The half-life of a substance (atoms, molecules, or ions) refers to the time it takes for half of its given amount to radioactively decay. The rate at which reactants are converted into products is always proportional to the concentration of reactants present. This study of the rate of decomposition of chemical species is called rate kinetics.

## Half-Life Calculation

In order to calculate the half-life of a chemical specie, integrated half-life equations are used according to the order of reactions. In nuclear reactions, this time period can also be calculated by a general half-life equation. This is because radioactive decays always follow first-order kinetics and they can be calculated with an integrated half-life equation:

The general equation of first-order kinetics;

aA   →  Product

The integrated rate law of first-order reaction is given as:

ln (No/Nt) = -λt    →    (i)

$\fn_jvn&space;\frac{N_{t}}{N_{o}}&space;=&space;e^{-\lambda&space;t_{1/2}}$     →     (ii)

where,

• Nt is the concentration of remaining species after time ‘t’
• N0 is the initial concentration of species
• λ is the decay constant
• t1/2 is the half-life

At half-life time (t = t1/2),

N = No/2

By putting these values in equation (i) we get;

ln ([N] / [No/2]) = k t1/2

ln (2) = k t1/2

$\fn_jvn&space;t_{1/2}&space;=&space;\frac{0.693}{k}\$

where,

### Activity or decay constant

The decay constant is the amount of radioactive substance that disintegrates in a unit of time. It is a constant value and a characteristic value based on the instability of nuclides. For example, if a sample contains 100 nuclides and 20 of them disintegrate in unit time, we can deduce that 20 decay events have occurred in a sample of 100 nuclides.

The rate of decay constant is given by:

The rate of the decay constant  $\fn_jvn&space;=-\frac{\Delta&space;N}{\Delta&space;t}$

It is directly proportional to the number of nuclides in a sample.

Rate = (-ΔN/Δt) ∝ N

The negative sign in the above equation indicates that the rate decreases with time.

Now using ‘k’ as a constant, the equation becomes;

Rate = (-ΔN/Δt) = k.N

## Graphical Illustration of Half-Life

The number of reactants decreases continuously through the decay process and it takes one half-life to become half of its original or starting value. After the first half-life, the remaining nuclei require the same time to become again one-half through disintegration.

The graph below shows the relative decrease in quantities with time.

It should be noted that the graph never touches the abscissa (x-axis) which represents the first-order reactions; the number of reactants goes near zero but never become zero.

## Examples of Half-life calculation

Example: Radon-220 decays to give Po-216 and beta particles. Calculate the fraction percent remains after 52 seconds if its decay constant is 1.33 x 10-2 s-1.

ln (No/N) = λ t

ln (No/N) = (1.33 x 10-2) (52)

= 6.9 x 10-1

By taking antilog on both sides;

No/N = 1.99

To calculate the remaining fraction N/No

⇒ 1/1.99

N/No= 0.5

0.5 or 50 % remaining after 52 seconds, thus, 52 seconds is the half-life of radon-220.

## Applications of Half-life

### Carbon-14 Dating

Carbon dating is the most commonly used method for the calculation of the time of dead remnants. The half-life method is implied in radioactive dating techniques to calculate the age of dead organisms. Carbon has a half-life of 5730 years, so this method is not applicable to calculating the ages of the cold age and Dinosaur age. The ‘uranium-thorium dating method’ is more relatable in this case.

### Radiation therapy

In advanced medical treatments, versatile radiation therapies are used to treat tumors, cysts, and stones in human bodies. Radioactive isotopes such as iodine-131 are commonly used in the treatment of Hodgkin lymphoma, a type of thyroid cancer. The half-life of a radioactive substance is very important in this case as the particles or rays emitted through disintegration are directed towards the target (unwanted substances) in the living organisms.

Cobalt-60 with a half-life of 5.26 years is used for food irradiation, a sterilization process, and, many other applications.

## Concepts Berg

What is the formula for calculating half-life?

Half-life can be calculated by the given formula:

Nt / No = e -λ t1/2

Why do we calculate half-life?

The half-life has importance for the quantification of a sample. It is fixed for radioactive species and it depends on the number of nuclides in a given sample. In order to evaluate the life of a wood piece, the quantity of carbon-14 radioactive isotope is measured and compared with the original value. As the half-life of 14C is calculated already, we can tell the life (age) of that wood.

Why does medication’s half-life matter?

Half-life is the time duration in which the species under consideration loses half of its original value. In medication, the time taken for a drug to diffuse in the body is important for treatment. For example, Epival is a sodium salt of Divalproex. It is used to treat epileptic seizures. This medicine is available in the pharmacies as Epival slow-release (S.R) and Epival Control Release (C.R).

Why is a pesticide’s environmental half-life important?

Pesticides are chemical compounds that are used to kill insects in shrubs and plants. They are harmful to alive organisms. During designing and preparation, a short life span for such compounds is important so that they diffuse in three to six days. Therefore, they would not be inhaled by human beings and animals during consumption.

References

• Radiochemistry and Nuclear Chemistry  3rd edition by Gregory Choppin (Florida State University), ‎Jan-Olov Liljenzin (Chalmers University of Technology, Sweden), ‎and Jan Rydberg (Chalmers University of Technology, Sweden)
• Chemistry| Fifth edition, by Steven S. Zumdhal and Susan A. Zumdhal (University of Illinois, Urbana Champaign, IL, USA)
• Essential of Physical Chemistry: 2nd edition By B.S Bahl (Gurdaspur, India) and Arun Bahl (RSC, UK) and G.D. Tuli (Delhi University, India)
• Half-life ( opentextbc.ca)
• Half-Life radioactivity (britannica.com)