Half life indicates the time required to reduce or disintegrate half of the original amount of a substance. Disintegration of a substance is taken place by radioactive decay. Generally radioactive nucleus are unstable. Since it is unstable, it emits the radiation and undergoes disintegration. The mathematical representation of Half life is given below.

**formula for half life** is,

Rutherford and Soddy found that the rate at which a particular radioactive material disintegrates was independent of physical and chemical conditions. The number of atoms that break up at any instant is proportional to the number of atoms present at that instant. Let N be the number of atoms present in a particular radio element at a given instant t. Then the rate of decrease `(-dN)/dt ` is proportional to N

(dN)/(dt) = lambda N`

where `lambda ` is the decay constant of the radioactive element.

`lambda` is defined as the ratio of amount of the substance which disintegrates in a unit time to the amount of substance present.

`lambda = -((dN)/(dt))/N`

Equation 1 is written as

`(dN)/N = - lambda t`

Integrating , `log_eN = -lambdat +C`

where Cis integration constant

Let the number of radioactive atoms at time t=0 is N0

log_e(N_0) = C`

Plugging in the value of C in 2

log N = -lambdat + log (N_0)`

or `log_e(N/(N_0)) = - lambda t`

N = N_0 e^(-lambdat)`

This equation shows the number of atoms of a given radioactive substance decreases exponentially with time.

We use this equation to derive the half life equation of radioactive elements.

The where

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Let us discuss the problems related to half life of radioactive substances.

### Solved Examples

**Question 1: **Calculate the half life of a radioactive substance whose disintegration constant is 0.002 years^{-1} ?

** Solution: **

Given quantities are,

$\lambda$ = 0.002years^{-1}

Half life equation is,

$t_{\frac{1}{2}}$ = $\frac{0.693}{\lambda }$

$t_{\frac{1}{2}}$ = $\frac{0.693}{0.002}$ = 346.5 years

**Question 2: **Determine the disintegration constant of a radioactive substance which has a half life of 1622 years?

** Solution: **

Given quantities are,

t_{1/2} = 1622 years

Half life equation is,

$t_{\frac{1}{2}}$ = $\frac{0.693}{\lambda }$

$\lambda$ = $\frac{0.693}{t_{\frac{1}{2}}}$

$\lambda$ = $\frac{0.693}{1622}$ = 4.27×10^{-4}years^{-1}

Given quantities are,

$\lambda$ = 0.002years

Half life equation is,

$t_{\frac{1}{2}}$ = $\frac{0.693}{\lambda }$

$t_{\frac{1}{2}}$ = $\frac{0.693}{0.002}$ = 346.5 years

Given quantities are,

t

Half life equation is,

$t_{\frac{1}{2}}$ = $\frac{0.693}{\lambda }$

$\lambda$ = $\frac{0.693}{t_{\frac{1}{2}}}$

$\lambda$ = $\frac{0.693}{1622}$ = 4.27×10