Question

(a) At time $t=0$, a sample of uranium is exposed to a neutron source that causes $N_{0}$ nuclei to undergo fission. The sample is in a supercritical state, with a reproduction constant $K>1$. A chain reaction occurs that proliferates fission throughout the mass of uranium. The chain reaction can be thought of as a succession of generations. The $N_{0}$ fissions produced initially are the zeroth generation of fissions. From this generation, $N_{0} K$ neutrons go off to produce fission of new uranium nuclei. The $N_{0} K$ fissions that occur subsequently are the first generation of fissions, and from this generation $N_{0} K^{2}$ neutrons go in search of uranium nuclei in which to cause fission. The subsequent $N_{0} K^{2}$ fissions are the second generation of fissions. This process can continue until all the uranium nuclei have fissioned. Show that the cumulative total of fissions $N$ that have occurred up to and including the $n$ th generation after the zeroth generation is given by $$N=N_{0}\left(\frac{K^{n+1}-1}{K-1}\right).$$ (b) Consider a hypothetical uranium weapon made from $5.50 \mathrm{~kg}$ of isotopically pure ${ }^{235} \mathrm{U}$. The chain reaction has a reproduction constant of $1.10$ and starts with a zeroth generation of $1.00 \times 10^{20}$ fissions. The average time interval between one fission generation and the next is $10.0 \mathrm{~ns}$. How long after the zeroth generation does it take the uranium in this weapon to fission completely? (c) Assume the bulk modulus of uranium is 150 GPa. Find the speed of sound in uranium. You may ignore the density difference between ${ }^{235} \mathrm{U}$ and natural uranium. (d) Find the time interval required for a compressional wave to cross the radius of a $5.50$ -kg sphere of uranium. This time interval indicates how quickly the motion of explosion begins. (e) Fission must occur in a time interval that is short compared with that in part (d); otherwise, most of the uranium will disperse in small chunks without having fissioned. Can the weapon considered in part (b) release the explosive energy of all its uranium? If so, how much energy does it release in equivalent tons of TNT? Assume one ton of TNT releases $4.20$ GJ and each uranium fission releases $200 \mathrm{MeV}$ of energy.

Name: Problem 67, Chapter 45: Physics for Scientists and Engineers with Modern Physics
Uploaded: 2020-06-05T07:36:01-08:00
Duration: 16 min 34 s
Channel: Ren Jie Tuieng

   (a) At time $t=0$, a sample of uranium is exposed to a neutron source that causes $N_{0}$ nuclei to undergo fission. The sample is in a supercritical state, with a reproduction constant $K>1$. A chain reaction occurs that proliferates fission throughout the mass of uranium. The chain reaction can be thought of as a succession of generations. The $N_{0}$ fissions produced initially are the zeroth generation of fissions. From this generation, $N_{0} K$ neutrons go off to produce fission of new uranium nuclei. The $N_{0} K$ fissions that occur subsequently are the first generation of fissions, and from this generation $N_{0} K^{2}$ neutrons go in search of uranium nuclei in which to cause fission. The subsequent $N_{0} K^{2}$ fissions are the second generation of fissions. This process can continue until all the uranium nuclei have fissioned. Show that the cumulative total of fissions $N$ that have occurred up to and including the $n$ th generation after the zeroth generation is given by $$N=N_{0}\left(\frac{K^{n+1}-1}{K-1}\right).$$ 
(b) Consider a hypothetical uranium weapon made from $5.50 \mathrm{~kg}$ of isotopically pure ${ }^{235} \mathrm{U}$. The chain reaction has a reproduction constant of $1.10$ and starts with a zeroth generation of $1.00 \times 10^{20}$ fissions. The average time interval between one fission generation and the next is $10.0 \mathrm{~ns}$. How long after the zeroth generation does it take the uranium in this weapon to fission completely?
(c) Assume the bulk modulus of uranium is 150 GPa. Find the speed of sound in uranium. You may ignore the density difference between ${ }^{235} \mathrm{U}$ and natural uranium. (d) Find the time interval required for a compressional wave to cross the radius of a $5.50$ -kg sphere of uranium. This time interval indicates how quickly the motion of explosion begins.
(e) Fission must occur in a time interval that is short compared with that in part (d); otherwise, most of the uranium will disperse in small chunks without having fissioned. Can the weapon considered in part (b) release the explosive energy of all its uranium? If so, how much energy does it release in equivalent tons of TNT? Assume one ton of TNT releases $4.20$ GJ and each uranium fission releases $200 \mathrm{MeV}$ of energy.

Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway,… 8th Edition

Chapter 45, Problem 67

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Solved by Expert Ren Jie Tuieng

06/05/2020

Step 1

The total number of fissions up to and including the $n$th generation is the sum of the fissions in each generation, which is given by $N = N_{0} + N_{0} K + N_{0} K^{2} + \ldots + N_{0} K^{n}$. Show more…

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(a) At time $t=0$, a sample of uranium is exposed to a neutron source that causes $N_{0}$ nuclei to undergo fission. The sample is in a supercritical state, with a reproduction constant $K>1$. A chain reaction occurs that proliferates fission throughout the mass of uranium. The chain reaction can be thought of as a succession of generations. The $N_{0}$ fissions produced initially are the zeroth generation of fissions. From this generation, $N_{0} K$ neutrons go off to produce fission of new uranium nuclei. The $N_{0} K$ fissions that occur subsequently are the first generation of fissions, and from this generation $N_{0} K^{2}$ neutrons go in search of uranium nuclei in which to cause fission. The subsequent $N_{0} K^{2}$ fissions are the second generation of fissions. This process can continue until all the uranium nuclei have fissioned. Show that the cumulative total of fissions $N$ that have occurred up to and including the $n$ th generation after the zeroth generation is given by $$N=N_{0}\left(\frac{K^{n+1}-1}{K-1}\right).$$ (b) Consider a hypothetical uranium weapon made from $5.50 \mathrm{~kg}$ of isotopically pure ${ }^{235} \mathrm{U}$. The chain reaction has a reproduction constant of $1.10$ and starts with a zeroth generation of $1.00 \times 10^{20}$ fissions. The average time interval between one fission generation and the next is $10.0 \mathrm{~ns}$. How long after the zeroth generation does it take the uranium in this weapon to fission completely? (c) Assume the bulk modulus of uranium is 150 GPa. Find the speed of sound in uranium. You may ignore the density difference between ${ }^{235} \mathrm{U}$ and natural uranium. (d) Find the time interval required for a compressional wave to cross the radius of a $5.50$ -kg sphere of uranium. This time interval indicates how quickly the motion of explosion begins. (e) Fission must occur in a time interval that is short compared with that in part (d); otherwise, most of the uranium will disperse in small chunks without having fissioned. Can the weapon considered in part (b) release the explosive energy of all its uranium? If so, how much energy does it release in equivalent tons of TNT? Assume one ton of TNT releases $4.20$ GJ and each uranium fission releases $200 \mathrm{MeV}$ of energy.

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Key Concepts

Nuclear Chain Reaction

This concept involves a self?sustaining process where each fission event releases neutrons that trigger additional fission events. In a supercritical state, as characterized by a reproduction constant greater than one, the chain reaction exhibits exponential growth, with each generation producing more fissions than the previous one until the available fissile material is exhausted.

Geometric Series and Cumulative Growth

The number of fission events over multiple generations follows a geometric series. The formula N = N?((K^(n+1) - 1)/(K - 1)) is derived by summing the fissions generated in each successive generation. This concept underlines how iterative multiplicative processes can be summed to obtain a closed-form expression for the total activity in chain reactions.

Criticality and Reproduction Constant

The reproduction constant, K, indicates the average number of neutrons from one fission that cause additional fissions. When K > 1, the system is supercritical and the chain reaction grows exponentially. Understanding criticality is essential for controlling chain reactions in both nuclear reactors and weapons, as it determines whether the reaction is self-sustaining, dying out, or exploding.

Fission Reaction Energy and TNT Equivalence

Each fission event releases a significant amount of energy (typically around 200 MeV per event), which when summed over the total number of fissions, results in a massive energy release. Converting this energy into TNT equivalent provides a familiar metric for the explosive yield, facilitating comparisons with conventional explosives.

Bulk Modulus and Speed of Sound in Materials

The bulk modulus is a measure of a material’s resistance to compression and is directly related to the speed of sound in the material via the equation v = ?(B/?), where ? is the density. This concept is crucial in understanding how quickly mechanical disturbances, such as shock or compressional waves, propagate through a medium like uranium.

Shock Wave Propagation and Time Scale Considerations

In explosive processes, the speed of sound sets the time scale for how rapidly pressure or shock waves travel through the material. This determines the time within which the material must undergo fission to release its energy before mechanical expansion or dispersal disrupts the chain reaction. Understanding these time scales is critical in ensuring that the energy release occurs explosively rather than dissipatively.

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Transcript

00:01 In the first part of the question, you're going to find what is the total number of fission that has occurred.

00:10 Now given that at the start, to start off the fission chain reaction, total of n -knot nuclei will be introduced to the radium sample.

00:22 So this is the 0th generation and it will cause n -knot number of fission reaction.

00:30 Now from this fusion reaction, there will be a reproduction constant that tells us how many neutrons are actually how many neutrons are produced.

00:45 And these neutrons will then go on and create more fusion.

00:49 So the reproduction constant is k.

00:52 And this tells us a total of k times n -0 number of neutrons will be produced.

00:59 And this k -n -0 will produce at k -n -0.

01:02 And not number of fission reactions.

01:08 And so this is the first generation of fission reaction, then so on and so forth, the second generation, and the third generation until the nth generation should be as such.

01:28 So we'll define what is the total number of fission reaction that has occurred until the nth generation.

01:40 Let us just say n, this big n is the total number.

01:53 It will be n0 plus nkk, plus nk squared, etc, until n0kk, power of n.

02:06 Let's factorize this out, the n0 out.

02:14 So we're going to do a small little trick over here after factorizing.

02:19 Out the n -0.

02:22 What we're going to do is we're going to multiply both left -hand side and right -hand side by k.

02:31 So 1 times k, you'll give you a k.

02:34 K times k, give you k squared, so on and so forth until k and plus 1.

02:45 And then after this we're going to add 1 to both sides plus 1 goes to 1 plus k plus k square plus k q plus until k n plus 1 now do you see that this until just before kn plus 1 which is until kn plus 1 so until k n right this is our n over n0 from here.

03:36 So we can replace this as n over n0 plus k n plus 1.

03:51 Now time to rearrange this equation a little bit.

03:55 Right, tender way and not going to bring it over to the other side.

03:59 Get n over n n0, k minus 1 equals to k of n minus 1, sorry n plus 1 minus 1 right this minus 1 is from the left hand side.

04:17 Finally we get n over n0 equals to k n plus 1 minus 1 over k minus 1 then we get n equals to n0 1 by k n plus 1 minus 1 over k minus 1.

04:39 This is our final equation that we needed to prove.

04:48 Now we're going to use this equation in a hypothetical situation, right, for a uranium weapon.

04:56 So we're given that uranium 2.

05:00 Uranium 235, we want to find out what is the time required to completely fission 5 .5 kilograms of uranium.

05:15 So 5 .5 kilograms, you first need to find how many nuclei there are.

05:21 So number of uranium 235 nuclei.

05:28 Take 5 .5 kilograms, divide by the mass of each individual nuclei.

05:35 That is 235, you convert that into kilograms.

05:49 That is 1 .41, 10 power 25 nuclei.

05:57 Next, we're going to use the equation that we previously solved to find out how many efficient generations is required in order to use up this number of nuclei.

06:14 So we equate our n to be equals to this 1 .41 times 7 part of 25, this equals to n n .0 times k and plus 1 this one or k minus 1.

06:42 Now we are given that the n0, the 0, the 0 generation is 10 to power of 20.

06:53 Our k is given as 1 .1.

07:04 All right, so we just have to substitute the equation in and rearrange a little bit.

07:10 So n divided by n0 equals to multiplied by k minus 1 which is 0 .1.

07:23 1 .0 minus 1 is 0 .1.

07:26 Goes to k and plus 1 minus 1.

07:31 I'm going to bring the minus 1 to the other side and take the natural log.

07:39 So long of k and plus 1.

07:43 Goes to 2 .1 over n0 plus 1.

07:58 So our n plus 1 will be long of the entire thing.

08:06 2 .1n over n0 plus 1 divided by lon k...

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