welcome to this video on dimensional analysis and unit conversions what we're going to do in this video is we're going to walk through some examples of using the dimensional analysis technique these are going to be the same problems that you saw in the powerpoint that were not done so what might be a good idea is if you want to pause the video pull up that slide or print out that slide that listed the problems because those are the problems we're going to do in this video and the very first one i want to do with you is a english to english conversion that you're probably already pretty familiar with but i want to make sure that we do this with the dimensional analysis technique because we're going to be using dimensional analysis for a whole bunch of problems this semester so i want to introduce in a way that you're already a little bit familiar with so the problem was we needed to convert 0.0067 miles and we needed to convert that to inches so that was our ultimate goal in this we would be provided some useful information in order to do that so some things that would be provided for us 12 inches and a foot 5280 feet in a mile those would be provided for us already because these are both english to english conversions you don't need to memorize any english to english conversions so the first step in using this technique is to go ahead and start with your given information .0067 miles and what we want to do with this is we want to convert these miles into inches this is going to be a two-step approach first we're going to convert from miles to feet well we already have the conversion how many feet equal a mile what we need to be able to do with this though is figure out what should go in the numerator and what should go in the denominator as we change this equality to a ratio i already have miles in the numerator so i want to put the miles in the denominator that way they're going to cancel i'm going to have the feet in the numerator so that's going to be my next step that i have the miles cancel remember we're focused on the units not necessarily the numbers we're not even pulling out our calculators at this point in order to calculate anything we make sure we get the units where we want them and then we do all the calculation all the math work at the end miles have cancelled numerator and denominator now i have feet now i can go from feet to inches i have the equality between feet and inches so now i need to use that equality figuring out where i want the feet to go feet in the numerator i want to go away from feet towards inches so the feet are going to go in the denominator one foot 12 inches feet cancel the unit i'm left with is the unit i want to be left with units of inches that was my goal all along to have units of inches so what the dimension analysis does is it sets it up such that i'm definitely going to cancel off the correct units and be left with what i need now dimensional analysis really just encodes a series of multiplications and divisions and this long horizontal line that's really one big division bar anything in the denominator i need to divide by anything up top i'm multiplying by so it's really a very efficient way to get a series of calculations in without really stressing about them you're just focused on the units and the numbers are going to take care of themselves so what you're going to end up doing is you're going to multiply across the top then anything you have in the in the denominator you divide by here it's just 1 so that's not going to change anything but if you had numbers down here you would just divide by that number divide by that number in order to actually calculate this through so when you do this you would get 424.5 inches now if you're worried about significant figures and rounding you would realize hey i got way too many significant figures there i only get two based upon my starting information so i'd have to round to this point if i was worried about significant figures if you weren't worried about significant figures you just leave it so i might round to that point that would round down the 420 but then i'd kind of recognize this zero is one of those kind of strange ambiguous zeros so to really be clear if you cared about the significant figures you would change this in the scientific notation and you get the 4.2 times 10 to the second for your final value for this so i wanted to do this first one as a way to introduce dimensional analysis to you are there other ways of going about these dimensional analysis problems yes but as we get more and more involved with chemistry problems and longer and more challenging chemistry problems we'll make sure we're able to use this technique to solve more complicated problems let's go ahead and do the next one next one we want to do is a metric to metric conversion and in this metric to metric conversion what we wanted to do is we wanted to change kilograms so we had 0.45 kilograms and we wanted to change that to milligrams that was our goal so here this is a metric to metric conversion being the symmetric to metric conversion none of the conversion factors are going to be supplied these come from you learning and memorizing those conversion factors those prefixes you need to supply metric to metric conversions for yourself that's part of being metric fluent so we're going to start this off the same way though 0.45 kilograms and we want to go at the very end so i'm going to draw this out a little bit farther we want to go to milligrams at the end sometimes you want to put down your final goal at the end just as a little subtle reminder that hey this is what i need to achieve before i'm actually done with this problem now my strong recommendation is anytime you are going from one metric unit to another metric unit so one prefix kilo to milli that you do not so do not try to do this in one step unless you are super great at metric you are much better off doing this in two steps and i want to show this in two steps so whenever you're going from one prefix to another first get rid of any prefix so what that means is i want to get rid of kilograms and go to grams i put kilograms in the denominator that way they're going to cancel now i need to understand and write my own conversion for this it's not supplied for me i need to come up with this on my own well kilo's thousand so what that means is one thousand grams in that one kilogram kilograms have cancelled now i have grams i want to go to milligrams well so i went from my old prefix to no prefix now i'm going to try to go to my new prefix which is milligrams i want the grams to go away i want to keep the milligrams so now i need a conversion factor between milligrams and grams and there's a couple you could write one you could write would be one thousand milligrams in one gram that's a true statement because milli is thousandth so it would take one 000 of them so that would be one conversion factor that you could write there's another one i'm going to put it up briefly in another color just so you can see that there's a different way of thinking about it instead of saying 1000 milligrams in one gram what you could also say is one milligram 0.001 grams kind of a different way of thinking about it but you see that it's mathematically going to be the same multiply by a thousand or divide by a thousandth it really is not going to matter mathematically most of us would probably tend to think about it in the purple way where we avoid the decimals and deal with the larger numbers as opposed to the smaller numbers but if you think about it this way that's perfectly good go ahead and use that you'll get to the same spot i'm going to take off that red so we're not too confused by that our grams have cancelled the unit we're left with milligrams is the unit we want to be left with so at this point it's about doing the calculation anything in the numerator is a multiplication anything on the bottom in the denominator is a division so to actually get the numerical value for this we'd go 0.45 times 1000 times one thousand divided by one divided by one you don't really have to do the divide by one that's not going to change the number but if you did have actual values here that weren't one you would just divide by those so when you go through that what you would get would be 450 000 milligrams if you cared about representing this in correct significant figures well you'd only get two you'd have to use scientific notation for that you would get 4.5 times 10 to the fifth one two three 1 2 3 4 5 milligrams for that final answer so metric conversions would work the same way in this dimensional analysis technique remember you have to be responsible for knowing the metric prefixes well enough to be able to write conversions like this for all of the ones that we asked you to memorize so the next conversion we want to do is kind of a mixture of the first two we kind of think about this as a mixture of the first two we would have some english and some metric units that we would be trying to convert so i want to do an example of that and the one that we listed on the powerpoint slides in the powerpoint video is we asked you to convert 0.048 gallons and you're trying to convert these to centiliters that was your goal now you would be provided some things to help you out with this so things you might be provided four cups in the court you might have that conversion available two cups in a pint you might have that one available you might have four quarts in a gallon available you might have some other ones available as well so far these are all english to english conversions remember you would need one way of getting between the english and the metric because that's what the problem's really asking gallons english centiliters metric so in order to do that you need at least one conversion for that property and this one the one that's supplied is 946 milliliters is one quart so these are what we have available to kind of work with it doesn't mean we have to use all of them sometimes you're given more than what you need but that's what we have to work with so even though this is using both english conversions and metric conversions using dimensional dimensional analysis it doesn't matter it's still the same process we start with what we know .0488 gallons now we know this is english and we want to convert it at the end and go ahead and do this at the end we know at the end we want to get centiliters out so what that means is some way i gotta go from english to metric so i'm going to look at what's provided and i think it's a good idea to identify the key translator equality the translator equality is that equality that you have available that relates the english to the metrics let's kind of pinpoint that cups and quartz no those are both english cups and pints no those are both english quarts and gallons no those are both english milliliters and quartz 946 milliliters equals one quart that is the translator equality that we have available for us that translator equality moves between english and metric now that doesn't mean i'm necessarily going to have to use that one first or last it could come somewhere in the middle but i identified that right away because i know at some point i have to use that there's no point but to and sometimes when you identify that you can also kind of see what your first couple steps might be so if i know i have to use this and i know the english parliament is in quartz i have to get these gallons to quartz so that kind of gives me a hint on where to start so i need to figure out can i go from gallons to quarts what do i have available that can do that well i can actually do that in one step four quarts equals one gallon i want the gallons to go away so they're going in the denominator one gallon four quarts my gallons have cancelled now i have corpse why did i do that well i did that because now i can use this key equality i can now do that conversion between the english and the metric so we're going to go ahead and use that i want the quartz to go away quartz numerator so i should put the quartz in the denominator 1 quart 946 milliliters quarts are going to cancel now i have milliliters that's great i'm in metric but i'm not quite where i need to be yet i need to be in centiliters and i'm in milliliters now it's entirely possible and okay to do this in one step i'm not going to advise it though because generally we're not super comfortable with the metric system everywhere else in the world we probably try to do this in one step but if you're not super comfortable in the metric system i highly recommend you do this in two steps and how can you tell if you're super fluent in metric well if it took you more than about a second to come up with the single conversion between milliliters and centiliters you had to think for more in a second you should not try to do this in one step instead you should try to do this in two steps so i'm not going to go directly between milliliters and center layers if you immediately saw that and knew how to convert that fine instead i'm going to use the two-step process whenever i'm going from one metric unit to another metric prefix i'm gonna first go through no prefix so old prefix no prefix new prefix again much safer unless you are super great at the metric system already so i'm going to show this in two steps milliliters and liters i want the milliliters in the denominator to cancel liters in the top well here's a metric to metric conversion these aren't provided for you need to come up with this for yourself so what's the relationship between milliliters and liters 1 000 milliliters in one liter so another way of thinking about this and you could do this this would be fine instead of saying one thousand milliliters in one liter you could also say one milliliter 0.001 liters that would also be okay milliliters have cancelled now i have liters now that i'm in liters i can go to centiliters well i want liters to go away so liters numerator liters in the denominator i want to be left with centiliters i should stick that into that numerator another metric to metric conversion well i need the conversion factor to go ahead and do this well conversion factor would be 100 centiliters in one liter you need to provide that for himself knowing what centi means another way of writing this and this would be just fine is instead of saying 100 centiliters per 1 liter you could also say 1 centiliter 0 liters in the denominator that's the same relationship just expressed a different way the liters cancel do you know i have left the centiliters well that's the unit i want so what this says is i have this set up correctly all my units have cancelled off now it's just about doing the math so anything in the numerator is a multiplication so .048 times four times 946 times one if you need to times a hundred divided by everything you see in the bottom so divided by one that's not gonna change the number divide by one divide by a thousand divide by the one if you needed to for that when you go ahead and go through that you're going to wind up with your final answer coming out at 18 centiliters once you kind of round that to two significant figures if you need to use significant figures for that so this is a one two three four five step problem so this technique really shows you how you can organize units in order to solve multi-step problems very cleanly and very effectively in order to do things the last problem i want to do for you is a density conversion generally if you can do a density unit conversion using dimensional analysis you definitely have dimensional analysis mastered so i want to do one of these for you i'm going to do the one that we put up in the powerpoint slide so in the powerpoint slide the one we put up is we went from an english density 6.23 ounces per cup and what were we trying to go to we are trying to go to a metric density decigrams per milliliter well we'd be provided some things to do this problem so some things you might have available to you would be something like 16 ounces in one pound you might have 2.2 pounds in a kilogram you might have something like 946 milliliters in a quart you might have four cups in one quart these are all things that would be provided for you in order to solve this you might even get more than that now the key thing in this is you're going to notice we start a little bit differently than we did before and it's key to do this this is a critical thing that you do this on the very first step because you can actually make a mistake how you key this information when you have 6.23 ounces per cup you just don't slam everything in that top box instead you have to make sure what goes in the top belongs in the top so i'm putting the ounces in the numerator the cups and just one 6.23 ounces per cup that's how you have to feed this information in to the dimensional analysis if not you're going to end up making errors same thing is kind of true for how we want to get our information out decigrams per milliliter so what this really is is this is really just both things rolled into one i need to change ounces to decigrams as well as changing cups to milliliters it doesn't matter where you start at most of us generally try to like to start at the top so i'm trying to change ounces to decigrams what i need is i need a way to get from the english ounces to the metric decigrams so i need a english to metric conversion factor for weight or mass ounces and pound that's both english that's not going to work milliliters and quartz well that's metric in english but that's volume here this is the one pounds in kilograms we're dealing with weight or mass and we have both english and metric that's that key conversion i know i have to use so being i know i have to use that that's going to kind of help tell me where to start this off at if i have to use this i need to be in pounds right now i'm in ounces so it's telling me first step or first couple steps ounces to pounds can i do that sure i have a conversion for that 16 ounces one pound well ounces in the numerator i need ounces in that denominator pounds will go on top one pound ounces have cancelled now i have pounds pd is kind of a short abbreviation for pounds make sure we have enough room to kind of fit this in 2.2 pounds one kilogram well now i'm able to use this already i'm going to go ahead and use this pounds in the numerator i need to put my pounds in the denominator in order to cancel so that's going to put kilograms up top pounds would cancel now i'm in metric i have kilograms right now but i'm not in the right metric i have kilograms i want to get to decigrams well to go from kilograms to decigrams you could try doing it in one step i really don't recommend it much safer to do it in two steps so instead of trying to do this in one step i'm going to do it in two first steps going to be to get rid of my old prefix which is the kilograms numerator denominator and just go to the base unit which is grams so now i need a conversion between grams and kilograms this is metric to metric this isn't provided for you you have to know the metric prefixes well enough in order to be able to solve these the relationship between grams and kilograms is one thousand grams in one kilogram kilograms have cancelled now i have grams now that i'm in grams i can go to decigrams all the prefix no prefix now i'm ready to go to my new prefix new prefix i want is decigrams i want to get rid of the grams so dust grams numerator grams in the denominator again a metric to metric relationship i need to know the prefixes well enough to be able to provide this so the relationship between decigrams and grams is 10 decigrams in one gram is there another way of thinking about this sure you could have had one decigram in the numerator and 0.1 grams in the denominator and that would be just fine grams would cancel now i have decigrams that's good that's what i want here's the other place you can be you can kind of fall into a little bit of a trap you want to make sure you're avoiding it's very tempting to stop here you see desk currents you're like great i'm done well you're not done yet because you also still have to go cups to milliliters you want to make sure you avoid that kind of trap and make sure you complete the problem well cups english milliliters metric in order to do this i need a english to metric volume conversion well that english to metric volume conversion is the 940 milliliters in one quart i'm in cups so before i can use this i have to go from cups to quarts now here's where you got to be careful sometimes we can get into kind of a rut or pattern kind of do something without thinking about it cups is in the denominator right here if i want cups to go away i can't put it in the denominator here i have to make sure i put it in the numerator such that it cancels correctly you have to be very careful of that cups in the denominator cups in the numerator those will cancel they don't have to be close to each other to cancel as long as one's in the numerator ones in the denominator now i have quarts now i can use my key conversion same thing i have quarts in the denominator so in order to cancel it my next quart needs to be in the numerator i'm going to put the milliliters in the denominator quarts in the denominator quartz numerator they'll cancel i have milliliters that's what i want if i do a quick double check of kind of what i canceled off and what i didn't i'm going to find there's two things i didn't cancel in the numerator the one thing i did not cancel was the decigrams in the denominator the one thing i did not cancel were the milliliters what am i looking for decigrams per milliliter so when you're trying to go through and solve this here's how you would mathematically compute this everything in the numerator is a multiplication so six point two three if you need two times one times one times a thousand times ten times four again remember you don't need to put the times one everything in the denominator is a divide by then divided by sixteen divided by two point two if you need to you can do the divide by ones divided by 946 when you go through and do this what you'll end up with is you should end up with 7.48 decigrams per milliliter for what you're looking for at the end and this has our three significant figures that matches our given thing if you're interested and need to care about significant figures if not kind of however you rounded was okay but if you need to report it in appropriate significant figures you get three and that's your three here so these techniques that we did in this video not only useful for doing these type of unit conversion problems we're going to see we're going to use this technique for the rest of the semester as we're doing kind of more actual chemistry problems instead of just unit conversion problems