The half-life of ²³⁸U is 4.5⨉10⁹ yr. A sample of rock of mass 1.6 g produces 29 dis/s. Assuming all the radioactivity is due to ²³⁸U, find the percent by mass of ²³⁸U in the rock.

Video transcript

Hello everyone. So in this video we're gonna calculate for the percent by mass of this in the rock. And the half life is 1.405 times 10 to 10 years. And we have a 25 sample of the rock. Alright so we're gonna go ahead and remember that whenever we're dealing with radioactive and nuclear decay of isotopes we're gonna go ahead and follow the fourth order of kinetics. So the half life of a first order reaction is given to us by this equation. So T to the half is equal to the natural log of two divided by K. So we're gonna go ahead and isolate R. K. To give us that K. It goes to the natural log of two divided by T. To the half. So plugging in the values the natural log of two is about 0.693 red tea. It's going to be 1.405 times 10 to the 10 year. So putting that into my calculator, I'll get the value of 4.93 to 4 times 10 to the negative 11 year, two negative one. Alright so next step here then We have our 40 six dis integrations per second. We want to convert that unit into our our seconds in two years. So doing some simple dimension analysis. We're starting off with the 46 here. We first want to convert our seconds into minutes. So 60 seconds for every one minutes. Then we're gonna go ahead and cover our minutes into hours. So every 60 minutes is one hour converting hours in today's for every 24 hours we have one day And last conversion is going from days to years. So every 365.25 days we have one year. So then you numerical value again from this is 1.4516 times 10 to the 9th power. Alright, so going back to our rate then. So we have that the rate is equal to K times N. Again, rearranging the equation to give us an isolate on one side of the equation We have. End equaling two rate divided by kate. So just plug in those values. The rate is 1.4516 times 10 to the nine And RK is going to be 4.93-4 times 10 to the -11. Alright, now, going ahead to put this into my calculator, I'll get the value of 2.94, 3 times 10 to the -9. Sorry to the tend to the 19. Okay, now we go ahead and solve for the mass of our element. So again, we're starting off with end value. Just 2.943 times 10 to the 90 power. Okay, now we're gonna go ahead and start our dimension analysis. Starting off with our avocados number. Now going ahead to convert our moles into grams. We use the molar mass. Alright, now the numerical value get is 0.1134 g of r t h a. Cackling for the percent. Finally, so we have the mass on top, Divided by the total amount of the sample that we have, giving us a percentage of 0.45%. And that is going to be my final answer to the question. Thank you all so much for watching.