Dimensional analysis is seriously one of my favorite things – it may sound boring, but sooooo much helpfulness it brings! Often in the lab you want to carry out reactions oh so fab but the units are all over the place and you have to rush to save face. Multiply? ✖️ Divide? ➗ What goes on which side? Eliminate any doubt, make sure your units cancel out!. Though if you’re in the U.S. you may sigh to find your units aren’t SI.
When I learned dimensional analysis (aka factor-label method or unit factor method), I thought it was like “training wheels” that “real scientists” grew out of. But now, as a “real scientist” I still use it ALL THE TIME & it is so great! It keeps me from making silly mistakes & helps me figure out how to get from A to B. And ❌ing out the units is oh so satisfying!
Basic concept: Take what you know & where you want to go, then multiply by “1” as many times as you need to get from where you are to your destination! Dimensional analysis is just a way to help organize & visualize these calculations by making a sort of table: A ✖️➖✖️➖✖️➖ = B (see pics)
Wait 🤚 Multiplying by 1 doesn’t change the value, right?! Right – and that’s the point! If you slice a pizza into 12 slices, you can call it 1 pizza or 12 slices – it’s still same amount of pizza. But slices might be a more convenient way to talk about it. So if you want to know how many slices you have in your 2 pizzas & you know there are 12 slices per pizza, you just multiply 2 pizzas by 12 slices per pizza -> the “pizzas” cancel out, leaving you with your answer: 24 slices.
Similarly, a person’s just as tall if you measure them in inches or centimeters – or even miles or kilometers. BUT it’s much more convenient to use the “finer slicing” of the smaller units (where we can talk in whole numbers) than it is to have to try to compare thousandths of the biggest numbers. So we can use dimensional analysis to help us convert between these different ways to describe the same thing – write down what you have and where you want to go and multiply by whatever versions of “1” (conversion factors) you need to get there.
Imagine the playground arguments: “I’m only 5 one-hundredths of a meter shorter than you, Bobby!” “Nuh uh – I’m 5 whole centimeters taller than you” “Children – knock it off – those are the same thing!”
And then the teacher brings out her pencil and paper and draws a dimensional analysis grid to put an end to the argument… See here, Bobby, “centi” means hundredth, so there are 1 hundred cm in a meter, so (100 cm / 1 m) = (1 m / 100 cm) = 1. So we can multiply that by anything without changing that thing.
So 5 cm x (1 m / 100 cm) = 0.05 m = 5 one-hundredths of a meter!
“How did you know what went on top or bottom?” “I knew I wanted my answer to be in units of m, so I set things up so that all the other units (in this case just cm) cancel out leaving m on top.” “Thanks teacher – now we’ll never argue again!”
Okay, maybe that’s a bit fanciful, but dimensional analysis really does seem magical – my notebooks and sticky note pads are chock full of scribbled tables and crossed-out units, and I have many a spreadsheet with similar things.
You might think I’m being overly dramatic – so far, I only changed the units – the value’s still the same. BUT I can also expand upon the method to solve problem by multiplying (or dividing which is really just flipping top & bottom & multiplying) by something *other than* 1 (because this is basically just adding to our starting information)
Want to know how many slices each person will get? Dividing that 24 by the # of people will tell you that. And if you want to know how many pizzas you need to buy so each person will get 2 slices, add a new “conversion factor” saying 2 slices = 1 person (you are what you eat?)
In the lab, we’re (perhaps unfortunately) more often confronted with situations like you measure a protein concentration in mg/mL, but you want to know its concentration in µM (µmol per liter).
µM Greek to you? me too! at least the µ! As a Greek letter, “μ” is pronounced “myew” – but when we use it as a prefix you say it as “micro” – as in micromolar (μM). And as biochemists we use it a lot, because it’s a metric prefix meaning “millionth” – it took me a really long time to remember all the prefixes – people often seem to be more interested in going bigger – kilo (thousand), mega (million), etc. so this is what I was mostly taught in school… BUT as a biochemist, I’m usually interested in going smaller – milli (m)(thousandth), micro(µ)(millionth), nano(n)(billionth), pico(p)(trillionth).
tip: I use Greek letters so much that I set up my computer (under keyboard settings) so that if I press “control+space” my keyboard switches to Greek. But if you don’t need Greek letters all that much you can still get “µ” with option+m
So, now that we know how to say it and how to type it, how do we use it? µM means 1 billionth of a mol per liter and a mole is 6×10^23 (Avogadro’s number) of something – anything – it’s the biochemist’s much much larger version of a dozen. We use these prefixes so we can work within the SI system but with more “reasonable” “step sizes” – it doesn’t make sense to measure intercellular distances or interstellar distances in meters!
SI stands for International System of units and it’s a standardized measurement system so everyone talks in the same scientific language.
If you’re in the US, where we use really inconvenient units, you may need to convert into SI from things like inches & pounds. It’s harder to remember inches to centimeters and pounds to grams than grams to kg but once you know the conversion factor, the process is the same. Basically, you just want to multiply by whatever conversion factors you need to get from where you start to where you want to go. And you can figure out what conversion factors you need by looking only at the units.
Sometimes it takes a bit of playing & flipping things around (12 slices/1 pizza is the same as 1 pizza/12 slices) to put the conversion factor puzzle pieces together so I often make my tables in a spreadsheet (or even an illustrator file) where it’s easy to flip them
See pics for real-lab examples & a chance to try it out yourself!
Moral of the story: a pizza by any other name is still a pizza & don’t be “embarrassed” of using any tools you need to make sure you do things right!
On a related note, I like watching and/or listening to recorded lectures and lessons (Kahn academy & MIT OpenCourseWare have some great ones) & I heard an upper chem class teacher refer to intro chem as “baby chem” which made me sad because chemistry (yes even intro chem) is hard! And I’d argue intro chem is even harder than upper chem because it’s like learning a whole new language & worldview rather than expanding your vocabulary. So if you are in intro chem, know that you’re doing hard work & you should be proud of all your accomplishments. I’m rooting for you! (& hope some of these posts may help you on your exciting journey!)