# Mass-Energy Conversion Example

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Fathers, like mothers, are not born. Men grow into fathers and fathering is a very important stage in their development.

— David Gottesman. His statement was true for me.

## Introduction

Figure 1: Photograph of Little Boy Bomb Casing.

Since 2015 is the 70th anniversary of the end of World War 2 (WW2), C-SPAN has been running a number of oral history interviews with people who worked on the Manhattan Project. I have found these interviews very interesting. You can find them on YouTube and watch them for yourself.

After watching one interview with a worker from the Y12 enrichment plant, I decide to look for some additional background material on the work done on the Manhattan Project. It was during this research that I watched a video in which engineer made the statement that only 900 grams of ${}_{{92}}^{{235}}U$ underwent fission in the Little Boy bomb. I thought this was an interesting number that I could show is consistent with the reported amount of energy released.

## Background

The following video states that 900 grams out of 60 kg of ${}_{{92}}^{{235}}U$ underwent fission in the Little Boy bomb. It was this statement that got me thinking about where the energy of these weapons come from – the binding energy that holds the uranium nucleus together.

## Analysis

Figure 2 shows my calculation for the energy released by splitting 900 grams of ${}_{{92}}^{{235}}U$. Most sources put the yield of the Little Boy device in the 16 kiloton range.

Figure 2: Energy Released From 900 Grams of ${}_{{92}}^{{235}}U$.

The video states that 900 grams of the 60 kilograms of ${}_{{92}}^{{235}}U$ underwent fission, which means about 1.5% of the fissile material was consumed. That is a very tiny amount of mass for a huge amount of energy.

It is useful to compare the energy released from the fission of single ${}_{{92}}^{{235}}U$ nucleus to that of the explosion of a single molecule of TNT (Figure 3). Observe that a single TNT molecule exploding is ~20 million times less energetic than a single ${}_{{92}}^{{235}}U$ fission.

Figure 3: Relative Energy Release of a ${}_{{92}}^{{235}}U$ Fission and TNT Reaction.

I should note that you will find a number of values listed for TNT's energy release – the value varies by how you define the release characteristics. For example, while the heat of combustion is listed as 14.5E6 Joules per kilogram, the energy released in the act of exploding is listed as 4.184E6 Joules per kilogram.

## Conclusion

It is hard to believe the amount of energy that is released by splitting atoms. An even more unimaginable number is the amount of energy released by the complete annihilation of matter, such as with matter/antimatter interactions.

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