CHEM 1053: Dimensional Analysis

Dimensional analysis is a method of solving problems using units to guide the process. In dimensional analysis, we can use conversion factors that relate two equivalent quantities with different units. In this example, we're going to use dimensional analysis to convert 15.5 feet to meters. We'll start by mapping out how we want to go from feet to meters using the conversion factors we're given. Starting with feet, we can convert to inches using 1 foot equals 12 inches. Then, we can convert from inches to centimeters using 1 inch equals 2.54 centimeters. After that, we'll convert from centimeters to meters using 1 meter equals 100 centimeters. Let's try it. We're starting with 15.5 feet. If we want to cancel out our units of feet, we need to multiply by a conversion factor that has feet in the denominator. Since 1 foot equals 12 inches, multiplying by 12 inches divided by 1 foot is like multiplying by the quantity 1. It doesn't change our length of 15.5 feet; it only changes the units from feet to inches. Here, our units of feet cancel, and we're left with just inches. Next, we want to cancel out our units of inches. Inches are in the numerator, so we need to multiply by a factor that has inches in the denominator. Thus, we multiply by 2.54 centimeters divided by 1 inch. Now, inches cancel, and we're left with centimeters. To cancel out centimeters, we can multiply by 1 meter divided by 100 centimeters, and we'll be left with meters, which is the unit we want. If you wanted to put this into your calculator to get the answer, you would multiply everything on top (in the numerators) and divide by everything on the bottom (in the denominators). So it would be 15.5 times 12 times 2.54 times 1 divided by the quantity (here you would use parentheses) 1 times 1 times 100. If you want, since multiplying or dividing by one doesn't change the number, you could leave them out of your calculation. You should get the number 4.72, so 15.5 feet is 4.72 meters. Now let's try a word problem. To solve a word problem with dimensional analysis, we're going to start by reading through the problem and determining if each piece of information we're given is a conversion factor or a standalone piece of information. Let's get started. Your kitten is very sick, and your vet prescribed medicine. You need to give your kitten 550 milligrams of medicine every 8 hours for the next 3 weeks. From this sentence, we know that 550 milligrams of medicine equals 8 hours; this is a conversion factor. We're also told "3 weeks," but this value isn't equated to any other value, so we'll write it down as a standalone piece of information. The medicine is only sold in 225 milligram tablets. So, 225 milligrams equals 1 tablet; this is another conversion factor. A bottle of 18 tablets costs $10.95, including tax. So, 1 bottle equals 18 tablets, which equals $10.95. How many bottles of medicine should you buy to save your kitten from the grip of death (no pressure), and how much will it cost you? Next, let's come up with a plan for our unit conversion. Our only standalone piece of information in this problem is 3 weeks. In general, it can be helpful to start your dimensional analysis with standalone information, so we'll start with 3 weeks and convert it eventually into bottles and then dollars. None of our conversion factors have weeks in them, but one does have hours. So we can convert our weeks into days and then days into hours. Then, we can convert from hours to milligrams of medicine using 550 milligrams equals 8 hours. Our other conversion factor with milligrams will let us convert from milligrams to tablets. And our last conversion factor will allow us to convert from tablets to bottles and bottles to dollars. Next, let's do the calculations. We'll start with 3 weeks. Since weeks are on top, we'll multiply by 7 days divided by 1 week. Putting weeks in the denominator of our conversion factor allows weeks to cancel, and we're left with days. Then, we can convert days to hours by multiplying by 24 hours divided by 1 day. Now that we're in hours, we can multiply by 550 milligrams of medicine per 8 hours, and we'll be left with milligrams. Next, we'll multiply by 1 tablet per 225 milligrams of medicine and finally 1 bottle divided by 18 tablets. If you put this in your calculator, you would enter it as 3 times 7 times 24 times 550 divided by (8 times 225 times 18). You should get 8.56 bottles. Since it's not likely that we can buy 0.56 bottles, we'll round up to 9 bottles. To determine the cost of 9 bottles, we'll multiply by $10.95 per bottle. So it should cost us $98.55 to save our kitten from the grip of death. I have tried recruiting other users to the platform but they are slow to use