# Calculating safe levels of radioactivity

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• Guest432
I don't know, but you can look it up on the internet. The activity of cesium in water depends on how much cesium is in the water and how long it has been there.f

#### Guest432

Hello!

I am not sure how I would go about calculating the time it takes for a radioactive element to decay to safe levels. I know of the decay formula (N=No*e^(-kt)) and K =ln2/t(1/2). I believe calculating the time until an atom has decayed to safe levels has to do with the activity (Bq) A=kN of the element. However, I do not have confirmation on this.

Take, Cesium 137 for example. It has a half life of 30 years. However, I would like to create a general formula for x kg of this element to decay to 'safe' levels.

Thanks,
Trontor

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You have to define "safe" levels, and you have to know the initial activity of the radioactive materials. Once you know the ratio between initial and "safe" levels (the latter is more a radiobiology question), everything else is just a matter of mathematics. The distribution of the material will be relevant, and if you consider long timespans, the migration of the material in the environment will be relevant as well (but that is yet another field of science).

You have to define "safe" levels, and you have to know the initial activity of the radioactive materials. Once you know the ratio between initial and "safe" levels (the latter is more a radiobiology question), everything else is just a matter of mathematics.

I am willing to do the mathematics myself, but I'm really not too sure what I should be doing... thanks for any help :)

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Well, if the spinach of 2011 had 1931 Bq/kg and the limit is 500 Bq/kg, you can calculate the worst case waiting time until spinach from the same place will be below the limit. About a factor of 4 in the activity, so roughly twice the half-life. The actual time will be significantly shorter as cesium gets spread out over time, but a proper estimate there would need a detailed study.

Well, if the spinach of 2011 had 1931 Bq/kg and the limit is 500 Bq/kg, you can calculate the worst case waiting time until spinach from the same place will be below the limit. About a factor of 4 in the activity, so roughly twice the half-life. The actual time will be significantly shorter as cesium gets spread out over time, but a proper estimate there would need a detailed study.
I have understood this. However, which formulae should I be using to get an accurate answer?

The decay formula also works as activity formula, both are proportional to each other, so both decay exponentially.

The decay formula also works as activity formula, both are proportional to each other, so both decay exponentially.
Ok, I've attempted a solution for this. Don't judge ;)

1. This seems way too long, for a GRAM of cesium.
2. The initial activity has no effect on my solution :/

I'm doing this wrong, I presume. What do I need to fix?

The decay formula also works as activity formula, both are proportional to each other, so both decay exponentially.
Actually, using this calculator confirms my answer is correct... I hope...

is there anything wrong guys?

This is the time you have to wait until 100% pure radioactive cesium* becomes safe drinking water. Well, if we replace the decay products by water.

*which would evaporate under its own heat in practice

This is the time you have to wait until 100% pure radioactive cesium becomes safe drinking water. Well, if we replace the decay products by water.
Do you mean the caesium IN contamined drinking water?

The concentration of cesium in drinking water is significantly below 1 gram per gram of drinking water (that would not be water at all, it would be a block of cesium). If you start with 0.1 picogram of cesium per gram of drinking water, you get a completely different result.

The concentration of cesium in drinking water is significantly below 1 gram per gram of drinking water (that would not be water at all, it would be a block of cesium). If you start with 0.1 picogram of cesium per gram of drinking water, you get a completely different result.
Oh. I'm confused now. Where does the per gram of water come into this. I just used the 0.2Bq/gram from the website as a 'safe level'. I didn't think of a Cesium/gram of water

Well, your water is mainly water, but with a small amount of cesium in it. The activity in 1 gram of water (=where the limit applies) depends on the amount of cesium in the water. This amount will go down over time.

Well, your water is mainly water, but with a small amount of cesium in it. The activity in 1 gram of water (=where the limit applies) depends on the amount of cesium in the water. This amount will go down over time.
I understand. Where would I find the value for the activity of a normal amount of cesium contaminated water?

There is no normal amount, every sample will be different.

There is no normal amount, every sample will be different.
OK,
Should I assume that after a nuclear meltdown, there could be 0.1 picograms of cesium in every gram of water, then calculate the activity from that.

No, that is a number I made up. The actual concentrations will vary wildly depending on where you take the water from. You have to measure it.

No, that is a number I made up. The actual concentrations will vary wildly depending on where you take the water from. You have to measure it.
So this can't be hypothesised accurately?

So far we didn't make any difference between water taken from the reactor pool in Chernobyl and water taken from some underground reservoir in the middle of nowhere in Africa. Clearly the activity will be different for those two cases. The same formula can be used in both cases (and the water in Africa is probably below the threshold), but the starting activity is massively different.

Guest432
So far we didn't make any difference between water taken from the reactor pool in Chernobyl and water taken from some underground reservoir in the middle of nowhere in Africa. Clearly the activity will be different for those two cases. The same formula can be used in both cases (and the water in Africa is probably below the threshold), but the starting activity is massively different.
I've thought about this and tried researching, however, I still do not fully understand this concept. I just want to show how long it takes for Cesium from say, a fallout to be rendered safe. I need this to make a statement highlighting the dangers of radioactive material.

I just want to show how long it takes for Cesium from say, a fallout to be rendered safe.
That is not the right question. If you accumulate pure Cs-137 in large amounts, it will always be dangerous, because it will have an activity of 3 TBq/g. But no one sifts through millions of tons of soil to extract and purify all the Cs-137 in it. Why should you.

The relevant question is "when will normal things (like our drinking water) have an activity that is low enough". And that depends on how much cesium initially gets into the drinking water, and how it gets distributed there.

That is not the right question. If you accumulate pure Cs-137 in large amounts, it will always be dangerous, because it will have an activity of 3 TBq/g. But no one sifts through millions of tons of soil to extract and purify all the Cs-137 in it. Why should you.

The relevant question is "when will normal things (like our drinking water) have an activity that is low enough". And that depends on how much cesium initially gets into the drinking water, and how it gets distributed there.

OK,
But where do I get these values? I don't know of any average grams per litre of contamination or something. I just want to create a scenario that is plausible :(

But where do I get these values?
After accidents, people take samples from different places and measure the activities of individual radioisotopes. Those values end up in publications, reports (academic/governments) and rarely the popular press. You cannot find a single number as answer. It is not that easy.

After accidents, people take samples from different places and measure the activities of individual radioisotopes. Those values end up in publications, reports (academic/governments) and rarely the popular press. You cannot find a single number as answer. It is not that easy.
Nice!

After looking, I found this table:

I am really curious as to why the concentration level (kBq/m^3) goes from 22.2 to 1.8 within a year?

I am really curious as to why the concentration level (kBq/m^3) goes from 22.2 to 1.8 within a year?
Because it goes down stream.

mfb and Guest432
Because it goes down stream.
Oh, haha.

Ok, so I did the following.

1. Found how many atoms of Cesium-137 is in a gram of water in 1986 (1.8 kBq)
2. Found how many atoms in the prescribed safe activity per gram (0.2Bq)
3. Used the half life formula to calculate the time taken for the initial activity to decay to the second activity.

Note the units: the values are in kBq/m3.

Note the units: the values are in kBq/m3.
Ahh yes,
I know its a river, however I calculated m^3 to grams as for water 1m^3 = 10^6 grams.

If I do not do it this way, then I cannot use the prescribed safe activity level I have on hand.

So what do you get for the activity per gram if you convert 1.8 kBq/m3 to Bq/g?

So what do you get for the activity per gram if you convert 1.8 kBq/m3 to Bq/g?
0.0018 Bq/g

Right. A factor 100 below the 0.2 Bq/g limit.

Guest432
Right. A factor 100 below the 0.2 Bq/g limit.
...

Help

What? You figured out that all the reported values in this thread were below the safety limits all the time.

If you want to calculate some time, you'll need water with more contamination. Or take something else, e. g. mushrooms in Bavaria.

What? You figured out that all the reported values in this thread were below the safety limits all the time.

If you want to calculate some time, you'll need water with more contamination. Or take something else, e. g. mushrooms in Bavaria.

OK. I have looked now at examples from Fukushima Daichii 2011, but they're reporting the levels to now be near identical to pre incidental levels. Considering this is the largest incident since Chernobyl, is it safe to assume there have been no other incidents where water contamination was worthy to investigate?

I am considering just making a scenario and analysing that.