Watch Free Classes

Half-life formulas are a decent metric to calculate the half-life of radioactive elements. The need for this formula occurred when the slowed rate of decay was found during the years. When an element is reduced, its decay rate also decreases, which might be challenging to calculate after some point. In simple words, the half-life is the time of decaying to half.

The Half-life of an Element

It is the time required to decay a radioactive element to its half. It can be alpha, beta or gamma decay in nature. 

Example: Uranium₂₃₈ has a half-life of 4.5 billion years.

Kinetics and half-life

  • The half-life of elements and kinetic order of reactions have some connection. 

  • The half-life of zero-order kinetic and second-order kinetic reactions depends on the rate of reaction constant and initial concentration.

  • The half-life of a first-order kinetic reaction depends on the reaction rate constant only.

Use of half-life formula

  • To know whether a radioactive object or place is safe.

  • To calculate the age of organic objects.

  • To calculate the age of old artefacts.

Half-life formulas

Various kinds of half-life formulas are present depending on the data provided or the nature of decay.

  1. Consider an original isotope of a radioactive element. To calculate the remaining amount of that isotope after a mentioned time interval can be found as:

m= 1/2ⁿ x original mass.

Where m is the mass remaining and 

n is the number of half-lives.

Example :

  • How much Np-240 will be remaining after 6 hours if there is 75g now? (Given that half-life of Np- 240 is one hour)

Answer: m= 1/2ⁿ x original mass.

=1/2⁶x 75

=1.1718

Suppose we know the half-life of an element and how much mass remains. To calculate the amount of time required to obtain the mentioned mass is,

  • T = n x t1/2
  • Example :

  • How long it will take an element ‘x’ to decay to 12.5g from 100g (half-life is 1200 years)

  • Answer: 100 to 12.5 will be a three half-life. 
  • Therefore
  •  T = n x t1/2
  • = 3 x 1200
  • =3600

An exponential decay can be calculated by 

  • N (t) = N0 e-λt
  • Where N0 is the concentration of element when t=0
  • λ is the decay constant 

From the exponential decay formula, we obtain another equation as

  • t1/2= 0.693/ λ
  • where t½ is the half lifetime and λ is the decay constant.
  • Example: 

  • if λ is 1.6, what is the half-life of the particular element?

Answer: t1/2 = 0.693/ λ

  • = 0.693/1.6
  • =0.433

Conclusion

The half-life is the time required to decay a radioactive element to its half. It can be alpha, beta or gamma decay in nature. Half-life formulas are a decent metric to calculate the half-life of radioactive elements. The need for this formula occurs after we find the slowed rate of decay during the years. 

  • When an element is reduced, its decay rate is also decreasing, which might be challenging to calculate after some point. In simple words, the half-life is the time of decaying to half. Various kinds of half-life formulas are present depending on the data provided or the nature of decay. 
  • The half-life formulas are used to calculate the age of old artefacts and organic products. It is also used as a measure of safety to handle a radioactive element and calculate the radioactivity of a place that previously encountered any radioactive accidents.