If i understand what you have said the expression [H+]= sqrt (Ka_1*C_1) is easier to get from the expression [H+]^2 = Ka_1*C_1 + Ka_2*C_2,already approximated, than the completed expression [H+]= ka_1/(Ka_1+ [H+]) *C_1 + ka_2/(Ka_2+ [H+])*C_2.
So the expression [H+]= sqrt (Ka_1*C_1) derives itself from the approximate expression [H+]^2 = Ka_1*C_1 + Ka_2*C_2 doing other (more) assumption:
Ka_1>>Ka_2 and C_1 >(or equal) C_2
1) So is correct think that the degree of accuracy of the expression [H+]^2 = Ka_1*C_1 + Ka_2*C_2 (one time approximated) is always > or = than the expression: [H+]= sqrt (Ka_1*C_1) (two times approximated)
2) looking this fact, the assumptions that must be verificated in order to use [H+]= sqrt (Ka_1*C_1) are not only:
Ka_1>>Ka_2 and C_1 >(or equal) C_2
But also:
[H+]>Ka_1, Ka_2
This last assumption because the approximate expression [H+]= sqrt (Ka_1*C_1) derives itself from [H+]^2 = Ka_1*C_1 + Ka_2*C_2 which has been got thanks to the assumption [H+]>Ka_1, Ka_2.
IS correct this??
Thanks very much