Problem 38 - College Physics

Question

At the beginning of Section \(30.6,\) a fission process is illustrated in which \(^{235} \mathrm{U}\) is struck by a neutron and undergoes fission to produce \(^{144} \mathrm{Ba}, 89 \mathrm{Kr}\) , and three neutrons. The measured masses of these isotopes are 235.043930 u \(\left(^{235} \mathrm{U}\right)\) \(143.922953 \mathrm{u}\left(^{144} \mathrm{Ba}\right), 88.917630 \mathrm{u}\left(^{89} \mathrm{Kr}\right),\) and 1.0086649 \(\mathrm{u}\) (neutron). (a) Calculate the energy (in MeV) released by each fission reaction. (b) Calculate the energy released per gram of \(^{235} \mathrm{U},\) in \(\mathrm{MeV} / \mathrm{g} .\)

Step-by-Step Solution

Answer

The energy released per fission reaction is 1067.73 MeV, and per gram of \(^{235} \text{U}\), it is approximately \(2.7355\times10^{24}\,\text{MeV/g}\).

Step 1: Write the Nuclear Reaction Equation

The reaction can be represented as: \( ^{235}\text{U} + n \rightarrow ^{144}\text{Ba} + ^{89}\text{Kr} + 3n \). We start with uranium-235 and a neutron, and it results in barium-144, krypton-89, and three more neutrons.

Step 2: Calculate Reactant Mass

Add the masses of the reactants: \( m_{\text{Reactants}} = m(^{235}\text{U}) + m(n) \). Using the given masses: \[ m_{\text{Reactants}} = 235.043930\,u + 1.0086649\,u = 236.0525949\,u \].

Step 3: Calculate Product Mass

Add the masses of the products: \( m_{\text{Products}} = m(^{144}\text{Ba}) + m(^{89}\text{Kr}) + 3m(n) \). Using the given masses: \[ m_{\text{Products}} = 143.922953\,u + 88.917630\,u + 3 \times 1.0086649\,u = 234.9065577\,u \].

Step 4: Calculate Mass Defect

The mass defect (\( \Delta m \)) is the difference between the reactant and product masses: \[ \Delta m = m_{\text{Reactants}} - m_{\text{Products}} = 236.0525949\,u - 234.9065577\,u = 1.1460372\,u \].

Step 5: Convert Mass Defect to Energy

Use \( E=mc^2 \) to find energy. The conversion factor is \( 1\,u = 931.5\,\text{MeV/c}^2 \):\[ E = \Delta m \times 931.5\,\text{MeV} = 1.1460372\,u \times 931.5\,\text{MeV}/u = 1067.728\,\text{MeV} \].

Step 6: Calculate Energy per Gram of \(^{235}\text{U}\)

First, find the number of \( ^{235} \text{U} \) atoms in 1 gram. The molar mass is approximately 235 grams per mole. \( \text{Number of atoms} = \frac{6.022 \times 10^{23}}{235}\,\text{atoms/gram} \).\[ \text{Energy per gram} = \frac{1067.728\, \text{MeV}}{1\, \text{atom}} \times \frac{6.022 \times 10^{23}\,\text{atoms}}{235\, \text{grams}} = 2.7355\times10^{24}\,\text{MeV/g} \].

Key Concepts

Energy Release

In a nuclear reaction, such as the fission of uranium-235, energy release is a crucial concept. Energy in nuclear fission is released due to the conversion of mass into energy, explained by Albert Einstein's famous equation, \( E = mc^2 \). This equation tells us how a change in mass, or mass defect, translates directly into energy. The result is a significant amount of energy compared to chemical reactions.
During nuclear fission, a heavy nucleus splits into lighter nuclei. For uranium-235, this process involves the absorption of a neutron, leading to the formation of barium-144, krypton-89, and three additional neutrons. An important point to note is that the energy released in a single fission event can be several million times more than typical chemical reactions. This high energy yield is why nuclear fission has attracted significant attention both for power generation and nuclear weapons.
By calculating how much mass is converted into energy, we can determine the energy released per fission reaction. This sheds light on the immense power of nuclear fission and factors heavily into the practicality and implementation of nuclear technology.

Mass Defect Calculation

The concept of mass defect is fundamental in understanding nuclear fission. Mass defect refers to the difference in mass between the reactants and products in a nuclear reaction.
In the case of uranium-235 fission, we begin by determining the total mass of the reactants, which includes the uranium atom and the incoming neutron. For uranium-235 and one neutron, the mass is 236.0525949 atomic mass units (u).
Next, we calculate the mass of the fission products, consisting of barium-144, krypton-89, and three neutrons. Their combined mass is 234.9065577 u.
The mass defect \( \Delta m \) is then the difference between these two totals:

• Reactant Mass = 236.0525949 u
• Product Mass = 234.9065577 u
• Mass Defect \( \Delta m = 236.0525949 \text{ u} - 234.9065577 \text{ u} = 1.1460372 \text{ u} \)

Understanding this mass defect is key to calculating the energy released during the reaction, using the conversion factor of 1 u being equivalent to 931.5 MeV.

Uranium-235 Fission

Uranium-235 is a crucial isotope in nuclear fission. It is unique because it can sustain a chain reaction once fission occurs. When a uranium-235 nucleus absorbs a neutron, it becomes unstable and splits into two smaller nuclei along with a few extra neutrons. These free neutrons can then go on to induce fission in other uranium-235 nuclei, sustaining a chain reaction.
This self-sustaining aspect of uranium-235 is what makes it suitable for nuclear power generation and nuclear weapons. In a controlled environment like a reactor, the chain reaction can be maintained at a steady rate to produce energy safely over time. In an uncontrolled situation, such as a nuclear bomb, the chain reaction leads to an explosive release of energy.
The fission of uranium-235 also results in valuable byproducts and must be carefully controlled to manage waste and prevent radioactive contamination.

Nuclear Reaction Equation

The nuclear reaction equation is a symbolic way of representing the nuclear fission process. In our example, the uranium-235 fission reaction can be symbolically represented as:\[ ^{235} \text{U} + n \rightarrow ^{144} \text{Ba} + ^{89} \text{Kr} + 3n \]This equation is a shorthand notation showing the reactant, uranium-235, interacting with a neutron to produce barium-144, krypton-89, and three additional neutrons.
Each component in this equation has an essential role:

• \(^{235} \text{U}\): Main fissionable material.
• \(n\): Neutron initiating the reaction.
• \(^{144} \text{Ba}\) and \(^{89} \text{Kr}\): Fission fragments or daughter nuclei.
• 3\(n\): Secondary neutrons that propagate the chain reaction.

These equations help scientists and engineers calculate energy outputs and determine conditions for controlling the reaction safely. Understanding nuclear reaction equations is crucial for working with nuclear processes and technology.