(a)

At time t=0, the activity (A) is given by:

A₀ = (N/2)⁰ = N

Where N is the initial number of radioactive atoms.

We are given a 500g sample. To calculate N, we need the molar mass of the isotope. Let's assume the isotope is Carbon-14 (¹⁴C) with a molar mass of 14g/mol.

N = 500g / 14g/mol = 35.71 mol

Number of atoms = N * Avogadro's number = 35.71 mol * 6.022 x 10²³ atoms/mol = 2.15 x 10²⁵ atoms.

Therefore, A₀ = 2.15 x 10²⁵ Bq.

At time t=20 years, the activity is:

A(20) = (N/2)^20/10 = (N/2)² = (N/4)

A(20) = (2.15 x 10²⁵ Bq) / 4 = 5.38 x 10²⁴ Bq.

(b)

The half-life of a radioactive isotope is the time it takes for half of the radioactive nuclei in a sample to decay. It is a useful property because it allows us to:

• Determine the age of ancient objects (radiocarbon dating).
• Monitor the rate of decay of radioactive materials.
• Predict the amount of a radioactive substance remaining after a certain time.

The half-life is constant for a given radioactive isotope and is independent of temperature, pressure, and chemical conditions. This makes it a reliable indicator of the age of a sample.

During beta decay, a neutron within the nucleus transforms into a proton. This process results in a reduction in the number of excess neutrons. Specifically, the number of neutrons decreases by one, and the number of protons increases by one. The particle emitted is a beta particle (β-particle), which is essentially a high-speed electron. The equation representing this change is: neutron → proton + electron. This process occurs to achieve a more stable neutron-to-proton ratio within the nucleus.

Isotopes can be radioactive due to two main reasons: an excess of neutrons in the nucleus or the nucleus being too heavy. An excess of neutrons destabilizes the nucleus. The strong nuclear force holds protons and neutrons together. Too many neutrons increase the repulsive force between the protons, making the nucleus less stable. This instability leads to radioactive decay, where the nucleus emits particles (alpha, beta, or gamma) to become more stable. A nucleus that is too heavy also suffers from instability. As the number of protons and neutrons increases, the repulsive electrostatic force between the protons becomes stronger, and the strong nuclear force may not be sufficient to overcome this repulsion. This leads to the nucleus being unstable and undergoing radioactive decay to reduce its mass and increase stability.

• Excess neutrons: Increases proton-proton repulsion, destabilizing the nucleus.
• Too heavy: Electrostatic repulsion between protons overwhelms the strong nuclear force.