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Rate of Decay Formula

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.
  1. The number of undecayed nuclei present in the sample on average doubling, the number of undecayed nuclei should double the rate of decay.
  2. 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

The half life (t1/2) is expressed as

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Rate of Decay Problems

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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 t1/2 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 Rate of Decay

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 14C analysis. New wood has a count of 15cpmg-1. The half-life of 14C is 5570years. Calculate the age of the old wood.
Solution:
 
Use t1/2 to find the rate constant

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

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