[DGauss]

When studying the formal theory of acids and bases I came across equations for pH of general acid-base titrations in the work of Robert de Levie.

All articles present the same formalism basically, a good succinct example which shows coverage of acids, bases, ampholytics, and salts, can be found in

• de Levie, Robert. "A general simulator for acid-base titrations." Journal of chemical education 76.7 (1999): 987.

I'm a newbie here so not sure if the better practice would be for me to provide pictures of the equations for example...

My question is, how general is this formalism. In particular, how to define 'total analytical concentration' C for components which are present from multiple different sources.  Say for example I mix $C_a$ (NH₄)2HPO₄ with $C_b$ NH₄H₂PO₄.

The analytical concentration starting for NH₄ is $2C_a+C_b$ right? But how do we handle the computation for HPO₄ and H₂PO₄, is it

[tex]
C_a* F(HPO₄) + C_b* F(H₂PO₄)
[/tex]

or is it

[tex]
(C_a+C_b) *(F(HPO₄) + F(H₂PO₄))
[/tex]

[Borek]

No idea what you mean by F.

Can't check the original de Levie paper at the moment, but total analytical concentration of 'phosphoric acid' is just Ca+Cb in this case.

[DGauss]

Since someone's biting maybe I am allowed to post pictures from the original paper.

Bit that got cut off saying 'here $p$ denotes the maximum number of dissociable protons an acid or base can accommodate...'.

From the examples de Levie gives I understand that p,q differ between salts and say acids. Phosphoric acid would have p=3, q=3 but if you are putting H2PO4- in solution you start with p=3, q=2 in the F equation.

My question is just, whether is valid to treat contributions from different sources of chemically identical species this way. If so that would be C_a* F(HPO4) + C_b* F(H2PO4) as I put in first post.

[Borek]

My question is just, whether is valid to treat contributions from different sources of chemically identical species this way.

Valid - I see no reason to doubt it. This paper lists function F as "defined", but most likely it was defined after someone (either deLevie, or one of authors of the cited papers) solved the problem starting from the first principles and found that this expression appears quite often, so defining it this way makes calculations easier. It wasn't taken from nowhere.

But it is not a total analytical concentration, more like an auxiliary function helping in calculations. Plus, as far as I am aware, it is not something universally used/taught as a part of the acid-base theory formalism.

[DGauss]

The total analytical concentration is (what de Levie defined as) C_s and C_t. I don't doubt the equation holds for simple cases like pure acids and bases, I've seen the derivation. It also makes sense for salts or mixtures of salts e.g. CaHPO4 + NaHSO4. In this case you just sum up contributions from Ca²⁺, Na⁺, HPO₄^-, HSO₄^- (plugging them into the F equation).

What I'm not sure about is specifically case where the same species is contributed by different salts. Can we sum up contributions from different salts independently given the same ions are introduced e.g. NH₄⁺?

I know you already said "Valid" but it felt like most of your post was justifying the equation should hold generally. I'm asking about whether having different sources of same ions is a special case.

Regarding comment about usability: the reason I am looking at de Levie is because it provides a single equation for all acid-base-salt systems. It's useful for work I'm doing regarding propagation of uncertainties to have only one equation describe the titration. I am not aware of any other such approaches, all the rest just use the whole set of equations.

[Borek]

I'm asking about whether having different sources of same ions is a special case.

I see what you mean. I haven't seen the derivation, so I don't know. As I wrote earlier, this is not a part of a standard approach to the AB calculations. A lot depends on the assumptions made and the description in the paper isn't clear. My gut feeling would be that titration of partially neutralized acid was not something taken into account (been there, done that, realized the limitations of that approach later).

Regarding comment about usability: the reason I am looking at de Levie is because it provides a single equation for all acid-base-salt systems. It's useful for work I'm doing regarding propagation of uncertainties to have only one equation describe the titration. I am not aware of any other such approaches, all the rest just use the whole set of equations.

I wonder if it will help you taking into account it is designed to work backwards - not pH=f(V_titrant) but V_titrant=f(pH).

Perhaps take a look at the ideas behind the engine described here: https://www.chembuddy.com/?left=Buffer-Maker&right=buffer-calculation

It is not designed for titration curves, but can be easily adapted. This approach uses total analytical concentrations of all acids and bases present (for example NaH₂PO₄ being equimolar mixture of H₃PO₄ and NaOH - you need to calculate their concentrations as if they were separately introduced into solution, summing over all substances present), so you don't have to worry about different forms.

[DGauss]

Perhaps take a look at the ideas behind the engine described here: https://www.chembuddy.com/?left=Buffer-Maker&right=buffer-calculation

It is not designed for titration curves, but can be easily adapted. This approach uses total analytical concentrations of all acids and bases present (for example NaH₂PO₄ being equimolar mixture of H₃PO₄ and NaOH - you need to calculate their concentrations as if they were separately introduced into solution, summing over all substances present), so you don't have to worry about different forms.

Interesting. So for every ion, the software recognises the original pure acid or base, and distinguishes forms by just stoichiometry. Other than that Q seems similar to de Levie's F.

Thanks for pointing me to it, is there a way to disable activity correction for now so it's just basically a fast way of solving equilibrium system? If so it would help me do some trial-and-error checks

[Borek]

is there a way to disable activity correction for now so it's just basically a fast way of solving equilibrium system? If so it would help me do some trial-and-error checks

In Buffer Maker? You have a choice of calculating, ignoring and forcing I, there is a selection list somewhere in the pH calculator dialog.

Other than that Q seems similar to de Levie's F.

When deriving solutions for AB equilibrium some combinations of H and K values show up quite often.

[DGauss]

I have been investigating and it seems De Levie's equation is general to any mixture with purely acid-base equilibria. Certainly, to salts. Of course Buffer Maker's approach is an alternative and it's certainly interesting they are equivalent.

Since Buffer Maker doesn't seem to have amino acids (?) I've been verifying the equation works for amino acids from scratch. I would note discrepancy with this thread where conclusion seems to be pH around 1.30-1.32, for solution of 0.19(047619) M glycine with 0.047(619048) M H2SO4. If I take thread values pKa1=2.3, pKa2=9.7 for glycine and pKa2=1.99 for sulphuric (assuming 1st diss. goes to completion), I get pH=2.44(6). De Levie equation seems to agree with solving the system of equations from scratch on this number.

(System is as follows, if I am not mistaken?)

a=[Gly-], b = [HGly], c = [H2Gly+], f = [SO4 2-], g = [HSO4-]
h = [H+]
10^(-2.3) = b*h/c
10^(-9.7) = a*h/b
10^(-1.99) = f*h/g
0.19047619 = a+b+c
0.047619048 = f + g
h+c = 10^(-14)/h + a + g + 2*f
Solve for: h, a,b,c,f,g

[Borek]

Since Buffer Maker doesn't seem to have amino acids (?)

At least glycine is there (plus you can add any number of acids to the database).

Buffer Maker yields 2.45.

[DGauss]

Thanks for confirming, yes I should have noticed glycine is there. Was wondering a couple other things:

(1) If I add a reagent, am I right in thinking the Ka values should be entered in order of deprotonation so e.g. for lysine, Ka1 is H₃Lys²⁺ to H₂Lys⁺, Ka2 H₂Lys⁺ to HLys, Ka3 is HLys  to Lys^-. Therefore the order is always strictly Ka1>Ka2>Ka3 (?). So we rearrange equilibria in descending order of K's magnitude. Rather than as here for example, where Ka3>Ka2 for aspartic acid or hystidine. If unclear what I mean, let me know, I can just calculate some examples and see if they agree with Buffer Maker (I can't seem to get lysine, aspartic acid etc working on it):

For example, 0.048 M of lysine + 0.022 M of aspartic acid + 0.065 M of histidine, using pKa values in the link (   2.18, 8.95, 10.53 for lysine; 1.88, 9.60, 3.65 for aspartic acid; 1.82, 9.17, 6.00 for histidine), should give pH=8.50(7) following this method (w/o activity correction). Or just 0.065 M histidine gives pH=7.58(5), or 0.048 M lysine gives pH=9.73(8 ), or 0.022 M aspartic acid gives pH=2.88(8 ). Are these correct?


(2) How does Buffer Maker extract the [H⁺]. If it solves the root of the system of equations, it would find a large number of roots, is it sufficient to select real positive root or does it have/need a protocol for choosing out of multiple real positive roots? (If this can ever happen chemically...) If it follows some other procedure it's clearly fast so am curious about that.



-----


Equation system for lysine (or histidine):

a=[Lys-], b = [HLys], c = [H2Lys+], d = [H3Lys 2+]
10^(-2.18) = c*h/d
10^(-8.95) = b*h/c
10^(-10.53) = a*h/b
0.048 = a+b+c+d
h+c+2d = 10^(-14)/h + a
Solve for: h, a,b,c,d

For aspartic acid:

a=[Asp 2-], b = [HAsp-], c = [H2Asp], d = [H3Asp+]
10^(-1.88) = c*h/d
10^(-3.65) = b*h/c
10^(-9.60) = a*h/b
0.022 = a+b+c+d
h+d = 10^(-14)/h + b + 2a
Solve for: h, a,b,c,d

[Borek]

If I add a reagent, am I right in thinking the Ka values should be entered in order of deprotonation so e.g. for lysine, Ka1 is H₃Lys²⁺ to H₂Lys⁺, Ka2 H₂Lys⁺ to HLys, Ka3 is HLys  to Lys^-.

Yes. Press F1 for help and you will learn that

each next formula must have charge smaller by 1

that forces consecutive dissociation steps.

Therefore the order is always strictly Ka1>Ka2>Ka3 (?).

I am actually not sure if that follows. This condition must hold for overall dissociation constants, but doesn't have to hold for stepwise ones. It probably typically does for electrostatic reasons, unless there are some additional stabilizing effects changing properties of the anion.

Data from the ucalgary page doesn't look correct to me, unless they use some strange numbering scheme. I just checked that my sources give a different order of values (monotone one).

How does Buffer Maker extract the [H⁺]. If it solves the root of the system of equations, it would find a large number of roots, is it sufficient to select real positive root or does it have/need a protocol for choosing out of multiple real positive roots? (If this can ever happen chemically...) If it follows some other procedure it's clearly fast so am curious about that./quote]

It selects the largest root. If memory serves me well all others are negative and just unphysical.