8.3.7 Nuclear Density

• Assuming that the nucleus is spherical, its volume is equal to:

• Where R is the nuclear radius, which is related to mass number, A, by the equation:

• Where R₀ is a constant of proportionality

• Combining these equations gives:

• Therefore, the nuclear volume, V, is proportional to the mass of the nucleus, A

• Mass (m), volume (V), and density (ρ) are related by the equation:

• The mass, m, of a nucleus is equal to:

m = Au

• Where:
  • A = the mass number
  • u = atomic mass unit

• Using the equations for mass and volume, nuclear density is equal to:

• Since the mass number A cancels out, the remaining quantities in the equation are all constant

• Therefore, this shows the density of the nucleus is:
  • Constant
  • Independent of the radius

• The fact that nuclear density is constant shows that nucleons are evenly separated throughout the nucleus regardless of their size

• Using the equation derived above, the density of the nucleus can be calculated:

• Where:
  • Atomic mass unit, u = 1.661 × 10^–27 kg
  • Constant of proportionality, R₀ = 1.05 × 10^–15 m

• Substituting the values gives a density of:

• The accuracy of nuclear density depends on the accuracy of the constant R₀, as a guide nuclear density should always be of the order 10¹⁷ kg m^–3
• Nuclear density is significantly larger than atomic density, this suggests:
  • The majority of the atom’s mass is contained in the nucleus
  • The nucleus is very small compared to the atom
  • Atoms must be predominantly empty space