Unit Conversion & Significant Figures: Crash Course Chemistry #2


A unit is the frequently arbitrary designation we have given to something to convey a definite magnitude of a physical quantity and every quantity can be expressed in terms of the seven base units that are contained in the international system of units. Hank thinks this is a thrilling subject, and while you may not agree, it is a subject that is very important if you want to be a scientist and communicate with accuracy and precision with other scientists. So listen up and learn something or Hank might have to kill you! (NOT REALLY!)


Transcript Provided by YouTube:

00:00
2640 lumens. 1 foot. 2.3 kilograms. 9 volts. Aaah!
00:08
I just closed the circuit with my tongue and I felt all 9 of the volts.
00:12
So what do all these things have in common?
00:15
They’re units. Yes, but they’re also absolutely, completely arbitrary.
00:20
[Theme Music]
00:29
You know who decides how much a kilogram weighs?
00:32
A hunk of platinum and iridium known as the International Prototype Kilogram or IPK.
00:38
The IPK isn’t just how much a kilogram weighs. In a very real sense the IPK is the kilogram.
00:45
Every other kilogram is exactly the same as the IPK,
00:48
and the IPK is the lump of metal that decides what that mass is.
00:53
A kilogram is defined as being the same mass as the IPK.
00:58
We made kilograms up just like we made up seconds and weeks and volts and newtons.
01:02
There’s nothing about these things that makes them them.
01:05
Someone just decided one day that that was a kilogram.
01:08
Now the fact that I find units fascinating probably says more about me then it does about units,
01:13
but I can talk about them all day.
01:14
For example, did you know that the International System of Units only includes seven base units
01:20
and every other unit is derived from those units?
01:23
Speed is length divided by time.
01:25
Acceleration is speed divided by time again, so meters per second per second.
01:29
Force is that acceleration multiplied by mass, cause F=ma remember?
01:35
Work done in joules is force multiplied by distance.
01:38
And power is work divided by time, so how much work can be done per unit of time. Makes sense.
01:43
It goes pretty deep, and it’s absolutely correct to say that there are an infinite number of possible derived units,
01:49
just most of them aren’t useful enough to name.
01:51
But here’s a bit of trivia for you. When I say watts or hertz, those things are just regular words.
01:55
No special capitalization necessary.
01:57
But Hertz and Watt, they were real people with like last names that were capitalized.
02:01
So what’s up with that? Well, getting a unit named after you is kind of the holy grail of science.
02:06
To quote Richard Hamming:
02:07
“True greatness is when your name – like hertz and watt – is spelled with a lowercase letter.”
02:13
Of course when these geniuses were first piecing together how the world works
02:15
they had no idea that there were fundamental basic units beneath it all.
02:20
They were basing all of their units on arbitrary values because, well,
02:23
how could there possibly be a fundamental amount of mass or distance.
02:27
Interestingly, one of the standard base units is derived from an actual value though not a universal one.
02:33
The second is 1/60th of 1/60th of 1/24th of the time it takes for the Earth to rotate a single time.
02:40
That’s something, at least but it also illustrates an interesting point.
02:44
As fundamental as that seems, when you get down to the dirty details things start to get kind of cloudy.
02:49
The Earth’s rotation for example is slowing down.
02:52
Does that mean that seconds should also slow down?
02:55
No. That would mess up every calculation ever.
02:58
So seconds are slowly becoming less and less based on reality.
03:01
Now don’t worry. It’s gonna take forever for the Earth to slow down noticeably.
03:04
And when it does we’ll just keep adding leap seconds to keep things balanced.
03:08
But units are extremely important in chemistry and in sciences in general,
03:12
as we learned when the Mars Climate Orbiter crashed into Mars
03:15
because instructions were inputted in the wrong units.
03:18
Next time you get a B instead of an A because you didn’t keep track of your units,
03:21
just remember at least you didn’t destroy a 300 million dollar mission to Mars.
03:26
But what do I mean when I say keep track of your units?
03:29
Well. I mean watch them. Do not let them do anything you didn’t tell them to do because they’re sneaky.
03:35
And a lot of chemistry is just converting between units.
03:38
So say you are in a car, and the car is going 60 miles per hour.
03:42
Now right now everyone who doesn’t live in America is like:
03:45
“Boo, miles are terrible. Convert to kilometers Hank!”
03:48
Well I’ll do you one better. From a scientific perspective, kilometers are terrible too.
03:53
They’re just as arbitrary. We should use something more universal.
03:56
Like lightyears. The amount of distance light can travel in a year. And hours, hours is no fun.
04:01
So let’s convert to lightyears per second. 60 miles per hour.
04:04
When you say it it sounds like a whole number with a single unit.
04:08
But it’s not. It’s actually a fraction. 60 miles over 1 hour.
04:12
Let’s start with the easy part. Getting to the seconds.
04:15
So first we’ve got to get to minutes. So there’s 60 minutes per hour. And also 1 hour per 60 minutes.
04:20
That fraction once we have it can flip either way.
04:23
We want it with the hours on the top, on the numerator. Why?
04:27
Because we want the units to cancel. We want to destroy the hours.
04:31
We don’t want them in our units when we’re done.
04:33
And then the same thing happens again with 1 minute per 60 seconds. Now we go to lightyears.
04:38
I asked Google, and there’s 1 light-year in every 5.9 * 10^12 miles.
04:43
Looking at this we see that the hours cancel and the minutes cancel and the miles cancel.
04:47
Leaving us with lightyears per second. That’s really what matters.
04:51
We’ve come out with the correct units.
04:53
The rest is just hammering at the calculator to discover that a car going 60 mph is also going
04:59
9.3 * 10^-12 lightyears per second.
05:02
Now we perform an important test. The “does this make sense?” test.
05:05
And yes indeed it does because 9.3 * 10^-12 is a very, very, very, very small number.
05:11
Which makes sense because when you’re traveling in a car you’re going
05:14
a very, very, very, very, very, very, very tiny fraction of a light-year every second.
05:19
Now there are probably gonna be fifty to a hundred thousand people that watch this video.
05:22
And I’m gonna guess that maybe a solid seven of you did the math along with me with your calculator out.
05:28
Now I’m not giving you a hard time. That’s just my guess.
05:30
If you want to follow along with your calculator in the future that might be helpful.
05:34
It would at very least be very nerdy.
05:35
But if you have been following along with your calculator, you might maybe have noticed something interesting.
05:40
I said 9.3 * 10^-12. When your calculator…
05:44
Your calculator probably said something like 9.3487658140029 * 10^-12.
05:53
So why, when I had so many more numbers to give, did I only give two? Was I trying to save time?
05:59
Well obviously not, because now I appear to be wasting time talking about it.
06:02
Do you think that it would be too hard for me to remember all those numbers?
06:05
Well obviously not, because I just did it. So I will tell you why.
06:08
When you’re doing experimental calculations, there’s two kinds of numbers. There’s exact and measured.
06:13
Exact numbers are like the number of seconds in a minute or the number of eggs in a dozen.
06:17
They’re defined that way and thus we know them in effect all the way out to an infinite number of decimal places.
06:23
If I say that there are a dozen eggs you know that that’s 12. It’s not 12.0000000001
06:31
or 11.9999999. It’s 12.
06:34
But that’s not true for the number of miles per hour my car was going.
06:37
That car wasn’t going 60.0000-out into infinity mph.
06:42
I only know the speed of my car to two decimal places because that’s all I get from the speedometer.
06:47
So the car could have been going 59.87390039 mph or 60.49321289 mph; the speedometer would still say 60.
06:57
And no matter how well I measure the car’s speed,
06:59
I will never know it at the same level of precision that I know the number of eggs in a dozen.
07:04
So that’s the second type of number, measured numbers.
07:06
Now the cool thing about measured numbers,
07:08
because you never ever know them exactly, is that they tell you two things at once.
07:12
First, they tell you the number that was measured.
07:14
And second, they tell you the precision at which that number was measured.
07:17
People often get their heads all tangled up about this,
07:19
but with a measured number you just have to remember that the actual number goes out to infinite decimal places,
07:24
you just never know all of them. You can’t. It’s impossible,.
07:28
So when my scale says 175 lbs, that doesn’t mean 175.000000 lbs. It means 175.something lbs.
07:37
And all those numbers after the five? We don’t know them.
07:40
And here’s the thing, a measured number can be pretty unhelpful if you don’t have knowledge
07:44
of the precision of the measurement.
07:46
So you have to conserve the precision through your calculations
07:49
or else you might end up killing someone with an imprecise dose of insulin or something.
07:53
So we have a set of rules for what are called significant figures:
07:55
these are the digits in your number that you actually know.
07:58
With my speedometer there are two: 6 and 0.
08:01
But 0 is weird, because sometimes it’s just used as a placeholder.
08:04
Like if I said that the fastest plane can go 13,000 mph, which it can by the way.
08:09
An unmanned military test glider did it in 2011.
08:12
That’s not an exact number, those zeroes are just placeholders.
08:15
So when a number ends in a zero, or two or three zeroes, it’s hard to tell if those zeroes are significant.
08:20
But this all gets so much simpler when you use scientific notation, which since it’s science we should.
08:26
So 60 mph would instead be 6.0 * 10^1. We get that zero is significant because we wrote it.
08:34
Otherwise it would just be 6 * 10^1. We keep that zero around because we actually know it.
08:39
Scientific notation is awesome by the way, once you get the hang of it.
08:42
If you’re having trouble you can always just type it into Google or your calculator to
08:45
see exactly what number we’re talking about,
08:47
but the number of the exponent just tells you how many places to move the decimal point.
08:51
So to the 1st power you move it one to the right and you get 60.
08:54
To the negative 1st power you move the decimal point one place to the left and you get 0.60.
08:58
To the fifth power, one, two, three, four, five, and you get six with five zeroes or 600,000.
09:03
Of course your significant figures get preserved, so 2.4590 * 10^-4 is 0.00024590 and you still
09:12
get the same five sig figs.
09:13
Now to the magic of figuring out how many sig figs your answer should have.
09:17
There are two simple rules for this.
09:19
If it’s addition or subtraction it’s only the number of figures after the decimal point that matters.
09:23
The number with the fewest figures after the decimal point
09:25
decides how many figures you can have after the decimal in your answer.
09:29
So 1,495.2+1.9903 you do the math.
09:34
First you get 1,497.1903 and then you round to the first decimal,
09:39
because that first number only had one figure after the decimal. So you get 1,497.2.
09:45
And for multiplication just make sure the answer has the same sig figs as your least precise measurement.
09:50
So 60 x 5.0839 = 305.034, but we only know two sig figs,
09:57
so everything after those first two numbers is zeroes: 300.
10:01
Of course then we’d have to point out to everyone that the second zero but not the third is significant,
10:05
so we’d write it out with scientific notation: 3.0 * 10^2. Because science!
10:10
Now I know it feels counterintuitive not to show all of the numbers that you have at your fingertips,
10:14
but you’ve got to realize: all of those numbers beyond the number of sig figs you have? They’re lies.
10:19
They’re big lying numbers. You don’t know those numbers.
10:22
And if you write them down people will assume that you do know those numbers.
10:26
And you will have lied to them. And do you know what we do with liars in chemistry? We kill them!
10:31
Thank you for watching this episode of Crash Course Chemistry.
10:33
Today you learned some keys to understanding the mathematics of chemistry,
10:36
and you want to remember this episode in case you get caught up later down the road:
10:40
How to convert between units is a skill that you’ll use even when you’re not doing chemistry.
10:44
Scientific notation will always make you look like you know what you’re talking about.
10:49
Being able to chastise people for using the wrong number of significant digits is basically
10:52
math’s equivalent of being a grammar Nazi.
10:54
So enjoy these new powers I have bestowed upon you, and we’ll see you next time.
10:58
Crash Course Chemistry was filmed, edited, and directed by Nick Jenkins.
11:01
This episode was written by me, Michael Aranda is our sound designer, and our graphics team is Thought Bubble.
11:07
If you have any questions, comments or ideas for us, we are always down in the comments.
11:11
Thank you for watching Crash Course Chemistry.


This post was previously published on YouTube.

Photo credit: Screenshot from video.