  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

Method 1

Cube each side of the equation, giving you:

R3 = r03A

You then need to calculate R3 in the table, and plot R3against A. That will give you a straight line graph (through the origin) with a gradient of r03.

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 Method 2

Calculate cube root of A for the table, and plot R against A1/3. That will give you a straight line graph (through the origin) with a gradient of r0.

 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)