Obtaining the Half-life equation experimentally

[sawhai]

Homework Statement

magine performing a counting experiment for 10 minutes, counting with a detector over 30 second intervals to determine the half-life of a radioactive sample. You obtain the data given in table 1. By approximating the activity at time ti, A(ti) by the counts measured over a minute interval Ci, show that the half-life can be obtained from the radioactive decay equation by: ln ci/delta(t) = ln(lambda*N0)-lambda*ti .
How can I obtain this equation from the original decay equation N(t)=N0*e^(lambda*t)?

Homework Equations

N(t)=N0*e^(lambda*t)?

The Attempt at a Solution

I am not sure how to use the general decay equation to obtain the equation in problem statement. I tried solving for lambda and plugging the half-life equation but not sure how to use it. Any help is appreciated.

 

[SammyS]

Homework Statement

Imagine performing a counting experiment for 10 minutes, counting with a detector over 30 second intervals to determine the half-life of a radioactive sample. You obtain the data given in table 1. By approximating the activity at time ti, A(ti) by the counts measured over a minute interval Ci, show that the half-life can be obtained from the radioactive decay equation by: ln ci/delta(t) = ln(lambda*N0)-lambda*ti .
How can I obtain this equation from the original decay equation N(t)=N0*e^(lambda*t)?

Homework Equations

N(t)=N0*e^(lambda*t)?

The Attempt at a Solution

I am not sure how to use the general decay equation to obtain the equation in problem statement. I tried solving for lambda and plugging the half-life equation but not sure how to use it. Any help is appreciated.

Hello sawhai. Welcome to PF !

First of all, the decay equation should have a negative sign in the exponent.

$\displaystyle N(t)=N_0e^{-\lambda t}$​

Let's initially assume that in your experiment you count all of the decays which occur. In order to derive the equation you will use to analyze your data, you need to understand all of the quantities in the decay equation.

Can you tell me what N(t) , N₀ , t, and λ are ?

 

[sawhai]

Surely, N(t) is the decay function, N0 is the initial value of the substance, t is the time and lambda is the decay constant. I just couldn't figure out how to get the equation
ln (ci/delta(t)) = ln(lambda*N0)-lambda*ti
from the original equation.

Thank you so much for your reply

 

[SammyS]

Surely, N(t) is the decay function, N0 is the initial value of the substance, t is the time and lambda is the decay constant. I just couldn't figure out how to get the equation
ln (ci/delta(t)) = ln(lambda*N0)-lambda*ti
from the original equation.

Thank you so much for your reply

You can also look at N(t) as the amount of substance remaining at time, t.

So the number of counts during a time interval from 0 to t, is N₀ - N(t) ... that's assuming we get a count for every atom which decays. Actually you will get some fraction of that, largely determined by geometry and the efficiency of your detector.

So C_i = N₀ - N(t_i) . Plug in N(t) from the decay equation.

Then $\displaystyle \frac{C_i}{\Delta t}$ is approximately equal to the derivative, (w.r.t. t_i) of N₀ - N(t_i).

See what you get putting all of that together.

(Next step: Take the log of both sides of the equation.)

 

[sawhai]

Thank you very much. I got the answer.