This question has gave me headaches so far due the problematic nature of how to get to the equation for finding the pH of the resulting solution from such salt.
Can someone help me with this?
The problem is as follows:
Find the pH of a solution of NH4CN 1M. Their respective hydrolisis constants are:
Kh=5.6×10−10
Kh=2.5×2.5×10−5
The alternatives given are as follows:
1.2.82.3.53.6.44.11.75.12.2
What I've attempted to do was to establish the following reactions:
NH+4+H2O⇌NH3+H3O+kh=KwKbCN−+H2O⇌HCN+OH−kh=KwKaH3O++OH−⇌2H2Okh=1Kw
This meant that:
K=KwKa×Kb
For the given information it can be calculated:
K=10−1410−145.6×10−10×10−142.5×10−5=1.4
All of this is translated into:
NH+4+CN−⇌NH3+HCN
From this equation it can be established the equilibrium equation:
K=[NH3][HCN][NH+4][CN−]
Returning to the previous equation I'm getting:
NH+4+CN−⇌NH3+HCNi11c−x−x+x+xe(1−x)(1−x)xx
Using this relation with what was established earlier it becomes into:
K=[NH3][HCN][NH+4][CN−]
1.4=[NH3][HCN][NH+4][CN−]
1.4=x2(1−x)2
Solving this for $x$ results into:
x=0.54196
and
x=6.45804
The thing here comes on which to choose for the concentration. I decided that the first seems more logical as it will not produce negative concentration in the denominator.
Therefore the concentration for x=0.54196
This means:
[NH3]=[HCN]=0.54196
[CN−]=[NH+4]=1−0.54196=0.45804
Then to get the pH can be established from the hydrolization of HCN:
HCN+H2O⇌CN−+H3O+
Then:
K=10−142.5×10−5
K=[CN−][H3O+][HCN]
4.0×10−10=[H3O+](0.45804)(0.54196)
[H3O+]=4.73286×10−10
pH=−log(4.73286×10−10)=9.3248
Which seems kind of logical that the pH is within the range of an alkaline solution as $Kb>Ka$.
But since this option does not appear within the alternatives. I'm not convinced whether my solution is right or not?.
I've found this equation which relates the pH with the pKw and pKa and pKb and the concentration:
It states:
pH=12[pKw−pKb+pKa+logC]
where c=concentration of the salt.
Inserting the given values in the above equation yields:
pH=(0.5)[−log(10−14)+log(10−145.6×10−10)−log(10−142.5×10−5)+log(1)]
pH=9.3248
which also checks with what was obtained previously.
But again none of these seem to check with any of the alternatives given. Could it be that my procedure was wrong.
The thing which it bothers me here is if this could be accepted as a general procedure to calculate the pH of a salt which has a weak acid and weak base ions. Can somebody help me here? and more importantly how to prove that equation for pH? I'm not sure if the hydrolization constants given in the problem are right or accurate. Does somebody has access to cross reference and check if those values are correct?.