Question

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

Approximately $940\mathrm{g}$ of uranium-235 undergoes fission in this atomic bomb.

Step-by-step solution

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 = $8\times {10}^{13}\mathrm{J}$

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 = $4\mathrm{mol}$

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.

Key concepts

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, $E=m{c}^2$, 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.