A laboratory rat is exposed to an alpha-radiation source whose activity is 14.3 mCi. (b) The rat has a mass of 385 g and is exposed to the radiation for 14.0 s, absorbing 35% of the emitted alpha particles, each having an energy of 9.12 * 10-13 J. Calculate the absorbed dose in millirads and grays.

Video transcript

All right. Hello everyone. So this question says that a radioactive isotope emits alpha radiation with an activity of 11.2 mile and each particle having an energy of 8.49 multiplied by tens of the negative 13th joules assume a person weighing 65.5 kg is exposed to this radiation for 20.0 seconds. And that 25.0% of the emitted alpha has been absorbed. Calculate the dose absorbed by the person in Milly rads and grapes. All right. So first and foremost, we have four different entry choices labeled A through D proposing different values for the dose in both milly rads and grays. But let's clarify a few things about these sets of units. First and foremost, recall it one curry or C for short is equal to 3.7 multiplied by 10 to the 10th power disintegrations per second. And so from here, we can also recall that one rad is equal to one multiplied by 10 to the negative fifth power joules per gram and one gray is equal to one jewel per kilogram and one gray is also equal to 100 rats. So here, our first step is to convert the activity given in mile curries into disintegrations per second. So starting with our given, we have 11.2 millicuries and to use the conversion factor that we used or that we discussed in the beginning of this video, we're going to have to convert milli curries into curries. So here recall that 1000 millicuries would equal one curry. So we're dividing 11.2 by 1000 to make sure that the units of Milly cures cancel out. From here, we can apply our conversion factor in which 3.7 multiplied by 10th, the 10th power disintegrations per second is on the top or in the numerator and one curry is in the denominator to make sure that curries cancel out. And so here, the total amount of disintegrations per second is equal to 4.144 multiplied by 10 to the eighth power. So this is the total amount of disintegrations per second. But recall that this is not the true amount of radiation that was absorbed by the person because the individual only absorbed 25.0% of this total radiation. So here to understand how much radiation the person actually absorbed, we have to take the total amount of disintegrations per second and multiply this by the percentage of or rather the percentage absorbed expressed as a decimal. So here that's going to be 4.144 multiplied by 10 to the eighth power disintegrations per second, multiplied by 0.25. And this equals 1.036 multiplied by 10 to the eighth power disintegrations per second, which is therefore what the person actually absorbed. And so now using the amount of radiation absorbed, we can convert this into joules per gram. So first, we're going to need the mass of the individual which was 65.5 kg. And we're going to convert this into grams by multiplying this number by 1000. Since 1000 g corresponds to 1 kg, this equals a mass of 65,500 g. All right. So now we have the mass of the individual in grams, which means that we're going to need the energy in joules. So here using the amount of radiation absorbed, that's 1.036 multiplied by 10 to the eighth power disintegrations per second. I can first multiply this by the time that was 20.0 seconds to cancel out our units of time. And recall that we can convert disintegrations into jewels using one of our other conversion factors, namely that 8.49 multiplied by 10 to the negative 13th power jules is equal to one disintegration. This cancels out disintegrations as a unit here. And so this simply ensures that our numerator is going to be in units of jewels. So this product is going to be divided by our mass, which was the 65,500 g. And so after evaluating this expression are answer in reds or excuse me, our answer in joules per gram is equal to 2.6857 multiplied by 10 to the negative eighth power. And so from here, we can convert jewels per gram into rads, later, Milly rads and then later grays. So let's go ahead and continue. We have 2.6857 multiplied by 10 to the negative eighth power joules per grams. And we can first convert this into rats because one rat in the numerator of my conversion factor is equal to one multiplied by 10th, the negative fifth power joules per grams in the denominator. This equals 2.6857 multiplied by 10 to the negative third rats. And to convert this into Miller rads, I can div divide this number by 1000. And after dividing by 1000, our answer rounded to three significant figures is 2.69 milly rat. And so now for our final step, using our value in rads, we can convert this into grace. So here that's 2.6857 multiplied by 10th, the negative third rads multiplied by our conversion factor in which one gray is equal to 100 rats. And this ultimately equals 2.69 multiplied by 10th, the negative fifth power grace and there you have it. So scrolling up here to consider our final answer, our final answer is going to be option D in the multiple choice because we have 2.69 millirads, which is equal to 2.69 multiplied by 10th of the negative fifth power grace. And with that being said, if you watch this video all the way through, thank you so very much and I hope you found this helpful.