The law of radioactivity states that the probability of a radioactive nucleus decaying per unit time is constant and independent of time.
In simpler terms, this means:
- Randomness: Radioactive decay is a random process at the atomic level. We cannot predict when a specific atom will decay.
- Constant Probability: For a given radioactive isotope, the probability of decay in a specific time interval is always the same, no matter how old the sample is.
- Decay Constant (λ): This constant probability is represented by the decay constant, denoted by the Greek letter lambda (λ). It is a characteristic property of each radioactive isotope.
- Exponential Decay: The rate of decay (number of decays per unit time) is proportional to the number of radioactive nuclei present. This leads to an exponential decay of the number of radioactive nuclei over time.
Mathematical Representation:
The law of radioactivity can be expressed mathematically as:
dN/dt = -λN
Where:
- dN/dt is the rate of change of the number of radioactive nuclei with respect to time (decay rate).
- λ is the decay constant.
- N is the number of radioactive nuclei present at time t.
The solution to this differential equation is:
N(t) = N₀e^(-λt)
Where:
- N(t) is the number of radioactive nuclei remaining at time t.
- N₀ is the initial number of radioactive nuclei at time t=0.
- e is the base of the natural logarithm.
Key Concepts Related to the Law of Radioactivity:
-
Half-life (t₁/₂): The time it takes for half of the radioactive nuclei in a sample to decay. It is related to the decay constant by the following equation:
t₁/₂ = ln(2) / λ ≈ 0.693 / λ
-
Activity (A): The rate at which radioactive decays occur in a sample. It is proportional to the number of radioactive nuclei present:
A = λN
Example:
Imagine you have 1000 atoms of a radioactive isotope. If the half-life of the isotope is 1 hour, then after 1 hour, you will have approximately 500 atoms remaining. After another hour, you will have approximately 250 atoms, and so on. The decay constant (λ) determines how quickly this process happens. An isotope with a large λ will decay quickly, while one with a small λ will decay slowly.
