The structure of tryptophan is shown at the right along with the pK values of the α-carboxyl and α-amino groups. (pK1=2.38; pK2=9.39)
Tryptophan is a diprotic acid which can exist in three protonated forms: H2A , HA0, and A–. (HA0 consists primarily of the zwitterionic form with a negatively charged carboxyl group and positively charged amino group, but a very small amount of the completely neutral species does exist). Calculate the fraction (α) of αH2A , αHA0, and αA– present at pH 2.60.
To calculate the fractions of αH2A, αHA0, and αA− present at pH 2.60 for tryptophan, we need to compare the pH (2.60) to the pK values (pK1=2.38, pK2=9.39) of the α-carboxyl and α-amino groups.
The αH2A form refers to tryptophan being fully protonated (having both the α-carboxyl and α-amino groups as H+ ions). The αHA0 form refers to tryptophan having a negatively charged carboxyl group and a positively charged amino group, while αA− refers to tryptophan being fully deprotonated (having both α-carboxyl and α-amino groups as -OH and -NH2 groups, respectively).
To calculate the fraction of each form, we can use the Henderson-Hasselbalch equation:
pH = pKa + log ([A-]/[HA])
For αH2A to αHA0, we have:
pH = pK1 + log ([A-]/[HA])
Substituting the given values:
2.60 = 2.38 + log ([A-]/[HA])
Rearranging the equation:
log ([A-]/[HA]) = 2.60 - 2.38
log ([A-]/[HA]) = 0.22
Using properties of logarithms, we can simplify the equation:
[A-]/[HA] = 10^0.22
Now, for αHA0 to αA−, we use the same equation with different pKa values:
pH = pK2 + log ([A-]/[HA])
Substituting the given values:
2.60 = 9.39 + log ([A-]/[HA])
Rearranging the equation:
log ([A-]/[HA]) = 2.60 - 9.39
log ([A-]/[HA]) = -6.79
Again, using properties of logarithms:
[A-]/[HA] = 10^-6.79
Now, we can calculate the fractions αH2A, αHA0, and αA− using the following formulas:
αH2A = [H2A] / ([H2A] + [HA] + [A-])
αHA0 = [HA] / ([H2A] + [HA] + [A-])
αA− = [A-] / ([H2A] + [HA] + [A-])
To calculate these fractions, we need to know the initial concentration of tryptophan ([H2A] + [HA] + [A-]). However, this information has not been provided. You would need to know the concentration or molarity of tryptophan to calculate the fractions precisely at pH 2.60.