 We've seen several types of equilibrium constant at this point. Molecule-based or mole-based or pressure-based equilibrium constants. We can think of them all as products over reactants, but it's molecules of product or moles of product or pressures of the product raised to the stoichiometric coefficients and likewise reactants raised to their stoichiometric coefficients which are negative. We've also seen relationships for how to convert back and forth between the molecule-based and mole-based equilibrium constant using some number of multiples of Apagadro's number and back and forth between the molecule-based and the pressure-based equilibrium constant converting molecules to molecules to pressures using the adiogast law, KT over V. But there are other forms of the equilibrium constant that are also convenient to use sometimes. For example, if we're not working with a gas phase reaction where pressures are the most convenient unit to use to describe the relative amounts of the species, we might be working in solution. Lots of reactions happen in solution. We're more interested in the concentration of the reactants and the products. So it's nice to be able to talk about a concentration-based equilibrium constant. Concentrations in molarities perhaps. If those concentrations are in molarities then that's just moles of each species divided by the volume of the solution. The volume is the same for each one of the species so there's no reason for that to be inside the product. I can pull that out and call that 1 over V raised to each stoichiometric coefficient added up. That's the thing we call this delta nu, the change in the stoichiometric coefficients for the reaction. And then what's left over is a product of moles raised to the stoichiometric coefficients but that's exactly the same as KN. So this is, I can rewrite this as 1 over V raised to the delta nu times K little n. Of course K little n is itself related to K big n, K little n is some factors of Avogadro's number times K big n so if I just take an extra Avogadro's number and insert it into this thing that I'm raising to the delta nu I can write this as K big n times 1 over not just the volume but Avogadro's number times the volume raised to the delta nu. So either way I can think of Kc as being related to K little n or to K big n by these relationships. The other form of the equilibrium constant that's pretty common to use is the equilibrium constant written in terms of mole fractions so that's another unit of concentration that might be convenient in a solution or we could also use mole fractions in the gas phase just as easily as we can use them in solution. So mole fractions raised to their stoichiometric coefficients, mole fractions are just moles of each species divided by the total number of moles. So as usual I can leave the species specific term inside the product, pull 1 over n total out, it's 1 over n total raised to each stoichiometric coefficient once for each species so that ends up being 1 over n total raised to the delta nu. That's fine with one caveat the total number of moles is likely going to be changing as the reaction proceeds that's not a constant in the same way that Avogadro's number is a constant or that the volume and the temperature might be constant if we're doing the reaction at constant volume. So we need to do something with this n total under some circumstances to make sure we're using something that's actually a constant and I guess let me go ahead and rewrite this one time and say that this is equal to since product of moles raised to their stoichiometric coefficient that's just my case of little n I can write this as 1 over total moles to the delta nu times k little n or again if I'd prefer to convert to k big n instead of k little n just introduce a factor of 1 over Avogadro's number inside the parentheses and I'll write instead of writing 1 over Avogadro's number times number of moles what that means Avogadro's number times number of moles that would be the number of molecules. So these expressions are correct as long as you remember that the total number of moles and the total number of molecules is changing as the reaction proceeds so what's often more convenient is to use convert this to a k p if I notice that k n is related to k p I'll rewrite this equation and say k n is equal to v over k t to the delta nu k p where I've just moved this conversion the k t over v over to the other side now I can replace this k n with that quantity v over k t k p so I'll write 1 over n total to the delta nu k n now looks like v over k t itself raised to the delta nu all times k p but notice what I've got here if I combine these two terms in parentheses the thing that I'm raising to the delta nu power is volume over n k t so if we trust the ideal gas law I can look at that as 1 over pressure to the delta nu times k p so this result k x is equal to 1 over p to the delta nu times k p that's a procedure for converting a pressure based equilibrium constant into a mole fraction based equilibrium constant if we're dealing with gases so that's a lot of equations a lot of things to remember or to organize to keep track of how to convert one type of equilibrium constant into another so let me see if I can make things a little more organized by writing out a little diagram now of how we convert various of these equilibrium constants into one another so let's start with our first one if I want to if I have a k big n and I want to convert it to a k little n I just multiply by 1 over Avogadro's number to the delta v or so k big n to k little n I can think of that as multiplying by Avogadro's number to the negative delta v or if I want to go in the opposite direction I multiply by Avogadro's number to the positive delta v likewise to get from if I have a k little n and I want to get a kc I multiply by 1 over volume to the delta v so in this direction multiply by volume to the negative delta n delta nu in this direction multiply by volume to the positive delta nu if I have a k big n and I want to obtain kc I multiply by 1 over n a times v to the big n to the delta nu so in this direction multiply by Avogadro's number times volume to the negative delta nu in this direction I multiply by Avogadro's number times volume to the positive delta nu next we have kp so that's one way to obtain previously if I have k big n and I want kp I multiply by kt over v to the delta nu in the other direction I would multiply by the reciprocal of that v over kt to the delta nu and lastly this final one we obtained if I have a kp to obtain kx I multiply by 1 over p to the delta nu or pressure to the negative delta nu in the opposite direction I would multiply by pressure to the positive delta nu so this diagram doesn't say anything that we hadn't already written in the board in terms of equations but perhaps it's an easier reference to look back on in your notes and decide if I have let's say if I have a kc and if I want to obtain a kp here are the conversions I need to do in order to make that conversion so now we know how to convert various different flavors of equilibrium constant into one another now we do have a few more things to say about where those equilibrium constants come from how we obtain the values for those equilibrium constants and that's coming up soon