Nuclear Radius
- The radius of a nucleus depends on the nucleon number A of the atom
- The greater the number of nucleons a nucleus has, the greater the space the nucleus occupies, 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:
- Where:
- R = nuclear radius (m)
- A = nucleon / mass number
- R0 = constant of proportionality = 1.20 fm
Nuclear Density
- Assuming that the nucleus is spherical, its volume is equal to:
- Combining this with the expression for nuclear radius gives:
- This tells us that the nuclear volume V is proportional to the mass of the nucleus m, which is equal to
- Where u = atomic mass unit (kg)
- Using the definition for density, nuclear density is equal to:
- Since the mass number A cancels out, the remaining quantities in the equation are all constants
- Therefore, this shows the density of the nucleus is:
- The same for all nuclei
- Independent of the radius
- The fact that nuclear density is constant shows that nucleons are evenly separated throughout the nucleus regardless of their size
- 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 which 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
Worked example
Determine the value of nuclear density.
You may take the constant of proportionality R0 to be 1.20 fm.
Answer:
Step 1: Derive an expression for nuclear density
- Using the equation derived above, the density of the nucleus is:
Step 2: List the known quantities
- Atomic mass unit, u = 1.661 × 10–27 kg
- Constant of proportionality, R0 = 1.20 fm = 1.20 × 10–15 m
Step 3: Substitute the values to determine the nuclear density