Radiation and differential equations

In summary: It should be $$x(t)=\frac{1}{6} \times 10^6+5 \times 10^5+ \lambda \exp(\frac{-R}{10^6}t)$$In summary, we have a room with a volume of 1 million liters that is being contaminated at a rate of 1/6 * 10^6 pCi/hr from the basement. The room also has a ventilation system that exchanges air with the outside at a rate of R liters/hr. The outside air has a concentration of 0.5 pCi/liter. We can set up a differential equation, dx/dt = a - bx, to represent the total radiation in the room, with a representing
  • #1
Pqpolalk357
13
0
Sam is seeping into a room from the basement at a rate of

$$\frac{1}{6} \times 10^6 \frac{\mathrm{pCi}}{\mathrm{hr}}$$

(pCi=picocuries). The room contains $$10^6$$ liters of air. (The rate was chosen so that the room reaches the EPA action level of $$4 \frac{pCi}{liter}$$ after $$24$$ hours.) Air in the room is being exchanged with the outside air at a rate of $$R$$ liters/hr. The outside air has a concentration of $$0.5$$ pCi/liter.
Set up an equation for the total radiation $x$ in the room in picocuries, assuming instantaneous uniform mixing, that is, the indoor concentration of radioactivity is $$\frac{x}{10^{6}}$$ pCi/liter.

Explain your equation using the notations $$\Delta x$$ pCi, $$\Delta t$$ hr, and display the units of every variable explicitly to show that they match (e. g., (pCi/liter)(liter/hr) = pCi/hr).

I wanted to try to express the differential equation under the form $$\frac{dx}{dt}=a-bx$$ such that x is the value of the radiation, but I am not sure how to use the remaining information.
 
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  • #2
The form of your target equation says radiation is entering at rate a and leaving at a rate proportional to the amount in the room. So far so good.
What are the sources of radiation entering the room. What is the rate for each source?
If the concentration in the room is x, at what rate is it leaving the room?
 
  • #3
So we have a=1/6*10^6. I still don't understand what is b ..
 
  • #4
I think I found. Is it: $$\frac{\Delta x}{\Delta t}=\frac{1}{6}\times 10^6+0.5R-R\frac{x}{10^6}$$
 
Last edited:
  • #5
I am having a problem now with the final question:

Find the rate R at which the equilibrium is 1.5 pCi/liter (well below the EPA action level). At this value for R, how many times per day is the total volume of air of the room exchanged?

I tried to solve the above differential equation and obtained: $$x(t)=\frac{1}{6} \times \frac{10^{12}}{R}+5 \times 10^5+ \lambda \exp(\frac{-R}{10^6}t)$$

Perhaps it is the wrong procedure, but I don't really know how to proceed with this question.
 
  • #6
I don't think you need to solve for x(t) at all. Equilibrium occurs when dx/dt = 0. So just plug that in and solve for R given that you want x/(10^6) = 1.5

The rest should be pretty trivial once you have R.
 
  • #7
We find R=$$\frac{1}{6} \times 10^6$$ but I don't understand the rest of the question.
 
  • #8
Well, in the simplest terms, that means that one-sixth of the air is exchanged every hour. So... how many times is the entire volume exchanged over 24 hours?
 
  • #9
4 times
 
  • #10
Yup.
 
  • #11
Ok Thank you very much for all your help. Have a nice day.
 
  • #12
No problem, and the same to you.
 
  • #13
Pqpolalk357 said:
$$x(t)=\frac{1}{6} \times \frac{10^{12}}{R}+5 \times 10^5+ \lambda \exp(\frac{-R}{10^6}t)$$
Turned out you didn't need it, but that is not a solution of the differential equation.
 

Related to Radiation and differential equations

1. What is radiation and how is it related to differential equations?

Radiation is the process of energy being emitted in the form of electromagnetic waves or particles. It is related to differential equations because these equations are used to model and describe the behavior of radiation, such as how it spreads through space and interacts with matter.

2. What are some common types of radiation and how do they differ?

Some common types of radiation include alpha, beta, and gamma radiation. These types differ in their ability to penetrate materials, with alpha radiation being the least penetrating and gamma radiation being the most penetrating. They also differ in their ionizing abilities, with alpha radiation being highly ionizing and gamma radiation being weakly ionizing.

3. How can we use differential equations to study the effects of radiation on living organisms?

Differential equations can be used to model the transfer of energy from radiation to living organisms. By using these equations, we can calculate the absorbed dose of radiation and its effects on different tissues and organs in the body. This can help us understand the potential risks of exposure to radiation and develop ways to protect against it.

4. What is the role of boundary conditions in solving radiation differential equations?

Boundary conditions are essential in solving radiation differential equations because they define the limits of a system and help us determine the behavior of radiation at the boundaries. These conditions can include factors such as the type and intensity of the radiation source, the properties of the material through which the radiation is passing, and the geometry of the system.

5. Are there any limitations to using differential equations to model radiation?

One limitation of using differential equations to model radiation is that they are based on assumptions and simplifications of the actual physical processes. These assumptions may not always accurately reflect the complex behavior of radiation in real-world scenarios. Additionally, differential equations can become computationally challenging to solve for more complex systems, which may require the use of numerical methods.

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