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# Relationship between Ka and Kb

## Video transcript

- We've already seen that NH4 plus and NH3 are a conjugate acid-base pair. Let's look at NH4 plus. The ammonium ion would function as an acid and donate a proton to water to form H3O plus. If NH4 plus donates a proton you're left with NH3. The Ka for this reaction is 5.6 times 10 to the negative 10. Now let's look at NH3 which we know is a weak base, and it's going to take a proton from water, therefore forming NH4 plus. If we take a proton from water we're left with OH minus. Since we talked about a base here we're gonna use Kb, and Kb for this reaction is 1.8 times 10 to the negative 5. What would happen if we add these two reactions together? We have two water molecules for our reactants, so let me go ahead and write H2O plus H2O here. What about ammonium? We have ammonium on the left side for reactant, we also have ammonium over here for our product. That cancels out. Same thing happens with ammonia, NH3. We have NH3 on the left. We have NH3 on the right. We have NH3 as a reactant, NH3 as a product. We can cancel those out too. Our only reactants would be two H2O. For our products we would get H3O plus and OH minus, so H3O plus, hydronium, and hydroxide. This reaction should sound familiar to you. This net reaction is the auto-ionization of water where one water molecule acts as an acid, one water molecule acts as a base. We get H3O plus and OH minus. The equilibrium constant for the auto-ionization of water you've already seen that Kw is equal to 1.0 times 10 to the negative 14. We added these two reactions together and we got this for our net reaction. What would we do with Ka and Kb to get Kw? It turns out that you multiply them, Ka times Kb for a conjugate acid-base pair is equal to Kw. Let's do that math. Ka is 5.6 times 10 to the negative 10. So 5.6 times 10 to the negative 10. Kb is equal to 1.8 times 10 to the negative 5, 1.8 times 10 to the negative 5, and let's get out the calculator and let's go ahead and do that math. We have 5.6 times 10 to the negative 10. We're going to multiply that by 1.8 times 10 to the negative 5. We get 1.0 times 10 to the negative 14. This is equal to 1.0 times 10 to the negative 14 which is our value for Kw. When you add reactions together to get a net reaction you multiply the equilibrium constants to get the equilibrium constant for the net reaction which in this case is Kw for the auto-ionization of water. Let's go ahead and go in even more detail here. Ka, that's your products over your reactants. That'd be H3O plus, the concentration of H3O plus times the concentration of ammonia. Let's go ahead and do that. The concentration of H3O plus times the concentration of ammonia, and that's all over the concentration of ammonium. This is Ka. This is all over the concentration of ammonium. This represents, let me go ahead and highlight this here. This represents Ka. Next let's think about Kb. Over here is Kb, that'd be the concentration of your products, so NH4 plus times OH minus. Let's go ahead and do that. Let's put this in parenthesis here. We have the concentration of ammonium, NH4 plus, times the concentration of OH minus. That's over the concentration of NH3. That's over the concentration of NH3 here. What do we get? The ammonium would cancel out. That cancels here. Then the NH3 cancels out. We're left with H3O plus times OH minus which we know is equal to 1.0 times 10 to the negative 14. Just another way to think about this. This can be important, relating Ka and Kb to Kw. If you know one you can figure out the other. Let's think about a strong acid for a second here. Let's think about HCl. The conjugate base to HCl would be Cl minus, the chloride anion here. Let's think about what this equation means. HCl is a strong acid which means a very high value for Ka. An extremely, extremely high value for Ka. What does that say about Kb for the conjugate base? The conjugate base here is the chloride anion. If Ka is very large then Kb must be very small for this to be equal to Kw. Kb is extremely small here, so a very small value for Kb. This mathematically describes what we talked about earlier the stronger the acid the weaker the conjugate base. HCl is a very strong acid, so it has a very, very high value for Ka. And the conjugate base is the chloride anion, and it must have a very, very, very low value for Kb which means it's an extremely weak base. This is mathematically how to think about that relationship. Next let's look at a problem where we're calculating one of those values. Methylamine is a weak base, and the Kb for methylamine is 3.7 times 10 to the negative 4. Our problem asks us to calculate the Ka value for the methylammonium ion which is CH3NH3 plus. We're talking about a conjugate acid-base pair. There's one proton difference between those. Therefore we can use our equation, Ka times Kb is equal to Kw. We can plug in Kb here. Now we have Ka times 3.7 times 10 to the negative 4 is equal to Kw which is 1.0 times 10 to the negative 14. Let's do the math and solve for Ka. 1 times 10 to the negative 14. We need to divide that by 3.7 times 10 to the negative 4. Ka is equal to 2.7 times 10 to the negative 11. Ka is equal to 2.7 times 10 to the negative 11. We're done. That's our answer, 2.7 times 10 to the negative 11 is the Ka value for the methylammonium ion. Let's go a little further. Let's take our equation here, Ka times Kb is equal to Kw. Let's take the log of both sides. That would be the log of Ka times Kb is equal to the log of Kw. The log of Ka times Kb is the same thing as the log of Ka plus the log of Kb equal to the log of Kw. If we take the negative of everything, let's go ahead and do that, the negative of everything, negative log of Ka. I'll put that in parenthesis, plus the negative log of Kb is equal to the negative log of Kw. The negative log of Ka, we know that this is equal to the pKa. The negative log of Ka was our definition for our pKa, and the negative log of Kb was our definition for pKb. pKa plus pKb is equal to finally the negative log of Kw. That would give you 14. 14.00. The negative log of 1.0 times 10 to the negative 14 is 14.00. Now we have something else that we can work with, so let me go ahead and box this right here. Let's take the Ka value that we just found. Let's find the pKa. The pKa would be equal to the negative log of 2.7 times 10 to the negative 11. Let's do that on our calculator here. Let's get some room. The negative log of 2.7 times 10 to the negative 11 gives us 10.57. We have to round that. 10.57. These are our two significant figures because we have two significant figures here. Let's go back up to our problem here, so that's the pKa for the methylammonium ion. Let's say you're given the pKa for the methylammonium ion and asked for the pKb for methylamine. What is the pKb for methylamine? All we have to do is plug in to our equation. 10.57 was our pKa value. Let's go ahead and write that in here. We have 10.57 plus pkb is equal to 14. When we solve for the pKb that would give us 3.43. So the pKb is equal to 3.43. We could double-check that. Let's go back up here and we could double-check that, because if we took the negative log of this number that's what we should get. Let's go ahead and do that. Let's take the negative log of 3.7 times 10 to the negative 4. We get 3.43. That's what we just calculated down here. The pKb is equal to 3.43. That's the relationship between Ka and Kb, and you can also talk about the relationship between pKa and pKb.