## Half-Life and Decay Constant

The** radioactive decay law** states that the probability per unit time that a nucleus will decay is a constant, independent of time. This constant is called the **decay constant** and is denoted by λ, “lambda”.

One of the most useful terms for estimating how quickly a nuclide will decay is the **radioactive half-life** (**t**** _{1/2}**). The

**half-life**is defined as the amount of time it takes for a given isotope to lose half of its radioactivity.

In calculations of radioactivity one of two parameters (**decay constant** or **half-life**), which characterize the rate of decay, must be known. There is a relation between the half-life (t_{1/2}) and the decay constant λ. The relationship can be derived from decay law by setting N = ½ N_{o}. This gives:

where *ln 2 *(the natural log of 2) equals 0.693. If the decay constant (λ) is given, it is easy to calculate the half-life, and vice-versa.

## Example – Radioactive Decay Law

A sample of material contains 1 mikrogram of iodine-131. Note that, iodine-131 plays a major role as a radioactive isotope present in nuclear fission products, and it a major contributor to the health hazards when released into the atmosphere during an accident. Iodine-131 has a half-life of 8.02 days.

**Calculate:**

- The number of iodine-131 atoms initially present.
- The activity of the iodine-131 in curies.
- The number of iodine-131 atoms that will remain in 50 days.
- The time it will take for the activity to reach 0.1 mCi.

**Solution:**

- The number of atoms of iodine-131 can be determined using isotopic mass as below.

**N**_{I-131}** = m**_{I-131}** . N**_{A}** / M**_{I-131}

**N**_{I-131 }**= (1 μg) x (6.02×10**^{23}** nuclei/mol) / (130.91 g/mol)**

**N**_{I-131}** = 4.6 x 10**^{15}** nuclei**

- The activity of the iodine-131 in curies can be determined using its
**decay constant**:

The iodine-131 has half-live of 8.02 days (692928 sec) and therefore its decay constant is:

Using this value for the decay constant we can determine the activity of the sample:

3) and 4) The number of iodine-131 atoms that will remain in 50 days (N_{50d}) and the time it will take for the activity to reach 0.1 mCi can be calculated using the decay law:

As can be seen, after 50 days the number of iodine-131 atoms and thus the activity will be about 75 times lower. After 82 days the activity will be approximately 1200 times lower. Therefore, the time of ten half-lives (factor 2^{10} = 1024) is widely used to define residual activity.