Intro To Chemistry - Key Conversion Problems

Let's do two conversion problems, and we'll start with one that is super intuitive, and then we'll use that structure and strategy for more complicated problems. So our first problem is to convert 1.5 years into seconds. A useful strategy for solving these types of problems is to restate it in a way that helps us better find the answer. How much time, or what quantity in seconds, equals the same amount of time in 1.5 years? What do we have, and what do we need? Well, we need the number of seconds, the amount of time in seconds, and we have the amount of time in years. So the question is, how do we bridge that gap and convert from one quantity to an equal quantity in a different set of units? Let's make an intuitive prediction, because there's always an aspect of chemistry where common sense serves your problem-solving efforts, and one year is far, far, far greater than one second, so it follows that the number of seconds should be way larger than 1.5. Now, with a workflow, we want to make sure that when we complete the problem, we do the calculations, we do the algebra, we do the steps, we're not lost in the nitty-gritty. We want to have an overview of how we think we should solve the problem. And it stands right now that it's kind of a far reach to go from years to seconds. So there are a bunch of equivalencies or equations that directly relate variables of time, such as one day is equal to 24 hours, one hour is equal to 60 minutes, and one minute is equal to 60 seconds. So these are all equivalent values, they're all equivalent statements, so we can use them as what are called conversion factors. And the thing about conversion factors is that essentially we take those two statements, like say that we have one year is equal to 365 days, we can divide one over the other and it's a form of one, where we go from years to days to hours to minutes to seconds. In order to start with the same quantity as we end with, because that's what conversion does, we have to multiply it by a form of one every single time. So if we have that our conversion factors are one year is equal to 365 days and so on, we want to convert these statements into sort of like ratios, where we convert between one unit into the next. Now specifically, we are choosing the ones in which we cross out the right units and we're left with the correct units. So if you look at this, we go from 1.5 years into however many seconds that is. We multiply it by one conversion factor and then another conversion factor and then another conversion factor and another conversion factor. But we do it in a way such that the years cross out with the years, the days with the day, hours with the hour, etc. etc. And we do that and we get 4.725 times 10 to the 7 seconds, or 4.7 times 10 to the 7 seconds accounting for significant figures. Now does this make sense? Does it make sense that 1.5 years is equal to 4.7 times 10 to the 7 seconds? Well, we stated earlier in our intuitive prediction that because one year is so much bigger than one second, then in order to have two equivalent times, one in years, one in seconds, the number, the quantity in front of seconds should be so much bigger. So it does make sense. Now we have our answer, but we need to state it in a way that meaningfully answers the question, especially since on an exam we may have to use our answer for further calculation. And there are ways of saying it. There are 4.7 times 10 to the 7 seconds in 1.5 years, or that same amount of seconds is equal to 1.5 years. Let's move on to a problem that is very specific to chemistry. We have to convert 100 grams of calcium to particles of calcium. What is this really asking? It's really asking how many atoms of calcium adds up to the mass of 100 grams. So we can rewrite this as 100 grams of calcium is equal to a certain number of atoms of calcium, and we can put that at the top of our paper. So what do we have? What do we need? So given the mass of all the calcium atoms together, we need the number of calcium atoms which correspond to that mass. Likewise, we have known unit relationships, equations relating different quantities of units that we can use to make conversion factors, a fraction by which definition is equal to 1 because the top and the bottom are equal quantities. So we have 1 mole of calcium atoms is equal to 40.08 grams of calcium, and you can look at the periodic table to figure this out, and we also know that a mole is the same thing as 6.02 times 10 to the 23. Now if we specify to this problem, if a mole is equal to 6.02 times 10 to the 23, then a mole of CA atoms is equal to 6.02 times 10 to the 23 atoms of CA. So why did we do the intuitive prediction at the beginning of the time problem, but we waited until now to do it here? Well, time is something that we encounter in our everyday lives, so it's really easy to know that one year, for example, is a lot larger than a second, but you would probably want to know how many grams are in a calcium atom or a mole of them, so leveraging these conversion factors is also useful for making these common sense predictions. In other words, because chemistry is inherently a little bit less intuitive, we want to gather more information before we make intuitive predictions. So we have more than 6.02 times 10 to the 23, there should be a 23 right here, atoms of calcium, because we have more than 40.0 grams of calcium. We have 100 grams of calcium, and if 40 grams of calcium is one mole, and we have more than 40 grams, we should have more than one mole of calcium, therefore more than 6.02 times 10 to the 23 atoms. From there, let's set up the problem. In these types of conversions, we must go through moles, so we go from mass of calcium to moles of calcium to atoms of calcium. Similar to the last problem, we have these conversion statements, these conversion factors where we say this is equal to this, and this is equal to this, and we know that we're going to have to take our original quantity and multiply it by the conversion factors as ratios. And also similar to the last problem, just because we have moles of atoms and 40 grams of calcium, we don't have to memorize that the mole goes on the top and the grams goes on the bottom, we can just look and see, okay we start with grams, we're going to have to cross that out, so the grams have to go on the bottom, and it follows that moles is going to go on the bottom because moles is the top right here. So we do that, so we calculate that, and we get 1.5 times 10 to the 24 atoms of calcium, and that's our answer with the correct number of significant figures. Likewise, there are different ways to report our answer, we have 100 grams of calcium have 1.50 times 10 to the 24 atoms of calcium, and we have 1.50 times 10 to the 24 atoms of calcium have a mass of 100 grams, and these are equivalent statements, so that could also be a way of reporting the answer. This value is equal to this value.