# Radioactive samples and differential equations

1. Oct 29, 2013

### fogvajarash

1. The problem statement, all variables and given/known data

A certain nuclear plant produces radioactive waste in the form of strontium-90 at the constant rate of 500kg. The waste decays exponentially with a half-life of 28 years. How much of this radioactive waste from the nuclear plant will be present after the following increments of time? Assume that initially 600kg of strontium-90 is in the nuclear plant.

a. N years
b. 140 years
c. To perpetuity

2. Relevant equations
-

3. The attempt at a solution

I found that the "rate in" was 500kg/year, and i'm not sure about the "rate out" (rate of depletion of the sample).

We are given that the sample decays exponentially, so we should have that y(t) = yo e-0.02475t. We can assume that yo is just y (the radioactive sample that is decaying). Then, the rate of depletion would be (I doubt that this is true):

y'(t) = -0.2475ye-0.02475t

This is because this is the rate of change of the sample. Then, i set up my differential equation to be the following (k is just the constant k = (ln2)/(half life)):

dm/dt = 500 - kme-kt

I rearranged the terms into the form dm/dt - P(x)y = f(t), i came up that the integrating factor was actually e-e^(-kt)), but this is not true (while differentiating the function I realized that it it was a completely new expression that was different to the one shown).

How can I proceed from this information or have i made a wrong set-up to the exercise?

Thank you.

2. Oct 29, 2013

### Simon Bridge

The decay rate depends on the current amount in the store.
That is what "exponential decay" means.

If y is the mass of 90Sr then:

$$\frac{dy}{dt} = -\lambda y$$

see:
http://en.wikipedia.org/wiki/Exponential_decay

The constant $\lambda$ is related to the half-life.

Last edited: Oct 29, 2013
3. Oct 29, 2013

### ehild

The differential equation has to be :

dm/dt=500-km,

as the amount of Sr-90 increase by 500 kg/year, and decreases proportionally with the amount present, that is by km. Solve, and fit the integration constant to the initial condition m(0)=600 kg.

ehild

4. Oct 29, 2013

### fogvajarash

I see. So the only way of doing this exercise is knowing the definition? Or is there a non definition way?

As well, the constant in the linear rate term (rate of decay) is just the k constant found in the exponent right?

5. Oct 29, 2013

### Simon Bridge

The only way to do a problem is to understand what the words mean - yeah.

If you don't know, you should solve the differential equation to find out what the constant is.

Have you not done a section of coursework that covers radioactive decay?

6. Oct 29, 2013

### ehild

That is true with a sample decaying only - with no input. Now you have input at constant rate. It is true that the rate of decay is proportional to the amount present : -0.2475y, but there is also input 500 kg/year. So the net rate of change of the amount of Sr-90 is dm/dt=500-0.2475m.

ehild

7. Oct 29, 2013

### fogvajarash

I have, but i just wanted to make sure if i'm correct or not (I'm sorry! I don't want to commit the same mistakes again).

I was thinking, as the rate of depletion is ry0e-rt, can we just say that it should be ry? (consider that m, the mass of the radioactive sample, is defined as y = y0e-rt).

After having this information it should just be a simple variables separable problem to get to the final answer. Say, if we had that the input wasn't constant (suppose it was 600t), then we would have to deal with a first order linear differential equation?

Thank you for your great patience Simon.

8. Oct 30, 2013

### ehild

The rate of depletion is is ry, but y is not equal y0e-rt.

We deal with a first order linear differential equation anyway.

ehild