Nuclear Chemistry: Forces, Decay, Stability, and Kinetics

Nuclear Chemistry and Radioactivity

Physical Forces in Nature

There are four fundamental physical forces that govern interactions in the universe, each with distinct roles in atomic and nuclear processes:

• Gravitation: The weakest force, responsible for the attraction between masses.

• Electromagnetic Force: Governs interactions between charged particles, such as electrons and protons.

• Weak Nuclear Force: Responsible for processes like beta decay, where a neutron transforms into a proton, emitting an electron and an antineutrino.

• Strong Nuclear Force: The strongest force, responsible for holding protons and neutrons together in the nucleus, providing nuclear binding energy.

Key Nuclear Particles and Notation

• Proton (p): $^1_1\text{H}$

• Neutron (n): $^1_0\text{n}$

• Electron (e-): $^0_{-1}\text{e}$

• Positron (e+): $^0_{+1}\text{e}$

• Antineutrino (\bar{\nu}_e)$

Isotopes are atoms with the same number of protons (Z) but different numbers of neutrons (A-Z), resulting in different mass numbers (A).

Radioactive Decay

Radioactive decay is the spontaneous decomposition of unstable atomic nuclei (nuclides). Of approximately 2000 known nuclides, only about 279 are stable with respect to radioactive decay.

• Alpha (α) Decay: Emission of a helium nucleus ($^4_2\text{He}$ or $\alpha$-particle).

• Beta (β) Decay: Transformation of a neutron into a proton (or vice versa), accompanied by the emission of an electron ($\beta^-$) or positron ($\beta^+$).

• Electron Capture: The nucleus captures an inner orbital electron, converting a proton into a neutron.

• Gamma (γ) Emission: Release of high-energy photons ($\gamma$-rays) from an excited nucleus.

General Decay Equations

• Alpha decay: $^A_Z\text{X} \rightarrow ^{A-4}_{Z-2}\text{Y} + ^4_2\alpha$

• Beta decay: $^A_Z\text{X} \rightarrow ^A_{Z+1}\text{Y} + ^0_{-1}\beta$

• Positron emission: $^A_Z\text{X} \rightarrow ^A_{Z-1}\text{Y} + ^0_{+1}\beta$

• Electron capture: $^A_Z\text{X} + ^0_{-1}\text{e} \rightarrow ^A_{Z-1}\text{Y}$

Conservation Laws: In all nuclear equations, the sum of atomic numbers (Z) and mass numbers (A) must be conserved on both sides.

Band of Stability and Nuclear Stability

The stability of a nucleus depends on its neutron-to-proton (N/Z) ratio. Stable nuclides reside within the band (or belt) of stability:

• For light elements (low Z), stability is achieved when N ≈ Z.

• For heavier elements, more neutrons are needed (N > Z) to offset proton-proton repulsion.

• Nuclides outside the band of stability undergo radioactive decay to move toward stability.

Decay Modes Relative to the Band of Stability:

• Nuclides above the band (high N/Z): Undergo β- decay (neutron → proton).

• Nuclides below the band (low N/Z): Undergo β+ decay (positron emission) or electron capture (proton → neutron).

• Very heavy nuclides: Undergo α-decay to reduce both protons and neutrons.

Table: Decay Modes and N/Z Ratios

Position Relative to Band	Decay Mode	Effect
Above (high N/Z)	β- emission	n ↓, p ↑
Below (low N/Z)	β+ emission or electron capture	n ↑, p ↓
Very heavy	α emission	Reduces both n and p

Radioactive Decay Kinetics

All radioactive decays follow first-order kinetics:

• Rate law: $\text{rate} = -\frac{dN}{dt} = kN$

• Integrated form: $\ln\left(\frac{N}{N_0}\right) = -kt$

• Half-life: $t_{1/2} = \frac{\ln 2}{k} = \frac{0.693}{k}$

Where:

• N = number of radioactive nuclei at time t

• N0 = initial number of nuclei

• k = decay constant

Example Calculation:

• For $^{238}_{92}\text{U}$: $t_{1/2} = 4.51 \times 10^9$ y, $k = \frac{0.693}{4.51 \times 10^9 \text{ y}} = 1.547 \times 10^{-10} \text{ y}^{-1}$

Decay Series and Half-Lives

Some heavy nuclides decay through a series of steps, forming a decay series until a stable nuclide is reached. Each step has its own characteristic half-life and decay mode.

Radiocarbon Dating (Carbon-14 Dating)

Radiocarbon dating is a method for determining the age of formerly living materials by measuring the ratio of $^{14}\text{C}$ to $^{12}\text{C}$.

• $^{14}\text{C}$ is produced in the atmosphere by cosmic rays and incorporated into living organisms.

• After death, $^{14}\text{C}$ decays with a half-life of 5730 years ($k = 1.21 \times 10^{-4} \text{ y}^{-1}$).

• Decay law: $\ln\left(\frac{N}{N_0}\right) = -kt$

Application: By measuring the remaining $^{14}\text{C}$ in a sample, the time since the organism's death can be estimated.

Summary Table: Key Nuclear Particles

Particle	Symbol	Charge	Mass (amu)
Proton	$^1_1\text{H}$	+1	1.0073
Neutron	$^1_0\text{n}$	0	1.0087
Electron	$^0_{-1}\text{e}$	-1	0.00055
Alpha particle	$^4_2\text{He}$	+2	4.0026

Additional info:

• The "magic numbers" (2, 8, 20, 28, 50, 82, 126) refer to numbers of protons or neutrons that confer extra stability to nuclei.

• 1 electron volt (eV) = 96 kJ/mol, a useful conversion for nuclear energy calculations.