[xshadow]

Hi!!
I have a problem with  one of the approximate formulas (for a mixture of two weak acids) which (it) derives from the expression:

[H+]= ka_1/(Ka_1+ [H+]) *C_1  +   ka_2/(Ka_2+ [H+])*C_2

At this point my textbook says that there are two main possible approximation:

1) If   [H+]>Ka_1, Ka_2   :
[H+]^2 = Ka_1*C_1 + Ka_2*C_2

2)  If   Ka_1>>Ka_2  and  C_1 >(or equal) C_2 :
[H+]= sqrt (Ka_1*C_1)

My PROBLEM is to understand how i can arrive at the expression  [H+]= sqrt (Ka_1*C_1) , using  the assumptions  Ka_1>>Ka_2  and  C_1 >(or equal) C_2, starting from the expression:
 [H+]= ka_1/(Ka_1+ [H+]) *C_1  +   ka_2/(Ka_2+ [H+])*C_2

Which are the  mathematical-logical steps  that i have to follow in order to get the approximate formula 2)


Thanks very much!!!!

[Hunter2]

It is going by skipping of terms.

So if you skip Ka1 and Ka2 in the denominator what will be left.

And if you skip C2 then what you get in second case.

[AWK]

 [H+]>Ka_1, Ka_2   :
[H+]^2 = Ka_1*C_1 + Ka_2*C_2

put
Ka_2*C_2 = 0.05*Ka_1*C_1

 then

[H+]^2 = 1.05*Ka_1*C_1

[H+] = 1.0247*Ka_1*C_1
and error in pH is 0.0106 which is negligible in most approximate calculations


 

[xshadow]

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

[AWK]

The exact equation is
[H+]= ka_1/(Ka_1+ [H+]) *C_1  +   ka_2/(Ka_2+ [H+])*C_2 + Kw/[H+]

The last part of equation (marked in blue) you need take into attount when lcalcualted pH is between 6.7 and 7.0.
When [H+] is at least 20 times greater then Ka_1 and Ka_2 simultaneously you  can neglect Kas' in denoninator and then

[H+]^2 = Ka_1*C_1 + Ka_2*C_2.

Note the factor 20 gives you an error of pH of order 0.01 unit.
In the last equation you can consider c_1 = c_2
then
[H+]^2 = (Ka_1+Ka_2)*C_1
or
Ka_1*C_1 >> Ka_2*C_2 (at least 20 times)
then
[H+]^2 = Ka_1*C_1 ie as if only the first acid is present

[xshadow]

thanks very much : )


...But when i get a pH of about 6.7-7.0 (without consider the H2O contribute)  I have always to consider the term Kw/[H+] or the range 6.7-7.0 is correct only in this case (mixture of two weak acids) ??

Thanks.

[AWK]

In this case without water contibute you can obtain pH sometimes greater than 7 (for acids!)
Lets try: acetic acid concentration 10^-10 M
and formic acid concentration 10^-11 M