Fick's law of diffusion

so I have a picture for you of Adolf Fick and this is probably the second most well-known ad off in history but this ad off was well known for his science he actually came up with some fantastic laws that we use in all sorts of different branches in science today and that we're going to talk about one of his laws right now so I drew for you a little box and I thought one of the the most fun ways to kind of think about some of these laws that mr. Fick came up with was to do a little game so I'm going to give you a challenge and the challenge is that let's say that you're a person standing right here maybe standing behind this box and the part of the box that's facing you there nearest you is that blue wall that I kind of shaded in this blue wall is the back wall of the box and on the front wall I'm actually going to put in some little molecules let's say there are some molecules I'll put in I don't know let's say three or four molecules here and the challenge is this if a molecule gets from the front of the box I'm going to call it one this first side is side one if it gets from side one which is here over to side two which is the back wall if it gets from side one to side two of the box then you get $5 for each particle that makes it over so to put that into words we're interested in the amount of particles amount of particles that move over some period of time and you know that period of time can be you know one hour 10 hours whatever period of time we want but I'm going to just for just for argument's sake let's just say that we're going to do this for one hour so let's say I do it for now and these molecules start kind of moving around or they're migrating around because of course they're they're bouncing these molecules don't stay stationary and I come back and it turns out that only one molecule ever kind of eventually made it over to this side so I say hey you know good job you get five bucks because you know I'd promised that to you and so of course you got your five dollars in here you're happy and smiling right but I'm feeling in a generous mood and I say you know what let's start this experiment over let's just start over and this time I'm actually going to give you a chance to tweak the experiment you're going to actually get a chance to modify the experiment and you can do whatever you want to try to maximize your profit so so think about this you want to try to maximize this right here and how do you do that how do you maximize the amount of particles that make it to that blue wall over some period of time and I'm going to write your good ideas down here so start brainstorming some good ideas for how you might want to tweak the game or play the game to maximize your profits well if you're if you're thinking about this you might think well maybe the first obvious thing is why do you make this so darn far right why not make it a little bit closer so let's get rid of this back wall and scooch it nearer so that the molecules don't have to go so far and that's a pretty good idea it seems to me right so let's just make this a little bit smaller that's your first idea and I would say that's a brilliant strategy right now let's just make it half the size so it's less thick and these molecules don't have to go nearly as far and actually let me maintain the blue wall so that we can kind of keep seeing properly what this should look like and this is going to be dashed back here and like that and of course the blue wall is going to look like that so now you just basically make it come closer and so the molecules don't have to go nearly as far so idea one is less thick less thick wall less thick wall what's another idea well you remember from Graham's law we learned that these molecules you know the big ones actually don't move their diffusion rate is not as quick and that that's the smaller molecules that actually have a faster diffusion rate so if I'm waiting at that back wall to see how many molecules can get over I want tiny little molecules to make their way over because they're going to have a faster diffusion rate when I say tiny I really mean a smaller molecular weight so small molecular weight molecules that's the second idea change that the molecules make the molecular weights smaller and as Graham's law tells us they'll move faster so what's a third idea well maybe you can just have more of them maybe in this point in this number one plane which is the leftmost plane of this box why don't you just am it full of more little molecules if you have more molecules moving around that's another way of saying just increase the pressure right that's increasing increasing the pressure at that position one then you're going to have a better chance of having molecules move across right so increased pressure at one and what's the fourth idea I'm just going to make a little bit of space well it's a fourth idea that we can maybe put on here well if you're thinking really outside the box and this is actually kind of literally outside the box then you might think well why not just expand this entire thing make it a larger a larger area you know what about that why not just make it a larger area so that's the last idea maybe you can actually just make it a bigger surface area so maybe something like this you can expand it in all directions and maybe you can do something like that I'm going to have to make sure I draw it carefully so I don't confuse you but basically something like this where you now have kind of the same thickness I'm not changing the thickness but basically you're going to make this wall bigger actually I screwed that up a little bit let me just fix that so this is what my new back wall is going to look like and maybe I should do it in blue just to make sure we stay consistent with the colors but of course this is just going to extend out like that and this is my new back wall right this whole thing is my back wall and so if I expand the area now I have of course much more chance of getting some of these molecules back there and you know the partial pressure is going to stay the same so if I expand the area I still have more molecules kind of on this initial leftmost face right and so the pressure is going to stay the same this is the p1 that we just got through talking about but because I have more area there's more chance that's somewhere along this entire area a molecule will make its way across the thickness and hit that back wall so something like that and let me fill this in so this is a fourth idea let me write that down as the fourth idea is increase the area increase the area so these are four good ideas right four good ideas for how you can maximize the amount of particles over time and hopefully make as much money as you can probably much more than five bucks so this is exactly what ficks law talks about it talks the idea of amounts of particles moving over time so let me write out ficks law and this is actually how you most commonly will kind of come across it although there are some kind of variations on it it's going to look something like this and I'm actually going to try to color code it to go along with the ideas that we kind of already presented right so we said you know there's a some things you can do with pressure some things you might do with the surface area and also remember we have that diffusion constant and you divide all this stuff by the thickness of the wall so it's very colorful but this is ficks law as you usually see it there are some other variations I'll talk about so to go through this kind of piece by piece this is V with a dot over this is kind of the rate of particles moving and when I say rate you know that that means that there's some time component so this gets to kind of what the challenge was right we said you know how many particles can you get to go to that blue back wall over some period of time and sometimes we when we talk about particles we can think of them as you know giving giving that in terms of an amount you know might think of that as like a moles or some you know numerical value or you know the volume of a gas that's moving across so that's why you sometimes you'll see it as V to refer to volume on the other side this bit makes perfect sense right if you have more molecules that's going to cause more pressure we said earlier and if there's a bigger pressure difference between what's on you know the first side versus what's on the second side and whether the second side is the back wall then of course that's going to mean that more of the molecules are going to move over so this is a bigger difference and so sometimes you'll see this as Delta P and Delta just means difference this a we said refers to kind of just surface area of course if you have a greater surface area that's going to allow for more of the molecules to get across this D that's an interesting one this is diffusion constant diffusion constant and remember when we think about diffusion constant there are two laws that might jump into your head in fact the first one was Henry's law remember we talked about solubility because of the amount of molecules that go from for example air to liquid this was Henry's law that kind of told us about that and of course if something is very soluble then maybe that would be an increased p1 going back to the idea of pressure at fate at the the first wall and then you have to divide divided by the molecular weight the square root in fact of the molecular weights that's the idea we had in this actually comes from Graham's law so whenever I talk about the diffusion constant remember there are two laws that are at play here Henry's law and Graham's law that are kind of coming together to offer us some information about the diffusion constant and that's actually why we said well if you have a small molecular weight molecule maybe that'll help you out because it's in the denominator it's going to cause the rate of particles moving across to go up and finally this t-this is thickness right this is the thickness of the wall and this is totally intuitive right if you have a thick wall it's going to be harder for molecules to make it across very quickly so without even knowing it you kind of derived ficks law all by yourself just kind of using intuition and that's kind of the best way to learn this stuff and sometimes as I mentioned you might see this formula written differently in fact let me actually just rearrange this formula in a different way I'm just going to kind of sketch out how you might also see it which is that sometimes you see area on this side of the equation of course that's just rearranging it right dividing both sides by area and then you might see you know the key actually up here like p1 and then minus p2 something like this and in the denominator over here you'll see the T so you'll get this and then very separately you'll see times D so this might not look so different but what happens is that then people lump things and that's when things get kind of tricky they'll say well let's lump this together and let's lump this together and they'll call this flux I'll call this flux and the second part they'll call gradient so you may see ficks law written this way where it says flux equals the gradient times diffusion constant because of course the diffusion constant hasn't changed it's the same thing it just carries on down and if you see it that way let me just give you an example of what these things mean let's start with flux which is basically the net rate net rate of particles moving moving through an area moving through some area and you can kind of follow through the equation that makes sense right and here the important part is the idea of a net rate so not total rate but you're actually looking for what is the net gain or net rate that you're seeing and this gradient over here this is just change in pressure change in pressure over / or over some distance or over a distance and occasionally you'll see pressure written out as particles in a volume alright which is kind of the same thing conceptually it's the same thing right particles in a volume of course are going to have or they're going to exert some pressure so sometimes you'll even see it written this way so I guess the point is that you'll see these different terms I just want you to be familiar with them but at least now you know that the most common way you'll see ficks law is written out this way and that it's completely intuitive in fact if you had to come up with it given some time you'd probably come up with ficks law yourself