# Half Life

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 To Calculate: Final Quantity Initial Quantity Half Life Decay Time Final Quantity: Initial Quantity: Half Life: Decay Time:

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## Half-Life Calculator

The half-life of a reaction is defined as the time taken by a reactant's concentration to get reduced to one-half the initial concentration. This principle is widely used in medicine and chemistry in predicting the concentration of a component.

Matching to its name, the half-life of a reaction is represented by t1/2

### The Phenomenon of Half-Life

Consider a sample of radioactive molecules ‘N’ in quantity exists initially. Let the half-life of the substance be t1/2 seconds.

• According to the half-life principle, after the first half-life t1/2 of the substance, the number of molecules left would be N/2.
• After another half-life, the number of molecules left would be N/4...and the process goes on.
• An interesting point to note about half-life is the time taken for N molecules to become N/2 is same as the time taken for N/28 molecules to become N/56. That is irrespective of the initial number of molecules, the time is taken for them to become half in number is the same.
• Though the name says ‘half’, the half-life phenomenon is an exponential decay one. New to exponential calculations? Check our exponent calculator here!

### How the concept of Half-Life came into the light?

Can we imagine any substance getting reduced to its half on its own? How would it have come into the light? Well, Ernest Rutherford, a physicist, is credited with the discovery of the half-life principle. He, along with physicist JJ Thompson worked together on the discovery of electrons. He also had contributed to the research in radioactivity and in the differentiation of alpha rays and beta rays, which he extended and gave the principle of half-life. Here he noticed that samples of radioactive materials took the same time to decay by half, the phenomenon which he termed as "atomic disintegration".

Ernest Rutherford was awarded Nobel in 1908 for his genius work regarding the discovery of atomic disintegration.

• The half-life of various nuclei ranges in the order of 10-23 to 1016.
• The lesser the half-life, the lease stable is the substance.
• For substances with least half-life, that is for those which disintegrate faster, the nuclear forces of attraction are very minimal.

### Applications of Half-life principle

Half-life finds its applications in many fields of study such as pharmacokinetics, particle physics, carbon dating, etc.

#### Pharmacokinetics

Half-life principle is widely used in pharma to predict the concentration of a reactant over time. In pharmacokinetics, this principle of a half-life has a key role in the drug administration tests into the target, especially in the phase of elimination, where half-life is used in the calculation of time as to how fast a drug volume decrease in the given target after the reactant has been absorbed. This is measured in the units of time like sec, minute, hour, day, etc.

##### Key aspects about Concentration of products in a reaction
• The order of reaction where the reactant takes different forms influences the half-life of the reaction.
• During a reaction, to observe the occurrences on a molecular level, differential rate laws are used.
• To determine the value of rate constant and the reaction order, integrated rate laws are used.
• Using equations of integrated rate, the concentration of a product or reactant a given moment can be determined if the time, rate constant and initial concentration of the reactant is known. Same way, time can be determined with the knowledge of concentration and rate constant.

Half-life varies between different types of reactions. In the coming section, let us understand the different types of reaction, and how to derive its half-life reaction.

##### Zero-order reaction
• In this type of reaction, substrate concentration does not influence the rate of a reaction.
• In other words, the saturation of the amount of reactant does not speed up or increase the rate of the reaction.
• As the concentration decreases, the half-life of the zero order reaction also decreases.

The zero order kinetic rate law can be shown as below,

 [A]
=
 [A]0 k × t
 ------ (1)
Where [A] = current concentration,
[A]0 = initial concentration,
k = reaction constant,
t = time

To determine half-life, dividing equation (1) by 2,

 t1/2
=
 [A]0 2 × t
 ------ (2)

From equation (2), it can be seen that a zero order reaction states that the half-life depends on rate constant and the amount of initial concentration.

##### First Order reactions

A reaction which takes at a rate depending linearly on the concentration of one reactant only, i, e. the rate of drug concentration is proportional to the rate of drug elimination.

The half-life of first-order reactions is determined by the rate law of the first-order reaction:

[A] = [A]0 × e − kt  ------ (3)

When we substitute A with [A]0/2, we get:

t1/2=ln2 × k ≈ 0.693 / k ------ (4)

Hence, it can be seen that the half-life of first-order reactions depends on the rate of reaction constant only.

##### Second order reactions

Half-life calculation on second order reactions results in concentration [A] vs. time (t), i.e., the length of half-life increases with the decrease of concentration of the substrate.

The rate law of a second order reaction is:

1[A] = k × t + 1[A]0 ------ (5)

Determining half-life t/2 from the above equation:

 t1/2
=
 1 k × [A]0
 ------ (6)

Thus derived equation represents half of the initial concentration at a given time, t/2.

Here we can see that the half-life of a second-order reaction depends on the rate constant and the initial concentration.

#### Particle Physics

In particle physics, the following equation gives the relation between the original number of atoms or nuclei present and the final number of nuclei remaining after a time t:

N = N0 × E - λt

This equation is critical in determining the decay of excited states in atoms and nuclei.

#### Carbon Dating

Carbon-14, popularly known as C-14 has a half-life of 5730 years. This concept is used in determining the age of fossils and dead bodies found during excavations, Egyptian mummies, etc. Carbon is the most widely present element in every living organism on this Earth. Thus, when they die, the percentage of carbon left over in their artifact indicates how older the organism is.

A similar concept is used to determine the age of trees. Termed as ‘tree ring counting’, the process is based on the half-life of carbon itself.

By the way, do you know some interesting concepts about the age of various living beings on this Earth? Check them here!

### How CalculatorHut’s Half-life calculator helps you?

CalculatorHut, the expert in simplifying calculations and delivering online calculators for free, offers free online half-life calculator that is user-friendly and accurate to many decimals. You can calculate any value – the initial quantity, final quantity, and the half-life, using this online half-life calculator for free. You would get the results in your desired units – days, hours and years.

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