Calculate and compare the energy released by a) fusion of 1.0 kg of hydrogen deep within Sun and b) the fission of 1.0 kg of 235U in a fission reactor.

To understand the energy released by nuclear fusion and fission, let's break down the calculations for both processes. We’ll start with the fusion of hydrogen in the Sun and then look at the fission of uranium-235 in a reactor.

Energy Released by Fusion of Hydrogen

In the core of the Sun, hydrogen nuclei fuse to form helium, a process that releases a substantial amount of energy. The primary reaction can be simplified as:

• 4 protons (hydrogen nuclei) → 1 helium nucleus + energy

The energy released from this reaction is approximately 26.7 MeV (mega-electronvolts) per reaction. To find the energy released from 1.0 kg of hydrogen, we need to calculate the number of hydrogen atoms in that mass.

Calculating the Number of Hydrogen Atoms

Using the molar mass of hydrogen (about 1 g/mol), we find that 1.0 kg of hydrogen contains:

• 1 kg = 1000 g
• Number of moles of hydrogen = 1000 g / 1 g/mol = 1000 mol
• Number of hydrogen atoms = 1000 mol × 6.022 × 10²³ atoms/mol ≈ 6.022 × 10²⁵ atoms

Energy Calculation for Fusion

Since each fusion reaction involves 4 hydrogen nuclei, the number of reactions that can occur is:

• Number of reactions = 6.022 × 10²⁵ atoms / 4 ≈ 1.505 × 10²⁵ reactions

The total energy released from these reactions can be calculated as follows:

• Energy per reaction ≈ 26.7 MeV
• Convert MeV to joules: 1 MeV ≈ 1.602 × 10⁻¹³ joules
• Total energy = 1.505 × 10²⁵ reactions × 26.7 MeV/reaction × 1.602 × 10⁻¹³ joules/MeV

Calculating this gives:

• Total energy ≈ 1.505 × 10²⁵ × 26.7 × 1.602 × 10⁻¹³ ≈ 6.037 × 10¹² joules

Energy Released by Fission of Uranium-235

Now let's look at the fission of uranium-235. When uranium-235 undergoes fission, it typically splits into two smaller nuclei along with several neutrons and releases energy. The average energy released per fission event is about 200 MeV.

Calculating the Number of Fission Events

Similar to the hydrogen calculation, we first determine the number of uranium-235 atoms in 1.0 kg of uranium. The molar mass of uranium-235 is approximately 235 g/mol:

• Number of moles of U-235 = 1000 g / 235 g/mol ≈ 4.255 mol
• Number of atoms = 4.255 mol × 6.022 × 10²³ atoms/mol ≈ 2.56 × 10²⁴ atoms

Energy Calculation for Fission

The total energy released can be calculated as follows:

• Number of fission events ≈ 2.56 × 10²⁴
• Energy per fission ≈ 200 MeV
• Total energy = 2.56 × 10²⁴ fissions × 200 MeV/fission × 1.602 × 10⁻¹³ joules/MeV

Calculating this provides:

• Total energy ≈ 2.56 × 10²⁴ × 200 × 1.602 × 10⁻¹³ ≈ 8.199 × 10¹² joules

Comparative Summary

Now we can compare the energy outputs:

• Energy from fusion of 1.0 kg of hydrogen: ≈ 6.037 × 10¹² joules
• Energy from fission of 1.0 kg of uranium-235: ≈ 8.199 × 10¹² joules

From this comparison, we see that the fission of uranium-235 releases more energy than the fusion of hydrogen when considering the same mass. However, it's essential to note that fusion is more efficient on a larger scale, as it powers stars like our Sun and has the potential to provide vast amounts of energy with minimal waste, while fission has its own challenges and risks associated with radioactive waste and safety.