Question

What is the pH of a solution of lysine if the epsilon-amino group is 40% dissociated?

As a resource:

Amino Acid	Alpha amino pK	Carboxy pK	R group pK
L	9.74	2.33
K	9.06	2.16	10.54
R	8.99	1.82	12.48

Answer 1

Amino Acid: K=Lysine
  

Alpha amino pK : 9.06
Carboxy pK:      2.16
R group pK:      10.54

Lysine has one alpha amino group having pK = 9.06 and
one espilon amino group having R group pK=10.54

Dissociation of epsilon-amino group from lysine

[N(epsilon)H3+] = [N(epsilon)H2] + [H+]

The lysine solution will be a buffer of lysine [N(epsilon)H3+] and
dissociated lysine [N(epsilon)H2]

According to Henderson-Hasselbach equation

pH = pK + Log([N(epsilon)H2]/[N(epsilon)H3+])

fraction of dissociation = 40/100 = [N(epsilon)H2]/([N(epsilon)H3+] + [N(epsilon)H2])

[N(epsilon)H2] = 0.4*([N(epsilon)H3+] + [N(epsilon)H2])

0.6*[N(epsilon)H2] = 0.4*[N(epsilon)H3+]

[N(epsilon)H2] = 2/3*[N(epsilon)H3+]

Substituting above relation into Henderson-Hasselbach equation,
pH = 10.54 + Log(2/3) = 10.54 - 0.18 = 10.36