Understanding Radioactivity Law

Radioactivity is a natural phenomenon exhibited by certain elements where they spontaneously emit particles or radiation as they decay into more stable forms. The study of radioactivity involves understanding the laws that govern this decay process. The fundamental law that describes the rate of radioactive decay is known as the Radioactive Decay Law.

Radioactive Decay Law

The Radioactive Decay Law states that the rate at which a radioactive substance decays is directly proportional to the number of undecayed nuclei present in the substance at any given time. This law can be mathematically expressed as:

[ \frac{dN}{dt} = -\lambda N ]

Where:

• ( N ) is the number of undecayed nuclei at time ( t )
• ( \lambda ) is the decay constant, characteristic of the radioactive substance
• ( \frac{dN}{dt} ) is the rate of decay, i.e., the number of nuclei decaying per unit time

The negative sign indicates that the number of undecayed nuclei decreases over time.

Integrated Form of Decay Law

By integrating the above differential equation, we can find the number of undecayed nuclei at any time ( t ):

[ N(t) = N_0 e^{-\lambda t} ]

Where:

• ( N_0 ) is the initial number of undecayed nuclei at ( t = 0 )
• ( e ) is the base of the natural logarithm

Half-Life

The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. It is denoted by ( T_{1/2} ) and is related to the decay constant by the formula:

[ T_{1/2} = \frac{\ln(2)}{\lambda} ]

Activity

The activity of a radioactive substance is the number of decays per unit time and is given by:

[ A(t) = \lambda N(t) = \lambda N_0 e^{-\lambda t} ]

The unit of activity is the becquerel (Bq), where 1 Bq corresponds to one decay per second.

Examples

Let's consider an example to illustrate the use of the Radioactive Decay Law.

Example 1: A sample of a radioactive isotope has an initial activity of 1000 Bq. If the half-life of the isotope is 2 hours, what will be its activity after 6 hours?

Solution: First, we calculate the decay constant ( \lambda ) using the half-life formula:

[ \lambda = \frac{\ln(2)}{T_{1/2}} = \frac{\ln(2)}{2 \text{ hours}} ]

Now, we can find the activity after 6 hours using the activity formula:

[ A(6 \text{ hours}) = 1000 \text{ Bq} \cdot e^{-\lambda \cdot 6 \text{ hours}} ]

Substituting the value of ( \lambda ) and calculating, we get the activity after 6 hours.

Differences and Important Points

Here is a table summarizing the key concepts related to the Radioactivity Law:

Concept	Description	Formula
Decay Constant	A constant that characterizes the rate of decay of a radioactive substance.	( \lambda )
Half-Life	The time taken for half of the radioactive nuclei to decay.	( T_{1/2} = \frac{\ln(2)}{\lambda} )
Activity	The number of decays per unit time.	( A(t) = \lambda N(t) )
Number of Undecayed Nuclei	The number of radioactive nuclei that have not decayed at time ( t ).	( N(t) = N_0 e^{-\lambda t} )

Understanding these concepts and their interrelations is crucial for solving problems related to radioactivity in exams. Remember that the half-life is independent of the initial number of nuclei and that the activity of a sample decreases over time as the number of undecayed nuclei decreases.

In conclusion, the Radioactivity Law provides a quantitative description of the decay process of radioactive substances. By using the decay constant, half-life, and activity formulas, one can predict the behavior of a radioactive sample over time, which is essential in fields such as nuclear physics, medicine, and environmental science.