Nuclear Radius
Q4. The radius of a nucleus, R, is related to its nucleon number, A, by the equation:
R = r_{0}A^{1/3}
where r_{0} is a constant.
The table lists values of nuclear radius for various isotopes.
Element |
R/10^{–15}m |
A |
R^{3}/10^{–45} m^{3} |
carbon |
2.66 |
12 |
18.8 |
silicon |
3.43 |
28 |
40.4 |
iron |
4.35 |
56 |
82.3 |
tin |
5.49 |
120 |
165 |
lead |
6.66 |
208 |
295 |
(a) Use the data to plot a straight line graph and use it to estimate the value of r_{0}
calculate data for table
plot graph: units on axes scales chosen to ensure points spread across more than 50% of page in both x and y directions
plot data (lose one mark for each error)
calculation of gradient of line = 1.41 × 10^{–45} m^{3}
calculation of r_{0} (cube root of the gradient)
quote of answer r_{0} = 1.1(2) × 10^{–15} m , given to 2 or 3 sf, with unit
Element |
R/10^{–15}m |
A |
A^{1/3}^{} |
carbon |
2.66 |
12 |
2.29 |
silicon |
3.43 |
28 |
3.04 |
iron |
4.35 |
56 |
3.83 |
tin |
5.49 |
120 |
4.93 |
lead |
6.66 |
208 |
5.93 |
calculate data for table
plot graph: units on axes scales chosen to ensure points spread across more than 50% of page in both x and y directions
plot data (lose one mark for each error)
calculation of gradient of line r_{0} = 1.12 × 10^{–15} m
quote of answer r_{0} = 1.1(2) × 10^{–15} m , given to 2 or 3 sf, with unit
(8 marks)
(b) Assuming that the mass of a nucleon is 1.67 × 10^{–27} kg, calculate the approximate density of nuclear matter, stating one assumption you have made.
Assuming that:
- the nucleus is spherical OR
- all nuclei have the same density OR
- that total mass is equal to the mass of constituent single nuclei (ignoring the mass difference)
OR ignoring the gaps between nucleons
any one assumption
ρ = M/V
V = ^{4}/_{3}(πR^{3})
- volume of a sphere
∴M = ^{4}/_{3}(πR^{3}ρ)
ρ = ^{3}/_{4}(M/πR^{3})
If A = 1 then R = r_{0} = 1.12 × 10^{–15} mand m = 1.67 × 10^{–27}kg
ρ = ^{3}/_{4}(1.67 × 10^{–27}/π{1.12 × 10^{–15}}_{}^{3 })
ρ = 2.8 × 10^{17} kg m^{–3}
(4 marks)
(Total 12 marks)