A sample of 37Ar undergoes 8540 disintegrations/min initially but undergoes 6990 disintegrations/min after 10.0 days. What is the half-life of 37Ar in days?

Video transcript

Hello everyone today we have the falling problem. Initially a sample of sodium 24 decays at a rate of disintegration per second after 200 minutes, the rate slows down to 385.59 disintegration per second, calculate the half life of sodium 24 in hours. So to do that, we first have to recall the integrated rate law which is the natural log of the amount of atoms that we have at a given time equal to negative the constant times the time plus the natural log of our initial number of atoms. And so we also have to recall the formula for our rate constant which is K. Is equal to the natural log of two divided by our half life. If we were to rearrange the integrated rate law and replace K. We would end up with the following, we would have our natural log of our in sub t. Over our in sub zero equal to negative Our natural log of two divided by our time over our half life. And so are in sub T. Is going to be this secondary disintegration per second. Our tea is going to be our 200 minutes. And our in sub zero or our initial number of disintegration per second is going to be this 450 disintegration per second or the initial rate. So we have to essentially solve for our half life or our T sub half. So what we're gonna do. We're gonna plug in our values. We're gonna have our 385 like sticks Divided by 4 50. We're gonna we're gonna equal that to our negative natural log of two times minutes over our half life. And so if we were to solve for our half life we would get our half life Equalling 897.45 minutes. However, we need this in hours. So what we're gonna do is we're gonna use the conversion factor that in one hour we have 60 minutes And so our units of minutes will cancel out. We'll be left with a half life of 14.96 hours as our final overall. I hope this helped. And until next time.