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Matter and Measurements

We use **D****imensional Analysis **as a fail proof process to convert from one unit to another.

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Dimensional Analysis

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so dementia analysis is seen as a failed proof process that allows you to convert from one unit to another. You designed the problems to begin with your given amount and to finish with the end amount of your unknown. Now just follow units to ensure the unwanted units are canceled out in the same way that we did metric prefix conversions where we lined things up on opposite levels to help them cancel out. We're gonna take that and apply it here to the mental analysis. Now the strategy is many. Conversion problems will utilize your given amount and conversion factors, and you're gonna use those together to help isolate your end amount. So if we take a look here in this general format, let's say we're given 32 inches so 32 inches air given amount, and they're asking us to identify how many centimeters this equals. So Centimeters is our end about what we're looking for. The conversion factor is a way of bridging these two ideas together, it's used to convert, are given amount into our end amount. Now to go from interest of centimeters means that we have to use a conversion factor that relates inches to centimeters. When we talked about conversion factors, we said that one inch is equal to 2.54 centimeters. I put inches here on the bottom so that they can cancel out with these inches up here on top, so you always want them on opposite levels. Doing that helps me isolate my end amount unit, which is centimeters. So then I would just multiply 32 times 2. Initially, I'll get 81 to 8 centimeters. But remember, when you're multiplying numbers together, your answer is the least number of sig figs. 32 has 26 figs in it because remember, when you don't have a decimal point, you move from right to left. You start counting once you get your first non zero number and count all the way through. So this has to sig figs. When you have a decimal point, you go the opposite way. Our first non zero number is this to you Start counting. Here you go all the way through. So we have three sig fix. So you go with the least number of sick figs, which would be too. So this would round 2 81 centimeters as our end amount. But let's say we had to do ah type of dementia analysis with way more steps. What do we do then? Well, let's say here were given 115 minutes. So that is our given amount. And we have to get two years. Years would be on our end amount to connect. Given to end. We have to utilize conversion factors. I need to cancel out minutes, so I'll put minutes here on the bottom. We know that minutes is connected to ours. We know that one hour has in it 60 minutes. And here this would represent our first conversion factor, which I'll abbreviate as conversion factor one. Sometimes it requires more than one conversion factor to get to our end amount. This is that example. Minutes would cancel out. Now I have hours. We know that hours and days are connected. We know that one day has 24 hours. This would be my second conversion factor or C f too. Hours cancel out. Now I have days and then finally we know that days and years could be connected as well. We put days on the bottom so we can cancel out with the days on top, and we know that one year has approximately 365 days. This is my conversion factor. Three. So days cancel out. And now I'm left with years. So what we're gonna have here is 1 15 on top, multiplying with a bunch of ones which doesn't change anything. But on the bottom we have multiplying 60 24 3 65. Now it's incredibly important. You know how to plug this into your calculator. My suggestion, when you have multiple things on the bottom multiplying, is to just multiply them all together and get that sum total. When we multiply 60 times 24 times 3 65 their total is 525, and we still have the 1 15 on top when we divide 1 15. By that total, we're going to get 2.19 times 10 to the negative four. We're going to stay here, put that so 2.19 times since the night before. We're going to say here that the number of sick figs within our final answer is based on our given information now are only given information was the 15. These other numbers were the ones who supplied that. Those numbers were not given to us in any way, so we can't use them to determine the number of sick fix. So again, it's all based on the information that's presented. Look at those numbers presented to you and use those to determine the number of sick fixing. Your final answer 1 15 has no decimal point. So we moved from right to left. It has in it three sig figs. So our answer should have three sick fix, which it does so 2.19 times 10 to the negative four years would be our answer here. So everything we've learned up to this point, we're gonna use in some way when it comes to dementia analysis and remember, our conversion factors are incredibly important because there are a way of connecting are given amount to our end amount. The whole point is toe cancel out units and isolate the unit that you're looking for at the end. Now that we've done these example questions, let's move on in our discussion of dimensional analysis,

Using Dimensional Analysis, start with the given amount and obtain the end amount by using conversion factors.

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Dimensional Analysis Example 1

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Here are example states a ta King. Great for Simon's per hour. If each assignment has 12 questions, how many questions can the th grade and 130 minutes? All right, So if we're gonna approach a question like this, let's look at our steps. If present, start with the given amount. That is not a conversion factor, remember, are given. Amount is when we have a single unit by itself that isn't tied toe another. We're going to stay within this question are 130 minutes is are given amount to identify the end amount. You want toe isolate for your unknown. So we're going to say, here, this is our given amount. We have to figure out what our end amount will be. So our end amount is over here. They're asking us how many questions How many questions. So questions will be our UN amount. Ah, step to write down all the conversion factors. So all our conversion factors Let's see, we have four assignments per hour. Remember, per is the word that connects different units together. So one is four assignments every per hour. So that's four assignments for every one hour they tell me. Also, each assignment has 12 questions, so that means one assignment is 12 questions. Okay, so 12 questions now this last part find the connection between the given amount and the conversion factors in orderto isolate the end amount. Okay, so let's look at the given amount given amount has minutes within it. But neither one of the conversion factors has minutes involved. What we do have is ours. That tells me there's a conversion factor that's even before either one of these two. So conversion factor one actually involves us first changing minutes into hours. So we have minutes here on the bottom hours here on top. One hour is 60 minutes minutes. Cancel out this way, and the reason we're doing that is because now that we have ours, we can connect it to the hours here within this conversion factor. So that will be my second conversion factor bringing the one hour for every four assignments. Now that we have ours lined up, they cancel out. Now we have assignments and we need to get the questions. Here is our last conversion factor. Now it has assignments and questions within it, but we need assignments to cancel out. So assignments need to be here on the bottom, right? And then we need questions. Questions go here on top and it is one assignment is 12 questions. Remember? One of the first things we said is that conversion factors we can flip them. They could be presented in two different ways. Here, we had to flip the initial conversion factor so that assignments can cancel one another out. They have to be on opposite levels to be able to do that. So in conversion factor three, what I'll have left at the end is questions. So what we do now is we're going to multiply times four times 12, divided by 60. So when we do all of that, we're gonna get here. Ah, 130 actually, 930. We're gonna get here as our final answer. Ah, 104 questions. So we have 104 questions as our end amount. But remember, we need to worry about significant figures. 1 30 has to sick fix four has won six fig 12 has to sick figs, 60 has won six fig. So we need to go and have only one sick thick as our final answer. So we say roughly about 100 questions is what the TA could do within the 130 minutes, which is a lot of questions. All right, so now that we've done this example where we've set up the basic principles behind dimensional analysis, let's continue onward and do some practice questions.

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Problem

If the distance between Washington, D.C. and New York City is 224.9 miles, determine the distance in centimeters.

A

B

C

D

E

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Problem

The average human body is composed of approximately 160 fluid ounces of blood. How many quarts of blood does the average human body possess? (1 gallon = 4 quarts, 1 pint = 2 cups, 1 cup = 8 fluid ounces, 1 quart = 2 pints).

A

7.3 quarts

B

11.8 quarts

C

4.13 quarts

D

5.0 quarts

E

0.85 quarts

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Problem

Lipitor, a pharmaceutical drug that has been shown to lower “bad” cholesterol levels, while boosting “good” cholesterol levels had over $12 billion in sales last year. Each pill weighs 2.5 g, which contains 4.0% of the active ingredient by mass. What mass in kg of the active ingredient is present in one bottle of 120 pills?

A

0.112 kg

B

0.020 kg

C

0.012 kg

D

0.024 kg

E

1.02 kg

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