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# Exponential Decay Formula

Exponential Decay is the rate of decay with respect to time. A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Exponential decreases start fast and get slower, hence the problem with radioactive waste hanging around for ages.  True exponential increases and decreases occur whenever the rate of change of something is proportional to the thing itself. The rate of change of decay is given by The Exponential Decay Formula is given by Where,
t is the time taken,
No is the initial no of atoms present at time t = 0,
N(t) is the no of atoms present after time t,
$\lambda$ is the decay constant. Exponential Decay Formula is used to calculate the amount of atoms remaining during the radioactive decay of any nucleus.

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## Exponential Decay Problems

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Below are problems on exponential decay which may be helpful for you.

### Solved Examples

Question 1: If 3 centigram of substance decays in 3s in 1 gm. Calculate the amount present in the nucleus.
Solution:

Initial atoms No = 1 gram,
No of atoms after time t, N(t) = 1 - 0.03 = 0.97 g.

Question 2: A gram of a radioactive substance loses 1 centigram in 50 s, calculate the decay constant?
Solution:

Initial atoms N0 = 1 gram,
No of atoms after time t N(t) = 1 - 0.01 = 0.99 g,
Time taken t = 50 s
$\lambda$ = $\frac{2.303}{t}$ log $\frac{N_{o}}{N_{t}}$
= $\frac{2.303}{50}$ log $\frac{1}{0.99}$
= 2.026 $\times$ 10-4 s-1.

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