# Law of decay

Exponential decrease of a quantity from the initial value N - z. B. the number of radioactive atomic nuclei in a given substance sample - with time t .

The law of decay is the name commonly used in physics for the equation that describes an exponential decrease in quantities over time. In nuclear physics , the law of decay indicates the number of atomic nuclei in a radioactive substance sample that have not yet decayed at a given point in time . This number is ${\ displaystyle N}$${\ displaystyle t}$

${\ displaystyle N (t) = N_ {0} \ cdot \ mathrm {e} ^ {- \ lambda t}}$,

where is the number of nuclei present at the beginning ( ) and the decay constant of the nuclide in question . ${\ displaystyle N_ {0}}$${\ displaystyle t = 0}$${\ displaystyle \ lambda}$

## Derivation

If one looks at a radioactive preparation with initially atomic nuclei and the activity , the following applies to the number of nuclei that have not yet decayed during the period : ${\ displaystyle N_ {0}}$ ${\ displaystyle A}$${\ displaystyle N}$${\ displaystyle t}$

{\ displaystyle {\ begin {aligned} A & = - {\ frac {\ mathrm {d} N} {\ mathrm {d} t}} \ qquad {\ text {with}} A = \ lambda \ cdot N \\ - \ lambda \ cdot N & = {\ frac {\ mathrm {d} N} {\ mathrm {d} t}} \\ - \ lambda \ cdot \ mathrm {d} t & = {\ frac {1} {N} } \ cdot \ mathrm {d} N \\\ int _ {0} ^ {t} - \ lambda \ cdot \ mathrm {d} t '& = \ int _ {N_ {0}} ^ {N} {\ frac {1} {N '}} \ cdot \ mathrm {d} N' \\ - \ lambda t - (- \ lambda \ cdot 0) & = \ ln (N) - \ ln (N_ {0}) \ \ - \ lambda t & = \ ln \ left ({\ frac {N} {N_ {0}}} \ right) \\\ mathrm {e} ^ {- \ lambda t} & = {\ frac {N} { N_ {0}}} \\ N (t) & = N_ {0} \ cdot \ mathrm {e} ^ {- \ lambda t} \ end {aligned}}}

So after the time there are still some starting cores left. ${\ displaystyle t}$${\ displaystyle N_ {0}}$${\ displaystyle N (t)}$

## Average lifespan

The decay constant ( lambda ) is the reciprocal of the mean life , i.e. the time after which the number of atoms has decreased by the factor . ( Tau ) differs from the half-life only by the constant factor : ${\ displaystyle \ lambda}$ ${\ displaystyle \ tau = 1 / \ lambda}$${\ displaystyle \ mathrm {e} = 2 {,} 71828 \ dotso}$${\ displaystyle \ tau}$ ${\ displaystyle T_ {1/2}}$${\ displaystyle \ ln 2}$

${\ displaystyle T_ {1/2} = {\ frac {\ ln 2} {\ lambda}} = \ tau \ cdot \ ln 2 \ approx 0 {,} 693 \ cdot \ tau}$

This results in the following form for the law of decay:

${\ displaystyle N (t) = N_ {0} \ cdot e ^ {- {\ frac {\ ln (2)} {T_ {1/2}}} t}}$