Answer:
pKa of the histidine = 9.67
Explanation:
The relation between standard Gibbs energy and equilibrium constant is shown below as:
[tex]\Delta{G^0} =-RT \ln \frac{[His]}{[His+]}[/tex]
R is Gas constant having value = 0.008314 kJ / K mol
Given temperature, T = 293 K
Given, [tex]\Delta{G^0}=15\ kJ/mol[/tex]
So, Applying in the equation as:-
[tex]15\ kJ/mol=-0.008314\ kJ/Kmol\times 293\ K\times \ln \frac{[His]}{[His+]}[/tex]
Thus,
[tex]15\ kJ/mol=-0.008314\ kJ/Kmol\times 293\ K\times \ln \frac{[His]}{[His+]}[/tex]
[tex]\frac{[His]}{[His+]}=e^{\frac{15}{-0.008314\times 293}[/tex]
[tex]\frac{[His]}{[His+]}=0.00211[/tex]
Also, considering:-
[tex]pH=pKa+log\frac{[His]}{[His+]}[/tex]
Given that:- pH = 7.0
So, [tex]7.0=pKa+log0.00211[/tex]
pKa of the histidine = 9.67
