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In summary, the conversation discusses the calculation of the activity, in Bq, produced by a nuclear power plant that breeds 1mg of 239Pu per week. The solution involves finding the number of moles and the initial activity of the sample, which requires the use of the decay constant λ. The half-life of Pu-239 is usually looked up in a table or can be found online. A reliable source for nuclear data is provided as well. The correct answer for the activity is 2.3x10^6 Bq.

## Homework Statement

A nuclear power plant breeds 1mg of 239Pu per week. What activity, in Bq, does that create?

## Homework Equations

$R_0 = \lambda N_0$ (initial activity of the sample)

$R = R_0 e^{-\lambda t}$ (exponential behavior of the decay rate)

$T_{1/2}=\frac{ln 2}{\lambda}$ (Half-life)

## The Attempt at a Solution

(a) First I find the number of moles in 1mg of 239Pu:

$n= \frac{1\times10^{-3}}{239}=4.184 \times 10^{-6}$

Now, I think to find N0 I have to times n by 6.02x1023 nuclei/mol. Which gives us N0=2.518x1018.

But how can I find the decay constant λ, if we are not given the half life?

I tried out the 1 week as the half life but I didn't get the correct answer:

$\lambda = \frac{0.693}{604800 \ s} = 1.1458 \times 10^{-6}$

$R_0 = \lambda N_0 = 2.88 \times 10^{12} \ Bq$

$R = R_0 e^{-\lambda t} = 1.44 \times 10^{12}$

So, what should I do? The correct answer must be: $2.3 \times 10^6 \ Bq$.

The half-life is usually something that you look up. Back in my time () we had a book of tables where you could find this, nowadays we have Wolfram Alpha.

Right, I see. So there's no equation or anything you can use in this situation if you don't have a table?

To find the decay constant, you can use the given information that the power plant breeds 1mg of 239Pu per week. This means that after one week, the initial activity (R_0) will decrease by half, giving a half-life of one week. Therefore, the decay constant can be calculated using the half-life equation, T_{1/2}=\frac{ln 2}{\lambda}. Plugging in the half-life of one week, we get λ = 0.693/604800 s = 1.1458 x 10^-6 s^-1.

Using this value for λ, we can now calculate the initial activity (R_0) using the equation R_0 = λN_0. As you correctly calculated, N_0 = 2.518 x 10^18 nuclei. Plugging in these values, we get R_0 = 2.88 x 10^12 Bq.

To find the activity after one week (R), we can use the equation R = R_0e^-λt, where t is the time in seconds. Since one week is equivalent to 604800 seconds, we can plug in these values and solve for R. This gives us R = 1.44 x 10^12 Bq.

Finally, to find the activity per week, we can simply multiply R by the number of weeks (1 week in this case). This gives us an activity of 1.44 x 10^12 Bq per week, which is equivalent to 2.3 x 10^6 Bq, the correct answer.

## 1. What is radioactivity and how does it work?

Radioactivity is the spontaneous emission of energy or particles from the nucleus of unstable atoms. It occurs when the nucleus of an atom is unstable and tries to reach a more stable state by releasing energy in the form of radiation.

## 2. What are the different types of radiation and how do they differ?

The three main types of radiation are alpha, beta, and gamma. Alpha radiation consists of high-energy helium nuclei, beta radiation is made up of electrons, and gamma radiation is a form of electromagnetic radiation. They differ in their ability to penetrate materials and their ionizing power, with alpha being the least penetrative and gamma being the most.

## 3. How can we protect ourselves from the harmful effects of radioactivity?

The best way to protect ourselves from the harmful effects of radioactivity is to limit our exposure. This can be done by maintaining distance from radioactive materials, shielding ourselves with lead or concrete barriers, and using protective equipment such as gloves and masks. It is also important to follow safety protocols and regulations when working with radioactive materials.

## 4. What is half-life and how is it related to radioactivity?

Half-life is the amount of time it takes for half of a radioactive substance to decay into a more stable form. This is a constant rate for each radioactive substance and is used to measure the rate of decay. It is related to radioactivity because as the radioactive substance decays, it releases energy in the form of radiation.

## 5. What are some real-world applications of radioactivity?

Radioactivity has many practical applications in fields such as medicine, energy production, and archaeology. In medicine, it is used for diagnostic purposes and cancer treatment. In energy production, nuclear reactors harness the energy released by radioactive decay to generate electricity. In archaeology, radiocarbon dating is used to determine the age of artifacts and fossils by measuring the amount of radioactive carbon-14 present in them.