons which can go on to trigger additional fission events and so on.

1) If the first event fissions 1 nucleus, and the second event fissions 2 nuclei, and the third fissions 4 nuclei, etc., how many nuclei fission during the 5 th event?

Nneutrons =

2)After the 5 th event, how many nuclei have been fissioned since the start of the reaction?

Nnuclei =

3)

In another chain reaction, each nucleus that fissions releases three neutrons (instead of two). Again, let the first event fission one nucleus. (Then the second event triggers three fissions, the third event triggers nine fissions, and so on.) How many nuclei fission during the 5 th event in this case?

Nneutrons =

4)After the 6 th event, how many nuclei have been fissioned since the start of the reaction?

Nnuclei =

Please, show work

Explanation / Answer

Each fission involves breaking of a nucleus into 2 other nuclei and release of 2 neutrons.

N = No. of nuclei fissoned by neutrons

n = no. of the event.

So,

1st event fissions N=1 nuclei

2nd fissions N = 2

3rd fissions N = 4

We can write this as : n = 1 N = 2, n = 2 N = 2, n=3 N=4....

in general : for any n, N = 2n-1

As a check, n =1, N = 20 = 1

                   n = 2, N = 21 = 2

                  n = 3, N = 22 = 4

Thus, for

1. n=5 , N = 25-1 = 24 = 16

2. After n = 5, No. of total nuclei that have fissioned since the start of 1st:

20 +21 +22 +23 +24 = 1+2+4+8+16 = 31 nuclei

3. In second event where each neutron fissions 3 nuclei, we have no. of nuclei fissioned in the nth event as: 3n-1

n =1, N =1 = 30

n = 2, N =3 = 31

n=3, N =9 = 32

So, for n = 5, N = 34 = 81 Nuclei

4. After n = 6, total no. of nuclei fissioned since the start of 1st event :

30 +31 +32 +33 +34 + 35 = 1+3+9+27+81+243 = 364 nuclei

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