This course explains the process of nuclear fission, focusing on the calculation of the mass defect (\(\Delta m\)) and the energy released from the fission of uranium-235 (U-235) after it absorbs a neutron. We will calculate the total energy released from 1 kg of U-235 using Einstein’s famous equation \(E = mc^2\).

1. Nuclear Fission of U-235

Nuclear fission occurs when a heavy nucleus, such as uranium-235, absorbs a neutron and becomes unstable, leading to the splitting of the nucleus into two smaller nuclei (called fission products), along with the release of additional neutrons and a large amount of energy.

1.1 Fission Reaction Equation

The general fission equation for U-235 when it absorbs a neutron is:

\[ \text{U}^{235} + \text{n} \rightarrow \text{Ba}^{141} + \text{Kr}^{92} + 3 \, \text{n} + \text{Energy} \]

Here, the fission products include the isotopes Ba-141 (Barium-141) and Kr-92 (Krypton-92), along with the release of 3 neutrons and energy.

2. Mass Defect (\(\Delta m\)) Calculation

The total mass before the fission (reactants) is greater than the total mass of the fission products. The difference in mass is called the mass defect, and this difference is converted into energy according to Einstein’s equation \(E = mc^2\).

2.1 Mass of Reactants

The mass of the reactants consists of the mass of the U-235 nucleus and the neutron that is absorbed:

\[ \text{Mass of Reactants} = \text{Mass of U-235} + \text{Mass of neutron} \]

Using the approximate values:

The total mass of the reactants is:

\[ \text{Mass of Reactants} = 235.0439299 \, \text{u} + 1.008665 \, \text{u} = 236.0525949 \, \text{u}. \]

2.2 Mass of Fission Products

The mass of the fission products includes the masses of Ba-141, Kr-92, and the 3 neutrons produced. Using the approximate masses:

The total mass of the products is:

\[ \text{Mass of Products} = 140.914411 \, \text{u} + 91.926156 \, \text{u} + 3 \times 1.008665 \, \text{u} = 235.866567 \, \text{u}. \]

2.3 Mass Defect Calculation

The mass defect (\(\Delta m\)) is the difference between the mass of the reactants and the mass of the products:

\[ \Delta m = \text{Mass of Reactants} - \text{Mass of Products} \]

Substituting the values:

\[ \Delta m = 236.0525949 \, \text{u} - 235.866567 \, \text{u} = 0.1860279 \, \text{u}. \]

3. Converting Mass Defect to Energy

Now, we convert the mass defect to energy using Einstein’s equation \(E = mc^2\), where:

First, we need to convert the mass defect from atomic mass units (u) to kilograms (kg). The conversion factor is:

\[ 1 \, \text{u} = 1.66053906660 \times 10^{-27} \, \text{kg}. \]

Therefore, the mass defect in kilograms is:

\[ \Delta m = 0.1860279 \, \text{u} \times 1.66053906660 \times 10^{-27} \, \text{kg/u} = 3.088 \times 10^{-28} \, \text{kg}. \]

Now, using \(E = mc^2\), the energy released is:

\[ E = (3.088 \times 10^{-28} \, \text{kg}) \times (3 \times 10^8 \, \text{m/s})^2 \]
\[ E = 2.7792 \times 10^{-11} \, \text{J}. \]

Thus, the energy released from one fission event is approximately \(2.78 \times 10^{-11} \, \text{J}\).

4. Total Energy Released from 1 kg of U-235

To find the total energy released from 1 kg of U-235, we first need to calculate the number of atoms in 1 kg of U-235:

\[ \text{Number of atoms} = \frac{1000 \, \text{g}}{235 \, \text{g/mol}} \times 6.022 \times 10^{23} = 2.56 \times 10^{24} \, \text{atoms}. \]

Now, multiplying the energy per fission event by the number of atoms in 1 kg of U-235:

\[ E_{\text{total}} = 2.56 \times 10^{24} \, \text{atoms} \times 2.78 \times 10^{-11} \, \text{J} \]
\[ E_{\text{total}} \approx 7.12 \times 10^{13} \, \text{J}. \]

Therefore, the total energy released from **1 kg of U-235** undergoing fission is approximately **\(7.12 \times 10^{13} \, \text{J}\)**.

5. Conclusion

We have now calculated the mass defect and the energy released from the fission of a U-235 nucleus after it absorbs a neutron. The total energy released from 1 kg of U-235 is approximately \(7.12 \times 10^{13} \, \text{J}\), which is a massive amount of energy and highlights the power of nuclear fission as a source of energy in nuclear reactors and weapons.