Hello everybody, this is a Let Your Topic video covering dimensional analysis. So what is dimensional analysis? It's essentially just a process.
It's a way for us to learn to solve problems systematically. And we've kind of touched on this already several times in previous videos. But essentially, what are your goals? What is the goal?
What's the question? asking, all right, where are you starting? What information do you already know?
These sorts of things kind of fall into that category of strategy, okay, strategy, all right, and then the question is like, how do you get there, okay? How do we take with what we started with and get to what the question is ultimately asking us, and that kind of falls into the setup, right? the setup. And sometimes, you know, you may think that you always need to work forward, which many times that will get us there, but sometimes you may have to work backwards if necessary.
Okay. And then of course, if we go beyond these points, these bullet points, all right, hopefully we'll come to some sort of solution. And then after we analyze that solution, hopefully it makes sense. All right. And, and again, figuring this out has a lot to do with developing that chemical intuition.
So. Again, does it make sense? Okay. All right.
So as we've already seen, a lot of times this takes into account conversion factors. All right. So we've already seen a lot of conversion factors related to the different prefixes. All right. When we were talking about magnitude or scale.
All right. But these guys right here. These are also conversion factors that you should know, okay? And these things are conversion factors. They are essentially equivalent to each other.
So what do I mean by that, all right? So much like when we were talking about going from to a micrometer, okay? So one meter, all right, is equal to one times 10 to the sixth.
micrometers, all right, or one times 10 to the minus six meter is equal to one micrometer, all right? It doesn't matter. They are both equivalent to each other.
So that's what I mean by these are based off of an equivalent statement. I'll put that kind of quotation marks here, okay? So these are conversion factors. The other thing you need to remember okay is that they do not go into determining sig figs all right so do not take into account when determining significant figures all right because these are known relationships they are defined relationships for example when we're talking about length Okay, I think we all know that one foot, all right, is equal to 12 inches.
Okay, just like one inch is equal to 2.54 centimeters. Okay, so these conversions are allowing us, especially this one inch equals 2.54 centimeters, it's allowing us to go from the imperial or the English system of units into the metric system. Okay, and you'll see that.
that happens a lot down through these different conversion factors. On the same token, one mile is equal to 5,280 feet. So these are units of conversion and conversion factors that deal with length. We also have some that deal with mass.
Again, allowing us to go between, in one case, the imperial system of units. into the metric system. So again, pound into grams, one pound equals 453.59 grams. And accordingly, one pound also equals 16 ounces. These are both imperial units.
Okay. So we got length, we got mass, we have volume. Okay.
One liter is equal to 1.0567 quarts. Again, we're going from the metric system. in this case, into the imperial system.
Or if we want to deal with something on a much smaller scale, we also have one milliliter is equal to one cubed centimeter. Okay, so going from kind of units of length, all right, into volumes as maybe at least me, I like to think about. Okay, so this could be something like a length times a width.
times the height. All right. So, but if that's in centimeters cubed, one centimeter cubed would be equal to one milliliter.
And then we have units of temperature. You'll see there's a slide at the end of this presentation that deals specifically with temperature, but we have a couple of formulas because we have three different units of temperature that you may see. Again, in the Imperial or the English system, we'd like to, or specifically in the United States, we'd like to think about degrees Fahrenheit, okay?
Where in the metric system, we're talking about degrees Celsius. So we have a very convenient formula that allows us to kind of toggle between the two, right? And that's degrees Celsius is equal to five ninths times the degrees Fahrenheit minus 32, okay? And then what we can also do is then take degrees Celsius and kind of convert it into the SI unit.
Okay, so remember the SI units are based off the metric system, and that is degrees Kelvin, all right? And this is a very simple and straightforward formula, all right? The degrees Kelvin is equal to the degrees Celsius plus 273.15, all right? So using these formulas, we can bounce back and forth between degrees Fahrenheit, degrees Celsius, or degrees Celsius and Kelvin, all right? This...
formula will be provided if we want you to use it. It will be provided, okay? Then we have units of energy.
Again, if you recall back from a couple of sets of slides ago, we talked about energy being measured in joules, okay? The kilocalorie is also a unit of energy, and it's usually what we think about when we're thinking about the calories, okay? from our that are coming from our food so one kilocalorie one uh kilocalorie is equal to one capital letter calorie okay so so this is talking about like the energy we get from the food well we can convert that into the si unit for energy which again is joules okay so one kilocalorie is equal to 4.184 kilojoules all right and so we'll see this a lot when we start talking about interactions between molecules and the energy that's released. So when we start talking about thermochemistry. And then we have some units of pressure, or some conversion factors for pressure.
So one millimeter of mercury is equal to one torr, whereas 760 torr is equal to one atmosphere. So these are some conversion units that... are conversion factors that we're going to see again when we start talking about gases and pressure.
So we're going to see this later in the semester. So again, these are some kind of standard or common conversion factors that we would like for you to know. All right, so let's do a sample problem. Okay, so gold can be hammered into extremely thin sheets called gold leaf.
So if a 200 milligram piece of gold is hammered into a sheet measuring 2.4 feet by 1.0 feet, what is the average thickness in nanometers of the sheet? And they give us the density. So again, let's go back.
What's our goal? What do we need? Well, we need to find the average thickness. And where are we starting and what do we already know?
Well, we know that we have a mass. We have a mass and that is 200. And notice the decimal places there. That's again, if that decimal place wasn't there, we could say that this value at least has one significant figure.
All right. But it's ambiguous if those two zeros are significant or not. But by putting the decimal place there.
That tells us that all three of those values are in fact significant. So we got a mass of 200 milligrams. All right. And we have a length.
Oops. We have a length. of 2.4 feet. And we also have a width, okay, or a height.
I will just say width doesn't really matter. And we'll say that that is 1.0 feet because that's provided for us. And then we also have the density, okay?
So 19.32. grams per centimeter cubed. All right. So, you know, what are, again, what's our goal? We want to get the average thickness and you can consider that to be, okay, the height, right?
And we want that in nanometers. All right. Well, we're given several things that can help us figure this question out, right?
So first off, what we can see is we have a density that's given in grams per centimeters cubed. Okay. So again, you can imagine that that would be like a length times a height times the width right there.
And using this, we should be able to calculate the height because we're given the length and the width. Right. And we also know that, again, that we want to solve for the average thickness. OK, so we need to think about how we can rearrange.
the formula for density to allow us to get the volume because again the volume which is represented by the centimeters cubed is the value that's going to help us figure out what the height is. So what we can do is we can start with density equals mass divided by volume right and we need to rearrange this okay so we can rearrange this very simply by cross multiplying and then dividing so we can get density times volume equals mass, and then we can get our volume is equal to the mass divided by the density, okay? Well, we have a mass that's in grams, but unfortunately our dense, or milligrams, excuse me, but our density is given in grams. So the first thing we need to do is convert our milligrams into grams, all right? And so now this is one of those prefix kind of questions, all right?
So remember, milla is 10 to the minus three. So 200 point, um, migs, and we want to convert that to grams. So we can multiply this by one milligram is one times 10 to the minus third grams. And that equals 0.2, zero, zero grams.
Okay. All right. So that's our mass. And now we need to start thinking about The fact that our volume portion of this density that was provided for us is in centimeters cubed, whereas our dimensions of this gold leaf is in feet.
So we need to convert our feet into centimeters. So how are we going to do that? Well, we all know that 12 inches is in one foot.
Notice that. We're starting with feet in the numerator. Therefore, when we use our conversion factor, we want to set it up so the feet will ultimately end up being canceled out.
So therefore, the feet within the conversion factor need to be in the denominator. And then, so we can cancel those out. And then remember from the previous page, all right, that one inch equals 2.54 centimeters. So one inch.
is equal to 2.5, excuse me, 2.54 centimeters, right? And now you see that our inches cancel out. And when we do this multiplication, 2.4 times 12 times 2.54, our final value, well, yes, we're gonna get a number, but it's gonna be in the correct units. Remember, with these sorts of things, we need a number and we need a unit. These are quantitative.
observations, quantitative calculations. Okay. And that ends up being 73.152 centimeters.
And remember, this is a measurement 2.4 feet. All right. That has two significant figures.
Well, this value here that we just calculated has five, but we're not going to round quite yet. Okay. So we're going to hold on to those extra values. Then we need to do the same thing with width.
So we got 12 inches. One foot, our feet cancel, times 2.54 centimeters, one inch. our inches cancel, and our answer will end up being in centimeters, 30.48 centimeters.
Okay. All right. So now we come back down here to our rearranged equation for density. Okay.
So now we want to solve for volume. So we have to, to do that, we need to divide the mass divided by the density that was provided. Okay. So our mass was equal to 0. 2.00 grams. And now our density that was given to us is 19.32 grams per centimeters cubed, right?
And that equals 0.010351967 centimeters cubed. Again, This is way in excess. We don't need all these.
All right. But we're going to hold on to all these values right now. OK.
And I also just want to point out one more time, density, that is an intrinsic property of of of of of a substance. All right. So it is a known value. This does not go into figuring out our significant figures in our final answer. So that is our volume.
OK. And so. volume, so this is equal to our volume, and volume is equal to length times width times height in this case.
So we want to solve for the thickness or the height. All right, so we need to rearrange this equation. So we can rearrange this into height is equal to our volume divided by our length times our width.
All right, and so... What does that equal? Well, we just calculated the volume, right?
0.010351967 centimeters cubed, right? And we have our length and our width in centimeters, the correct units to allow us to calculate this question. And so 73.152. centimeters times 30.48 centimeters. And what does that equal?
Okay, that equals 4.6 times 10 to the minus 6 centimeters. All right, we have our height, we have our thickness, if you will, in centimeters. Well, what's the problem? Well, the problem is Okay.
The problem is we need it in our nanometers. All right. So we need to convert centimeters into nanometers.
All right. So let's do that. Okay.
So what we can do is we can take four. Oops. Let's erase that real quick. and take 4.6 times 10 to the minus 6 centimeters.
And let's take it to the base unit first, which is just meters, right? So 1 centimeter is equal to 1 times 10 to the minus 2 meters, right? Times 1 times 10 to the minus 9 meters is equal to 1. nanometer.
All right. So we do all this math out again, do all the multiplication and the numerator first, and then divide by one, take that value and divide it by one times 10 to the minus nine meters. And what you can see is we set this up perfectly so that our units cancel, our meters cancel, and we're left with our desired units of nanometers.
And that ends up being 46 nanometers. And so now you may be saying, well, Dr. W, how did you know that it was 4.6 centimeter, 4.6 times 10 to the minus six centimeters in our original calculation? And I say, that's because I looked at the significant figures that I was working with, like the precision of my measurements.
So here, our mass, we had three sig figs, the two, zero, zero, because of that decimal place, all three are significant. The length, 2.4, both non-integer values. Okay, so that's two sig figs. And then the width was 1.0 feet.
All right, and that zero is following a decimal place. And so therefore, it is also okay, significant. So two two, and then three. So my final answer, okay, because density does not go into determining significant figures here.
It's an intrinsic property of the material, okay? So our measurements say that our final answer should only have two significant figures. So that is why I reported my value in centimeters to two.
So 4.6 times 10 to the minus six centimeters. And therefore, when I convert that... to meters and then subsequently to nanometers, my final answer is also in two significant figures.
So if you have questions about that, again reach out and we can talk about it at office hours or via email. So here's a participation question. Okay, so convert pressure from 4.45 times 10 to the fifth uh teragrams per micrometer. divided by second squared to pound per feet, pound feet per hours squared.
Okay. So you're going to have to go back and kind of look at these prefixes, look at the conversion factors that have recently been provided to you. And this sets a slide to help you solve this. So you're going to want to take your teragrams, okay, go to the base unit first, which is grams, and then convert that.
using some of the new conversion factors that were provided. We can take that to pounds. Okay. We can take our micrometers to the base unit, which is meters. And then that can be taken to centimeters.
Okay. And then we can use some of the new conversion factors that take centimeters into inches and finally into feet. All right. And then we can think about the the time unit, okay, which is second squared. So remember, second squared is equal to second times second.
So we're going to have to do these conversion factors when we're going from second squared to hour squared twice each, or we're going to have to remember to square them. Okay. Again, if you have questions on this, please do reach out. All right.
So now let's talk a little bit about temperature. Again, there's three different. units that you've probably heard of. Okay, so we like to think about things in terms of the Fahrenheit scale, okay? Where zero degrees, if we're looking, so here's our Fahrenheit right here, Fahrenheit scale.
If we look at zero degrees, it's somewhere right in here. Okay, so what you can already tell is that it is below the freezing point of water, all right? So this is really cold outside, okay? All the way up to, 100 degrees.
Well, our body is 98.6, so 100 is somewhere around here. And of course, we all live here in Texas, at least during the academic year, all right? And we all know that this 100 degrees Fahrenheit is actually pretty uncomfortably hot outside, okay? Now, this is versus the Celsius scale, okay?
Again, we have a nice formula that allows us to convert from degrees Fahrenheit into degrees Celsius. So let's remember that that's T, temperature in degrees Celsius, it's equal to 5 9th times the temperature in degrees Fahrenheit minus 32. And again, this would be provided for you if we want you to do this conversion. But this is a really nice conversion to allow us to calculate degrees Celsius from degrees Fahrenheit. And again, if you want to know how this is derived, We can talk about that during like office hours or via email.
Okay. All right. So, but let's see where this falls.
Okay. So zero degrees Celsius, all right, is actually above where zero degrees Fahrenheit is. So while this is kind of cold, it's not so bad.
So it's fairly cold, but it's not terrible. All right. Versus let's see here. These scales don't scale. These two temperature units of temperature don't scale with each other.
All right, clearly you can see that from that formula. So 100 degrees Celsius, okay, is actually really, really quite hot. That is at the boiling point of water, which would correspond to 212 degrees Fahrenheit.
So 100 degrees Celsius does not represent very well 100 degrees. Fahrenheit. Okay. So while it's really hot for a hundred degrees Fahrenheit, if we're at a hundred degrees Celsius. more than likely you are dead, okay?
So they don't scale with each other, all right? And then there's the Kelvin. Again, the Kelvin scale is our SI unit, okay, for temperature, all right?
And what you can see here is that zero degrees Kelvin, all right, this is called absolute zero, all right? It is the lowest temperature possible, okay? It's the lowest temperature possible. All right.
There is no motion at zero degrees Kelvin. There is no heat at zero degrees Kelvin. It is cold.
So if you look, all right, so zero degrees Kelvin is much, much lower because again, these scales are broken right here. Okay. That's what those break points are for. You can see that at zero degrees Kelvin, we are dead.
Okay. We're not there. So we get a big frowny face. Um, now If we go up to 100 degrees Kelvin, okay, so we're somewhere in here, somewhere in there, we are still likely dead, right?
Okay, so Celsius and Fahrenheit don't scale with one another, but Celsius and degree Celsius and Kelvin actually do, okay? So there's equivalent spacing, if you will, between a degree Celsius and a degree Kelvin. All right. But at zero degrees Kelvin or 100 degrees Kelvin, we are dead. And remember, there is a very convenient formula that allows us to go from degrees Celsius into the Kelvin scale.
And that's the temperature Kelvin is equal to the temperature degrees Celsius plus 273.15. And we'll expect you to know that. All right, so here's a participation question. Okay, last one for the set of slides, all right?
Convert the temperature from 315 degrees Kelvin to degrees Fahrenheit. So using those two equations, so temperature Kelvin equals temperature in degrees Celsius plus 273.15, you can take degrees Kelvin to degrees Celsius. And then what you can do is you can take degrees Celsius using the other equation that connects degrees Celsius with degrees Fahrenheit to degrees Fahrenheit. Okay. So you're going to have to use both of these equations and remember your orders of operations.
So please excuse my dear aunt Sally. So please excuse my dear aunt. Sally, this is telling us the order of operations. So we do things in parentheses first, then we do our exponents if they're there, and then we do our multiplication divisions next, followed by additions and subtractions. All right, if you have any questions, please don't hesitate to reach out, and I hope you're all well.
Take care.