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A small atomic bomb releases energy equivalent to the deto- nation of \(20,000\) tons of TNT; a ton of TNT releases \(4 \times 10^{9} \mathrm{J}\) of energy when exploded. Using \(2 \times 10^{13} \mathrm{J} / \mathrm{mol}\) as the energy released by fission of \(^{235} \mathrm{U}\) , approximately what mass of \(^{235} \mathrm{U}\) undergoes fission in this atomic bomb?

Short Answer

Expert verified
Approximately \(940 \, \mathrm{g}\) of uranium-235 undergoes fission in this atomic bomb.

Step by step solution

01

Calculate the total energy released by the atomic bomb

We are given that the bomb releases energy equivalent to 20,000 tons of TNT. To find the total energy released in joules, we can use the information provided: Energy released per ton of TNT = \(4 \times 10^9 \mathrm{J}\) Total energy released = (Energy released per ton of TNT) \(\times\) (Number of tons of TNT) Total energy released = \((4 \times 10^9 \mathrm{J}) \times 20,000\) Total energy released = \(8 \times 10^{13} \mathrm{J}\)
02

Calculate the number of moles of uranium-235 used in fission

Now, we will determine how many moles of uranium-235 were used in the fission process to release the calculated energy. We are given the energy released per mole of uranium-235 (\(2 \times 10^{13} \mathrm{J} / \mathrm{mol}\)). We can use this information to find the number of moles of uranium-235 used in the atomic bomb: Number of moles of uranium-235 = (Total energy released) / (Energy released per mole of uranium-235) Number of moles of uranium-235 = \((8 \times 10^{13} \mathrm{J}) / (2 \times 10^{13} \mathrm{J} / \mathrm{mol})\) Number of moles of uranium-235 = \(4 \, \mathrm{mol}\)
03

Calculate the mass of uranium-235 used in fission

Finally, we will convert the number of moles of uranium-235 to mass using the molar mass of uranium-235, which is approximately 235 g/mol: Mass of uranium-235 = (Number of moles of uranium-235) \(\times\) (Molar mass of uranium-235) Mass of uranium-235 = \(4 \, \mathrm{mol} \times 235 \, \mathrm{g} / \mathrm{mol}\) Mass of uranium-235 = \(940 \, \mathrm{g}\) Thus, approximately \(940 \, \mathrm{g}\) of uranium-235 undergoes fission in this atomic bomb.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

uranium-235
Uranium-235 is a heavy isotope of uranium that is crucial in the process of nuclear fission. Found naturally in uranium ore, and alongside other isotopes like Uranium-238, Uranium-235 is particularly important for nuclear reactors and atomic bombs due to its ability to easily undergo fission.
When a Uranium-235 nucleus is struck by a neutron, it splits into two smaller nuclei, releasing additional neutrons and a significant amount of energy. This released energy is what powers nuclear reactors and makes atomic bombs so destructive.
  • Uranium-235 is a rare isotope, comprising only about 0.7% of natural uranium.
  • The fission process with Uranium-235 releases a tremendous amount of energy, making it a potent fuel for nuclear reactions.
Understanding the role of Uranium-235 in energy production helps us appreciate the science behind both constructive uses like nuclear power plants and destructive applications like weapons.
energy release
In nuclear fission, the energy release is the driving force behind the phenomenon. When Uranium-235 undergoes fission, a large amount of energy is released. This energy primarily depends on the interactions within the nucleus of Uranium-235.
The energy released during fission can be harnessed for two main purposes: generating electricity in power plants or as an explosive force in atomic bombs.
  • The conversion of a small amount of mass into energy in alignment with Einstein's famous equation, \(E=mc^2\), explains the significant energy yield.
  • The step-by-step chain reactions initiated by the released neutrons cause a cascading effect, amplifying the total energy released.
By studying how energy is released in these reactions, we learn more about harnessing nuclear energy safely and efficiently.
molar mass
Molar mass is a key concept in chemistry that helps us understand the mass of a given substance based on the amount of substance in moles. For Uranium-235, the molar mass is approximately 235 grams per mole.
This value allows us to convert between the number of moles of Uranium-235 undergoing fission and the actual mass of Uranium-235 used.
  • The molar mass provides a bridge between microscopic quantities (moles) and macroscopic quantities (grams).
  • It is used to determine the mass of a compound from the amount of substance used in reactions.
Knowing and using the molar mass is essential for accurately measuring and understanding chemical reactions, especially those involving fission.
joules
Joules are the standard unit of energy in the International System of Units. When discussing nuclear fission, joules are used to quantify the vast amounts of energy released.
This unit helps us gauge the energy produced in practical and meaningful terms, whether in scientific calculations or comparisons with everyday energy outputs.
  • In the context of the given problem, one can see how various energy sources, such as TNT, are converted into joules to standardize calculations and comparisons.
  • The joule serves as a reliable means of comparing different energy sources and activities on a common scale.
By using joules, scientists and engineers can communicate and quantify energy release or consumption clearly and consistently.

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Most popular questions from this chapter

Rubidium- 87 decays by \(\beta\) -particle production to strontium- 87 with a half-life of \(4.7 \times 10^{10}\) years. What is the age of a rock sample that contains 109.7 \mug of \(^{87} \mathrm{Rb}\) and 3.1\(\mu \mathrm{g}\) of \(^{87} \mathrm{Sr} ?\) Assume that no \(^{87}\) Sr was present when the rock was formed. The atomic masses for \(^{87}\mathrm{Rb}\) and \(^{87} \mathrm{Sr}\) are 86.90919 \(\mathrm{u}\) and 86.90888 u, respectively.

Strontium- 90 and radon-222 both pose serious health risks. \(^{90}\) Sr decays by \(\beta\) -particle production and has a relatively long half-life \((28.9 \text { years). Radon-2222 decays by } \alpha \text { -particle production }\) and has a relatively short half-life \((3.82 \text { days). Explain }\) why each decay process poses health risks.

During the research that led to production of the two atomic bombs used against Japan in World War II, different mechanisms for obtaining a supercritical mass of fissionable material were investigated. In one type of bomb, a gun shot one piece of fissionable material into a cavity containing another piece of fissionable material. In the second type of bomb, the fissionable material was surrounded with a high explosive that, when detonated, compressed the fissionable material into a smaller volume. Discuss what is meant by critical mass, and explain why the ability to achieve a critical mass is essential to sustaining a nuclear reaction

Assume a constant \(1^{14} \mathrm{C} /^{12} \mathrm{C}\) ratio of 13.6 counts per minute per gram of living matter. A sample of a petrified tree was found to give 1.2 counts per minute per gram. How old is the tree? (For \(^{14} \mathrm{C}, t_{1 / 2}=5730\) years.)

Breeder reactors are used to convert the nonfissionable nuclide 238 \(\mathrm{U}\) to a fissionable product. Neutron capture of the 238 \(\mathrm{U}\) is followed by two successive beta decays. What is the final fissionable product?

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