**Definition:**

The half-life of a radioactive substance is the time it takes for half of the radioactive nuclei in a sample to undergo radioactive decay. It is a characteristic property of each radioactive isotope and is independent of the initial quantity of the substance.

**Key Points**

**1. Mathematical Representation**

The equation gives the mathematical expression for the decay of a radioactive substance:

Where:

N(t) is the remaining quantity at time *t*,

N*0* is the initial quantity

T*1/2 *is the half-life.

**2. Dependence on Isotope**

Each radioactive isotope has its characteristic half-life. Some isotopes have very short half-lives (fractions of a second), while others have extremely long half-lives (millions or billions of years).

**3. Constant Fraction Decay**

The half-life represents a constant fraction of the radioactive material that decays during each interval. Regardless of the initial amount, the same fraction will decay in each successive half-life.

**4. Decay Curve**

A decay curve can graphically represent the decay of a radioactive substance. The curve shows an exponential decrease in the quantity of the substance over time.

**5. Practical Applications**

**Medicine:** Used in nuclear medicine for planning the timing of diagnostic procedures and treatments involving radioactive isotopes.

**Archaeology:** Used in radiocarbon dating to estimate the age of organic materials.

**Environmental Science**: Used to determine the age of rocks and minerals.

**6. Parent and Daughter Isotopes**

In radioactive decay, the original radioactive isotope is referred to as the parent isotope, and the stable isotope resulting from the decay is the daughter isotope.

**7. Stability and Instability**

Isotopes with short half-lives are generally more unstable and undergo rapid radioactive decay, while those with long half-lives are more stable and decay slowly over extended periods.

**8. Half-life and Radioactive Decay Series**

Some radioactive isotopes are part of a decay series where multiple isotopes in the series decay in a sequential manner, each with its characteristic half-life.

Understanding the half-life of a radioactive isotope is crucial for various scientific disciplines, including nuclear physics, medicine, archaeology, and environmental science. It provides valuable information about the decay rate, the stability of isotopes, and the timing of processes involving radioactive materials.