Weak acid-base solution assumption

[Conductivity]

I have been using a way to verify the assumptions.

For example let's say, If we add 200 mL of NaOH to 400 mL of HNO2 2M, The pH become 1.5 + ($pH_i$)
calculate the concentration of NaOH.

The usual way of doing is this:
$K_a = (0.02+x)[H^+]/(4/3-x)$ then find the x which most of the time is equal to the initial concentration of OH- after addition which you can convert back to the original solution and find the required information.

But the way I see how these approximations work is that you have to also think about how the additional OH can interact with H+ (or H3O+ if you like) to form water. So you really need simultaneous equations
$K_w = ([H+]_i -y)([OH-]_i-x-y)$
Where X is from the original acid reaction and Y comes from forming water reaction.

Now usually Y is negligible so I can assume the OH- fully reacts with the acid but here in this situation Y isn't

I did the math and found that x = 0.41345 and y = 0.01917811 which is not negligible ( about 4.63% mistake, However I am talking about more extreme version of this) . Is this way of thinking is true?

 

[Borek]

The pH become 1.5 + ($pH_i$)

$K_a = (0.02+x)[H^+]/(4/3-x)$

No idea where these equations and numbers come from nor what the i means, care to elaborate?

In general when it comes to dealing with such calculations it is best to approach the system in a way described here: http://www.chembuddy.com/?left=pH-calculation&right=toc

 

[Conductivity]

Sure sorry,
First the concentration of NO2- in the acid solution alone is
$Ka * [HNO3]_i = x^2$ solve for x, I get x = 0.03 (assuming [HNO3] doesn't change)

initial [NO2-] in the combined solution is 0.02 M
Initial [HNO3] let's assume it doesn't change so it is 4/3 M
Now when you add OH- the reaction will go to the right:
HNO2 (-----) H+ + NO2-
4/3-x $[H+]_f$ 0.02+x
Using the equilibrium constant equation we get the first equation.
After finding x which is 0.41345 M, It means that we had 0.41345 M of OH- from NaOH in the combined solution. Going back to the NaOH solution alone we get 0.620175

Note: Prepare for the mess! I know there are much simpler ways.

The other way that I am concerned about is this:
$[OH-]_i$ represents the initial concentration of OH- that comes from NaOH in the combined solution
$[H+]_i$ represents the initial concentration of H+ that comes from the disassociation of HNO2 in the combined solution before any reaction happen in the combined solution
If I form two simultaneous equation concerning that OH- might react with H+ or HNO2, I get
$\frac{K_a}{K_w} = \frac{(0.02+x)}{(4/3-x)([OH-]_i -x -y)}$
Where x results from HNO2 and OH- reaction

and Y results from this

$K_w = ([H+]_i -y)([OH-]_i-x-y)$

Usually, Y is negligible but in this specific question it is not entirely negligible.
I found X = 0.41345 and Y = 0.019178
First, Is all this true?
and I am asking about if there was a extreme version of this, Where Y is not negligible at all. I have to do this in order to get an exact solution right.

 

[Borek]

Sure sorry,
First the concentration of NO2- in the acid solution alone is
$Ka * [HNO3]_i = x^2$ solve for x, I get x = 0.03 (assuming [HNO3] doesn't change)

That is already based on an approximation, and you are mixing HNO₃ with NO₂^- as if these were related - basically they are not.

I strongly suggest you read the linked pages before digging deeper, as it seems like you are just juggling equations, no wonder you are getting strange results.

 

[Conductivity]

Ah, I meant HNO2 with a ka of 4.8 * 10^(-4)
I made 2 parts one with an approximation and one without

Please see the second part. It must be true because I used it multiple times on very weak acid reaction and it gave exact answer. I want to know if this approach is true.

Yes I have seen some of topics you provided before. I also said in the beginning there is another way
Thanks in advance

 

[Borek]

You can't assume your[OH^-]_i (nor [H⁺]_i) to be the only source of OH^- (H⁺), once their concentrations go down acid (and base) will dissociate further (even if you ignore base dissociation here).

I told you to approach the problem in a more rigorous way, you are wasting time ATM.

 

[Conductivity]

I tried both ways and it gave the same answer.
So I guess both ways are true

I also found a way to simplify all what I wrote to 2 lines.
and I derived a equation to solve these kind of questions from your link

Thanks borek