Nuclear Radius
Q4. The radius of a nucleus, R, is related to its nucleon number, A, by the equation:
R = r0A1/3
where r0 is a constant.
The table lists values of nuclear radius for various isotopes.
Element |
R/10–15m |
A |
R3/10–45 m3 |
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 r0
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 m3
calculation of r0 (cube root of the gradient)
quote of answer r0 = 1.1(2) × 10–15 m , given to 2 or 3 sf, with unit
Element |
R/10–15m |
A |
A1/3 |
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 r0 = 1.12 × 10–15 m
quote of answer r0 = 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(πR3)
- volume of a sphere
∴M = 4/3(πR3ρ)
ρ = 3/4(M/πR3)
If A = 1 then R = r0 = 1.12 × 10–15 mand m = 1.67 × 10–27kg
ρ = 3/4(1.67 × 10–27/π{1.12 × 10–15}3 )
ρ = 2.8 × 1017 kg m–3
(4 marks)
(Total 12 marks)
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