# Radioactive samples and differential equations

• fogvajarash
In summary: I have just found out that the constant in the rate term is actually the rate constant k. So now I can do the problem!
fogvajarash

## Homework Statement

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

-

## 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.

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:
fogvajarash said:
dm/dt = 500 - kme-kt
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

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?

I see. So the only way of doing this exercise is knowing the definition? Or is there a non definition way?
The only way to do a problem is to understand what the words mean - yeah.

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

fogvajarash said:

## Homework Statement

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

-

## 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) = -0.2475y0e-0.02475t=-0.2475 y

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

Simon Bridge said:
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?
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.

fogvajarash said:
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.

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

fogvajarash said:
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?

We deal with a first order linear differential equation anyway.

ehild

## Related to Radioactive samples and differential equations

Radioactive samples are materials that contain unstable atoms that emit radiation in the form of particles or electromagnetic waves. These samples are used in various scientific and medical fields for research and treatment purposes.

## How are radioactive samples measured?

Radioactive samples are measured using a technique called radioactivity measurement, which involves using a Geiger-Muller counter or a scintillation counter to detect and measure the radiation emitted from the sample.

## What are differential equations?

Differential equations are mathematical equations that describe how a quantity changes over time or space. They involve derivatives, which represent the rate of change of a quantity, and are used to model various natural phenomena, including radioactive decay.

## How are differential equations used in the study of radioactive samples?

Differential equations are used in the study of radioactive samples to model and predict the rate of decay of the unstable atoms in the sample. This allows scientists to determine the half-life of the sample and the amount of radiation emitted over time.

## What safety measures should be taken when handling radioactive samples?

When handling radioactive samples, it is important to follow proper safety protocols, including wearing protective gear and handling the samples in a designated area with proper shielding. It is also crucial to dispose of the samples properly to avoid any potential harm to humans and the environment.

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