Atomic Masses and Composition of Nucleus

Nucleus

Atomic Nucleus

Terms Related to Nucleus:

(I) Atomic Number: The number of protons in the nucleus of an atom of the element is called atomic number (Z) of the element.

(II) Mass Number: The total number of protons and neutrons present inside the nucleus of an atom of the element is called mass number (A) of the element.

(III) Nuclear Size: The radius of the nucleus R ∝ A^1/3

⇒ R = R_o A^1/3

where, R_o = 1.1 * 10^-15 m is an empirical constant.

(IV) Nuclear Density: Nuclear density is independent of mass number and therefore same for all nuclei.

ρ = mass of nucleus / volume of nucleus ⇒ ρ = 3m / 4π R³_o

where, m = average mass of a nucleon.

(V) Atomic Mass Unit: It is defined as 1 / 12th the mass of carbon nucleus.

It is abbreviated as amu and often denoted by u. Thus

1 amu = 1.992678 * 10^-26 / 12 kg

= 1.6 * 10^-27 kg = 931 Me V

Isotopes:

The atoms of an element having same atomic number but different mass numbers are called isotopes.

e.g., ₁H¹, ₁H², ₁H³ are isotopes of hydrogen.

Fig: Isotopes of hydrogen

Isobars:

The atoms of different elements having same mass numbers but different atomic numbers, are called isobars.

e.g., ₁H³, ₂He³ and ₁₀Na²², ₁₀Ne²² are isobars.

Isotones:

The atoms of different elements having different atomic numbers and different mass numbers but having same number of neutrons, are called isotones.

e.g., ₁H³, ₂He⁴ and ₆C¹⁴, ₈O¹⁶ are isobars.

Isomers:

Atoms having the same mass number and the same atomic number but different radioactive properties are called isomers.

Nuclear Force:
The force acting inside the nucleus or acting between nucleons is called nuclear force. Nuclear forces are the strongest forces in nature.

• It is a very short range attractive force.
• It is non-central. non-conservative force.
• It is neither gravitational nor electrostatic force.
• It is independent of charge.
• It is 100 times that of electrostatic force and 10³⁸ times that of gravitational force.

According to the Yukawa, the nuclear force acts between the nucleon due to continuous exchange of meson particles.

Mass Defect:

The difference between the sum of masses of all nucleons (M) mass of the nucleus (m) is called mass defect.

Mass Defect (Δm) = M - m = [Zm_p + (A - Z)m_n - m_n]

Nuclear Binding Energy:

The minimum energy required to separate the nucleons up to an infinite distance from the nucleus, is called nuclear binding energy.

Nuclear binding energy per nucleon = Nuclear binding energy / Total number of nucleons

Binding energy, E_b = [Zm_p + (A - Z) m_n - m_N]c²

Binding Energy Curve (BE/A vs Mass Number A)

This is the graph of binding energy per nucleon (BE/A) plotted against mass number (A).

Key points from the graph:

• For very light nuclei (A < 20), BE/A is low and rises sharply - this means fusion of light nuclei releases energy
• BE/A rises and reaches a maximum of about 8.8 MeV at A ≈ 56 (Iron - Fe-56)
• Fe-56 is the most stable nucleus in nature
• For heavy nuclei (A > 56), BE/A slowly decreases - this means fission of heavy nuclei releases energy
• The curve explains why both fission and fusion release energy

Conclusion from the curve:

• Fission: Heavy nucleus (U-235) → medium nuclei → BE/A increases → energy released
• Fusion: Light nuclei (H) → heavier nucleus → BE/A increases → energy released
• Both processes move towards the peak (Fe-56) and release energy

Nuclear Stability - N/Z Ratio Rule

The stability of a nucleus depends on the ratio of neutrons (N) to protons (Z):

• For light nuclei (A < 40): Stable nuclei have N/Z ≈ 1
• For heavy nuclei (A > 40): Stable nuclei need N/Z > 1 (more neutrons needed to overcome proton repulsion)
• If N/Z is too high → nucleus undergoes β⁻ decay (neutron converts to proton)
• If N/Z is too low → nucleus undergoes β⁺ decay or electron capture (proton converts to neutron)
• If A > 83 → nucleus undergoes α decay to reduce both N and Z

Magic Numbers: Nuclei with proton or neutron number equal to 2, 8, 20, 28, 50, 82, or 126 are exceptionally stable. These are called magic numbers.

Packing Fraction (P)

p = (Exact nuclear mass) - (Mass number) / Mass number = M - A / M

• The larger the value of packing friction. greater is the stability of the nucleus.
• The nuclei containing even number of protons and even number of neutrons are most stable.
• The nuclei containing odd number of protons and odd number of neutrons are most unstable.

Solved Examples

Q1. The radius of a nucleus of mass number A is R. The radius of a nucleus of mass number 8A will be:

Solution:

R ∝ A^(1/3) 
R₁/R₂ = (A₁/A₂)^(1/3)
R/R₂ = (A/8A)^(1/3) = (1/8)^(1/3) = 1/2 R₂ = 2R 

Q2. If the nuclear radius of Al-27 is 3.6 fm, find the approximate nuclear radius of Cu-64.

Solution:

R ∝ A^(1/3) 
R(Cu)/R(Al) = (64/27)^(1/3) = (64/27)^(1/3) = 4/3 R(Cu) = 3.6 × 4/3 = 4.8 fm 

Q3. Which of the following has the highest binding energy per nucleon?
(A) ²H 
(B) ⁵⁶Fe 
(C) ²³⁵U
 (D) ⁴He

Answer: (B)

Solution:

Fe-56 has the highest BE/nucleon of approximately 8.8 MeV. It sits at the peak of the binding energy curve and is the most stable nucleus. 

Q4. The mass defect of a nucleus is 0.04 u. Find the binding energy in MeV.

Solution:

BE = Δm × 931.5 BE = 0.04 × 931.5 = 37.26 MeV 

Q5. Which of the following statements is correct about nuclear forces?
(A) Nuclear force is a long range force 
(B) Nuclear force is charge dependent
 (C) Nuclear force becomes repulsive at very short distances (< 0.8 fm)
 (D) Nuclear force follows inverse square law

Answer: (C)

Solution:

Nuclear force is attractive between 0.8 fm to ~2-3 fm but becomes repulsive below 0.8 fm. This repulsion prevents the nucleus from collapsing. It is charge independent and short range, and does NOT follow inverse square law.