The Mathematics of Instability
Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation (such as alpha particles, beta particles, or gamma rays).
This process is fundamentally random; it is impossible to predict exactly when a single specific atom will decay. However, when dealing with trillions of atoms, the overall rate of decay follows a very precise mathematical curve called exponential decay.
The Decay Constant (λ)
The rate at which a substance decays is determined by its Decay Constant ($\lambda$).
- A large decay constant means the substance is highly radioactive and decays very quickly.
- A small decay constant means the substance is relatively stable and decays very slowly over thousands or billions of years.
The Formula
Example Calculation
You start with $1,000,000$ atoms of a radioactive isotope ($N_0$) that has a decay constant of $0.05 , \text{s}^{-1}$. How many atoms remain after $20 , \text{seconds}$?
- Multiply Constant by Time: $-0.05 \times 20 = -1.0$.
- Calculate Exponential ($e^{-1}$): $e^{-1} \approx 0.3678$.
- Multiply by Initial Amount: $1,000,000 \times 0.3678 = 367,800$.
After 20 seconds, only $367,800$ unstable atoms remain. The rest have decayed into a stable element.