6.7.5 Nuclear Radius & Density

• The radii of some nuclei are shown in the table below:

• In general, nuclear radii are of the order 10^–15 m or 1 fm

• The nuclear radius, R, varies with nucleon number, A as follows:

• The key features of this graph are:
  • The graph starts with a steep gradient at the origin
  • Then the gradient gradually decreases to almost horizontal

• This means that
  • As more nucleons are added to a nucleus, the nucleus gets bigger
  • However, the number of nucleons A is not proportional to its size r

Calculating the Nuclear Radius

• The radius of nuclei depends on the nucleon number, A of the atom
• This makes sense because as more nucleons are added to a nucleus, more space is occupied by the nucleus, hence giving it a larger radius
• The exact relationship between the radius and nucleon number can be determined from experimental data
• By doing this, physicists were able to deduce the following relationship:

R = r₀A^1/3

• Where:
  • R = nuclear radius (m)
  • A = nucleon / mass number
  • R₀ = constant of proportionality = 1.2 fm = 1.2 x 10⁻¹⁵ m (the radius of a proton)

Equation for 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:

• This shows that 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

Step 1: Write down the equations: 

• Density: 
• Volume of a sphere: 
• From the data booklet: nuclear radius, 
  • Where:
    • r_o = constant = 1.2 x 10^-15 
    • A = mass number = 7 for lithium

Step 2: Combine equations: 

• V = π R³  = π (r_o A^1/3)³ = π r_o³ A

Step 3: Calculate the volume of the nucleus: 

• V = πr_o³ A = π x (1.2 x 10^-15)³ x 7 = 5.07 x 10^-44 m³ 

Step 4: Calculate mass of lithium nucleus: 

• Mass = Au = 7 x (1.661 x 10^-27) = 1.1627 x 10^-26 kg

Step 5: Calculate the density: 

• Density =  =  = 2.29 x 10¹⁷ kg m^-3

Step 6: Finalise your answer: 

• The density of a lithium nucleus is 2.3 x 10¹⁷ kg m^-3 (2 s.f.)