 Så det turns out that those peaks in the energy landscape are important because they will teach us about how fast things happen. Let's look at little on that from another point of view. And I'm going to be looking at processes where I specifically consider how long it takes to get over the barrier, not just whether we get over them. So let's assume that I have a process that happens along some sort of reaction coordinate going from an initial state zero to an end state one. And here we have some sort of free energy, we can be physicists here, F. There is a starting free energy here, there is a barrier and there is an end free energy that's lower, so I would like to be in the lower one. But for this to happen I first need to get to cross the barrier here. And that barrier relative to the starting state has a free energy barrier that says delta F hash. You can call it anything you want. Now, it's not entirely trivial to calculate how fast that process will happen, but I can reason about it if we think about orders of magnitude and don't worry about the details. If I have a total of capital N molecules in my system, at any point in time the fraction of molecules that are going to be up here relative to the total amount of molecules in the system. Well, let's call that N hash divided by N. We already know that relative populations don't require a constant and this corresponds to e raised to minus delta F hash divided by kT, right? Plain and simple. Once we are at this peak, half the molecules are going to fall down to the left and half to the right, so some of them will now be at the end state, that's great. In principle I should include this factor too, but there will be worse approximations I make, so I will just say that if we are up here, they are all going to fall down in a tau, time tau. Time, tau reaction, time for a single molecule. How long that is depends. If it's diffusion it might be nanoseconds or so. If it's galaxy formation, well, then you should ask somebody else. And I would now like to know that how long does it take before this N hash might be tiny here, this might be one in a billion of molecules. So how long does it take before roughly all the molecules or at least half of them or something have reacted? Well, I can calculate that. If I have N molecules in total and each time this happens N hash of them have gone up. The time it will roughly take for all of them to go through that change would be say time t or t0. So that would then be roughly N divided by N hash because N hash would go through in each batch and each batch would take in the ballpark of this time tau. This is then simply the time for the elementary step multiplied by e raised to in this case it's going to be plus delta F hash divided by kT. This makes a lot of sense and this is also the reason why I don't worry too much about tau. If I look at the same process but this barrier is suddenly a factor too higher, I know here roughly how that is expected to change the time. Or if I make tau significantly lower, sorry delta F significantly lower. I can also reason about this relative to the temperature scale I have. If the free energy barrier here delta F is significantly smaller than kT. Remember what was that at room temperature? 0.6 kK per mole. So if delta F is significantly smaller than 0.6 kK per mole that means that these barriers are just going to be like gravel in the road. I can just bike over them, I won't notice it. But on the other hand if the energy barriers are significantly larger than kT then it's going to be like brick walls. They will really stop me and it will take a very long time to go over them and they're going to dominate my process. And armed with this I can actually at least estimate roughly how fast reactions will happen and I can also in particular compare different reactions when the tau is roughly the same. The books goes through some details to argue what tau should be for chemical reactions based on diffusion and it can be instructive to follow that. But I'm going to do something else. I will directly move to slightly more complicated processes. Because let's face it, there are very few processes in life that are so simple that you just have one step. What if I have a process that first is one step and then a second step we need to go through. Well, in this case first I have to go across the first barrier here. And then you might think that the second step can start from this intermediate point. But no it can't. It has to start from the starting point. The reason for that is that once I'm sitting here and you keep adding energy, what's first going to happen is that I will fall down here again. But if I just use those as delta F1 and delta F2. Then I can just adapt the reaction times. Because first my molecules will have to go through F1 and then they will have to go through F2. Technically you can make this more complicated by considering backflow and everything. But trust me it really becomes that simple. This will of course get super complicated if you have 55 reactions here. But look at the exponential dependence here. If I have two barriers and one is say a factor of three higher. It turns out that the higher barrier is completely going to dominate that. That's going to be the rate limiting factor, the time limiting factor here. So compared to the highest barrier, if the other ones are significantly smaller and I need to go through them sequentially. I can forget about the other ones. Only the highest barrier will matter. So I can approximate a multistate sequential process by only the highest barrier. And then I can use that very simple expression. Which is of course the reason why we are introducing it. Now all processes are in sequential. What if somebody tells you that there is free candy on the other side of the building or so. So half the students might take the path to the left. Unfortunately you will meet the dean there. So there is going to be a high free energy barrier there. While the other half of the students they will take the path to the right. You are lucky. You will only meet the vice dean. So it's not quite as much as an obstacle. But in this case both processes can happen in parallel. And then we are going to need a slightly different way to treat that. And what we do then is that we introduce something called reaction rates. Which is just one over the reaction time. Literally, it's very simple. So the reaction time describes that it would take this molecule one millisecond to go through a barrier. The reaction rate would then say well that means that there are 1000 molecules per unit of time that go through the reaction. And as for a serial process sequential you add up reaction times. For a parallel process you add up rates. If 50 students per unit of time go to the left. And another 100 students per unit of time go on the right. There will be 150 students in total getting to the candy. And this too becomes simpler. The lower the barrier is the higher the rate. And that will go as e to the minus exponent right. So there will be a very strong dependence that almost all the reactions will happen across the lowest barrier. Can you imagine any way of simplifying that? Yes, we simply take the lowest barrier and ignore the other ones. And we can simplify that too with one of these so-called first order processes. So to sum that up, for a sequential serial process we can simplify it by just considering the highest barrier. For a parallel process with many paths that are independent of each other we can simplify it and only study the lowest free energy barrier.