Half-life
Understanding Half-life
Half-life is a concept widely used in physics, chemistry, and biology to describe the time required for half of the unstable nuclei in a sample of a radioactive substance to undergo radioactive decay. It is a measure of the stability of a radioactive isotope and is a fundamental property of each radioactive species.
Definition of Half-life
The half-life (denoted as $t_{1/2}$) of a radioactive substance is the time taken for the number of nuclei of the isotope in the sample to reduce to half its initial value. This is a probabilistic measure since it is based on the likelihood of decay of individual atoms, which is a random process.
Mathematical Representation
The decay of radioactive isotopes can be described by the following first-order differential equation:
[ \frac{dN}{dt} = -\lambda N ]
Where:
- $N$ is the number of undecayed nuclei,
- $\lambda$ is the decay constant (probability of decay of a nucleus per unit time),
- $t$ is time.
The solution to this differential equation gives us the decay law:
[ N(t) = N_0 e^{-\lambda t} ]
Where:
- $N_0$ is the initial number of undecayed nuclei at time $t=0$,
- $e$ is the base of the natural logarithm.
The half-life is related to the decay constant by the following equation:
[ t_{1/2} = \frac{\ln(2)}{\lambda} ]
Table of Differences and Important Points
| Property | Description |
|---|---|
| Definition | Time taken for half the nuclei in a sample to decay |
| Symbol | $t_{1/2}$ |
| Unit | Usually seconds (s), but can be in any unit of time |
| Relation to Decay Constant | Inversely proportional; as decay constant increases, half-life decreases |
| Dependence | Unique to each isotope; does not depend on the amount of substance or its physical state |
| Calculation | Can be calculated using the decay law or directly from the decay constant |
Examples
Example 1: Calculating Half-life from Decay Constant
Suppose a radioactive isotope has a decay constant $\lambda = 0.693 \text{ day}^{-1}$. The half-life can be calculated as follows:
[ t_{1/2} = \frac{\ln(2)}{\lambda} = \frac{0.693}{0.693 \text{ day}^{-1}} = 1 \text{ day} ]
Example 2: Using Half-life to Determine Remaining Nuclei
If we have a sample with an initial number of nuclei $N_0 = 1000$ and a half-life of 2 hours, how many nuclei will remain undecayed after 6 hours?
Using the decay law:
[ N(t) = N_0 e^{-\lambda t} ]
First, we find the decay constant using the half-life:
[ \lambda = \frac{\ln(2)}{t_{1/2}} = \frac{0.693}{2 \text{ hours}} = 0.3465 \text{ hour}^{-1} ]
Now, we can calculate the number of nuclei remaining after 6 hours:
[ N(6 \text{ hours}) = 1000 \cdot e^{-0.3465 \cdot 6} \approx 1000 \cdot e^{-2.079} \approx 1000 \cdot 0.125 \approx 125 ]
So, after 6 hours, approximately 125 nuclei will remain undecayed.
Conclusion
Half-life is a crucial concept in understanding radioactive decay and is used in various fields such as radiometric dating, nuclear medicine, and environmental science. It provides a measure of the rate at which a radioactive substance transforms and is independent of the initial amount of the substance. By understanding half-life, we can predict the behavior of radioactive materials over time and utilize this knowledge in practical applications.