# 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, as follows: • The key features of this graph are:
• The graph starts with a steep gradient at the origin

• 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

• 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 = r0A1/3

• 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)

## Mean Densities of Atoms and Nuclei

#### 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

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

#### Worked example

Calculate the approximate density of a lithium nucleus.

Assume the atomic mass of lithium to be 7u.

Step 1: Write down the equations:

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

Step 2: Combine equations:

• V π R π (rA1/3)3 π ro3 A

Step 3: Calculate the volume of the nucleus:

• V = πro3 A = π x (1.2 x 10-15)x 7 = 5.07 x 10-44 m

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

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

#### Exam Tip

Don't let all the powers and letters confuse you. Work through each step of a question one by one. It is just mass/volume to get the density with a little bit of substitution! ### Get unlimited access

to absolutely everything:

• Unlimited Revision Notes
• Topic Questions
• Past Papers 