Unit Conversion & Significant Figures: Crash Course Chemistry #2

Summary

This Crash Course Chemistry segment delves into the fascinating world of units, emphasizing their arbitrary nature and the International System of Units' seven base units. It humorously highlights the significance of keeping track of units, as exemplified by the Mars Climate Orbiter disaster, and teaches viewers how to convert between units, the importance of significant figures, and the utility of scientific notation. The episode is both educational and engaging, aiming to empower viewers with a better understanding of chemistry's mathematical foundations.

Takeaways

• 🔍 Units of measurement are arbitrary and man-made, such as the International Prototype Kilogram defining the kilogram.
• 🌐 The International System of Units (SI) comprises seven base units, with all other units derived from these.
• ⏱️ The second is an example of a base unit derived from the Earth's rotation, highlighting the complexity of defining units.
• 🚀 Units are crucial in scientific calculations, as demonstrated by the Mars Climate Orbiter disaster due to unit conversion errors.
• 🔢 The concept of significant figures is vital for maintaining the precision of measured numbers in scientific calculations.
• 📐 Scientific notation simplifies the handling of large or small numbers and aids in preserving the significance of digits.
• 🌌 Converting between units, like from miles per hour to lightyears per second, illustrates the importance of unit consistency in calculations.
• 🧮 Exact numbers, such as the number of seconds in a minute, are known to an infinite precision, unlike measured numbers which have limits to their known precision.
• 📉 Rules for significant figures in calculations involve considering the least number of significant figures in the measurements to determine the precision of the result.
• 🛠️ The importance of unit tracking and precision is emphasized, as inaccuracies can have significant real-world consequences.

Q & A

What is the International Prototype Kilogram (IPK) and why is it significant?

The International Prototype Kilogram (IPK) is a hunk of platinum and iridium that serves as the physical standard for the kilogram. It is significant because it defines the exact mass of one kilogram, and all other kilograms are compared to it.

How many base units are there in the International System of Units (SI)?

There are seven base units in the International System of Units (SI).

What is the relationship between speed, acceleration, force, work, and power as described in the script?

Speed is defined as length divided by time. Acceleration is speed divided by time again, resulting in meters per second per second. Force is calculated as mass times acceleration (F=ma). Work, measured in joules, is the product of force and distance. Power is the rate at which work is done, which is work divided by time.

Why are units like watts and hertz not capitalized, but Hertz and Watt are capitalized?

Watts and hertz are lowercase because they are units of measurement derived from the names of scientists. However, Hertz and Watt are capitalized because they refer to the scientists themselves, Heinrich Hertz and James Watt.

What is the significance of the second as a base unit in the International System of Units?

The second is significant as it is defined as 1/60th of 1/60th of 1/24th of the time it takes for the Earth to rotate once on its axis. This definition provides a practical basis for the unit of time.

Why are the Earth's rotations relevant to the definition of a second?

The Earth's rotations are relevant to the definition of a second because the second was historically defined based on the Earth's rotation. However, due to the Earth's rotation slowing down, the second is now defined independently to maintain consistency in scientific measurements.

What is the importance of keeping track of units in scientific calculations?

Keeping track of units is crucial in scientific calculations because incorrect unit conversions can lead to significant errors, as exemplified by the Mars Climate Orbiter crash, which was caused by miscommunication of units.

How does the script explain the conversion of units from miles per hour to lightyears per second?

The script explains the conversion by first converting miles per hour to seconds and then to lightyears, emphasizing the cancellation of units to arrive at the final unit of lightyears per second.

What is the concept of significant figures and why are they important in scientific calculations?

Significant figures are the digits in a number that are known to be accurate, reflecting both the number measured and the precision of the measurement. They are important because they prevent the misrepresentation of precision in scientific calculations, which can lead to errors.

How does scientific notation simplify the handling of significant figures?

Scientific notation simplifies the handling of significant figures by clearly indicating which digits are significant. The exponent part of the notation does not count towards significant figures, making it easier to maintain the correct level of precision throughout calculations.

Why is it important not to show more digits than the number of significant figures you have?

It is important not to show more digits than the number of significant figures because doing so can imply a false level of precision. This can lead to misinterpretations and errors in scientific communication and calculations.

Outlines

00:00

🔍 The Nature of Units in Science

This paragraph delves into the concept of units in science, emphasizing their arbitrary nature. It introduces the International Prototype Kilogram (IPK) as the physical standard for the kilogram unit. The narrator explains that units like seconds, weeks, volts, and newtons are human constructs, highlighting the International System of Units (SI) which comprises seven base units from which all others are derived. The paragraph also touches on the idea that while there are infinitely many possible derived units, only a few are practically useful. It ends by discussing the significance of units in scientific calculations, referencing the Mars Climate Orbiter disaster as an example of the consequences of unit conversion errors.

05:02

📏 Understanding Significant Figures and Scientific Notation

The second paragraph focuses on the importance of significant figures in scientific measurements and calculations. It distinguishes between exact numbers, which are defined with infinite precision, and measured numbers, which are known only to a certain degree of precision due to the limitations of measurement tools. The narrator illustrates the concept of significant figures with examples, explaining how they reflect both the measured value and the precision of the measurement. The paragraph further clarifies the use of scientific notation, which helps in maintaining the precision of significant figures throughout calculations. It provides rules for determining the number of significant figures in the results of addition, subtraction, multiplication, and division, and emphasizes the importance of not including non-significant figures to avoid misrepresentation of data.

10:05

🚀 Practical Applications and the Importance of Precision

The final paragraph wraps up the discussion by stressing the practical applications of understanding units and significant figures. It suggests that these skills are not only essential in chemistry but also in everyday life. The narrator advocates for the use of scientific notation as a tool to enhance clarity and precision in scientific communication. The paragraph concludes with a humorous note on the importance of using the correct number of significant digits, likening it to being a 'grammar Nazi' in mathematics. The video credits are also mentioned, acknowledging the production team behind Crash Course Chemistry, and inviting viewers to engage with the content through comments and questions.

Keywords

💡Lumens

Lumens refer to a unit of measurement for the total quantity of visible light emitted by a source. In the context of the video, '2640 lumens' is used to illustrate how units are arbitrary but standardized. The video uses lumens as an example of a unit that might seem random but is agreed upon for consistency in measuring light output.

💡Kilograms

Kilograms are units of mass in the metric system, defined by the International Prototype Kilogram (IPK), a physical object made of platinum and iridium. The video explains that the IPK is the reference point for all kilograms, emphasizing the human-made nature of units and how they are based on a single, agreed-upon standard.

💡Volts

Volts are a unit of electric potential difference or electromotive force. The video uses the experience of feeling 9 volts with a tongue to highlight the tangible nature of units and how they represent real-world phenomena. It also serves as a reminder of the importance of units in understanding and applying scientific principles.

💡International System of Units

The International System of Units (SI) is the modern form of the metric system and is the dominant system of units used in science and technology. The video mentions that the SI includes seven base units from which all other units are derived, illustrating the foundational role of these units in scientific measurement.

💡Derived Units

Derived units are units of measurement that are defined in terms of the base units. The video explains concepts like speed (length divided by time) and acceleration (speed divided by time) as examples of derived units, showing how they are constructed from base units to quantify various physical phenomena.

💡Significant Figures

Significant figures are the digits in a number that carry meaning contributing to its precision. The video emphasizes the importance of significant figures in scientific calculations, explaining how they reflect the precision of a measurement and how they should be preserved in calculations to avoid misrepresentation.

💡Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is used in the video to simplify calculations and to clearly indicate which zeros are significant, making it easier to manage precision in scientific measurements.

💡Arbitrary

Arbitrary in the video refers to the idea that many units of measurement are not based on any inherent property but are instead agreed-upon standards. This concept is used to explain how units like the kilogram or the meter are chosen somewhat arbitrarily but serve a crucial role in providing consistency and reliability in scientific measurements.

💡Units Conversion

Units conversion is the process of changing the units of a quantity from one unit to another. The video uses the example of converting miles per hour to lightyears per second to demonstrate the importance of understanding and managing units in scientific calculations, emphasizing the need for precision and consistency.

💡Measurement Precision

Measurement precision refers to the degree of exactness of a measurement. The video discusses how measured numbers provide both the value and the precision of a measurement, which is crucial for accurate scientific work. It contrasts this with exact numbers, which are defined with infinite precision.

💡Leak Seconds

Leap seconds are a correction applied to Coordinated Universal Time to compensate for the slowing of the Earth's rotation. The video mentions leap seconds to illustrate how units like the second, while seemingly fundamental, can become less based on 'reality' over time and require adjustments.

Highlights

Units are arbitrary and man-made, like lumens, kilograms, and volts.

The International Prototype Kilogram (IPK) defines the kilogram.

Units like seconds, weeks, volts, and newtons are human constructs.

The International System of Units (SI) has seven base units.

Derived units are created from base units, like speed from length and time.

Force is calculated as mass times acceleration (F=ma).

Work done in joules is force multiplied by distance.

Power is work done per unit of time.

There are an infinite number of possible derived units, but most are not practical.

Units named after people, like watts and hertz, are lowercase because they are units.

Early scientists based units on arbitrary values due to the unknown fundamental units.

The second is derived from the Earth's rotation, which is slowing down.

Seconds are decoupled from the Earth's rotation to maintain consistency.

Units are crucial in science, as seen in the Mars Climate Orbiter crash due to unit errors.

In chemistry, keeping track of units is vital to avoid errors.

Conversion between units is common in chemistry, like miles to kilometers.

Scientific notation simplifies unit conversions and maintains precision.

Measured numbers have limited precision, unlike exact numbers.

Significant figures are crucial for maintaining the accuracy of measured numbers.

Rules for significant figures in calculations ensure precision is not lost.

Scientific notation helps in determining the correct number of significant figures.

Understanding and using significant figures correctly is essential in chemistry.

Transcripts

2640 lumens. 1 foot. 2.3 kilograms. 9 volts. Aaah!

I just closed the circuit with my tongue and I felt all 9 of the volts.

So what do all these things have in common?

They're units. Yes, but they're also absolutely, completely arbitrary.

[Theme Music]

You know who decides how much a kilogram weighs?

A hunk of platinum and iridium known as the International Prototype Kilogram or IPK.

The IPK isn't just how much a kilogram weighs. In a very real sense the IPK is the kilogram.

Every other kilogram is exactly the same as the IPK,

and the IPK is the lump of metal that decides what that mass is.

A kilogram is defined as being the same mass as the IPK.

We made kilograms up just like we made up seconds and weeks and volts and newtons.

There's nothing about these things that makes them them.

Someone just decided one day that that was a kilogram.

Now the fact that I find units fascinating probably says more about me then it does about units,

but I can talk about them all day.

For example, did you know that the International System of Units only includes seven base units

and every other unit is derived from those units?

Speed is length divided by time.

Acceleration is speed divided by time again, so meters per second per second.

Force is that acceleration multiplied by mass, cause F=ma remember?

Work done in joules is force multiplied by distance.

And power is work divided by time, so how much work can be done per unit of time. Makes sense.

It goes pretty deep, and it's absolutely correct to say that there are an infinite number of possible derived units,

just most of them aren't useful enough to name.

But here's a bit of trivia for you. When I say watts or hertz, those things are just regular words.

No special capitalization necessary.

But Hertz and Watt, they were real people with like last names that were capitalized.

So what's up with that? Well, getting a unit named after you is kind of the holy grail of science.

To quote Richard Hamming:

"True greatness is when your name - like hertz and watt - is spelled with a lowercase letter."

Of course when these geniuses were first piecing together how the world works

they had no idea that there were fundamental basic units beneath it all.

They were basing all of their units on arbitrary values because, well,

how could there possibly be a fundamental amount of mass or distance.

Interestingly, one of the standard base units is derived from an actual value though not a universal one.

The second is 1/60th of 1/60th of 1/24th of the time it takes for the Earth to rotate a single time.

That's something, at least but it also illustrates an interesting point.

As fundamental as that seems, when you get down to the dirty details things start to get kind of cloudy.

The Earth's rotation for example is slowing down.

Does that mean that seconds should also slow down?

No. That would mess up every calculation ever.

So seconds are slowly becoming less and less based on reality.

Now don't worry. It's gonna take forever for the Earth to slow down noticeably.

And when it does we'll just keep adding leap seconds to keep things balanced.

But units are extremely important in chemistry and in sciences in general,

as we learned when the Mars Climate Orbiter crashed into Mars

because instructions were inputted in the wrong units.

Next time you get a B instead of an A because you didn't keep track of your units,

just remember at least you didn't destroy a 300 million dollar mission to Mars.

But what do I mean when I say keep track of your units?

Well. I mean watch them. Do not let them do anything you didn't tell them to do because they're sneaky.

And a lot of chemistry is just converting between units.

So say you are in a car, and the car is going 60 miles per hour.

Now right now everyone who doesn't live in America is like:

"Boo, miles are terrible. Convert to kilometers Hank!"

Well I'll do you one better. From a scientific perspective, kilometers are terrible too.

They're just as arbitrary. We should use something more universal.

Like lightyears. The amount of distance light can travel in a year. And hours, hours is no fun.

So let's convert to lightyears per second. 60 miles per hour.

When you say it it sounds like a whole number with a single unit.

But it's not. It's actually a fraction. 60 miles over 1 hour.

Let's start with the easy part. Getting to the seconds.

So first we've got to get to minutes. So there's 60 minutes per hour. And also 1 hour per 60 minutes.

That fraction once we have it can flip either way.

We want it with the hours on the top, on the numerator. Why?

Because we want the units to cancel. We want to destroy the hours.

We don't want them in our units when we're done.

And then the same thing happens again with 1 minute per 60 seconds. Now we go to lightyears.

I asked Google, and there's 1 light-year in every 5.9 * 10^12 miles.

Looking at this we see that the hours cancel and the minutes cancel and the miles cancel.

Leaving us with lightyears per second. That's really what matters.

We've come out with the correct units.

The rest is just hammering at the calculator to discover that a car going 60 mph is also going

9.3 * 10^-12 lightyears per second.

Now we perform an important test. The "does this make sense?" test.

And yes indeed it does because 9.3 * 10^-12 is a very, very, very, very small number.

Which makes sense because when you're traveling in a car you're going

a very, very, very, very, very, very, very tiny fraction of a light-year every second.

Now there are probably gonna be fifty to a hundred thousand people that watch this video.

And I'm gonna guess that maybe a solid seven of you did the math along with me with your calculator out.

Now I'm not giving you a hard time. That's just my guess.

If you want to follow along with your calculator in the future that might be helpful.

It would at very least be very nerdy.

But if you have been following along with your calculator, you might maybe have noticed something interesting.

I said 9.3 * 10^-12. When your calculator...

Your calculator probably said something like 9.3487658140029 * 10^-12.

So why, when I had so many more numbers to give, did I only give two? Was I trying to save time?

Well obviously not, because now I appear to be wasting time talking about it.

Do you think that it would be too hard for me to remember all those numbers?

Well obviously not, because I just did it. So I will tell you why.

When you're doing experimental calculations, there's two kinds of numbers. There's exact and measured.

Exact numbers are like the number of seconds in a minute or the number of eggs in a dozen.

They're defined that way and thus we know them in effect all the way out to an infinite number of decimal places.

If I say that there are a dozen eggs you know that that's 12. It's not 12.0000000001

or 11.9999999. It's 12.

But that's not true for the number of miles per hour my car was going.

That car wasn't going 60.0000-out into infinity mph.

I only know the speed of my car to two decimal places because that's all I get from the speedometer.

So the car could have been going 59.87390039 mph or 60.49321289 mph; the speedometer would still say 60.

And no matter how well I measure the car's speed,

I will never know it at the same level of precision that I know the number of eggs in a dozen.

So that's the second type of number, measured numbers.

Now the cool thing about measured numbers,

because you never ever know them exactly, is that they tell you two things at once.

First, they tell you the number that was measured.

And second, they tell you the precision at which that number was measured.

People often get their heads all tangled up about this,

but with a measured number you just have to remember that the actual number goes out to infinite decimal places,

you just never know all of them. You can't. It's impossible,.

So when my scale says 175 lbs, that doesn't mean 175.000000 lbs. It means 175.something lbs.

And all those numbers after the five? We don't know them.

And here's the thing, a measured number can be pretty unhelpful if you don't have knowledge

of the precision of the measurement.

So you have to conserve the precision through your calculations

or else you might end up killing someone with an imprecise dose of insulin or something.

So we have a set of rules for what are called significant figures:

these are the digits in your number that you actually know.

With my speedometer there are two: 6 and 0.

But 0 is weird, because sometimes it's just used as a placeholder.

Like if I said that the fastest plane can go 13,000 mph, which it can by the way.

An unmanned military test glider did it in 2011.

That's not an exact number, those zeroes are just placeholders.

So when a number ends in a zero, or two or three zeroes, it's hard to tell if those zeroes are significant.

But this all gets so much simpler when you use scientific notation, which since it's science we should.

So 60 mph would instead be 6.0 * 10^1. We get that zero is significant because we wrote it.

Otherwise it would just be 6 * 10^1. We keep that zero around because we actually know it.

Scientific notation is awesome by the way, once you get the hang of it.

If you're having trouble you can always just type it into Google or your calculator to

see exactly what number we're talking about,

but the number of the exponent just tells you how many places to move the decimal point.

So to the 1st power you move it one to the right and you get 60.

To the negative 1st power you move the decimal point one place to the left and you get 0.60.

To the fifth power, one, two, three, four, five, and you get six with five zeroes or 600,000.

Of course your significant figures get preserved, so 2.4590 * 10^-4 is 0.00024590 and you still

get the same five sig figs.

Now to the magic of figuring out how many sig figs your answer should have.

There are two simple rules for this.

If it's addition or subtraction it's only the number of figures after the decimal point that matters.

The number with the fewest figures after the decimal point

decides how many figures you can have after the decimal in your answer.

So 1,495.2+1.9903 you do the math.

First you get 1,497.1903 and then you round to the first decimal,

because that first number only had one figure after the decimal. So you get 1,497.2.

And for multiplication just make sure the answer has the same sig figs as your least precise measurement.

So 60 x 5.0839 = 305.034, but we only know two sig figs,

so everything after those first two numbers is zeroes: 300.

Of course then we'd have to point out to everyone that the second zero but not the third is significant,

so we'd write it out with scientific notation: 3.0 * 10^2. Because science!

Now I know it feels counterintuitive not to show all of the numbers that you have at your fingertips,

but you've got to realize: all of those numbers beyond the number of sig figs you have? They're lies.

They're big lying numbers. You don't know those numbers.

And if you write them down people will assume that you do know those numbers.

And you will have lied to them. And do you know what we do with liars in chemistry? We kill them!

Thank you for watching this episode of Crash Course Chemistry.

Today you learned some keys to understanding the mathematics of chemistry,

and you want to remember this episode in case you get caught up later down the road:

How to convert between units is a skill that you'll use even when you're not doing chemistry.

Scientific notation will always make you look like you know what you're talking about.

Being able to chastise people for using the wrong number of significant digits is basically

math's equivalent of being a grammar Nazi.

So enjoy these new powers I have bestowed upon you, and we'll see you next time.

Crash Course Chemistry was filmed, edited, and directed by Nick Jenkins.

This episode was written by me, Michael Aranda is our sound designer, and our graphics team is Thought Bubble.

If you have any questions, comments or ideas for us, we are always down in the comments.

Thank you for watching Crash Course Chemistry.