It takes 4 h 39 min for a 2.00-mg sample of radium-230 to decay to 0.25 mg. What is the half-life of radium-230?

Video transcript

Hello. Everyone in this video being told that in about two hours and seven minutes the mass of americium is going to be decreasing from five mg to 1.5 mg via a radioactive decay. In this question where we want to determine the half life of americium 37. So the radioactive decay follows the first order of kinetics. So we have the equation natural log of the concentration of N. T. You're going to a negative K. T. Plus the natural log of our N. Initial and T. Half is going to be equal to the natural log of two divided by R. K. Constant. So T. Is equal to. So we have two hours. We're gonna convert this in two minutes. Which is going to be of course that for every one hour we have minutes and we'll go ahead and add this to our seven minutes. This gives us our total time. We see for the kind of um setup that we have here that the hours units will cancel. So once we put all this into a calculator, get that T. Is equal to then 1 27 minutes. Now you need this information we can go ahead and solve for R. K. Value again, I'm just gonna rewrite this in our purple color. So the natural log of N. T. Is equal to the negative K. T. Plus the natural log of the initial concentration of N. So we'll just plug in these values then. So we have the natural log of their point or rather let's see we have a natural log of the value 1.50. This equals to negative K. And then we said is 1 27 minutes plus the natural log of five mg. So once you put this in the calculator we just need to simplify. So we get 0.405465. Mingling to negative K. Times 1 27 minutes Plus 1.6909. Let's see we have 1.609438. And of course we can see here now we isolate four K. Once we do so we get that K then is equal to 9.480102 times 10 to the negative three minutes to the negative one. So one over minutes and now four my T half is equal again to the Ln of two divided by K. We already have K. So let's plug this in. So this is just the natural log of two divided by 9.480102 times 10 to the negative three minutes to the negative one. Let's put this into the calculator. If we do so we get the numerical value to be 73. units being minutes. So then my final answer for the half life of a mere cm to is equal to 73.1 minutes. And that is my final answer for this question