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calculate the pH at the half equivalence point so let's just remind ourselves what the half equivalence point even is the equivalence point is when the tight rent in this case the hydrochloric acid completely reacts with the potassium sorbate the thing that we are titrating now the half equivalence point is the point at which half of the potassium sorbate has been converted to the sorbic acid or another way of thinking about it is the concentrations of the potassium sorbate and the sorbic acid are equivalent well how do we relate that to pH and what other information have they given us to actually solve this and a good thing to do whenever you are if you feel a little bit stuck here say well what other information has they given have they given us well they gave us the KA of sorbic acid as being one point seven times 10 to the negative fifth so some how can we connect the KA of sorbic acid to the ph at the half equivalence point well the other thing that they give you is they give you a whole series of formulas in fact they give you a lot all the formulas that I'm using here at the the first couple of pages of the free response section and even a whole bunch of formulas on equilibria and on on all of these different notations they use and the one that might show up that looks interesting is this one right over here that you might recognize as the henderson hasselbalch equation and it's actually not hard to prove it it comes straight out of the definition of K a and then rearranging things and then taking taking the negative log of both sides and doing things like that and I encourage you to watch those videos on Khan Academy if you are curious but what's neat here is it connects pH it connects pH connects pH to PKA and the concentrations of the of an acid and its conjugate base so how do we how do we make a relationship here well at the half equivalence point the acid and the conjugate base are going to their concentrations are going to be equivalent so this so this and this are going to cancel out you're just going to get 1 + log of 1 is just going to be zero so at the half equivalence point the pH is going to be equal to the PK a and so what is the pKa here well they told us that the KA the K a is equal to and this is the KA of sorbic acid ka of sorbic acid is one point seven times ten to the negative fifth one point seven times ten to the negative fifth one point seven times ten to the negative fifth and once again when we're thinking about ka we're thinking about the disassociation constant for the acid and so that's why we used sorbic acid there and either way these two concentrations are going to be the same and if I want to find the pKa I take the negative log of I take the negative log of this so P K a which is equal to the negative log base 10 of the KA and they actually give you all of these formulas at the on the first page this is going to be equal to the negative log base ten of one point seven times ten to the negative fifth power and what is that going to be let me get a calculator out the zoo so let's see I'll write one point seven this these two capital e's that's times ten to the so times ten to the naught v the negative fifth power now I want to take log base ten that's this button here if your calculator just has a log button that's going to be that defaults to base 10 so I'll take log base ten of that and then I want to take the negative event and so this is going to be for point approximately four point seven seven so this is approximately approximately four point seven seven so the pH pH is equal to the pKa which is equal to that right over there