Question

The fission of \(_{92}^{235} \mathrm{U}\) releases \(2.1 \times 10^{13}\) joules per mole of \(_{92}^{235} \mathrm{U} .\) Calculate the energy released per atom and per gram of \(\frac{235}{92} \mathrm{U}\)

Short answer

The energy released per atom of \(_{92}^{235}\mathrm{U}\) is approximately \(3.49 \times 10^{-11}\,\text{J/atom}\), and the energy released per gram of \(_{92}^{235}\mathrm{U}\) is approximately \(8.94 \times 10^{10}\,\text{J/g}\).

Step-by-step solution

Calculate the energy released per atom

We are given the energy released per mole of \(_{92}^{235}\mathrm{U}\), which is \(2.1 \times 10^{13} \,\text{J/mol}\). To find the energy released per atom, we will divide this value by the Avogadro's number \(N_A= 6.022 \times 10^{23}\, \text{atoms/mol}\): \[\text{Energy per atom} = \frac{2.1 \times 10^{13}\, \text{J/mol}}{6.022 \times 10^{23}\, \text{atoms/mol}}\]

Calculate the achieved result

Performing the above division, we get: \[\text{Energy per atom} = \frac{2.1 \times 10^{13}}{6.022 \times 10^{23}}\, \text{J/atom} \approx 3.49 \times 10^{-11}\, \text{J/atom}\] So the energy released per atom of \(_{92}^{235}\mathrm{U}\) is approximately \(3.49 \times 10^{-11}\,\text{J/atom}\).

Calculate the energy released per gram

Now, we will find the energy released per gram of \(_{92}^{235}\mathrm{U}\). First, we need the molar mass. Since the atomic mass of \(_{92}^{235}\mathrm{U}\) is 235 g/mol, we can now find the energy released per gram by dividing the energy released per mole by the molar mass: \[\text{Energy per gram} = \frac{2.1 \times 10^{13}\, \text{J/mol}}{235\, \text{g/mol}}\]

Calculate the achieved result

Performing the above division, we get: \[\text{Energy per gram} = \frac{2.1 \times 10^{13}}{235}\, \text{J/g} \approx 8.94 \times 10^{10}\, \text{J/g}\] So the energy released per gram of \(_{92}^{235}\mathrm{U}\) is approximately \(8.94 \times 10^{10}\,\text{J/g}\).

Key concepts

Energy Conversion Calculations

Energy conversion calculations are essential for understanding how much energy one mole of a substance can release. Here, we focus on the fission of uranium-235 (\(_{92}^{235}\mathrm{U}\)).
Fission releases a large amount of energy - in this case, \(2.1 \times 10^{13}\) joules per mole. To appreciate this massive figure, we need to break it down into more familiar units, like energy per atom and per gram.

• Energy per Atom: Start by dividing the total energy per mole by Avogadro's number to find the energy released by a single atom. This value helps us understand the microscopic perspective of nuclear reactions.
• Energy per Gram: By using the molar mass, we can translate the energy into a per gram basis. This is done by dividing the energy per mole by the molar mass of the element. It's particularly useful for practical applications, as substances are often measured in grams.

These calculations allow scientists and engineers to predict performance and efficiency in nuclear reactors.

Avogadro's Number

Avogadro's Number is a fundamental constant that makes chemistry calculations much easier. It's a specific number: \(6.022 \times 10^{23}\) atoms, molecules, or particles per mole. But why is it important?
First, it connects the atomic scale to the macroscopic world, meaning we can translate atomic-level occurrences to real-world quantities like moles and grams.
In the context of fission, using Avogadro's Number allows us to calculate how much energy a single uranium atom releases. This is essential because it connects the energy released in gigantic scales (like moles) to the energy observed per individual particle.

• Atomic to Macroscopic: Avogadro's Number helps transform energy per mole into energy per atom, providing a clearer understanding of individual processes.
• Universal Constant: It's used across various scientific fields, not just chemistry, showcasing its broad relevance.

Whether you're studying reactions on tiny or large scales, Avogadro's number is your bridge between worlds.

Molar Mass

Molar mass serves as the conversion factor between the weight of a substance and the number of moles. It's given in units of grams per mole (g/mol). Here, the molar mass of uranium-235 (\(_{92}^{235}\mathrm{U}\)) is 235 g/mol.
In energy calculations, knowing the molar mass allows us to find out how energy is distributed per gram of a substance. It's as simple as dividing the total energy per mole by the molar mass.

• Definition: Molar mass shows the mass of one mole of atoms or molecules, acting as an essential conversion tool.
• Practical Use: When you know the molar mass, you can easily convert between mass and moles, crucial for calculating energy per gram in reactions.

Understanding molar mass makes handling complex chemical equations manageable and practical, especially in energy-related applications.