The nucleus: charge, mass, size and notation
A nucleus holds Z protons and A − Z neutrons, packed so tightly that its radius is R = r₀∛A and every nucleus has nearly the same huge density.
What is inside a nucleus
A nucleus contains protons (charge +e) and neutrons (no charge). Together they are called nucleons. James Chadwick found the neutron in 1932. The atomic number Z is the number of protons; it equals the number of electrons in the neutral atom. The mass number A is the number of nucleons. So the number of neutrons is N = A − Z. A nucleus is written with A raised and Z lowered on the left of the symbol. For example, ³⁵₁₇Cl has 17 protons and 35 − 17 = 18 neutrons.
Mass, charge and the atomic mass unit
The charge of a nucleus is +Ze, with e = 1.6 × 10⁻¹⁹ C. A neutron is slightly heavier than a proton, and almost all the mass of an atom is in its nucleus. Nuclear masses are given in atomic mass units (u): 1 u = 1/12 of the mass of one carbon-12 atom = 1.66 × 10⁻²⁷ kg. On this scale a proton is 1.00727 u and a neutron 1.00865 u. By E = mc², 1 u is equivalent to about 931 MeV (1 MeV = 1.6 × 10⁻¹³ J).
Size and density of a nucleus
Many nuclei are nearly spheres. Experiments show that the radius is R = r₀∛A, with r₀ = 1.2 fm (1 fermi = 10⁻¹⁵ m).
- 1Volume of the nucleus: V = (4/3)πR³ = (4/3)πr₀³A. So the volume is proportional to A.
- 2Mass of the nucleus: M ≈ A × m, where m is the mass of one nucleon.
- 3Density ρ = M/V = Am ÷ (4/3)πr₀³A.
- 4A cancels: ρ = 3m/(4πr₀³).
- 5★So, every nucleus has nearly the same density, about 2.3 × 10¹⁷ kg m⁻³, whatever its size.
★ key step
Worked example: radius and density
Find (a) the radii of the ⁸₄Be and ²¹⁶₈₄Po nuclei, (b) the density of nuclear matter. (r₀ = 1.2 × 10⁻¹⁵ m, mass of a nucleon m = 1.67 × 10⁻²⁷ kg, π = 3.14)
- (a) R = r₀∛A. For A = 8: ∛8 = 2, so R = 1.2 × 2 = 2.4 fm.
- For A = 216: ∛216 = 6, so R = 1.2 × 6 = 7.2 fm.
- (b) ρ = 3m/(4πr₀³). r₀³ = (1.2 × 10⁻¹⁵)³ = 1.728 × 10⁻⁴⁵ m³.
- 4πr₀³ = 4 × 3.14 × 1.728 × 10⁻⁴⁵ = 2.17 × 10⁻⁴⁴ m³.
- ρ = 3 × 1.67 × 10⁻²⁷ ÷ 2.17 × 10⁻⁴⁴ = 2.3 × 10¹⁷ kg m⁻³.
(a) 2.4 fm and 7.2 fm; (b) about 2.3 × 10¹⁷ kg m⁻³, about 10¹⁴ times the density of water
Worked example: counting particles
For the atom ²²⁶₈₈Ra, find the number of protons, neutrons, nucleons and electrons, and the charge on its nucleus. (e = 1.6 × 10⁻¹⁹ C)
- Protons Z = 88; nucleons A = 226.
- Neutrons N = A − Z = 226 − 88 = 138.
- A neutral atom has as many electrons as protons: 88.
- Charge on the nucleus = Ze = 88 × 1.6 × 10⁻¹⁹ C = 1.41 × 10⁻¹⁷ C.
88 protons, 138 neutrons, 226 nucleons and 88 electrons; charge ≈ 1.41 × 10⁻¹⁷ C
Picture the scale
- A nucleus is about 10⁻¹⁵ to 10⁻¹⁴ m across; an atom is about 10⁻¹⁰ m. So an atom is about 10⁵ times wider and about 10¹⁵ times larger in volume. (Your NIOS lesson gives 10⁵ for the volume; that figure is really the ratio of sizes.)
- Nuclear matter is about 10¹⁴ times denser than water.
- If the Earth (6 × 10²⁴ kg) were squeezed to nuclear density, it would be a ball only about 180 m in radius.
- A teaspoon (5 cm³) of nuclear matter would have a mass of about 10¹² kg.
Water has a density of 1000 kg m⁻³ and iron about 7900 kg m⁻³, but a nucleus is about 2.3 × 10¹⁷ kg m⁻³. Neutron stars, the collapsed cores of giant stars, are about as dense as a nucleus.
2 marks: “Show that the density of nuclear matter is the same for all nuclei.” R = r₀∛A, so V = (4/3)πr₀³A; mass M = Am. ρ = M/V = 3m/(4πr₀³), which does not contain A.
2 marks: “Find the radius of a ¹²⁵₅₂Te nucleus. (r₀ = 1.2 fm)” R = 1.2 × ∛125 = 1.2 × 5 = 6.0 fm = 6.0 × 10⁻¹⁵ m.
1 marks: “Define the atomic mass unit.” 1/12 of the mass of one carbon-12 atom, 1.66 × 10⁻²⁷ kg.
Radius by cube root, density stays put: R = 1.2∛A fm and ρ ≈ 2.3 × 10¹⁷ kg m⁻³ for every nucleus.