Cobalt-60 is a strong gamma emitter that has a half-life of 5.26 yr. The cobalt-60 in a radiotherapy unit must be replaced when its radioactivity falls to 75% of the original sample. If an original sample was purchased in June 2016, when will it be necessary to replace the cobalt-60?

Video transcript

Welcome back everyone. We're told that your opium 152 has the potential to be used in external beam radiotherapy but is not yet developed for clinical use. We're told that European 152 has a half life of 13.5 years, assuming that your opium also needs to be replaced when it's activity drops to 75% of the original sample. Like in cobalt 60 machines determine when it will be replaced if the machine is developed on January 2030. So what we should first recall is that for radioactive or nuclear Dick clay? We will always follow first order kinetics meaning we'll follow the following rate law where we have the natural log of our concentration of our isotope. In this case your opium represented by E. U. This is going to be our concentration of your opium at the given time. Which is divided by our concentration of your opium initially and set equal to negative one times our rate constant K. Times the amount of time that we also want to follow. The half life. Where we have our half life equal to 10.6 93 divided by our rate constant K. And so our first step is to actually figure out what this rate constant K. Is going to be. So we would isolate so that we have our rate constant K equal to 930.6 93 divided by our half life. And sorry there should be T 1/2. And so what we would have is K equal to .693 divided by our half life given in the prompt as 13.5 years, meaning that we would have a rate constant equal to a value of 0.051333 in verse years. And this is our rate constant. When are your opium activity drops to 75% of the original sample. But to answer our prompt we want to calculate the amount of time that it takes for the activity dropped to reach 100% of Its potential if the radiotherapy machine isn't developed until 2030. And so we're going to be solving for T in our formula for first order kinetics. So what we're going to have is the natural log of our initial concentration of your opium which is given in the prompt as .75. And rather not initial concentration but the concentration of your opium when its activity drops to 75%. This is then divided by our concentration of your opium at 100% capacity. So this would just be one This is going to be set equal to negative one times our rate constant K. Which we can actually plug in. Since we determined that above as 0.051333. In verse years. Multiplied by time. T. Which is what we're solving for here. So simplifying this, we would have the natural log of .75 equal to negative 0.051333. In verse years times time. And simplifying this further, we would take the natural log of .75 and get negative 0.287682 equal to 0 -0.051333 inverse years multiplied by time, simplifying this so that we can divide both sides by negative 0.051333 in verse years. This cancels out on the right hand side and we would get that the amount of time that passes is going to be 5.6042 years. Where we're going to need to convert this amount of time. Today's so we're going to begin with this value 5.6042 years. We're going to convert from one year. So we'll have sorry one year To an equivalent of 12 months and converting from months We would be able to go to and rather let's end this conversion here to make things simple, we'll calculate our months first which will give us a result of 67.2, 5 months. However, when we want to calculate days, we go from 5.6042 years and we multiply to go from one year to 365.25 days where we will cancel out years as we did before. And this is going to yield 2047 days And so 2047 days after January 2030 is going to be approximately september 2035 we would say. Which is the difference of 67.25 months. And so for our final answer that are your opium machine, our european radiotherapy machine would need to be replaced by september of 2035 approximately. For it to be back to 100% functional capacity. So I hope that everything I reviewed was clear. If you have any questions, please leave them down below and I will see everyone in the next practice video.