The decay of any particular nucleus is unpredictable and unlike chemical reactions decay is not affected by physical conditions such as temperature. The rate at which an isotope decays depends on.

- The number of undecayed nuclei present in the sample on average doubling, the number of undecayed nuclei should double the rate of decay.
- The stability of the isotope, some isotopes decay much more rapidly than others.

**"The rate of decay is the number of nuclei that decay each second. It is measured in becquerel(Bq) where 1Bq = 1 decay/s."**

**Rate of Decay Formula** is expressed as

The above equation on integration gives

Related Calculators | |

calculate radioactive decay | half life decay calculator |

air flow rate calculator | Average Rate of Change Calculator |

Solved problems based on rate of decay are given below. ### Solved Examples

**Question 1: **The half-life of 226-radium is 1622years. Calculate how long it will take for a sample of 226-radium to decay to 10% of its original radioactivity.

** Solution: **

Use t_{1/2} to find the rate constant.

k = $\frac{0.693}{1622}$ = 4.27 $\times$ 10-4year-1

Insert the value for k into the integrated form of the rate equation

4.27 $\times$ 10$^{-4}$ $\times$ t = ln($\frac{100%}{10%})$

t = 5392years

It will take 5392 years to decay to 10% of its original activity.

**Question 2: **A piece of old wood was found to give 10 counts per minute per gram of carbon when subjected to ^{14}C analysis. New wood has a count of 15cpmg-1. The half-life of ^{14}C is 5570years. Calculate the age of the old wood.

** Solution: **

Use t_{1/2} to find the rate constant

k = $\frac{0.693}{5570}$ = 1.24 $\times$ 10^{-4}years^{-1}

Insert the value of k into the integrated form of the rate equation.

1.24 $\times$ 10^{-4} $\times$ t = ln($\frac{14C\ content\ in\ new\ wood}{14C\ content\ in\ old\ wood}$) = ln($\frac{15}{10}$) = ln1.5

t = 3270years

The wood is 3270 years old.

Use t

k = $\frac{0.693}{1622}$ = 4.27 $\times$ 10-4year-1

Insert the value for k into the integrated form of the rate equation

4.27 $\times$ 10$^{-4}$ $\times$ t = ln($\frac{100%}{10%})$

t = 5392years

It will take 5392 years to decay to 10% of its original activity.

Use t

k = $\frac{0.693}{5570}$ = 1.24 $\times$ 10

Insert the value of k into the integrated form of the rate equation.

1.24 $\times$ 10

t = 3270years

The wood is 3270 years old.