# Obtaining the Half-life equation experimentally

• sawhai
In summary, you can use the decay equation to find the half-life of a radioactive sample by solving for lambda.
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.

sawhai said:

## 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 ?

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

sawhai said:
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 Ci = N0 - N(ti) . Plug in N(t) from the decay equation.

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

See what you get putting all of that together.

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

Thank you very much. I got the answer.

## 1. What is the purpose of obtaining the half-life equation experimentally?

The half-life equation is used to determine the rate of decay of a radioactive substance. This is important in various fields of science, such as nuclear physics, chemistry, and medicine, in order to accurately measure the amount of time it takes for a substance to decay.

## 2. How is the half-life equation obtained experimentally?

The half-life equation can be obtained experimentally by measuring the decay rate of a radioactive substance over a period of time. This involves taking multiple measurements of the substance's activity and using this data to calculate the half-life.

## 3. What factors can affect the accuracy of obtaining the half-life equation experimentally?

There are several factors that can affect the accuracy of obtaining the half-life equation experimentally. These include the purity of the substance, external factors such as temperature and pressure, and the method of measurement used.

## 4. Can the half-life equation be obtained for all radioactive substances?

No, the half-life equation can only be obtained for radioactive substances that follow first-order kinetics. This means that the rate of decay is directly proportional to the amount of substance present. Substances that do not follow this pattern cannot have their half-life equation determined.

## 5. How is the half-life equation used in practical applications?

The half-life equation has many practical applications, such as in nuclear power plants to monitor the decay of radioactive materials, in medical imaging techniques using radioactive tracers, and in radiocarbon dating to determine the age of organic materials. It is also used in research to study the properties of radioactive substances and their behavior over time.

### Similar threads

• Advanced Physics Homework Help
Replies
9
Views
1K
• Engineering and Comp Sci Homework Help
Replies
1
Views
1K
• Introductory Physics Homework Help
Replies
3
Views
818
• Advanced Physics Homework Help
Replies
6
Views
2K
• Advanced Physics Homework Help
Replies
2
Views
2K
• Advanced Physics Homework Help
Replies
2
Views
1K
• Advanced Physics Homework Help
Replies
11
Views
2K
• Advanced Physics Homework Help
Replies
6
Views
2K
• Advanced Physics Homework Help
Replies
9
Views
1K
• Advanced Physics Homework Help
Replies
4
Views
3K