Differential equation - radioactive decay

In summary: AmBn?halflife = ln 2 / lambdaThe Attempt at a SolutionWell this isn't like any other radioactive decay problem I've seen. Usually we're given initial conditions, or the half life, or something! I don't even know where to begin... I'll just try to solve the differential equation for da/dt and see where that takes me.solve for da/dt:da/dt = -k1 awhere k1 is the decay rate.equation for db/dt:db/dt = kAmBn?
  • #1
phil ess
70
0

Homework Statement



Suppose that a given radioactive element A decomposes into a second radioactive element B, and that B in turn decomposes into a third element C. The amount of A present is initially a0. The amounts of A and B present at a later time t are a and b respectively. If k1 and k2 are the reaction rates for the two chemical reactions, find b as a function of t.

Hint: Consider k1 and k2 to be positive, so, for example, one of the equations that you need is da/dt = rate in − rate out = −k1a. Also, you may assume k1 =/= k2.

Homework Equations



rate = kAmBn?

halflife = ln 2 / lambda ?

The Attempt at a Solution



Well this isn't like any other radioactive decay problem I've seen. Usually we're given initial conditions, or the half life, or something! I don't even know where to begin...

please help!
 
Physics news on Phys.org
  • #2
phil ess said:

Homework Statement



Suppose that a given radioactive element A decomposes into a second radioactive element B, and that B in turn decomposes into a third element C. The amount of A present is initially a0. The amounts of A and B present at a later time t are a and b respectively. If k1 and k2 are the reaction rates for the two chemical reactions, find b as a function of t.
Does the problem statement really say these are chemical reactions? I ask because they are not -- they are nuclear reactions.

Hint: Consider k1 and k2 to be positive, so, for example, one of the equations that you need is da/dt = rate in − rate out = −k1a. Also, you may assume k1 =/= k2.
Okay, for starters, do you understand that radioactive decay is described by an equation like
da/dt = -k1 a​
where k1 is the decay rate?

The next steps are:
1. Solve the differential equation for da/dt
2. Then write an equation for db/dt

Homework Equations



rate = kAmBn?

halflife = ln 2 / lambda ?

The Attempt at a Solution



Well this isn't like any other radioactive decay problem I've seen. Usually we're given initial conditions, or the half life, or something! I don't even know where to begin...

please help!
You are given initial conditions: the amount of A is initially a0. There is no B or C present initially.
 
  • #3
Can someone finish up the problem and get an equation for db/dt?
 

1. What is a differential equation?

A differential equation is an equation that relates a function to its derivative. In other words, it describes the rate of change of a quantity in terms of its current value.

2. How is a differential equation used to model radioactive decay?

In radioactive decay, the rate of decay of a radioactive substance is proportional to the amount of the substance present. This can be represented by a differential equation, where the rate of change of the quantity of the substance is equal to a constant multiplied by the quantity itself.

3. What is the half-life of a radioactive substance?

The half-life of a radioactive substance is the amount of time it takes for half of the substance to decay. It is a characteristic property of each radioactive substance and can be calculated using the differential equation for radioactive decay.

4. How can differential equations be solved?

There are various techniques for solving differential equations, including separation of variables, substitution, and using integrating factors. In the case of radioactive decay, the solution can be found by using the exponential function.

5. Can radioactive decay be modeled by more complex differential equations?

Yes, in some cases, more complex differential equations may be used to model radioactive decay. For example, if the rate of decay changes over time, a first-order differential equation with a variable coefficient may be used. Additionally, higher-order differential equations may be used to model multiple substances decaying at different rates.

Similar threads

  • Biology and Chemistry Homework Help
Replies
2
Views
1K
  • MATLAB, Maple, Mathematica, LaTeX
Replies
18
Views
3K
  • Biology and Chemistry Homework Help
Replies
11
Views
2K
  • Engineering and Comp Sci Homework Help
Replies
1
Views
876
  • Introductory Physics Homework Help
Replies
4
Views
1K
  • Biology and Chemistry Homework Help
Replies
5
Views
2K
  • Introductory Physics Homework Help
Replies
3
Views
833
  • Precalculus Mathematics Homework Help
Replies
5
Views
2K
  • Introductory Physics Homework Help
Replies
4
Views
2K
  • Advanced Physics Homework Help
Replies
5
Views
1K
Back
Top