[NP1957]

When pH of 0.1000 M H₂CO₃  is calculated  regarding only the first dissociation (Ka1=4.45 x 10^-7) the  proton concentration,  H1 = 2.11 x 10^-4 M and pH = 3.68
or if using quadratic equation,  also get H1= 2.11 x 10^-4 M. (H2 would be very small as Ka2=4.70 x 10^-11)

Note: H1 = Proton from first dissociation and H2 = Proton from second dissociation

But if the first and second dissociation equations are combined and the overall equilibrium constant would be 2.09 x 10^-17 for H₂CO₃   2H⁺ + CO₃^2- and then used for the calculation assuming  the acid is completely ionized, will get [H+] = 8.07 x 10^-7 M which is very much less than calculation from only the first dissociation. In theory it should be equal to H1+H2. Why?

[AWK]

But if the first and second dissociation equations are combined and the overall equilibrium constant would be 2.09 x 10-17 for H2CO3   2H+ + CO32- and then used for the calculation assuming  the acid is completely ionized, will get [H+] = 8.07 x 10-7 M which is very much less than calculation from only the first dissociation. In theory it should be equal to H1+H2. Why?

Weak acid cannot be completely ionized - look at the equatuion K₁ x K₂.
Moreover, assuming the equlibrium concentration of H₂CO₃ is equal to its stoichiometric concentration you have still two unknown values - [H₃O⁺] and [CO₃^2-] - how you can calculate pH?

[NP1957]

Thank you for your response.
I know weak acid cannot be completely ionized. Anyhow I just assume if it could be and calculate pH from the stoichiometric total reaction which [H₃O⁺] = 2x and [CO₃^2-] = x . Then x is solved from equilibrium equation.

My confusing is [H₃O⁺] from total equilibrium constant should be the sum of [H₃O⁺] from the first and the second ionized but it is much less than only calculation from fist ionized.

[Borek]

I just assume if it could be and calculate pH from the stoichiometric total reaction which [H₃O⁺] = 2x and [CO₃^2-] = x.

Which basically means you are assuming

[H₃O⁺] = 2[CO₃^2-]

which in turn is equivalent to the assumption that there is no HCO₃^- in the solution. This is wrong.

[H₃O⁺] = [HCO₃^-] + 2[CO₃^2-]

[AWK]

[H₃O⁺] = [HCO₃^-] + 2[CO₃^2-]

Moreover [HCO₃^-] is much, much greater than 2[CO₃^2-].

[NP1957]

Now I would understand that for this case, weak acid, I cannot assume if the acid could be completely ionized. Because the assumption for calculation should be true and  can be proved.
Mass-balance expression is very important.
Do I correctly understand?

Sometimes I found in chemistry textbook, when they have more than 2 equilibrium equations, the equations are combined and get new total equilibrium constant. After that the concentration of chemical species can be calculated. Any suggestion?

[AWK]

look at: Solution of Acid-Base Equilibria by Succesive Aproximation by Alejandro C. Olivieri
Journal of Chemical Eductation 67229-231 (1990),

[Borek]

Now I would understand that for this case, weak acid, I cannot assume if the acid could be completely ionized. Because the assumption for calculation should be true and  can be proved.
Mass-balance expression is very important.
Do I correctly understand?

Looks like.

Sometimes I found in chemistry textbook, when they have more than 2 equilibrium equations, the equations are combined and get new total equilibrium constant. After that the concentration of chemical species can be calculated. Any suggestion?

It is all about finding which simplifications to original equations are justified. In this particular case we can at first step assume that [HCO₃^-] >> [CO₃^2-], so the sum presented eariler can take a form

[H₃O⁺] = [HCO₃^-]

(we can ignore [CO₃^2-] in this sum, as it will be most likely several orders of magnitude smaller). There is so called 5% rule, which tells that in such sum you can ignore numbers that are less then 5% of the other. But - you have to check (after calculations) if this assumption holds.

[NP1957]

Borek and AWK

Thank you very much for an important fundamental on equilibrium calculation. I always assume when Ka /Kb is less than 10^-5 without concerning original concentration in order to simplify the equation but did not check after the calculation.
Please suggest the literature which I can refer to 5% rule.

Unfortunately my university does not subscribe for J. of Chemical Education. I will try to get the suggested paper. May I find in textbook?

[Borek]

Please suggest the literature which I can refer to 5% rule.

I am afraid the only source (other than my web pages) that I am aware of is in Polish, so it will be of no use for you.

Plese consult pH cheat sheet and analyse way of calculating pH suggested there, also take a look at plot shown here - it is obvious what are ranges of applications of different equations, this is in practice equivalent to the use of the 5% rule.

Unfortunately my university does not subscribe for J. of Chemical Education. I will try to get the suggested paper.

I doubt it will be very useful, IMHO medicine it proposes is worse than the illness

[NP1957]

Hi Borek,
Thanks again for all what you have suggested, it is very helpful.
But please explain if there is any message behind “medicine it proposes is worse than the illness”.

[Borek]

But please explain if there is any message behind “medicine it proposes is worse than the illness”.

What I mean is that approach they propose is so convoluted, that it doesn't have practical value for me. They propose to replace one difficult multistep calculation with other difficult multistep calculation and they claim the latter will be easier to grasp for students. Only bored postdoc can write something like that 

[NP1957]

I think so as it is hard to follow.
Thank you very much for the motto. I like it.