Happy Mole Day! What better way to celebrate the biochemist’s dozen than with a discussion of one of my favorite places it often comes into play – one of my all-time favorite things. Dimensional analysis! It may sound boring, but it’s sooooo helpful because 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. And if that’s all too geeky for you, it can help you order pizza (also, if it’s too geeky for you you’re probably in the wrong place…)
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?)
Note that this conversion factor has enabled us to enter a new “dimension” of sorts, going from pizzas to people. (dimension is typically something like length or time or mass – basically different things you’re measuring that aren’t necessarily inherently related).
Speaking of more conventional uses of dimensional analysis conversions… in the lab, instead of pizzas, 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 millionth of a mol per liter and a mole is 6×10²³ (Avogadro’s number). A mole (mol) is like the biochemist’s “dozen” – it is just a set number of things – anything – but if you ordered a mole of bagels you’d get 6.02 x 10²³ of them… (that’s 602 and and then 21 0’s…) which, if you were to divide up among the human population would be about 86 trillion bagels per person. Talk about carbo-loading!
A mole is defined as the number of atoms in 12 grams of carbon-12 (the main isotope (version) of carbon). But normally we don’t really care about that part… Instead, we use it more for “accounting purposes” so that we can talk in terms of groups of really really tiny things like atoms and molecules. Kinda like when you’re keeping a tally and you make marks in groups of 5) – or how you might talk about 1 dozen bagels instead of “12 bagels” – but we need a much bigger grouping than 5 or 12. Because molecules are really tiny. Like really, really tiny – so you’re usually dealing with tons and tons of them. ⠀
if you were to coat the continental US (sorry Alaska and Hawaii) with 1 mole of M&M’s, that coating would be 84km (52 miles) deep! http://bit.ly/33ha24m⠀
Yet, 1 mole of water molecules only takes up 18mL (at normal temp) & weighs just 18g. As I like to say, “Dream big, think really really really small!” (and use Avogadro’s help to keep track of it all!) ⠀
For example, there are ~8.4 septillion (10²⁴) water molecules in 1 cup of water. This is where that whole “molarity” thing comes in – it’s a measurement of how many of a thing there are in one liter. And for tiny things you have a lot of, the mole’s really good for this. Since liquid water has a molecular weight of 18.02 g/mol and a density of 1g/mL and for molarity we want moles per liter, we can do our dimensional analysis to figure out the molarity of water⠀
1 mol/18.02g * 1g/1mL * 1000mL/1L = 1000/18.02 = 55.49 M⠀(the capital M stands for “molar”)
But what about the stuff you mix with water to form cool solutions? And all the proteins, ions (charged particles), etc. hanging out in your watery cells. Their concentrations aren’t gonna be that high. They can’t be because the water hasn’t left them enough space! So often, even for really tiny things, you have less than a mole or mole/L of them you need to do calculations with. Thankfully, just like you can use centimeters when a meter’s too big and millimeters when a cm is too big, you can use metric prefixes to talk about smaller numbers while still using mole as your base unit. So, for example, a millimole is 1 thousandth of a mole and 1 millmole per liter gives you a concentration of 1 millimolar (1mM).
You might be a biochemist if you see mm and think milimole instead of milimeter! Because we use mM a lot – the mM range is common for a lot of stuff in your body. Like blood sugar(plasma glucose concentration), which ranges from about 3mM (after fasting or heavy exercise) to about 9mM after you eat. Sodium and potassium are also in the mM domain, whereas proteins tend to be more in the nM-uM range. If anyone’s curious about concentrations of things in your cells, check out the bionumbers website. Really cool stuff https://bit.ly/3kr7ABE
Anyways, it’s not just molarity we need to “smallify.” We can use these same prefixes so we can work anywhere 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!)
This post is part of my weekly “broadcasts from the bench” for The International Union of Biochemistry and Molecular Biology. Be sure to follow @the_IUBMB if you’re interested in biochemistry! They’re a really great international organization for biochemistry.
You can learn a lot more about concentrations here: http://bit.ly/solutionconcentrations
more math-y stuff on a new blog page I added: http://bit.ly/mathyposts
more on topics mentioned (& others) #365DaysOfScience All (with topics listed) 👉 http://bit.ly/2OllAB0⠀