The maximum level of radon in drinking water is 4.0 pCi (4.0 x 10^-12Ci) per milliliter. (a) How many disintegrations occur per minute in 1 mL of water at the maximum radon level? (b) If the radioactive isotope is 222Rn(t1/2 = 3.8 days), how many 222Rn atoms are present in 1 mL of the water?

Video transcript

Welcome back, everyone. We're told that the radio activity in a soil sample that comes from the decay of uranium isotopes is 6 62 picocuries per milligram. Calculate the number of disintegration in one mg of soil in one hour. Assuming that the major radio isotope is uranium 2 38 with a half life of 4.468 times 10 to the ninth power years. How many atoms of uranium 2 38 are there in 2.5 mg of soil? All right, we're going to begin by recalling some conversion factors Where one Becquerel, which is typically our units for the activity of a radioisotope is equivalent to one disintegration per second. We also want to recall that a disintegration Is for one Atom. Next, we want to recall that one Curie is equivalent to 3.7 times 10 to the 10th Power Becquerels. And then note that were given the activity of uranium in PICO curies per milligram. And so we're going to recall that one PICO Curie are prefixed, PICO tells us is equivalent to 10 to the negative 12 power curies. Now, with these in mind, we're going to calculate the activity And we can use a as shorthand for activity of uranium 2 in our proper units as disintegration, sorry disintegration In one hour as the prompt asks. And so we're going to begin with the sample size of our soil, which we're told is one mg of soil which will multiply by the activity of the uranium in the soil being 662 PICO Curies as given in the prompt per milligram. And so this allows us to cancel out milligrams and we want disintegration per hour as our final unit. So let's move from PICO curies in the denominator to curious in the numerator where we recall that as we stated, one PICO curie is equivalent to 10 to the negative 12th of our base unit. Curie canceling PICO curies will now move from curies in the denominator to Becquerel is using our other conversion factor where one curie is equivalent to 3.7 times 10 to the 10th Power Becquerels. Next, we want to cancel out curious and move from Becker Ells, which we understand are equivalent to one disintegration which will just use as dis for shorthand per second. We're going to continue our study geometry and we need to get rid of seconds because we want ours. So we're going to recall that we have an equivalent of 60 seconds to one minute. So canceling out seconds will now go from 60 minutes in the numerator equivalent to one hour. So canceling out minutes were left with disintegration per hour as our final unit, which are equivalent to one becquerel in this case. And so we would get an activity of uranium 2 38 in disintegration per hour in one hour of time being 88,178 10. disintegration per hour. And so this would be our first answer as the activity of uranium 2 38 in one hour. Now, we're going to move forward. And for part two were asked to determine the number of atoms of uranium 2 38 and 2.5 mg of soil. And so to find our atoms of uranium 2 we're going to recall our rate law of decay where the rate of decay, sorry, the rate of decay is directly proportional to R D K constant K. So K is R D K constant multiplied by our atoms of our radio isotope. And so in this case, N is Adams of uranium 2 38 which is what we're solving for. However, we don't know our decay constant, but we should recall that we can calculate it by taking the Ln of two divided by the given half life for our radio isotope of uranium in the prompt. And so what we would find is that we want to isolate for N to solve for the second answer of our prompt. So we're going to take the rate equal to the Ln of two results in a value of 0.6 93 divided by our half life, which will just say is T one half for now times N. So isolating for N, we want to divide both sides by the reciprocal fraction of our decay constant. And so we would divide by T one half over 10.6 93 on both sides. And rather, this is actually multiplication here since we're taking the reciprocal. And so what we would find is that our atoms of our radio isotope and are equal to our rate multiplied by the half life of our radio isotopes. So T one half Divided by 0.693 in the denominator. And now with this expression written out, we want to make note of the fact that we're given our half life of our uranium isotope Were given T 1/ As 4.46, 8 times 10 to the ninth power years note that the units are in years. And so we're going to need our rate of decay in years. So that's what we're going to calculate next before we can find n. And so to find our rate, we would say that our rate of decay is equal to our activity because they are actually interchangeable, They just have different units activity as we discussed above is expressed in disintegration per hour. And our rate is going to be in respect of inverse time. And so we would incorporate our activity as we calculated above being 88,178.4 dis integrations per hour In which now we want to incorporate the unit years because we have years for our half life unit. And so we need to cancel hours. So we're going to begin with the conversion factor where 24 hours is equivalent to one day, canceling out hours. We're now going to get rid of days where we recall that we have 365 days in one year. And so canceling out days were left with disintegration per year as our final unit. And this will result in a rate of decay equal to 7.72443 times 10 to the eighth power disintegration per year. Now, with this rate of decay defined, we can now go back to our equation to solve for N are atoms of uranium 2 38 in which will plug in our rate, which we just found as again 7.72443 times 10 to the eighth power disintegration per year, which is multiplied by R D K constant, which we found above as the value or sorry correction. We are multiplying by the half life T one half. So we're multiplying our rate of decay by T one half. And that was given to us as this value here. So we're multiplying by our half life given as 4.468 times 10 to the ninth power years. And so now we Will divide by from our equation 0.693. Now, we need our final unit to be Adams since we understand that N represents our atoms of our radio isotope uranium 2 38. And so we're going to include another conversion factor where we recognize that as we stated earlier, one disintegration is for one atom. In this case of uranium 2 38 allowing us to now cancel out our units as follows where we can get rid of disintegration here with disintegration. In the denominator, we can cancel out inverse years with years here. And we're left with Adam as our final unit, which is what we want. And what we'll find is that N is equal to a value of 4. times 10 to the 18th power atoms of uranium 38 in 2.5 milligrams of soil. And this would be our second final answer. So what's highlighted in yellow are our two final answers? First, the activity of uranium 2 38 as 88,178 380. disintegration per hour. And then our atoms of uranium 2 38 as 4. times 10 to the 18th power atoms of uranium 2 38. And this will correspond to choice be in our multiple choice. I hope that everything I went through is clear if you have any questions, just leave them below and I'll see everyone in the next video.