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 × 10–15m and 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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