hey class welcome to our final lecture here in chapter one uh in this chapter we're going or sorry in this chapter in this video we're going to really focus on problem solving this is the most important skill that you could possibly have and take from this course we're going to be doing tons of unit conversions we're going to use dimensional analysis to solve problems so right now where we stand in the course i think probably the most important thing for you to do is just to practice converting from one unit to another so that's what we're going to practice here today because the skills we learned doing that are the exact same skills that we'll need to solve chemical problems down the road so we're going to perform these unit conversions using a method called dimensional analysis so really all dimensional analysis is is a way for us to keep track of what unit we're in currently it also is going to be really helpful in kind of checking our work if you work a problem and the units work out then you've most likely done that problem in the correct manner so it's really important to learn this method to be very intentional about keeping track of your units and that's what we're going to focus on here but in general what is dimensional analysis let's say i had a 4.2 centimeters and i wanted to know that in inches well i know that there's an equality that exists that says that one inch is equal to 2.54 centimeters since these two are the same thing one inch is exact you know the same thing as 2.54 centimeters because that's true i can interconvert between them and the way i would set that up is in a way that's going to cancel out this unit of centimeters that is in my given information and over here in the answer i want my units to be inches so the way i would apply this equality is one inch over the 2.54 centimeters and you'll notice like i said before those centimeters cancel out and then you know you get some answer in inches so it would be 4.2 divided by 2.54 and you get some answer in inches but this dimensional analysis process is this process of canceling out essentially units that you don't want anymore to give you units that you do want and really in chemistry i mean that is first semester that is essentially all that will do right mathematically this is probably 90 of the type of the math that we'll be doing in first semester what is a conversion factor as i said before it's simply an equality so you anytime you take two things that are equal right and i apply that so really what we're saying if i have this equation 1 inch equals 2.54 centimeters really what we're saying is that one inch over 2.54 centimeters they equal one right i could rearrange this to show that so when i convert from centimeters to inches in a sense it's kind of like i'm just just multiplying by one and by that i mean i get the same answer right it's just in a different form and that's all we're doing here with conversion factors so write two conversion factors relating cents to dollars so what i would do is start from the equality that relates cents and dollars so we know that there are 100 cents in every dollar so 100 cents is equal to one dollar now depending on the scenario that we have we might want to convert from cents to dollars or from dollars to cents and depending on which one we're trying to do we would apply this equality or this conversion factor in opposite manners so let's say we start out with x number of cents and we want to end up in dollars we're going to have to apply this equality or this conversion factor in a manner where the cents are in the denominator and the dollar is in the numerator so one dollar over 100 cents and obviously cents would cancel out and we would be left so if i said we start with x cents we end in y dollars right so whatever that is but this is our conversion factor that will allow us to convert from cents to dollars now if i'm given some value in dollars so we'll say we're given x dollars when we convert that we're going to have to apply this equality this conversion factor in the opposite manner to get the result that we want so in this case we would put a hundred cents over one dollar and in the end our dollars cancel out and we get some answer in cents now these are relatively simple right they're one step but some things that we'll ask it's going to reply involve finding multiple conversion factors and applying them in kind of a linear fashion to get to your desired answer but one thing i would i would note like if you have a conversion factor table so i have one here at the end in a sense what you're trying to do is just kind of find a root through that conversion table that is connecting things right so if i want to convert from inches to yards for example well i start in inches i can convert to feet now my next conversion factor that i use has to have feet in it right because i would end that first mathematical operation in feet so then i could say um is it in here i don't yeah so i could get the feet and then i could go from feet to yards right so you would just connect those conversion factors with each other and we'll practice some of that a bit later all right using a dimensional analysis convert 50 cents to dollars so our given in this case is the 50 cents we know that a hundred cents is equal to a dollar but really you know if we want to think about this in a very general way maybe there were lots of uh conversion factors that take me from cents to something else that's fine but really what you're looking for what you're thinking about is which one of those conversion factors contains cents because i got to get to some other unit right in this case it's very simple there is a inequality or a conversion factor that directly relates dollars to cents and we covered that on the last slide so one dollar is equal to 100 cents uh one thing that i would note in this particular example is that all of these are exact numbers right uh we don't have fractional cents right the the smallest unit would be a single cent we can count cents exactly and also this equality is an exact equality right one dollar is exactly 100 cents so we can really state this to whatever level of precision we want right 50 divided by 100 is 0.5 so we could say that is 0.5000 and actually you could say that that's repeating in this particular case because it is exactly 0.5 dollars all right so that's how you would do that conversion right about as easy as it could possibly be here we have another example we have so a sample of sodium metal is burned in chlorine gas producing uh 573 milligrams of sodium chloride how many grams and how many kilograms is this so for this one you're going to have to remember the definitions of your metric prefixes and then apply those in the correct order to get to your answer now one thing one piece of information that i will give you about converting within metric prefixes you probably don't know for example how many milligrams are in a kilogram right that's not one of our equalities that we have learned when you have situations like this where you're converting from one metric prefix to another metric prefix it's always easier just to go through the base unit so if i want to get to kilograms i would convert from milligrams to grams which is the base unit and then i would convert from grams to kilograms so the first one where we're asked just to go from milligrams to grams that one's going to be fairly straightforward so remember there's a thousand milligrams in one kilo or sorry in one sorry a thousand milligrams and one gram so we have that equip that equality right a thousand grams or sorry a thousand milligrams eventually i'll stop screwing this up is equal to one gram that's exact right that's a definition so we can use this equality either to convert from grams to milligrams or milligrams to grams in this case just remember we're going to want milligrams in the denominator and grams in the numerator so we would apply it as such right one gram over a thousand milligrams now if we divide by a thousand that's the same as moving that decimal place over three positions so we would expect or we would get an answer of 0.573 grams as i said before there is no direct can well there is okay but you probably don't have it in your prefix list right of how to go from milligrams to kilograms so i would go milligrams to grams i know that conversion and then i would go from grams to kilograms because i know that conversion as well so the second conversion i would need is a thousand grams is equal to 1 kilogram and when you're looking at these conversion factors or you're imagining these conversion factors right because you won't always have them just written out on paper but what i'm thinking is if i want to get from grams to kilograms or sorry from milligrams to kilograms i know there's an equality that goes to grams and then if i know grams i can get to kilograms so we're using grams kind of as a link between milligrams and kilograms and you're going to do this a lot okay where you kind of have to think of some intermediate that you'll have to go through in order to get to your final answer so to answer that last one the way that we would work that out we would first apply the same conversion we used before to get from milligrams to grams so the way you're thinking about this okay i did that first step i've canceled out milligrams if i stopped this problem right now i would be in grams right i did the exact same thing i did in the first one but i don't want to stop at grams i want to take this one step further and end up in kilograms so whatever conversion i use here has to have grams in the denominator and then of course we know that this equality exists and we can apply it 1000 grams in one kilogram so what we're going to end up with is just this divided by a thousand again right so it's going to have three zeros right so if you answered this it'd be perfectly correct but generally right we would write that in proper scientific notation so we're going to move that decimal place until we have some integer here in the ones place like i've done here and then i'll just count how many positions you have to slide that over so one two three four times ten to the negative four right because we're talking about a number that's smaller than one all right this one's going to be a bit more challenging because we're going to have to work with an exponential in this case so we got 54.3 mils of ethanol and we want to get that to cubic meters so 54.3 milliliters and we want to get that to meters cubed right the first thing that we need to recognize is that a milliliter is the exact same thing as a centimeter cubed so if we can get to there then we know the relation between centimeters and meters and we can apply that conversion factor but there is a bit of a trick when you're dealing with cubes or exponentials in general so we start with that 54.3 milliliters we know so we know this is going to be like this what's the quality well those two are the same thing right so i could just say one over one one cubic centimeters the same as one milliliter my units cancel and leaving me in cubic centimeters this is where things get a little bit tricky uh when you're dealing with an exponential what is the equality that relates centimeters to meters so there's a hundred centimeters in one meter but there is not a hundred centimeters cubed so this is not equal to one meter cubed okay because of the cube uh we we actually have to take that in into account okay so i'll show you how to do that so the way i do that is i write my equality the same as i always would a hundred centimeters over one meter but remember that's not what we're talking about right we're talking about centimeters cubed and meters cubed so you've got to take this whole thing and cube it you got to take this whole thing and cube it so uh you know there's actually a whole boatload of centimeters cubed in a meter cubed okay and from here or maybe i'll write it a bit of a different way just so you have a reminder of how this cubed is distributed so i'll rewrite the the whole thing here cancel those units all right when we distribute this cube we end up with 1 cubed meters cubed we distribute this cube we get 100 cubed centimeters cubed so now my centimeters cubed can cancel out and i will be left in meters cubed in this calculation so i'll pause this just for a second and get the answer and i'll be right back all right plug that into your calculator you get 5.43 times 10 to the negative 5. what are my units they are those meters cubed right the one unit that was not cancelled out in that process next question the star of asia sapphire weighs 330 carats what is its weight in grams 1 carat equals 200 milligrams so sometimes right you'll be given that or a or that conversion factor within the problem so in this one the statement one carat equals 200 milligrams that's going to be a conversion factor for you so we can write one carat equals 200 milligrams we're told that these are exact so we don't have to concern or sorry we're told this one's exact we're told this one that that zero remember we said trailing zeros can be ambiguous we're actually told in this case uh that it's not right sometimes you'll see it written like this with a decimal place to show you that that zero is significant anyhow our given is in carats we're given this equality we know we can get to that mass in milligrams but we're asked for it in grams well of course we can get from milligrams to grams through the equality relating the two right a thousand milligrams equals one gram all right so let's go through that 330 carats first thing we'll do is convert that to milligrams and it was 200 yeah so 200 milligrams and one carrot carrots cancel out if i stop this problem right now i would get an answer in milligrams so what i would suggest and i i see this a ton like when i get exams back and stuff so this semester you guys are doing multiple choice but uh sometimes you know when we're doing paper exams i see this a lot where students will stop the problem they'll answer it here and then use that number to start a new dimensional analysis problem if you do that you are almost always going to get silly rounding errors okay and it's completely avoidable and really who wants to be typing all this stuff in their calculator all the time it's easier just to work it as one big dimensional analysis problem right we were asked to give this answer in grams do we know how to get from milligrams to grams we do we have an equality for that and it is a thousand milligrams in one gram that's going to cancel out the units of milligrams we're left in our desired unit which is grams so if we plug this into our calculator 330 times 200 divided by a thousand we get an answer of 66. so is that correct not not quite right because in the problem we were told this number 330 that that zero was significant it had three sig figs and we were told that this equality was exact we know that this equality is exact so the only sig figs we need to worry about are the ones that come along with this 330 carats it has three significant figures we need to add that third significant figure so the correct answer here would be 66.0 grams not just 66 grams you don't want to shortchange yourself right if we know this to three sig figs we want to report that number to the same number of significant figures all right here i just want to do a little bit of practice doing some of these conversions so some that might take a few steps so the first one i have here is kilometers to inches so how would we do that let's say we have 2.76 kilometers and we want to convert that to inches for some reason i don't know why you'd ever want to do it but we're doing it we've got to find some connection between those kilometers and those inches so what things can we convert kilometers to well if we look here we can see that we can convert kilometers to meters so the next conversion factor i need to be looking at is from meters to something else right over here i see that one meter is equal to 39.4 inches so what i know is i need to use this conversion and then i need to use this conversion right that's one way i can go from kilometers to meters and then meters to inches there are probably other ways that we could do this yeah there is i could also go from kilometers to meters meters to centimeters and then i see here there is a conversion from centimeters to inches so we'll do both of those all right and i can show you that there is more than one way to do this let's start with the first one we need kilometers in the denominator we need meters in the numerator so a thousand meters in one kilometer again if we stopped this problem right now we would be left in meters so now i i live in a world where i'm in meters what can i do from meters that might get me closer to inches well in this case we have this conversion available 1 meter to 39.4 inches and beware these metric to us conversions these are in general uh measured quantities right these are usually measured so you kind of you do have to keep track of the sig figs there so the one is assumed to be exact this is measured okay so the 39.4 we need to keep track of the sig figs there all right cancel our units this last step is going to cancel meters and leave us in inches so 2.76 times a thousand times thirty nine point four uh that gives us an answer of a hundred and eight thousand seven hundred and forty four now i can't report it that way right because this number has three significant figures this number has three significant figures i got to answer this to three sig figs so we have to round it here and usually the way to do this so it's not ambiguous as to the precision is to write it in scientific notation so i can move this decimal place over to the one spot so we get 1.0 i'm going to round that up 1.09 times 10 this is going to be a positive number because this is greater than 1. it's going to be 10 to the 3 4 5 10 to the 5 inches all right i said i'd go through another one we'll do the same one but we'll just take a different route and see if we get to the same answer 2.76 sorry kilometers um the only jumping off point that we have when we try to get two inches let's go let's do something crazy okay i'll circle everything in red that i want to use so we got to start well we don't have to start there right we can wherever we see kilometers we can start so let's start with this let's convert to miles first right so we go from kilometers to miles now i got to go from miles to something else i'm probably going to be working in the english system right since we want to end up here in inches so kilometers to miles i got to look for miles somewhere here we got 1 mile equals uh 5280 feet well and then right if i know how many feet there are i should know how many inches there are and we have the conversion there for that so here i'll write this in red so it's distinct distinguished so 2.76 kilometers first conversion factor we decided to apply this time is this miles to kilometers 0.621 kilometers for every or sorry miles for every one kilometer that cancels kilometers out we're left in miles uh here we decided to convert from miles to feet so we've got 500 200 5280 feet in one mile the units of miles cancel out and then the last one we can convert to inches from feet so 12 inches and one foot feet cancel and we will be left in inches so i just want you to notice like we took two totally different routes here right but we should hopefully come up with the correct answer really it depends on how accurate this uh sorry it depends on how accurate our metric to us conversion is right they're both the three sig figs so hopefully they're good but right there's some rounding that takes place there so we got 2.76 times 0.621 times 5280 times 12. i got a hundred and eight thousand five hundred and ninety six point five yadda yadda yadda inches we got to do the same deal we got a round there right so we're going to get the same answer right 1.09 times 10 to the 5 inches but look at this right that this end of this is not the same all right it's because these measured numbers are ever so slightly different when we're converting between metric and uf so it's a good reminder right of why we consider significant figures right if i didn't consid consider significant figures we would have reported two different answers right but because we know those numbers are only so precise and we can only answer this to three significant figures we actually end up with the same answer there all right i have one more here gallons to liters we we just as well uh do that as well in addition so let's do a third color for that one we'll do purple so we're going to hang out here in the volume portion notice metric to us we don't have any direct way to get from [Music] gallons to liters so we're going to have to kind of go in a roundabout way um let's see how can we do this right the only thing that has gallon in it is this one right so that's where we're going to have to start with our conversion so i can get from gallons to chords and so now is there a way to get from quartz to liters or milliliters or something yes there is so over here we see that 1.06 quarts equals 1 liter so we can do this conversion in two steps so let's say we had 32.5 gallons um we can convert that to quartz using this equality four quarts in every gallon gallons cancel we can then get from quarts to liters using this equation uh the courts need to go in the denominator so 1.06 quart over 1 liter and bada bing bada boom we've got our answer here in liters now in terms of significant figures this is three significant figures this is a definition so this is infinite this one's between metric and us right that one's going to be a measured quantity so we have three sig figs here we have three here so our answer should be to three significant figures 32.5 times 4 divided by 1.06 gives us 122.64155 yada yada and that is liters we need to round to three significant figures which would be right there so we're going to round up so you could report this as 123 liters or you could do 1.23 times 10 to the 2 leaders either one of those would be perfectly acceptable as an answer for that problem so i encourage you at this point just pick two random units come up with a number like i did here and practice that practice navigating through these uh through these equalities to arrive at the answer that you're looking for now remember on a test you're not going to have these okay these metric so the conversions within the metric system so essentially knowing your metric prefixes you won't have this on a test okay these other ones you would have okay i'm not going to make you memorize how many feet are in a mile or how many centimeters are in it or in an inch or any of that you just need to know your prefixes right so you need to be able to pull those from your memory and use them but these other ones right you would be given all right that's a good place to leave off we're done with chapter one i'll see you next time bye