Nuclear Physics reaction equilibrium

In summary, the decay chain of 238U with a half life of 4.5e9 years involves multiple alpha and beta decays, with the final product being stable 206Pb. To determine the number of beta decays involved, a system of equations can be used, with the change in neutrons and protons being equal to the number of alpha and beta decays. However, this method may not yield a whole number solution. In part b, the rate of alpha particle production in a rock containing 1.0g of 238U can be determined by solving a large differential equation, taking into account the equilibrium of production and decay for all members of the decay chain. For part c, the global inventory
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Homework Statement



238U with half life of 4.5e9 years decays by alpha and beta(-) to stable 206Pb in a decay chain

a) how many beta decays are involved
b) What is the rate of alpha particle production in a rock containing 1.0g of 238U with all of the daughters in equilibirum?
c) Calculate the global inventory of 222Rn with half life 3.8 days if there are 4.9e9tons of 238U

Homework Equations


[tex]\alpha[/tex] = [tex]\lambda[/tex]n(t)
n(t) = n0e^-[tex]\lambda[/tex]t

The Attempt at a Solution



I am sort of stuck on the first one which is crtical for part B. I know this is probably a popular decay series but my instructor said you should be able to solve without looking it up. My problem is that there are a couple combos of alpha and beta decays that would work right, just that one is more popular than the other. Anyway I was trying to do something where I use 2 equations one for the change in neutrons and the other for the change in protons and solve a simple linear system but that has not work

my equations were
4 [tex]\alpha[/tex] - [tex]\beta[/tex] = 22 = neutrons lost
2 [tex]\alpha[/tex] + [tex]\beta[/tex] = 10 = protons lost

but i get a fractional answer for this so this is obviously not correct.

Far as part b goes the decay chain is going to be large, is the solution just going to be one huge differential equation? Equilibrium means that the production and decay for all the members of the chain is balanced. I am not sure how to solve such an equation. Each daughter will havea different [tex]\lambda[/tex] but somehow you need to solve to get the same activity.

Any help much appreciated
 
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. For part C I was thinking of using the equation for activity \lambdaN = \lambdaA / (1-e^-\lambdaT) where \lambdaT is the half life of the daughter, but I am not sure how to solve this either. Thanks!
 

1. What is nuclear physics reaction equilibrium?

Nuclear physics reaction equilibrium is a state in which the rates of nuclear reactions occurring within a system are equal, resulting in no net change in the composition of the system over time.

2. How is nuclear physics reaction equilibrium achieved?

Nuclear physics reaction equilibrium is achieved when the rate of forward reactions equals the rate of reverse reactions. This can be achieved by controlling the temperature, pressure, and concentrations of reactants in a system.

3. What factors affect nuclear physics reaction equilibrium?

The temperature, pressure, and concentrations of reactants all affect nuclear physics reaction equilibrium. Additionally, the presence of a catalyst, the nature of the reactants, and the energy of the reacting particles can also impact the equilibrium state.

4. What happens to a system at nuclear physics reaction equilibrium if the conditions are changed?

If the conditions of a system at nuclear physics reaction equilibrium are changed, the equilibrium state will shift in order to maintain equal rates of forward and reverse reactions. This is known as Le Chatelier's principle.

5. How is nuclear physics reaction equilibrium used in practical applications?

Nuclear physics reaction equilibrium is used in various practical applications such as nuclear power plants, nuclear weapons, and radiocarbon dating. It is also used in the production of certain chemical compounds and in the study of nuclear reactions in the universe.

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