# Half life - Calculate the length of time

• Faiq
In summary: The radioactive source emits alpha particles at a constant rate 3.5x10^6. The particles are collected for a period of 40 days.In summary, the radioactive source will emit alpha particles for a total of 40 days and there will be a 3.5x10^6 Bq of alpha particles emitted over that time period.
Faiq

## Homework Statement

A source having a half life of 5.27 years is calibrated and found to have an activity of 3.5*10^5 Bq.The uncertainty in the calibration is +- 2%

Calculate the length of time in days after the calibration has been made for the stated activity to have a maximum possible error of 10%

## Homework Equations

$$A=A_oe^{-\omega t}$$
$$t_{1/2} = \frac{ln2}{\omega}$$

## The Attempt at a Solution

I can't find a possible correlation between the uncertainty and activity.

Currently the actual activity can be up to 2% above or 2% below the given value.

Can the actual activity get 10% higher than the given value?
Can the actual activity get 10% lower than the given value?
If yes, when does that happen the earliest (what is the worst case)?

One way to approach this is to plot the activity over time using the measured ##3.5*10^5Bq## value and then also plot the activity over time using plus and minus 2% of this value. Track how these curves diverge from each other over time.

TJGilb said:
and then also plot the activity over time using plus and minus 2% of this value. Track how these curves diverge from each other over time.
I'm quite sure the 2% uncertainty refer to the activity, not the half-life. The curves won't diverge.

mfb said:
I'm quite sure the 2% uncertainty refer to the activity, not the half-life. The curves won't diverge.
I don't think @TJGilb thought it referred to the half-life. Rather, it looks like TJ thought the 10% referred to the discrepancy between the actual acitivity at a later time and the predicted activity at that time.

haruspex said:
I don't think @TJGilb thought it referred to the half-life. Rather, it looks like TJ thought the 10% referred to the discrepancy between the actual acitivity at a later time and the predicted activity at that time.

Yeah, I think I misinterpreted what the problem was asking him to find. It looks like it may be asking him to find when the activity will be potentially 10% less than the original calibration. Of course, you still want to use the lower curve and the higher curve for the worst case scenario.

The main problem I am having is I can't see why the activity and the uncertainty are related by the equation A = U + 2 rather than A = 2U

Faiq said:
The main problem I am having is I can't see why the activity and the uncertainty are related by the equation A = U + 2 rather than A = 2U
It's neither.
If the measured activity is A and the uncertainty is ±2% then the actual activity is anything from A(1-.02) to A(1+.02).

haruspex said:
It's neither.
If the measured activity is A and the uncertainty is ±2% then the actual activity is anything from A(1-.02) to A(1+.02).
I know that
What I am asking is, it is given in the answer that the source has to decay a further 8% to reach the uncertainty of 10%. My question is how is that relation derived?

Faiq said:
What I am asking is, it is given in the answer that the source has to decay a further 8% to reach the uncertainty of 10%. My question is how is that relation derived?
The source could have started at 98% the quoted activity, it will exceed the 10% deviation once it reduces its activity to 90% of the quoted activity, which is a ##\frac{0.08}{0.98} \approx 0.08## relative reduction.

So for every percent decay, there is a 2% error?

Oh so wait a minute are they asking length of time after which there would be a decrease of 10% from the original value?
Due to which we will subtract 2% since there is already a deviation of 2% present?

Faiq said:
Oh so wait a minute are they asking length of time after which there would be a decrease of 10% from the original value?
Not a 10% decrease the original activity (because that is not known exactly), but a decrease to 10% below the measurement value.
Faiq said:
Due to which we will subtract 2% since there is already a deviation of 2% present?
It is not exactly 2%, but to a good approximation: Yes.

Yes thank you very much.
So if they want to ask "Calculate the length of time in days after the calibration has been made for the stated activity to have a maximum possible error of x%"
All I have to do is use x-2/100 = e^-xt
Right?

Faiq said:
All I have to do is use x-2/100 = e^-xt
What is x? No matter what it is it should not appear both in an exponential and outside.

Oh sorry the first x was an arbitrary number, the second is decay constant

With a lot of interpretation and a new variable name, that might lead to a possible solution.

Okay thank you .

Which question?

A radioactive source emits alpha particles at a constant rate 3.5x10^6. The particles are collected for a period of 40 days.
BY reference to the half life of the source, suggest why it may be assumed that rate of emission of alpha particles remain constant?

It depends on the source. Questions unrelated to this homework problem should go in a new thread.

The half life can be determined by
A=ln2/t * 6.03*10^23
Where A is given in the question

Should I start a new thread?

Units are s^-1

## 1. What is half life and how is it calculated?

Half life is the amount of time it takes for half of a substance to decay. It is calculated using the formula T1/2 = ln(2)/λ, where T1/2 is the half life, ln is the natural logarithm, and λ is the decay constant.

## 2. What is the significance of half life in radioactive decay?

Half life is significant because it allows scientists to predict how long it will take for a substance to decay and become inactive. This is especially important in the study of radioactive materials and their potential hazards.

## 3. How do you determine the length of time using the half life formula?

To determine the length of time using the half life formula, you will need to know the initial amount of the substance, the decay constant, and the amount of the substance remaining. Plug these values into the formula T1/2 = ln(2)/λ and solve for T1/2.

## 4. Can the half life of a substance change?

The half life of a substance is a constant value that cannot be changed. It is determined by the type of substance and its decay constant, which are both inherent properties of the substance.

## 5. How is half life used in carbon dating?

In carbon dating, the half life of carbon-14 is used to determine the age of organic materials. By measuring the ratio of carbon-14 to carbon-12 in a sample, scientists can calculate how many half lives have passed and estimate the age of the sample.

• Introductory Physics Homework Help
Replies
3
Views
773
• Introductory Physics Homework Help
Replies
10
Views
2K
• Introductory Physics Homework Help
Replies
4
Views
2K
• Introductory Physics Homework Help
Replies
1
Views
1K
• High Energy, Nuclear, Particle Physics
Replies
3
Views
1K
• Biology and Chemistry Homework Help
Replies
2
Views
1K
• Introductory Physics Homework Help
Replies
23
Views
1K
• Introductory Physics Homework Help
Replies
1
Views
871
• Introductory Physics Homework Help
Replies
2
Views
1K
• Introductory Physics Homework Help
Replies
4
Views
4K