# Half life from two identical radioactive sources cpm

• Hercuflea
In summary, the problem presents two identical radioactive samples with different measurement results of decays during a certain period of time. The goal is to find the half-life of the radioactive sample. By setting up equations for the number of decays in each source and manipulating them, a cubic equation is obtained. However, it is unclear if this is the correct approach to finding the half-life.
Hercuflea

## Homework Statement

" Independent measurements are taken on two identical radioactive samples. On the first radioactive sample, a radiation detection system measures N decays on a radioactive sample during a total period of T. On the second radioactive sample, a radiation detection system measures 2N decays on a radioactive sample during a total period of 3T. What is the half-life of the radioactive sample? "

## Homework Equations

Number of particles decayed ## = \Delta N = N_0 (1-e^{-\lambda t}) ##
Half life = ## \frac{ln2}{\lambda} ##
decay constant = ## \lambda ##

## The Attempt at a Solution

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My reasoning is, since the question states that the sources are identical, this means that the isotopes are identical. I assume the measurements are begun at the same time i.e. ## T_0 = 0## for both of them. The number of particles cannot then be the same because the counts per minute for source 1 is ## cpm_1 = \frac{N}{T} ## and the counts per minute for source 2 is ## cpm_2 = \frac{2}{3} \frac{N}{T} = \frac{2}{3} cpm_1##, and therefore the sources must have different activities since they are otherwise identical, and therefore the only thing that can be causing the change in count rate is that there is a different number of particles ## N_0 ## in each source.

My attempt at the solution was this: the number of decays in each source is proportional to ## 4\pi r^2 ## times the counts detected. ## k ## is the proportionality constant which divides out.

# of decays in source 1:

$$N = k{N_0}_1 (1- e^{- \lambda T})$$

# of decays in source 2:
$$2N = k{N_0}_2 (1- e^{- \lambda (3T)})$$

Plugging the equation for ## N ## into the second equation:

$$2k{N_0}_1 (1-e^{-\lambda T}) = k{N_0}_2 (1-e^{-\lambda 3T})$$

rearranging for ##e^{- \lambda T}## I get a cubic equation:

$$(e^{-\lambda T})^3 - 2\frac{{N_0}_1}{{N_0}_2} e^{-\lambda T} + (2\frac{{N_0}_1}{{N_0}_2} -1) = 0$$

I guess if I had numbers i could plug this into a calculator to get the zeros but I'm not really sure if this is the right thing to do anyways.

Last edited:
Sorry about the equations, I put  around the equations like the LAtex Primer said, but they don't seem to be showing correctly.
edit: fixed

## 1. What is the concept of "half life" in relation to radioactive sources?

The half life of a radioactive substance is the amount of time it takes for half of the original amount of the substance to decay. This means that after one half life has passed, there will be half of the original amount of the substance remaining, and after two half lives, there will be one fourth of the original amount remaining, and so on.

## 2. How is the half life of a radioactive substance calculated?

The half life of a radioactive substance can be calculated using the formula t1/2 = ln(2)/λ, where t1/2 is the half life, ln(2) is the natural logarithm of 2, and λ is the decay constant of the substance.

## 3. How does the number of cpm (counts per minute) change over time in relation to the half life?

The number of cpm from a radioactive source will decrease over time as the substance decays. This decrease follows an exponential decay curve, meaning that the rate of decay slows over time. The cpm will decrease by half after each half life has passed.

## 4. Can the half life of a radioactive substance be affected by external factors?

No, the half life of a radioactive substance is a constant property of the substance and is not affected by external factors such as temperature, pressure, or chemical reactions. The half life is solely determined by the type of substance and its decay constant.

## 5. How is the concept of half life useful in scientific research and applications?

The concept of half life is important in various fields of science, including nuclear physics, chemistry, and medicine. It allows scientists to determine the rate of decay of a substance and to predict how long it will take for the substance to decay to a certain level. This information is used in various applications, such as determining the safety of radioactive materials, dating archaeological artifacts, and diagnosing and treating certain medical conditions.

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