Question

At the beginning of Section 43.7 the equation of a fission process is given in which \(^2$$^3$$^5\)U is struck by a neutron and undergoes fission to produce \(^1$$^4$$^4\)Ba, \(^8$$^9\)Kr, and three neutrons. The measured masses of these isotopes are 235.043930 u (\(^2$$^3$$^5\)U), 143.922953 u (\(^1$$^4$$^4\)Ba), 88.917631 u (\(^8$$^9\)Kr), and 1.0086649 u (neutron). (a) Calculate the energy (in MeV) released by each fission reaction. (b) Calculate the energy released per gram of \(^2$$^3$$^5\)U, in MeV/g.

Short answer

The energy released per fission is 1036.08 MeV and per gram of U-235 is approximately 2.65 x 10^24 MeV/g.

Step-by-step solution

Understanding the Reaction

The fission reaction given \[ ^{235}\text{U} + ^{1}\text{n} \rightarrow ^{144}\text{Ba} + ^{89}\text{Kr} + 3 \times ^{1}\text{n} \] involves a neutron striking a uranium-235 nucleus, which splits into barium-144, krypton-89, and three neutrons.

Calculate Total Mass Before Fission

Before the fission, the total mass is the mass of \[^{235}\text{U} \] plus one neutron: \[ 235.043930\ \text{u} + 1.0086649\ \text{u} = 236.0525949\ \text{u}. \]

Calculate Total Mass After Fission

The total mass after the fission can be calculated as the sum of the masses of \(^{144}\text{Ba}\), \(^{89}\text{Kr}\), and 3 neutrons: \[143.922953\ \text{u} + 88.917631\ \text{u} + 3 \times 1.0086649\ \text{u} = 234.9405797 \ \text{u}. \]

Determine the Mass Defect

The mass defect is the difference between the initial mass and the final mass: \[\Delta m = 236.0525949\ \text{u} - 234.9405797\ \text{u} = 1.1120152\ \text{u}. \]

Convert Mass Defect to Energy

Use Einstein's equation \(E=\Delta m c^2\) to calculate the energy released, where \(c\) is the speed of light. We first convert the mass defect to energy in MeV by using the conversion factor 1 u = 931.5 MeV/c²: \[E = 1.1120152\ \text{u} \times 931.5\ \text{MeV/u} = 1036.08\ \text{MeV}.\]

Calculate Energy Released per Gram of U-235

Find the energy released per gram of \(^{235}\text{U}\):First, calculate the number of atoms in 1 gram:\[\text{Number of atoms} = \frac{\text{Avogadro's number}}{\text{Molar mass of } \text{U}=235.043930\ \text{u/g/mol}} = 2.56 \times 10^{21} \ \text{atoms/gram}.\]Then multiply this by the energy per fission reaction:\[E_{\text{gram}} = 2.56 \times 10^{21} \ \text{reactions/gram} \times 1036.08\ \text{MeV} = 2.65 \times 10^{24}\ \text{MeV/gram}.\]

Key concepts

Mass-Energy Equivalence

Einstein's groundbreaking equation, \(E=mc^2\), plays a crucial role in explaining nuclear fission. This principle, known as mass-energy equivalence, links mass and energy, suggesting that they can be converted into one another. In nuclear reactions, the difference in mass before and after the reaction—called the mass defect—translates into energy release. This energy release is the foundation of nuclear power and atomic bombs.
For instance, in our Uranium-235 fission reaction, the mass of the reactants is slightly more than the mass of the products, and this "lost" mass is where the energy comes from. By calculating the mass defect and using the conversion factor between atomic mass units and energy (931.5 MeV/c² per 1 u), we can determine the energy released during the fission. This is an impressive demonstration of how small amounts of mass can produce vast amounts of energy.

Uranium-235

Uranium-235 is a fascinating isotope of uranium and one of the few materials that can sustain a nuclear chain reaction. Found naturally, U-235 is utilized in nuclear reactors and weapons due to its ability to undergo fission when hit by a neutron. It's significantly less stable compared to other isotopes like Uranium-238. This instability arises because when a neutron interacts with U-235, it can split into smaller atoms like barium and krypton, releasing additional neutrons and a considerable amount of energy in the process.
U-235's fission process is instrumental in both energy generation and scientific research. The fission products are a complex cocktail of radioactive isotopes, which continue to decay, providing additional energy even after the initial reaction. The ability to initiate chain reactions makes U-235 highly significant in both energy production and nuclear weaponry.

Nuclear Reaction Calculations

When calculating nuclear reactions, determining the total mass before and after the reaction is key. First, identify the masses of all reactants and products using known values in atomic mass units (u). Begin by summing the atomic masses of the fissile material (e.g., Uranium-235) and any interacting particles (e.g., neutrons). In our scenario, we calculate the mass before fission by adding the mass of one U-235 nucleus and one neutron.
After the reaction, calculate the total mass of the fission products, including all atoms and neutrons emitted. The difference between the mass before and after the reaction gives the mass defect \(\Delta m\). Finally, use this mass defect in Einstein’s equation \(E=\Delta mc^2\) to calculate the energy produced in the process. This step requires converting the mass defect from atomic mass units to energy units like MeV. Accurate nuclear reaction calculations help in predicting how much energy a particular fission process will release, a crucial component for both safety and effectiveness.

Energy Conversion in MeV

The energy released in nuclear reactions like fission is immense but often expressed in very specific units. Mega-electron volts (MeV) are used to measure the energy released per individual nuclear reaction. To convert a mass defect into energy, we typically use the conversion factor: 1 atomic mass unit (u) equals 931.5 MeV/c².
To apply this in calculations, multiply the mass defect by this conversion factor. For example, in our Uranium-235 fission, a mass defect of 1.1120152 u results in \(E = 1.1120152\ \text{u} \times 931.5\ \text{MeV/u} = 1036.08\ \text{MeV}\). This conversion is crucial for understanding energy scales in nuclear physics and comparing them to more familiar forms of energy. Therefore, energy conversion allows us to appreciate just how potent nuclear reactions are, even when initiated by small particles like neutrons.