pKa and pKb relationship for amino acids

[circulus vitios]

"What is the relationship between the pKa and pKb for lysine?

A. pKa1 + pKb1 = 14
B. pKa1 + pKb2 = 14
C. pKa2 + pKb2 = 14
D. pKa2 + pKb3 = 14"

Part of the reasoning in the solution says that the first proton released from the most acidic group (the carboxylic acid terminal) corresponds to the third proton gained by the conjugate base. How do you get "third proton gained?" Which of the groups is gaining the proton and what do I assume the starting structure looks like before I add the protons?

edit: Really, could someone tell me where I could find a source explaining pka and pkb value assignment for amino acids? I tried Googling for lecture notes but I'm having trouble finding what I want. Another question that shows what I'm having trouble with:

"At pH=7, what are the applicable values for histidine?

A. pKa1=1.81, pKa2=6.05, pKb3=4.85
B. pKa3=9.15, pKb2=7.95, pKb1=4.85
C. pKb2=7.95, pKa2=6.05, pKa3=9.15
D. pKa3=9.15, pKb2=7.95, pKb3=12.19"

I understand how to determine which groups are protonated or deprotonated at the given pH but I have no idea what goes into deciding that "since the carboxyl terminal is deprotonated, we consider pKb3 rather pKa1."

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[FormerPhysicist]

For the first question, when increasing the pH (say from 1 to 14) the carboxylic acid is the first to deprotonate. Therefore, if you were to take the pH from 14 back to 1, the carboxylic acid would be the last to regain its proton. pKa 1 will then correspond to pKb 3 in the conjugate base.

It's basically the same principal for your second question. Since the carboxylic acid is deprotonated, it has become a conjugate base, which makes it more appropriate to use pKb 3 rather than pKa 1.

 

[gettheleadout]

So I'm getting that the "conjugate base" that the pKb3 is referring to isn't just the carboxylate group, right? Since a RCOO- can't gain three protons...

 

[circulus vitios]

So I'm getting that the "conjugate base" that the pKb3 is referring to isn't just the carboxylate group, right? Since a RCOO- can't gain three protons...

Exactly. Which group is gaining that proton? Is it the side chain of lysine going from NH-, NH2, NH3+? Is it the amine terminal? What I'm asking is: how the hell do I do these problems? I've never learned amino acid acid/base reactions and my review book does a terrible job of covering it.

 

[gettheleadout]

Exactly. Which group is gaining that proton? Is it the side chain of lysine going from NH-, NH2, NH3+? Is it the amine terminal? What I'm asking is: how the hell do I do these problems? I've never learned amino acid acid/base reactions and my review book does a terrible job of covering it.

Okay, I get it. Lysine has two amine groups and one carboxylic acid group. The amine groups can exist protonated as NH3+, having gained one proton each, or NH2, having lost one proton each. The carboxylic acid group can do similarly. The "conjugate base" the question refers to is the entire amino acid molecule. The three protons gained go one each to an amine group and then one to the carboxylate.

We know that the carboxylic acid group has the lowest pKa of the three, so for the entire amino acid, its proton corresponds to pKa1. Because this amino acid is essentially a polyprotic acid, the pKa and pKb values go opposite directions in terms of pKa1-3 and pKb1-3; this is due to the order of protonation and deprotonation as FormerPhysicist explained. We know then that the pKa of the carboxylic acid group, pKa1, must correspond to pKb3. We can also infer that pKb1 must belong to either of the amine groups, however I don't believe there is a way to figure out which one without knowing their pKa/pKb values.

 

[Laureatebarrel]

Can you give answer and explanation?

 

[circulus vitios]

Okay, I get it. Lysine has two amine groups and one carboxylic acid group. The amine groups can exist protonated as NH3+, having gained one proton each, or NH2, having lost one proton each. The carboxylic acid group can do similarly. The "conjugate base" the question refers to is the entire amino acid molecule. The three protons gained go one each to an amine group and then one to the carboxylate.

We know that the carboxylic acid group has the lowest pKa of the three, so for the entire amino acid, its proton corresponds to pKa1. Because this amino acid is essentially a polyprotic acid, the pKa and pKb values go opposite directions in terms of pKa1-3 and pKb1-3; this is due to the order of protonation and deprotonation as FormerPhysicist explained. We know then that the pKa of the carboxylic acid group, pKa1, must correspond to pKb3. We can also infer that pKb1 must belong to either of the amine groups, however I don't believe there is a way to figure out which one without knowing their pKa/pKb values.

With which group is pKb3 associated?

 

[gettheleadout]

With which group is pKb3 associated?

The carboxylic acid group; the idea is that the group to most easily lose a proton will the least easily gain one (stronger acid = weaker conjugate base and vice versa, you know?). Since pKa1 denotes the favorability of first loss of a proton, and the carboxylic acid group is clearly the most acidic of the three (one COOH and two NH2), the pKa for the loss of a proton from the carboxylic acid group is pKa1. Because of the stronger ____ = weaker conjugate ____ rule, we can then infer that the pKb value of the carboxylate (the deprotonated conjugate of the carboxylic acid) is going to be the highest (least favorable) of the three pKb values the acid-base-reactive groups on the molecule will have. In the same way that pKa1 refers to the first proton lost, pKb1 refers to the first proton gained, which would go to the group that is the strongest base. The least favorable, highest pKb then, pKb3, must go to the conjugate of the group that holds pKa1: the carboxylic acid.

In summary:

For a triprotic acid (which is essentially what lysine is; it's fully protonated at a low enough pH):

pKa1 belongs to the same group as pKb3 - this will be the most acidic group
pKa2 belongs the same group as pKb2
pKa3 belongs to the same group as pKb1 - this will be the most basic group

CV: Are these answers correct? Answers in white: 1) C 2) B

 

[circulus vitios]

#1 is C.

#2 is D. Explanation:

"According to the table in the passage, pKas for histidine:

pKa1: 1.8-2.6
pKa2: 6.05
pKa 3: 8.8-10.6

At pH 7.0, the carboxyl group and side chain are deprotonated; the amino group is protonated. This is because pKa3 > pH > pKa2 > pKa1. Since the carboxyl group is deprotonated, we consider pKb3 rather than pKa1. Since the side chain is deprotonated, we consider pKb2 rather than pKa2. Since the amino group is protonated, we consider pKa3 rather than pKb1."

 

[Laureatebarrel]

So basically you need to know whether the group/side chain is protonated/deprotonated at that pH and choose pKa if it ia protonated and pKb if it is deprotonated right?

 

[MDhopeful74]

So I'm a little confused by this - if pKa1 represents the first proton lost by lysine, the carboxyl terminal, can someone explain how this means its pKb3? If its the first proton lost by the acid, wouldn't it be the first proton gained by the conjugate base? I understood gethteleadout's acid/conjugate base approach I think but I'm still struggling with this.