Finding the half-life of an unknown substance

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The discussion focuses on calculating the half-life of an unknown radioisotope that decays from 1000 particles to 472.37 particles in 50 days. Participants confirm the use of the decay formula and the relationship between the decay constant and half-life. One user calculates the half-life to be approximately 46 days and seeks verification for additional calculations regarding the remaining particles after 75 days and the time to decay to a single particle. The responses validate the calculations, confirming that 324.656 particles remain after 75 days and that it takes approximately 460.521 days to decay to one particle. The thread emphasizes the importance of applying logarithmic functions in decay calculations.
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1000 particles of an unknown radioisotope decays to 472.37 particles in 50 days.

(a) What is the half-life of this substance?


Any problems that I had done previously, the half-life was given.

Any help will be greatly appreciated.

Well, I know a few things, but don't know if they apply to this problem:

The decay constant = .693/T base 1/2
t 1/2 = half-life

Time to decay to a single particle = LN(Nbase0)/decay constant

N = number of particles remaining at some elapsed time
Nbase0 = number of particles we started with
e = a symbol that represents the irrational number 2.718281828...
lambda = decay constant
x = elapsed time

The formula to determing how much of the substance will be remaining at any particular time is: N = Nbase0 e^-lambda x
 
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Hi Sam,
you give the correct formulae
N = N0e-[lamb]x
and
[lamb]=.693/T1/2.
You could combine these, solve for T1/2, and plug in N0, N, and x.

Or, more instructive, you could use the equation
N = N0 * (1/2)x/T1/2.

In both cases, the important step is applying the ln() to both sides of the equation.
 
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Thank you.

I'll give it a try.
 
Is this correct?

I used, based on the above:

N(t) = N base0^(-lambda x /T),
where T is the half-life
t = 50 days
N base0 = 1000

I rounded to the nearest whole number for days and came up with 46.

Is that correct?
 


Originally posted by Sam
I used, based on the above:

N(t) = N base0^(-lambda x /T)
I think it should read
N(x) = N0eln(1/2)x/T, but that's probably just a typo, since
T is the half-life
t = 50 days
N base0 = 1000
is correct (except x=50 days), and your answer is also correct, although they probably expect you to come up with some more decimals...:wink:
 
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More Help, Please...

I have two more parts to this problem:

(a) How much will be left after 75 days?
(b) How long will it take the 1000 particles to decay to a single particle?

For (a), I came up with: 324.656 particles
For (b), I came up with: 460.521 days

Will you please verify my answers?

Thank you!
 
Both of your answers are correct.
 

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