Nuclear Physics - Uranium decay chain and Bateman equation

This means that the activity of each decay chain member is the same as the activity of the parent isotope. So you don't need to use the Bateman equation here. Just calculate the activity of U-238 (using the given decay constant) and then use the decay chain to calculate the activity of Radon.
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Homework Statement



Calculate the activity of ##^{222}Rn## in an ore sample containing 5g of natural uranium.

Homework Equations



##^{238}U## decay chain (to Radon): ##^{238}U\rightarrow^{234}Th\rightarrow^{234}Pa\rightarrow^{234}U\rightarrow^{230}Th\rightarrow^{226}Ra\rightarrow^{222}Ra##

Bateman equation (for activity of a daughter isotope after a series of decays):

##A_n = N_o\sum\limits_{i=1}^{n} c_ie^{-\lambda_it}## where ##c_i=\frac{\lambda_1\lambda_2 ... \lambda_n}{(\lambda_1 - \lambda_i)(\lambda_2 - \lambda_i) ... (\lambda_n - \lambda_i)} \qquad \left(i \neq n\right)##

The Attempt at a Solution



Well, I used the known decay constants for each isotope in the decay chain and plugged them into the Bateman equation. Unfortunately, ##c_i## turns out to be negative, giving me a negative number of atoms, which is obviously incorrect. The portion where I simply convert 5g of uranium into total number of atoms is fine. Anyone know where I'm going wrong? I'm assuming ##t=0## since time is seemingly irrelevant to the problem as stated.

EDIT: I guess I should add that I am getting my decay constants from Wolfram Alpha, but I don't see why it would be giving me anything but the correct values, as they match with other available literature.
 
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  • #2
If you start with pure uranium at t=0 then your Radon decays at t=0 will be zero.
U-238 should be by far the most long-living isotope in your sample. If the sample is old enough (no time is given, so I guess you have to assume that), all other isotopes should be very close to their equilibrium concentrations.
 

1. What is the uranium decay chain?

The uranium decay chain is a series of radioactive decays that occur in the element uranium, beginning with the isotope uranium-238 and ending with a stable isotope of lead. This process involves the emission of alpha particles, beta particles, and gamma rays, and can take thousands of years to complete.

2. What is the Bateman equation?

The Bateman equation is a mathematical formula used to describe the rate of change in the concentration of a radioactive nuclide over time. It takes into account the decay constant of the nuclide, the production rate from its parent nuclide, and the number of decay steps in the decay chain.

3. How does the uranium decay chain affect the environment?

The uranium decay chain can have significant impacts on the environment, as it produces radioactive elements that can persist for thousands of years. These elements can contaminate soil, water, and air, and can have harmful effects on plants, animals, and humans. Proper disposal and management of radioactive waste from the decay chain is crucial to minimize these impacts.

4. Can the Bateman equation be applied to other decay chains?

Yes, the Bateman equation can be used to model the decay of any radioactive nuclide, as long as the necessary parameters are known. It is commonly used in nuclear physics and environmental studies to track the behavior of different radioactive elements and their decay products.

5. How is the Bateman equation used in nuclear research?

The Bateman equation is a fundamental tool in nuclear research, as it allows scientists to predict the behavior of radioactive elements and their decay products over time. It is used to study the production and decay of isotopes in nuclear reactors, to understand the environmental impacts of nuclear waste, and to develop strategies for nuclear waste management.

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