Nuclear Density (AQA A Level Physics): Revision Note

Constant density of nuclear material

• Assuming that the nucleus is spherical, its volume is equal to:

$V=\frac{4}{3}\pi R^3$

• Where $R$ is the nuclear radius, which is related to mass number, $A$, by the equation:

$R=R_0{A}^{\frac{1}{3}}$

• Where $R_0$ is a constant of proportionality

• Combining these equations gives:

$V=\frac{4}{3}\pi {\left(R_0{A}^{\frac{1}{3}}\right)}^3=\frac{4}{3}\pi {R_0}^{3}A$

• Therefore, the nuclear volume, $V$, is proportional to the mass of the nucleus, $A$

• Mass ($m$), volume ($V$), and density ($\rho$) are related by the equation:

$\rho =\frac{m}{V}$

• 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:

$\rho =\frac{Au}{\frac{4}{3}\pi {R_0}^{3}A}=\frac{3u}{4\pi {R_0}^3}$

• 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

  • independent of the radius

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

Nuclear density

• Using the equation derived above, the density of the nucleus can be calculated:

$\rho =\frac{3u}{4\pi {R_0}^3}$

• Where:

  • Atomic mass unit, $u=1.661\times {10}^{-27}\mathrm{kg}$

  • Constant of proportionality, $R_0=1.05\times {10}^{-15}\mathrm{m}$

• Substituting the values gives a density of:

$\rho =\frac{3\times \left(1.661\times {10}^{-27}\right)}{4\pi {\times \left(1.05\times {10}^{-15}\right)}^3}=3.4\times {10}^{17}\mathrm{kg}{\mathrm{m}}^{-3}$

• The accuracy of nuclear density depends on the accuracy of the constant $R_0$

  • As a guide, nuclear density should always be of the order 10¹⁷ kg m^–3

• Nuclear density is significantly larger than atomic density

• This suggests that:

  • the majority of the atom’s mass is contained in the nucleus

  • the nucleus is very small compared to the atom

  • atoms must be predominantly empty space