 Okay, so this lesson we're going to talk about mineral-distortion precipitation in one dimension, like 1D columns. So this is different from the previous lessons that we had before mineral-distortion precipitation, which was in batch reactor well-mixed systems. So there's really no dimension, or you can call that zero dimension, because it's in every spatial point they are the same. So here we are talking about 1D. What that means is that essentially you have a column. So we're talking about processes like, for example rainfall comes in, hitting, pentaroc, and things dissolving out, and then over a long period of time, these chemical species release out and rock becomes soil. So I'm going to use this as one example to talk about mineral-distortion precipitation in 1D columns and how we set up these equations. But it's not really only about chemical weathering. It's really about these mineral-distortion precipitation reaction in general. In the short term, these reactions actually change. For example, water chemistry, if you think about groundwater system, it actually tends to, we can see the change if this mineral-distortion precipitation happened pretty fast. But in order to see change in soil, in the solid phase, we do need to wait a bit longer for enough change to happen. So today I'm going to use this system. So I talked about this, if we're thinking about rock transformation to become soil, we're setting up a system that has rainfall comes in at a rate maybe, for example, some rate that is about annual precipitation minus ET, something like that at this flow velocity. And usually we would have unsaturated system in these topsoil, but for simplicity we would just think about everything at the fully saturated system. So it's easier at the beginning. So we have these systems that have essentially two minerals as pen rock. One is quartz and dissolved pretty slowly. Everyone knows about that. The other is K-Feldzbar, which is a very common silicate in rocks. K-Feldzbar has a formula of K-A-L-S-I and then O-3. So essentially you can think about this system. If you think about rainfall comes in, it's dissolving out, and then release the chemical species that you can think about several processes that's happening at the same time. One is advection. So this is the same advection we talked about in one unit said with a chaser. Essentially the flow brings out to the chemicals, but also the dispersing diffusion. This again is the same as what we talked about last time in the AD equations. But then the last one that is different from the previous one are the reaction, which is mineral dissociation precipitation reaction. So essentially this is like combining the AD equation unit with the mineral dissociation precipitation reaction unit. If you need to go back, you're welcome to look at, to review these materials. But essentially this, because these three different processes happen at the same time, but then we also have multiple mineral, we know that quads dissolve pretty slowly, and we can more or less think about this as almost non-reactive because compared to K-Feldzbar it's a relatively very slow reaction. So when we think about the reaction that's happening, the K-Feldzbar will be dissolving out. So you would have this K-Feldzbar, let's say, dissolving out to become release calcium, which is very important to nutrients, metals, cations, and then it will be also released as aluminum plus SiO2. So that's the aqueous. These are species that are going to be coming out. But at the same time, you also think about some of these chemical species when they reach certain level, actually they can precipitate out as solid phase again. For example, aluminum plus SiO2 actually will become another mineral, which we call kaolinite. Essentially it will have the formula of AIL2, Si2, O5, OH4. This is kaolinite. This is one type of very common clay in our system, or if you go digging some soil you will very easily see this type of clay. Essentially you can think about the whole process. You have parent-bed rock, rainfall comes in, release some of the chemical species, but then some of the species also reprecipitate out. So it's almost like a redistribution of mineral. But at the same time, because some of the species dissolving out over a long period of time will change the property of the system, will make this solid phase more permeable and have more pore space for water to flow through. So there is a change in physical property as well. One question we often ask is how does this water and rock change over time and also over space? We want to know how fast things change and how much they change and what they have become. A lot of times we want to be able to predict that. So in order to do that we have to come set up equations, reactions and all that, putting everything together and think about how we solve them. So again, with the system, every time we need to think about the chemical system and how they evolve over time and space, the first thing we need to think about is the chemical species first. So you have several primary species. Again, if you think about, if you review back some of the material we covered before with the primary species, then this system, because we have an aluminum silica, they have to be there. This has to be part of the building block of the system. You have potassium too. And when the rainfall comes in, usually it brings some acidity. So H plus should be there. You would also have CO2AQ. In addition, a lot of times these rainfalls have a little bit of sourcing. So to be representative, we are also putting a bit of sodium chloride in that example, sodium chloride as part of the rainfall. That means also in the secondary species, so we need to decide what complexities will be formed, how much complexity reaction has been, and things like that. So to keep it simple, I'm going to just have potassium and then HCO3AQ and maybe KClAQ as two complexities. And then of course, we also need other species, for example, Hminers. We need the other form of bicarbonate and carbonate. So it's quickly become a long list of species. So if we are talking to these chemical species, we will be essentially solving for 1, 2, 3, 4, 5, 6, 7. We have 7 primary species. So that essentially means we would have 7 independent equations to solve for. So what I'm writing here is a general type of governing equation for one species I. And again, this first term is we call mass accumulation term that we discussed before. It's the summation of overall change, and then you have the advection term, diffusion term, or dispersion term. This doesn't change because of the system. But it's like some of the primary will change, but the terms wouldn't change. You could have different parasitic phi, or velocity, or diffusion, dispersion coefficient for a different chemical species, but the term itself wouldn't change. And then the last term is new. It's a reaction term. So it would take into account essentially the rates of different type of reactions that this species I is involved in. And you think about how fast these different reactions would change the concentration of species I. And of course, this I need to be written from 1 to NP, which is a number of primary species. So you need to write for this system 7 different equations to solve. But then on top of that, again, you have this number of secondary species you would need to solve for in the form of algebraic relationships. So how does this R-I-G look like? These are terms different from the previous AD equations that we saw this term. Now here this I is a rect term, so you have that. So for this R, it needs to be, if we think about this particular I, let's make this I is potassium. So for potassium means this system, you can think about two reactions involved. One is that the K felts by dissolving out to release out calcium. And then it comes out and it goes. So essentially actually one reaction. So it would be just K felts by dissolving out. So if it's K, then you would just have this rate and then you follow TST rate. So you have probably this complicated K H plus dependence on H plus whatever term empirical numbers K H 2 O and then a water plus K OH minus and then activity OH minus some empirical exponent here as well. So these are the rate law part, early part that depends on pH. But then you again you have surface area. And the last term would be 1 minus IAP over KEQ. Again, this is how fast this is close to equilibrium or far away from equilibrium. So that's the general form of reaction rate law. So when we solve this, this term will be put in there. So this is going to be a complicated term. But also if you think about for example, especially like aluminum or silica, then they're actually involving two different reactions. One is K felts by dissolving out to release these. But a kaolinite also precipitated out. So there's one source structure term and R I term. And the other would be a sink term for kaolinite precipitated out. So it's a sink for aluminum or silica. So we solve all these equations then. What do we come up with is all these C I as a function of time and space. Right. So you have X versus. Essentially once we know all the parameters, initial conditions, boundary conditions, water water comes in, water water come out at the beginning. What's the power water composition in the rock and all that. This will give us this distribution of concentration of functional time and space. And over geological time, when this changes line, when I say geological time, I mean at least thousands of years, thousands or tens of thousands or close to many years, things like that. So over this time scale, you would have these things keep on changing. And then the rock becomes soil property of the prism media change. You have pricity increase typically. And you have also pulmonary change because the water, the resistance of this prism media to water become lower and lower. And you have pulmonary increase as well. So this will be the chemical weathering over long term. Over short term, all these reactions change water chemistry. But the last part of it I want to mention a little bit about the dimensionless number from the mass point of view, how these would change, how we analyze these systems. So last time when we talk about this equation of AD without the reaction term, we talk about the PE or Peckling number, which is the tau advection. It's a tau diffusion dispersion over tau advection. Tau is a time scale. So the time scale of diffusion and dispersion, which is L squared over D. The length of this column over diffusion comes as a time scale for diffusion and dispersion divided by a time, if I do it this way, times 1 over tau A, which is the opposite of length over advection. So that cancels out. It becomes Lv over D. So that's the Peckling number that we talked about before. So this term essentially compares the relative magnitude of advection versus diffusion dispersion, which one is dominant under fast flow or slow conditions. But then because we have the reaction term, we have two more dimensionless numbers. We call them cola numbers. So the first term, because think about it, we have two transfer presses. One is diffusion dispersion of the advection. So the first one is comparing the time scale of reaction, the relative time scale of advection to the relative time scale of reaction. And in that case, again, tau A will be the same as L over V. But then you'll be, this is over tau reaction. For reaction, we are thinking about how much things can dissolve and eventually reach equilibrium. So you would have volume, total bulk volume times prosity, which is how much space you have, post-space you have that can hold water. So that's the water volume times the maximum concentrations that you can dissolving out, divided by the rate constant as the maximum, K times A. So this would be essentially the reaction time, the characteristic time for reaction. And in notes that I have simplified this and you can have that. You can look at these. For the second term cola number, you will be comparing the time scale of diffusion dispersion over tau R. And again, this will be going back to L squared over D, divided by this same thing, because the reserve reaction term will be the same. So if you think about, we typically talk about under fast reaction system, fast compared to transfer versus when you have very slow reaction system. So when you have very fast reaction system, the system tends to reach equilibrium quickly. And it tends to be transport controlled. It will be determined by the bottleneck of all processes, which means it's going to be transport controlled. If the reaction is actually slower, so you will see by this analysis, you will be able to tell how the system would behave, whether it's transport controlled or surface reaction controlled and how these will be changing over distant time will give us some kind of grouping analysis that we can do. So I'm going to end here. You can go back and look at notes. I went pretty fast. So look at notes we have. Do some homework and you will realize these analysis will really help you to understand how things are different under different conditions.