 OK, so in this video, we're going to talk about mineral distortion precipitation, like the equation involved and the reaction involved. This is different from the previous lesson on aqueous complexation or speciation, because the previous reaction only involves reaction thermodynamics, meaning we care about the end point of the reaction. When we have mineral distortion precipitation, because reaction usually occurs between the solid phase at the interface of solid phase and water, reaction occurs much slower, so we often need to consider the kinetics of the reaction, meaning we care about how long it takes for the reaction to occur to reach the end point. So what I'm going to do today here is using the carbonate dissolution system in a batch reactor, so batch reactor, meaning it's closed, well mixed, so that it does not have concentration gradient in different parts of a reactor. So essentially, you're solving for one concentration for each species for the whole reactor. So that's our system. So if you think about it, if you want to draw whatever is something relevant to what you do in the lab, if you think about these water and then you have these calcite grains and you have these mixers that will keep the system well mixed. And again, the reaction involving this system, it's similar to what we talked about last time for the aqueous speciation, the carbonate water. For example, all these three reactions are the same as what I wrote before in lesson one. But the key thing is we're adding this calcite dissolution, which is calcite dissolving out to become calcium and carbonate. Calcite, or carbonate in general, is a very common type of rock or mineral face on Earth's surface. So this is a reaction. I mean, compared to the aqueous speciation or complexation, it's much slower. But compared to other type of mineral dissolution, it's actually relatively fast. So we have these four reactions here. You can think about the calcite dissolution, essentially. It can be releasing out the elements from the solid face to the water face, right? So it dissolves, so add to the water with calcium species and carbonate species. But then once carbonate is released into water, it quickly goes through the speciation reaction to become either bicarbonate or carbonic acid, depending on the pH of the system, as we talked about last time. So if you think about the speciation in this system, now that the species we have, in addition to the five species for carbonic acid, bicarbonate, carbonate, this is wrong. So this should be carbonate. And this should be two. So you have these three carbonate species. You have hydrogen ion, OH minus, all these five as before. But on top of that, you're adding calcium. So essentially, now you have six species, right? You would have CA2 plus, carbonic acid, carbonate, bicarbonate, carbonate, and then you also have H plus or H minus. So that's six species. So what does that mean? It is that we essentially have six unknowns to solve for. And we will need six equations to solve for all these species. But notice that one thing that we need to pay attention to is that we have these three reactions that occur fast. So you have these three equations already there. We can think about as one, two, three. These are the three relationships. But we cannot use this one because this is not a faster reaction. These are faster actions. This is a slow reaction, which means we need to care about its kinetics, like how far it releases these chemical species over time. So when we solve these equations, we actually need to consider the time component. In this case, we will need to think about the mass balance of system. So these cells are dissolving out and release out, for example, CA2 plus and releasing out CO3, carbonate, and then these will be exchanged into bicarbonate and exchanged into carbonic acid. So these three species are essentially exchangeable. So when we think of the mass balance of system, you would think about the concentration of calcium first. Volume, this is the volume of the water in the system. And you have the concentration of calcium. Ultimately, we're all solving for concentration, not activity. So this will give you the mass. So concentration of the unit is mass per volume. And then we actually will need to solve for ordinary differential equation now, because there's time component here. Now the source, so in this kind of system, we don't really have flow transport. So the only source or sink of this calcium species is through these reactions. So let's say we have the rate constant of this reaction is Ksp. So that's the rate constant. And we talk about the TSC radar, which is transition state rate theory. In the online materials, you can go look this up. So this Ksp and this rate constant times the surface area of the mineral. And then times, for example, how far away the reaction is from equilibrium, which will depend on the activity of Ca2 plus activity of carbonate. And then divided by Ksp, which is how far away this is from equilibrium. So this is IAP. Now when we have dilute system, if we have dilute system, then this activity will be equal to C. So you can replace every A with C for simplicity. So essentially, this equation is saying mass of calcium is added to the system, to the water, by having this rate law for calcium. But we also will have total, adding total carbon to the system. So similarly, you would have Vd. And then we call total carbon Ct dt. It will have the same expression here as KspA1 minus activity of Ca. Because essentially, it shows the same reaction. So this reaction essentially adding calcium total carbon is Ct would be equals to, again, concentration of carbonic acid plus concentration of bicarbonate and concentration of carbonate. But then we need one more equation, right? 1, 2, 3, and then this is 4. This is 5. We need the sixth equation, which if you think about it, when this mineral dissolving out to form calcium and carbonate is actually changing the pH of the system. So the hydrogen ion, or sometime we also call the measure of acidity of system, will also change. And this will be, I'm not going through the detailed derivation of the whole process, how we come up with. Let's call this dC, which is the concentration of Ct, concentration of hydrogen ion total dt. And by dissolving out to calcium, it actually decreases acidity of system. So either I will actually have minus Ksp, and then activity 1 minus AcA2 plus. So again, it's the same rate law, but it will actually have the minus signs there, meaning it's decreasing the total acidity of system. So the expression for the total acidity will be different if we define different system, different combination primary species. But in general, if we define this carbonate as one there as a primary species, we should have HT equal to hydrogen, concentration of hydrogen ion, OK, this should be C HT, and to hydrogen ion plus concentration of H2CO3 minus, concentration of bicarbonate and minus concentration of OH minus. You can actually derive this equation from the Tableau method. And these expressions, actually, you can look at Tableau and by combining different terms, you come up with this. So essentially, now we have six unknowns, and you have six equations to solve with that. But on top of that, one thing we need to pay attention is that these are ODE equations. So we call it ordinary differential equation, meaning we have one independent variable which is time in these equations. So typically, the code will be solving these equations first using time stepping. And then once we get the concentration, calcium, concentration of Ct, concentration of C HT, you have three kind of variable in particular time step. And then we solve for algebraic relationship with these numbers together with these three relationships. And by doing that, we essentially get the concentration of all the six species as a function time in these aqueous species. And what do you see, for example, your homework is essentially generated by solving these equations.