To find the nuclear energy required to separate all the neutrons and protons in a coin made entirely of copper-63 (63Cu), we need to follow a few logical steps. This involves calculating the number of atoms in the coin, determining the total number of nucleons (protons and neutrons), and then using the binding energy per nucleon to find the total energy required for separation.
Step 1: Determine the Number of Atoms in the Coin
The mass of the coin is given as 3.0 grams. First, we convert this mass into atomic mass units (u) since the mass of a single copper atom is given in atomic mass units. The conversion factor is:
- 1 g = 1000 mg
- 1 u = 1.66053906660 x 10-27 kg
Now, we can calculate the number of moles of copper in the coin:
Number of moles = mass (g) / molar mass (g/mol)
The molar mass of copper-63 is approximately 63 g/mol. Thus:
Number of moles = 3.0 g / 63 g/mol ≈ 0.04762 mol
Next, we convert moles to atoms using Avogadro's number (approximately 6.022 x 1023 atoms/mol):
Number of atoms = Number of moles × Avogadro's number
Number of atoms ≈ 0.04762 mol × 6.022 x 1023 atoms/mol ≈ 2.86 x 1022 atoms
Step 2: Calculate the Total Number of Nucleons
Each copper-63 atom contains 29 protons and 34 neutrons (since 63 - 29 = 34). Therefore, the total number of nucleons in the coin can be calculated as follows:
Total nucleons per atom = protons + neutrons = 29 + 34 = 63
Total nucleons in the coin = Number of atoms × Total nucleons per atom
Total nucleons ≈ 2.86 x 1022 atoms × 63 nucleons/atom ≈ 1.80 x 1024 nucleons
Step 3: Calculate the Binding Energy
The binding energy per nucleon for copper-63 is approximately 8.6 MeV. To find the total binding energy for all nucleons, we multiply the binding energy per nucleon by the total number of nucleons:
Total binding energy (in MeV) = Binding energy per nucleon × Total nucleons
Total binding energy ≈ 8.6 MeV/nucleon × 1.80 x 1024 nucleons ≈ 1.55 x 1025 MeV
Step 4: Convert MeV to Joules
To express this energy in joules, we use the conversion factor where 1 MeV = 1.602 x 10-13 joules:
Total energy (in joules) = Total binding energy (in MeV) × Conversion factor
Total energy ≈ 1.55 x 1025 MeV × 1.602 x 10-13 J/MeV ≈ 2.48 x 1012 J
Final Thoughts
The nuclear energy required to separate all the neutrons and protons in a 3.0 g coin made entirely of copper-63 atoms is approximately 2.48 x 1012 joules. This calculation illustrates the immense energy associated with nuclear forces, which hold the atomic nucleus together.
