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Calculate time constant of an exponential decay

Hi *,

This is a simple question; however, while looking for the answer I have come across different answers.

I have some data which decays exponentially and would like to calculate the time constant of this decay.How can I do this?

I have come across the following which apparently is calculating the half life (not sure its relation to time constant) of of radioactive decay (which I think is exponential):

-t ln(N / No)

where No is the initial amount, N is the final amount and t is the time taken to go between the two amounts. This however gives me funny results that do not seem logical. For example, -100 ln(8/10) gives 22, however -100 ln(5/10) gives 69. A longer time constant for something that has decayed more in the same amount of time.

Many thanks,
James

d-glitch

Time constant is exponential with base e=2.718

Half-life is exponential with base 2.

If something has a half-life of 1 year, it has a time constant of (1 year)*ln(2) = 0.693 years.

http://en.wikipedia.org/wiki/Half-life

Member_2_2394978

ASKER

Hmm ok, that might explain the complication when searching for answers. I would like to calculate the time constant (with natural base). How can I do this?

Cheers,
James

d-glitch

Time_Constant = Half_Life * ln(2) = (0.693)* Half_Life

There is a section on the Wikipedia page that does the derivation.
It is hidden, you have to hit Show to see it.

Member_2_2394978

ASKER

Ok, I do the sums and the same thing happens as in my original example. Things which logically should have longer time constants have smaller ones.

Can you go through calculating the time constant of, for example, a drop from 10 to 8 that takes 182 seconds.

Cheers,
James

d-glitch

8  =  10*(1/(2^(182/HL))

Log10 of both sides

  log(8) = Log(10) - (182/HL)*log(2)

            log(10) - log(8)
  182/HL = ------------------  = 0.3219
                log(2)

                 182 s
  Half-Life  =  -------  =  565.3 s
                0.3219

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Member_2_2394978

ASKER

Thanks.

So all together?

time constant = ( t / ( (log(No)-log(N(t)) / log(2) ) ) * ln(2)

Member_2_2394978

ASKER

This all seems way too complicated. Do I have to go through the half-life. Can't I just go straight to the time constant for the exponential decay?

d-glitch

Yes you can solve for Time Constant the same way:

8  =  10*(1/(e^(182/TC))

Ln of both sides

  ln(8) = Ln(10) - (182/HL)*log(e)

  182/TC =  ln(10/8)  = 0.2231

                     182 s
  Time Constant  =  -------  =  815.5 s
                    0.2231

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d-glitch

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Member_2_2394978

ASKER

Brilliant that looks good. Thanks very much.

Cheers,
James

d-glitch

You can use XL to check your thinking (and math) on these sort of problems.

I you know that 10 decays to 8 in 182 seconds, you can use the XL Solver to find the
decay rate per second.

I get 0.998775.

You can extend the spreadsheet to find the half-life (N=5) and the
Time Constant (N = 1/e = 3.679).

The values match my earlier calculations.
Exponential-Decay.pdf