In Radioactive decay, half life and mean life are terms used to represent the time taken by radioactive substances/elements to disintegrate to a certain proportion. Half life and mean life both depends on the nature of radioactive elements. They are **independent **of external physical factors like temperature, pressure, humidity etc. They are also **constant** for a particular substance or elements.

## Half Life (T_{1/2})

It is defined as the time taken by radioactive substance to reduce to half of its initial concentration. Let us suppose, a radioactive substance initially containing ‘No’ number of atoms. Then in a half life time, it reduces to N_{o}/2.

If ‘T’ is the time taken by substance/element to disintegrate completely, then

Mathematically,

When t= T_{1/2}

N= N_{o}/2

Applying laws of radioactive disintegration,

N= N_{o}e^{-λt}

Or, N_{o}/2 = N_{o}e^{-λT1/2 }[N= N_{o}/2, t= T_{1/2}]

Or, 1/2 = e^{-λT1/2}

Or, 2 = e^{+λT1/2}

Or, ln2 = λT_{1/2}

T_{1/2} = ln2/λ = log_{e}2/λ = 0.693/λ

Where, λ = decay constant.

## Half Life Calculation

Radioactive disintegration is an exponential, spontaneous(natural radioactivity) process. In exponential process, a substance decay only half of its initial concentration in a single half life.

**For example**: If a Uranium has a half life of 24 days then it reduces to 50% in 24 days. For next half life (i.e another 24 days) 50% will be the initial concentration and reduces to 25% in this second half life duration. So, at the end of 2 half life, 50% + 25%=75%(total) percentage of atoms will decay. And this process goes on.

From above conclusion, some relations can be drawn for solving numerical easily.

Number of Half life | Decomposed percentage(%) | Remaining Percentage(%) |
---|---|---|

0 | 0 | 100 |

1 | 50 | 50 |

2 | 75 | 25 |

3 | 87.5 | 12.5 |

4 | 99.75 | 0.25 |

10 | 99.9 | 0.1 |

∞ | 100 | 0 |

**In short**,

T_{50%}(50% decay)=T_{1/2(}1 half life)

T_{75%}=2 T_{1/2}

T_{87.5%}=3 T_{1/2}

T_{99.75%}=4 T_{1/2}

T_{99.9%}=10 T_{1/2}

T_{100%}= ∞

For disintegration of substance comletely, it takes infinite time period. It means radioactive substance never decays completely. (It is found that the clothes, notebooks, furniture etc used by Madam Curie are still radiating rays in present time).

Read Also: Units of Radioactivity

## Average Life (T_{av})

It is also called as **Mean life**. It is defined as time taken by an atom to disintegrate on average. It Is represented by T_{av} and given by,

T_{av} = sum of life time of all atoms/total nos. of atoms

Average life is also defined as reciprocal of disintegration (decay) constant, ‘λ’.

i.e. T_{av} = 1/λ

or, T_{av }= T_{1/2}/0.693 = 1.44T_{1/2}

## Difference between Half Life and Average life

S.N | Half life | Average(Mean) life |
---|---|---|

1. | It is the time taken by radioactive element to disintegrate half of its atoms present initially. | It is the time taken by radioactive element to disintegrate on average. |

2. | It is given by formula T_{1/2} = 0.693/ λ. |
It is given by formula T_{av} = 1/ λ |

3. | Half life means 50% of original atoms decayed. | Average life (Mean life) means 63% of original atoms decayed. |

4. | Since T_{av} = 1.44 times T_{1/2}. It takes less to achieve half life. |
It takes more time to achieve average life as compared to half life. |

## FAQs

**a) What is half life in radioactivity?**

Ans: Half life is the time taken by a radioactive substance to reduce to half of its initial concentration. It is represented by T_{1/2} and given by T_{1/2} = 0.693/ λ.

**b) Define Average life(T _{av})?**

Ans: Average life is the sum of life times of all atoms to the total number of atoms present initially. It is also defined as the reciprocal of decay constant(λ). It Is represented by T

_{av}and given by T

_{av}= 1/ λ.

**c) What is the relation between half life & average life?**

Ans: Relation between half life and average life is given by relation T_{av} = 1.44 T_{1/2}. Average life is equal to 1.44 times half life. Half life is 50% decay of original atoms present initially while average life is 63% decay of initial atoms.