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A small atomic bomb releases energy equivalent to the detonation 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 grams of ${ }^{235} \mathrm{U}$ undergoes fission in the atomic bomb.

Step by step solution

01

Calculate the energy released by the atomic bomb

To calculate the energy released by the atomic bomb, we need to multiply the energy released by the detonation of 1 ton of TNT (4 × 10^9 J) by the total number of equivalent tons of TNT (20,000 tons). This can be written as: Energy released = (20,000 tons)(4 × 10^9 J/ton)
02

Convert the energy released to moles of uranium-235

Now that we know the energy released by the atomic bomb, we want to find out how many moles of uranium-235 would release this energy when undergoing fission. We do this by dividing the total energy released by the bomb by the energy released per mole of uranium-235 fission (2 × 10^13 J/mol). Let this value be N: N (moles of uranium-235) = (Energy released) / (2 × 10^13 J/mol)
03

Calculate the mass of uranium-235

Now that we know the number of moles of uranium-235 that undergoes fission, we need to find the mass of this amount of uranium. To do that, we multiply the number of moles (N) by the molar mass of uranium-235 (235 g/mol). This can be written as: Mass of uranium-235 = N × (235 g/mol) Combine all steps to find the solution: Energy released = (20,000 tons)(4 × 10^9 J/ton) = 8 × 10^13 J N = (8 × 10^13 J) / (2 × 10^13 J/mol) = 4 moles of uranium-235 Mass of uranium-235 = 4 × 235 g/mol = 940 g Hence, approximately 940 grams of 235U 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 key element in nuclear fission, which is the process used in nuclear reactors and atomic bombs to release energy. Uranium has different isotopes, and Uranium-235 is one of the most important because it can undergo fission. When Uranium-235 nuclei are struck by a neutron, they can split into smaller nuclei, releasing a large amount of energy in the process.
This reaction not only releases energy but also emits more neutrons, which can further cause even more fission reactions.
This chain reaction is what allows both controlled nuclear reactors and explosive nuclear bombs to function.
In nature, Uranium-235 is relatively rare compared to the more common Uranium-238, which is why enrichment is necessary for uranium to be used as a fuel in nuclear reactors.
The ability to sustain a chain reaction makes Uranium-235 highly valuable for both energy generation and military purposes.
Energy Conversion
The energy released during nuclear fission is incredibly high, especially when compared to chemical reactions like combustion. To put this into perspective, a single fission of Uranium-235 releases roughly 200 MeV, which is significantly more energy than released in typical chemical reactions.
To understand the scale of this energy, consider that during the explosion of an atomic bomb, energy equivalent to thousands of tons of TNT can be released.
In the exercise, the atomic bomb has an energy release equivalent to 20,000 tons of TNT. Given a ton of TNT releases about \(4 \times 10^9\) joules, the energy conversion and release in nuclear fission is enormous compared to these usual benchmarks.
This massive energy release is what drives the massive destructive power of nuclear weapons and also symbolizes the great potential of nuclear energy as a power source.
Moles Calculation
To understand the process of fission in quantitative terms, calculating the number of moles involved is essential. A mole is a fundamental concept in chemistry that helps bridge the atomic scale reactions to measurable laboratory amounts.
In this exercise, once the total energy released by the bomb is known, moles of Uranium-235 can be calculated using the given energy per mole of Uranium-235.
This is done by dividing the total energy released by the bomb by the energy released per mole, which helps convert energy into moles. For example, if \(8 \times 10^{13}\) joules of energy are released, the number of moles, \(N\), would be calculated as:
  • \(N = \frac{8 \times 10^{13} \text{ J}}{2 \times 10^{13} \text{ J/mol}}\)
  • This equals 4 moles of Uranium-235 fissioning in the bomb.
Understanding the concept of moles helps us make sense of how much material is needed or used in a chemical process.
Molar Mass of Uranium
The molar mass of Uranium, specifically Uranium-235, is key to converting between moles and grams, making calculations feasible. Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol).
For Uranium-235, the molar mass is 235 g/mol. This is a useful measurement for linking microscopic reactions to real-world quantities.
Once the number of moles of Uranium-235 is calculated, using its molar mass, we can easily determine the actual mass of Uranium needed or used in a reaction.
In our example, after finding out that 4 moles of Uranium-235 are involved, we multiply this by the molar mass of Uranium-235:
  • \(\text{Mass} = 4 \text{ moles} \times 235 \text{ g/mol}\)
  • The final result is 940 grams.
Thus, molar mass provides the bridge from theoretical chemistry calculations to practical and actionable information.

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Most popular questions from this chapter

The curie (Ci) is a commonly used unit for measuring nuclear radioactivity: 1 curie of radiation is equal to \(3.7 \times 10^{10}\) decay events per second (the number of decay events from \(1 \mathrm{~g}\) radium in \(1 \mathrm{~s}\) ). A 1.7-mL sample of water containing tritium was injected into a 150 -lb person. The total activity of radiation injected was \(86.5 \mathrm{mCi}\). After some time to allow the tritium activity to equally distribute throughout the body, a sample of blood plasma containing \(2.0 \mathrm{~mL}\) water at an activity of \(3.6 \mu \mathrm{Ci}\) was removed. From these data, calculate the mass percent of water in this 150 -lb person.

Assume a constant \({ }^{14} \mathrm{C} /{ }^{12} \mathrm{C}\) 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? \(\left(\right.\) For \({ }^{14} \mathrm{C}, t_{1 / 2}=5730\) years. \()\)

The binding energy per nucleon for magnesium- 27 is \(1.326\) \(\times 10^{-12} \mathrm{~J} /\) nucleon. Calculate the atomic mass of \({ }^{27} \mathrm{Mg} .\)

The bromine- 82 nucleus has a half-life of \(1.0 \times 10^{3}\) min. If you wanted \(1.0 \mathrm{~g}{ }^{82} \mathrm{Br}\) and the delivery time was \(3.0\) days, what mass of NaBr should you order (assuming all of the \(\mathrm{Br}\) in the \(\mathrm{NaBr}\) was \({ }^{82} \mathrm{Br}\) )?

Zirconium is one of the few metals that retains its structural integrity upon exposure to radiation. The fuel rods in most nuclear reactors therefore are often made of zirconium. Answer the following questions about the redox properties of zirconium based on the half-reaction \(\mathrm{ZrO}_{2} \cdot \mathrm{H}_{2} \mathrm{O}+\mathrm{H}_{2} \mathrm{O}+4 \mathrm{e}^{-} \longrightarrow \mathrm{Zr}+4 \mathrm{OH}^{-} \quad \mathscr{E}^{\circ}=-2.36 \mathrm{~V}\) a. Is zirconium metal capable of reducing water to form hydrogen gas at standard conditions? b. Write a balanced equation for the reduction of water by zirconium. c. Calculate \(\mathscr{E}^{\circ}, \Delta G^{\circ}\), and \(K\) for the reduction of water by zirconium metal. d. The reduction of water by zirconium occurred during the accidents at Three Mile Island in \(1979 .\) The hydrogen produced was successfully vented and no chemical explosion occurred. If \(1.00 \times 10^{3} \mathrm{~kg}\) Zr reacts, what mass of \(\mathrm{H}_{2}\) is produced? What volume of \(\mathrm{H}_{2}\) at \(1.0 \mathrm{~atm}\) and \(1000 .{ }^{\circ} \mathrm{C}\) is produced? e. At Chernobyl in 1986, hydrogen was produced by the reaction of superheated steam with the graphite reactor core: $$ \mathrm{C}(s)+\mathrm{H}_{2} \mathrm{O}(g) \longrightarrow \mathrm{CO}(g)+\mathrm{H}_{2}(g) $$

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