Decay constant
Decay Constant
The decay constant, often denoted by the symbol $\lambda$, is a parameter that characterizes the rate at which a radioactive substance undergoes radioactive decay. It is a fundamental property of each radioactive isotope and is inversely proportional to the half-life of the substance.
Understanding Radioactive Decay
Radioactive decay is a random process at the level of single atoms, in that, according to quantum theory, it is impossible to predict when a particular atom will decay. However, for a large number of atoms, the decay rate is predictable. This rate is expressed in terms of the decay constant.
Decay Rate and Activity
The decay rate, or activity, of a radioactive substance is defined as the number of decays per unit time. It is given by the equation:
$$ A(t) = \lambda N(t) $$
where:
- $A(t)$ is the activity at time $t$,
- $\lambda$ is the decay constant,
- $N(t)$ is the number of undecayed atoms at time $t$.
Relation to Half-Life
The half-life ($t_{1/2}$) of a radioactive isotope is the time required for half of the radioactive atoms in a sample to decay. The decay constant is related to the half-life by the following equation:
$$ t_{1/2} = \frac{\ln(2)}{\lambda} $$
where $\ln(2)$ is the natural logarithm of 2 (approximately 0.693).
Exponential Decay Law
The number of undecayed atoms as a function of time is described by the exponential decay law:
$$ N(t) = N_0 e^{-\lambda t} $$
where:
- $N_0$ is the initial number of atoms,
- $e$ is the base of the natural logarithm,
- $t$ is the time elapsed.
Table of Differences and Important Points
| Property | Symbol | Relation to Decay Constant | Units |
|---|---|---|---|
| Decay Constant | $\lambda$ | Fundamental property of a radioactive isotope | s-1 |
| Half-Life | $t_{1/2}$ | $t_{1/2} = \frac{\ln(2)}{\lambda}$ | seconds, years, etc. |
| Activity | $A(t)$ | $A(t) = \lambda N(t)$ | decays per second (Becquerel, Bq) |
| Initial Activity | $A_0$ | $A_0 = \lambda N_0$ | Bq |
| Number of Atoms | $N(t)$ | $N(t) = N_0 e^{-\lambda t}$ | dimensionless |
Examples
Example 1: Calculating Half-Life from Decay Constant
Suppose a radioactive isotope has a decay constant of $\lambda = 0.01 \text{ day}^{-1}$. To find the half-life, we use the relation:
$$ t_{1/2} = \frac{\ln(2)}{\lambda} = \frac{0.693}{0.01 \text{ day}^{-1}} = 69.3 \text{ days} $$
Example 2: Determining the Remaining Number of Atoms
If we start with $N_0 = 1000$ atoms of a radioactive substance with a decay constant of $\lambda = 0.001 \text{ s}^{-1}$, the number of atoms remaining after 1000 seconds can be calculated as:
$$ N(1000) = N_0 e^{-\lambda t} = 1000 e^{-0.001 \times 1000} \approx 367.88 $$
Therefore, approximately 368 atoms remain undecayed after 1000 seconds.
Example 3: Finding Activity After a Certain Time
Given an initial activity of $A_0 = 500 \text{ Bq}$ and a decay constant of $\lambda = 0.002 \text{ s}^{-1}$, the activity after 2000 seconds is:
$$ A(2000) = A_0 e^{-\lambda t} = 500 \text{ Bq} \times e^{-0.002 \times 2000} \approx 67.23 \text{ Bq} $$
In conclusion, the decay constant is a crucial parameter in describing the radioactive decay process. It is directly related to the half-life and activity of a radioactive substance and is used to calculate the number of undecayed atoms over time. Understanding the decay constant is essential for fields such as nuclear physics, radiometric dating, and nuclear medicine.