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# Decay Formula

We know that radioactive element decays with time. The decay law calculates the number of undecayed nuclei in a given radioactive substance.The decay formula is given by

Where,
No
= Initial amount of radioactive substance,
$\lambda$ = decay constant
t = time taken
Nt = Amount of radioactive substance present after time t.

Decay constant ($\lambda$) gives the ratio of number of radioactive atoms decayed to the initial number of atoms. It is given as

Decay Law is used to find the decay rate of the given radioactive substance if the above quantities are known.

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## Decay Problems

Problems based on decay are mentioned below.

### Solved Examples

Question 1: The half life period of yttrium is 20 min. If there are 1000 gm of yttrium present, how much will be left after 30 min?
Solution:

Given: Initial no of radioactive elements No = 1000 gms,
Half life t1/2 = 20 min,
Time taken t = 30 min,
Decay Constant is given by
$\lambda$ = $\frac{0.693}{t_{\frac{1}{2}}}$
= $\frac{0.693}{20 min}$
= 0.03465
The Decay rate is given by
Nt = No e-$\lambda$ t = 1000 e-0.03465(30)
= 1000 $\times$ 0.3536 = 353.63 gms.
353.63 gms of atoms will be present after 30 min.

Question 2: If you are given with 500 gms of radioactive carbon. Calculate the decay rate after 60 min if the half life period is about 30 min.
Solution:

Initial no of atoms N0 = 500 gms,
Half life period t1/2 = 30 min,
Time taken t = 60 min
The decay constant is $\lambda$ = $\frac{0.693}{t_{\frac{1}{2}}}$
= $\frac{0.693}{30 min}$
= 0.0231
The decay rate is given by
Nt = No e-$\lambda$ t = 500 e-(0.0231)60
= 500 $\times$ 0.25
= 125.04 gm.

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