Americium-241 is used in smoke detectors. It has a first-order rate constant for radioactive decay of k = 1.6 * 10-3 yr-1. By contrast, iodine-125, which is used to test for thyroid functioning, has a rate constant for radioactive decay of k = 0.011 day-1. (b) Which one decays at a faster rate?

Question

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Welcome back everyone in this example, we have a first order rate constant for the radioactive decay of Kelly um to 48 and seaboard GM to 60 as the following values. First being 0.292 inverse hours and 0.173 inverse milliseconds, respectively. We need to determine which of these isotope samples decays at a faster rate. So because we're concerned at of the sample with the faster decay rate, we want to recall the formula for the calculation of half life and we know that we have a first order rate constant. And so we want to recall our first order half life formula. So we should recall that that formula is that our half life is equal to the natural log of two divided by our rate constant K. And so to simplify this, we can say that our half life is equal to when we take in our calculators, the natural log of two. We should get a value of 0.693 divided by and sorry, that dividing sign should be bolder, so divided by our rate constant K. And so what we're going to do is solve for the half life for each of our isotopes. And depending on our value that we get for half life for each of the isotopes, whichever value is the smaller amount of time will be the sample that decays at a faster rate for our final answer. So for our first isotope will use Broccoli into 48. So we have And we'll just use the symbol here, so to 48 b. K. We want to calculate its half life and we would say that its half life is equal to 0.693 divided by the rate constant. And that is given according to our prompt as a value of 0.02 92 inverse hours. Now we want to go ahead and recall that we're going to be expressing half life in milliseconds Right now in our prompt were given in verse hours and in verse milliseconds but we want half life to be in just milliseconds. So we'll just place that in parentheses here for this example. And so we would go ahead and here we get a value equal to 23.73 29. And now that we don't have inverse hours in the denominator, we're back in the numerator, we just have units of ours. And so we can focus on the conversion factor to convert from hours into minutes And then from minutes we can get into seconds where from seconds we will go into milliseconds. And so we're going to follow the following dimensional analysis. So in our denominator we want to recall the conversion factor to go from hours to minutes and recall that in one hour we have 60 and sorry this is 60 minutes moving forward in our conversion now that we can cancel out ours here, we want to focus on going from minutes to seconds by recalling the conversion factor that in one minute we have 60 seconds. So now we're able to cancel out minutes and now we want to focus on the conversion factor which we recall to go from seconds to milliseconds. Our prefix milli tells us that for one millisecond we have 10 to the negative third power of our base unit seconds. And so now we're able to cancel out seconds leaving us with milliseconds as our final unit for half life, which is what we want and what we're going to get here is a value for 48 broccoli um Equal to 8.5438 and will include one more sigfig being four times 10 to the seventh power and we have units of milliseconds left. And so this is our half life for 2 48 per kelly um to decay. And so now we're going to compare this to our calculation for the half life of our second isotope which is for To 60 C Board GM. And so beginning its half life calculation, we would take in our numerator our natural log of two which we determined is .693. And this is then divided by our rate constant given in the prompt for seaboard gm to as 0.1 73 inverse milliseconds. And so solving for this quotient in our calculators we should get a value equal to 4.578. And we have units of just milliseconds here because it's in the numerator now. And so now we have both of our half life values for the each of our isotopes in our derived unit milliseconds. Now we want to compare these two values and based on these two values here, we can say that therefore our half life of Seaborg EM 260 is less than the half life. I'm sorry, that's T one half of our first isotope broccoli, um 2 48. And so thus we can say that Seaboard Gm 2 60. And sorry about that. So seaboard GM to 60 has a faster or decays at a faster rate. And so this statement here is going to be our final answer to complete this example. So I hope that everything that I reviewed was clear. If you have any questions, please leave them down below and I will see everyone in the next practice video.

Americium-241 is used in smoke detectors. It has a first-order rate constant for radioactive decay of k = 1.6 * 10-3 yr-1. By contrast, iodine-125, which is used to test for thyroid functioning, has a rate constant for radioactive decay of k = 0.011 day-1. (b) Which one decays at a faster rate?

Verified Solution

Key Concepts

Radioactive Decay

First-Order Kinetics

Unit Conversion