Uranium-238 has a half-life of 4.47 * 109 years and decays through a series of events to yield lead-206. Estimate the age of a rock that contains 105 mmol of 238U and 33 mmol of 206Pb. Assume all the 206Pb is from the decay of 238U.

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Welcome back, everyone. We're told CCM 1 37 decays mainly by beta emission and yields barium 1 and has a half life of 30 point oh five years. A sample of salt contains 156 million molds of cesium, 1 37 and 55 million moles of barium 1 37. How old is our salt sample? We're going to begin by recalling that whenever we have radioactive decay, we have a reaction that is first order, sorry. This says first order here. And we want to recall the following rate law which we should use where we would take the natural log of N over N zero. Defining N. We want to recall that N represents our number of our radio isotope at a given time. T And, and zero represents our amount of radio isotope at N. We're sorry, at T equals zero. With these notes in mind, we would finish our formula which is set equal to negative one times our decay constant K multiplied by the time that passes T recall that our decay constant is calculated by taking the Ln of two Over are given half life. And so we can also understand that our half life is calculated by taking the result of Ellen of to being 0.693, which is divided by our decay constant K. And so we can incorporate this into our integrated rate law where we can say the Ln of N over N zero is equal to negative one times 0.6 93 which is multiplied by the time that passes T Divided by our half life, 2.5 of our radio isotope. So the isotope that is decaying is our cesium 1 37. And note that we are given The amount that sassy makes up of our salt sample as 156 mil um als, we are also given an amount of our salt which is made up of 55 mil um als of barium 37. And so adding these two isotopes together, we would have an initial amount as 211 Mila Mel's. And so we can say that this is our calculation for N0. Now, with this in mind, we can go back to our formula and say that we have the L N of N where N is our amount of radioisotope at time T which according to our prompt, Our radio isotope Cesium 137 At the time the sample is gathered is 156 mm. So we can plug that in as N and then, and zero as we determined is the sum of our isotopes in total that make up our salt, which we calculated to be 211 mil. Um Als This is set equal to negative one times 0.693 multiplied by T Which we need to find divided by our half life given in the prompt for barium 137 as 30. years. So we want to continue and simplify this. So on the left hand side, when we take the Ln of our quotient, we have a result of negative 0.30-002. Our units of millennials cancel and this is equal to the right hand side which will be negative 0.6 93 T divided by 30.5 years. Now we need to isolate for time. So we can go ahead and recognize that time is being multiplied by negative 0.6 93. So on both sides, we want to divide by negative 0.6 93. And so on the left hand side, the result of this division Is going to be a value of negative 0.4357893 set equal to our right hand side which is now just T over 30.05 years and sorry, this cancels out here when we divide. So we now need to multiply both sides by our denominator 30.05 years so that it cancels out on the right hand side. And what we would find is that T is equal to a value of our product of the left hand side, which results in a value of 13.9 years in which we can round to about 13. years as three sig figs minimum for our final answer as the age of our salt. And this will correspond to choice D or choice C in the multiple choice. So I hope that everything I explained was clear. If you have any questions, please leave them down below. And I'll see everyone in the next practice video.

Uranium-238 has a half-life of 4.47 * 109 years and decays through a series of events to yield lead-206. Estimate the age of a rock that contains 105 mmol of 238U and 33 mmol of 206Pb. Assume all the 206Pb is from the decay of 238U.

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Key Concepts

Half-life

Radioactive decay

Isotope ratio