Talk:Acid dissociation constant/Archive 4

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Clarification not needed

"...This is the definition in common usage. For many practical purposes it is more convenient to discuss the logarithmic constant, pK_a

$\mathrm {p} K_{\mathrm {a} }=-\log _{10}\left(K_{\mathrm {a} }\right)$clarify|reason=According to the previous equation, K_a has a dimension of concentration, so taking its logarithm is meaningless. If the log is taken of the numerical value of K_a in some specific units, the pK_a value will depend on the choice of the units, so this must be stated explicitly. Or, if K_a is actually normalized to [H2O], this also must be explained rather than saying "ignored".|date=February 2019"

The basis for the omission of [H₂O] from the equilibrium expression is sound: the "thermodynamic" definition of K is as a quotient of activities and the concentration of water is essentially the same as in pure water so that the value of its activity coefficient is, by definition, one. Therefore the logarithm of the thermodynamic equilibrium constant always differs from the logarithm of the numerical value of the concentration quotient with water omitted by the same amount, $log(\{H_{2}O\})$. The convention that the concentration (or activity) of H₂O is omitted from expressions which define equilibrium constants has been observed for many, many years. Petergans (talk) 10:52, 5 February 2019 (UTC)

I'm not questioning that "the concentration of water is essentially the same". The problem is that whereas the expression $K_{\text{a}}^{\text{I}}={\frac {[{\ce {A^-}}][{\ce {H+}}]}{{\ce {[HA]}}{\ce {[H2O]}}}}$ is dimensionless and does not depend on the concentration units, the expression $K_{\text{a}}^{\text{II}}={\frac {[{\ce {A^-}}][{\ce {H+}}]}{{\ce {[HA]}}}}$ has the dimension of concentration, like mol/L or cm⁻³ or m⁻³ or so on, and its numerical values in these different units will be different by many orders of magnitude. And since taking a logarithm of a dimensional quantity is meaningless, to define $\mathrm {p} K_{\text{a}}$ in terms of $K_{\text{a}}^{\text{II}}$, it must be divided by some concentration. To be consistent with $K_{a}^{\text{I}}$, this must be actually $\mathrm {p} K_{\text{a}}=-\log _{10}{\frac {K_{\text{a}}^{\text{II}}}{55~{\text{M}}}}$. However, if we believe the reference cited in the Equilibrium constant section, IUPAC recommends the expression $K_{\text{a}}^{\text{III}}={\frac {[{\ce {A^-}}][{\ce {H+}}]}{{\ce {[HA]}}c^{\circ }}}$, where the standard concentration $c^{\circ }=1~{\text{mol/L}}$. So $K_{\text{a}}^{\text{III}}$ is 55 times larger than $K_{\text{a}}^{\text{I}}$, and thus $\mathrm {p} K_{\text{a}}^{\text{III}}=-\log _{10}K_{\text{a}}^{\text{III}}$ is offset from $\mathrm {p} K_{\text{a}}^{\text{I}}=-\log _{10}K_{\text{a}}^{\text{I}}$ by $\log _{10}55\approx 1.7$. Defining $K_{\text{a}}$ in terms of activities but without water does not have the dimensional problem, but has the same problem with the value, since "it is important to note that the activity depends on the choice of standard state such that changing the standard state will also change the activity". — Mikhail Ryazanov (talk) 19:44, 8 February 2019 (UTC)

I think we should remember that casual readers may be unfamiliar with the conventions in a given subject. Petergans is correct in saying that the convention used in the article has been used for many years in chemistry books and articles. However Wikipedia wants the articles to be intelligible to readers who may not have read a chemistry book (or article!) for many years, and some such readers may well worry about the units. So I suggest following the IUPAC recommendation cited by Mikhail Ryazonov and quoting (once) the equation with the c₀ in the denominator so that K_a is unitless, and then immediately pointing out that the factor c₀ is normally omitted in the chemical literature and will be omitted for the rest of the article. This will explain to the reader what value of c₀ has been assumed, and also reassure him or her that the missing factor is not an error. Dirac66 (talk) 23:46, 8 February 2019 (UTC)

Because this has been common practice for so long, it deserves no more than a footnote.Petergans (talk) 13:59, 9 February 2019 (UTC)

OK, I have added two sentences to footnote 2 to explain this point.Dirac66 (talk) 23:33, 10 February 2019 (UTC)

Thank you Petergans (talk) 10:18, 11 February 2019 (UTC)

This is better, but I would say that this note should be added earlier, where K_a first appears (near "defined for a simplified reaction equation in which the solvent H₂O is ignored").

And what is going on with different solvents? The two formulas $K_{{\ce {a}}}={\frac {[{\ce {A^-}}][{\ce {SH+}}]}{{\ce {[HA][S]}}}}$ and $K_{{\ce {a}}}={\frac {[{\ce {A^-}}][{\ce {H+}}]}{{\ce {[HA]}}}}$ (with an implicit c₀) contradict each other, even "when the concentration of solvent molecules can be taken to be constant", since generally that constant ${\ce {[S]}}\neq c_{0}=1~{\text{mol/L}}$. — Mikhail Ryazanov (talk) 21:46, 11 February 2019 (UTC)

I placed the comment in the note after the definition of pK_a because this is the equation which includes the logarithm of a quantity which is not a pure number, so this is the point at which readers may require discussion of the units. The previous equation which defines K_a is adequate if one only uses it to calculate concentrations without taking a logarithm, which is no doubt why it is widely used as is.

As for nonaqueous solutions, I agree that the two equations are inconsistent, but I don't know which one is generally used in chemistry books and articles. I think it will be necessary to survey the chemical literature before straightening this out. Dirac66 (talk) 21:51, 12 February 2019 (UTC)

The solvent as reactant is omitted from equilibrium expressions in all but very advanced treatments, so I think it would be better omit it here. The reason for doing so can be mentioned in the article, but does not need to be in the lead. Petergans (talk) 22:20, 12 February 2019 (UTC)

Well, the article is actually called "Acid dissociation constant" (K_a), not "pK_a". So it would be very appropriate to at least give a correct and explicit definition of what K_a is before using it to define pK_a and in other parts of the article. As I understood, [A] in this article means the normalized (to c₀ = 1 mol/L) dimensionless concentration. If it is defined this way, then K_a is also dimensionless and well-defined, and there are no problems. This convention, however, must be stated explicitly, since the notation [A] in other cases (kinetics, for example) means the absolute concentration (in cm⁻³, mol/L or whatever convenient units), and dimensional equilibrium constants K are also used in practice (see Equilibrium constant#Dimensionality).

The new lead (after this) is even worse than before. In particular, the reaction ${\ce {HA <=> A^- + H^+}}$ should be given before discussing its properties ($K_{{\ce {a}}}=[{\ce {A^-}}][{\ce {H+}}]/[{\ce {HA}}]$). And the role of the solvent, hidden in the footnotes, is surely more relevant to this topic that explaining the common notions of quasistatic processes and dynamic equilibrium. — Mikhail Ryazanov (talk) 01:21, 21 February 2019 (UTC)

I don't disagree with the points you make. They are all dealt with in the body of the text. The lead is intentionally simplified because this is what the general reader will see first.

The issue of dimensionality is very difficult to handle. On the one hand it is irrelevant in the context of common practice, on the other hand it is essential in the context of thermodynamics. A further complication is that it has become accepted in biochemical publications to state the dimension of the concentration quotient. The need for this arises, not from thermodynamics, but from the possible use of molar, millimolar or micromolar scales for measured concentrations. Petergans (talk) 09:29, 21 February 2019 (UTC)

Phosphoric acid pKa graph wrong?

It shows the falling populations as continuous with adjacent populations, rather than reaching zero and ending. The citric acid diagram below shows them stopping at zero and NOT connected to the next species. It's a nit, but it could be misleading. Riventree (talk) 09:54, 5 September 2019 (UTC)

Inclusion in Nernst equation

The dissociation of acids can happen in an electrochemical cell where at one of the electrodes there is a solution of an acid with a sodium salt of the acid and, say, sodium chloride. Such a cell allows the determination of the ionization constant/acid dissociation of the acid by using the Nernst equation. Therefore the ionization constant can be included in the Nernst equation expression. A short mentioning of this aspect is necessary to the lede of this article.--109.166.139.150 (talk) 19:53, 16 September 2019 (UTC)

The Nernst equation applies only under "ideal" conditions. The response of a glass electrode deviates from ideality to an increasing extent with time, so it needs to be calibrated frequently. There are two ways to deal with the deviation from ideality.

(i) Calibrate the electrode with standard pH buffers. This procedure implies that the electrode has a linear response to hydrogen ion activity.

(ii) Calibrate the electrode by means of a strong acid/strong base titration. With this procedure it is assumed that the electrode has a linear response to hydrogen ion concentration, but a "slope factor" may be determined as part of the calibration procedure which may not be exactly equal to 1, the value implied by the Nernst equation.

These issues are dealt with in the article on the glass electrode. This level of detail is not warranted in an article on dissociation constants. Petergans (talk) 11:29, 17 September 2019 (UTC)

The level of detail as a short mentioning as inserted before (2-3 sentences) is useful to article, to specify a method of determination and link to thermodynamic activity expression of this ionization constant.--109.166.139.150 (talk) 18:08, 17 September 2019 (UTC)

This mentioning of Nernst equation involvement is also useful to specify the determination of inductive effects.--109.166.139.150 (talk) 18:12, 17 September 2019 (UTC)

I don't disagree with these points. It's just that the insert was not in the right place. Inductive effects are detailed in the section Factors that affect pKa values, including reference to the Hammett equation. Petergans (talk) 20:54, 17 September 2019 (UTC)

A short mention in the lede is useful to article overview. Inductive effects become real due ionization constants, not viceversa.--109.166.129.52 (talk) 13:49, 18 September 2019 (UTC)

Discussion of dimension

I think the recently added discussion would be improved by explaining the usual method 2 in more detail, and deleting methods 1 and 3 which are not really used for aqueous acids which are the subject of this article. For method 2 I think the following points would be helpful to some readers:

1. The thermodynamic equilibrium constant K is a ratio of dimensionless activities so it must be dimensionless.
2. K_c is obtained by replacing each activity by its ideal dilute solution value c/c⁰ and then neglecting c⁰ for convenience. Therefore K_c has units mol/L (for HA = H⁺ + A^-), which are usually retained in simple calculations of equilibrium concentrations.
3. However pH and thermodynamic calculations require a value of log K_c, which is only mathematically defined when K_c is a pure number. For these purposes, the units of K_c are usually ignored which is equivalent to replacing each c by the ideal solution activity c/c⁰. Dirac66 (talk) 01:35, 1 August 2019 (UTC)

Variation of pK_a of acetic acid with ionic strength.

I am not happy with these changes. The use of c/c⁰ is an arbitrary and artificial device which is not needed.^§

When the quotient of activity coefficients, Γ, can be assumed to be a constant, the numerical value of the concentration quotient, K, is proportional to the numerical value of K⁰, the thermodynamic constant

K = K⁰ / Γ.

There a many examples in the literature where a K value is determined with media of different ionic strengths, allowing for extrapolation to zero ionic strength. The assumption here is that Γ has a constant value over the range of experimental conditions, such as pH, used in each determination. The data used for the plot at the right show that assumption is justified. This is not just of theoretical interest. In applications such as blood chemistry and seawater chemistry the ionic strength of the medium is not necessarily the same as that used in the determination of a value of K.

§ I believe that this device is used in some text books. The underlying assumption is that the single-ion activity coefficients in the standard state are numerically equal to 1. For me that's a stretch too far because the standard states themselves will be dependent on the ionic strength.

Note also that there is a section in the article on non-aqueous solutions.Petergans (talk) 08:56, 16 August 2019 (UTC)

Perhaps we should have a subsection Dimensionality with explanations at both elementary and advanced levels. Yes, it is found in some general chemistry textbooks so I believe it should be presented first; I will add one textbook as a source. But I will not object if you want to add the above criticism of its rigour. And you might think about restoring the other two methods which I deleted: mole fractions as the second method (not the first since it is a less common method) and activity coefficients with units as the third. But I am not familiar with either method so it would be better that you decide whether to restore them. Dirac66 (talk) 01:39, 17 August 2019 (UTC)

The fact is that the product of concentration quotient, Q, and activity coefficient quotient, Γ, must be a pure number. Therefore, either both quotients are dimensionless or their dimensional units are reciprocals of one another. The use of c/c⁰ is both arbitrary and unnecessary. When the value of a concentration quotient is quoted, the underlying assumption is that the numerical value of Γ is equal to 1, regardless of the unit associated with the concentration values. So, what we have here is, in my opinion, a classic textbook error. As such it deserves only a footnote. I don't have a textbook where c/c⁰ is used. Please supply a citation.

Another example of an error which is in common usage is to be found in the analysis of absorption spectra. The process of decomposing a spectrum into a sum of component spectra is often wrongly named as deconvolution. When such errors are commonplace in the literature, they need to be identified as errors WP. Petergans (talk) 10:04, 17 August 2019 (UTC)

I have added the specification re the reciprocal concentration dimension possibility for each activity coefficient.--109.166.139.150 (talk) 19:51, 17 September 2019 (UTC)

I have added the text by Petrucci et al. as a citation since they have a whole 3/4-page box to explain this point to students. Note that they describe c⁰ as a reference state equal to 1 mol/L by definition, introduced solely to fix the units problem. It is not a thermodynamic standard state dependent on activity coefficients, ionic strength etc., as students at this level know nothing of non-ideal thermodynamics. This seems to be the standard general chemistry textbook treament, although this book is more explicit than some, so I think it should be included in Wikipedia. If you have other sources which explain that this commonly used treatment is oversimplified or incorrect, that point(s) can be added afterward. Dirac66 (talk) 16:50, 17 August 2019 (UTC)

The new section Dimensionality by Petergans is much better, and I have only made one small change (1 → 1 mol/L) to conform to the source. However I also notice that the other subsections of Equilibrium constant (3.1 Monoprotic acids to 3.7 Temperature dependence) have been deleted at the same time with no explanation. Was this an error? Dirac66 (talk) 15:47, 18 August 2019 (UTC)

Does the section Dimensionality make the footnote 2 partially redundant, as claimed by Petergans in the recent edit? The WP:Lede says that very important aspects (ratio of concentrations for non-dimensionalization) should be also present/summarized in the lede, not just in a section below.--185.53.198.72 (talk) 22:09, 19 August 2019 (UTC)

Yes, I think Petergans was correct to shorten this footnote, as it does make the same point as the section Dimensionality now. And the point is not really important enough to be in the TEXT of the lede, which is why it was actually in a footnote to the lede which many readers would skip. I think it is more visible now in the actual text further down in the article, so I think the change is an improvement. Dirac66 (talk) 00:51, 20 August 2019 (UTC)

Since the logarithmic form is mentioned in the lede, this automatically implies the necessity of a short specification of the non-dimensionalization of the quantity under the logarithm. They are equally important. The short specification should include the words see the Dimensionality section below. I see that Gibbs free energy is present in the lede, so this could be linked to the necessity of non-dimensionality.--185.53.199.17 (talk) 16:32, 23 August 2019 (UTC)

The issue of dimensionality is ignored in general practice. For example, the popular text book "Fundamentals of Analytical Chemistry", by Skoog, West, Holler and Crouch, 8th Edn. Ch. 9, does not even mention it. The need to include it here arises from the practice, which has become common with biochemists, of reporting equilibrium constants in the form of a value and a dimension, with alternatives for the dimension of M, mM or uM. Further elaboration in the lead is not warranted. Petergans (talk) 09:35, 24 August 2019 (UTC)

Ignoring the non-dimensionality is not quite a valid option. I think that further elaboration in the lede could and must be done, perhaps as another footnote, including the connection to the Gibbs energy, which requests the non-dimensionality.--185.53.199.134 (talk) 14:35, 17 March 2020 (UTC)

larger_than_about_12_is_more_than_99%_dissociated_in_solution

If pK_a = 13, then pK_b ~ 1, so K_b ~ 0.1, which corresponds to much less than 99% dissociation in anything other than dilute solutions.
For example, the concentration of the two ions could be 0.1 M each, and the concentration of the undissociated base would also be 0.1 M. Then 0.1 × 0.1 / 0.1 = 0.1, corresponding to 0.1/0.2 = 50% dissociation.
On the other hand, the concentration of the two ions could be 0.01 M each, and the concentration of the undissociated base would have to be 0.001 M. Then 0.01 × 0.01 / 0.001 = 0.1, corresponding to 0.01/0.011 = 90.91% dissociation.
And so on. —DIV (1.129.106.124 (talk) 12:53, 24 August 2019 (UTC))

Plot of buffer capacity (beta) as a function of pH for a 0.1 M weak acid with a pKa of 7.00.

That's not how it works. With a moderately strong base (pK_a > ca.11) in very alkaline solutions there are two simultaneous equilibria to consider, that can be written as:

BH⁺ + OH^- ⇌ B + H₂O (1)

H⁺ + OH^- ⇌ H₂O (2)

Arguments based on only the first equilibrium are not valid. Because of the second equilibrium, the buffer capacity of water rises very steeply at high pH independently of the presence of a solute. Petergans (talk) 22:24, 24 August 2019 (UTC)

I must disagree with this reasoning. If there are more than one equilibrium in a system, each reaction is in equilibrium separately, and it is often possible to reach conclusions using only one (or some) of the equilibrium conditions.

There are also questions which should be answered about the argument in the section Strong acids and bases. First, what is meant by "too low to be measured" and "below the detection limit". Modern analytical methods have a wide range of detection limits, and we would not want to re-classify weak acids as strong every time someone develops a new method with a detection limit lower by a factor of 10, say. It would be better to just say that an acid is strong if the concentration of undissociated acid is very small under given conditions (pH 0?) and not try to be more precise than "very small". What do recent books say?

Also, the section refers to the pKa value of a base. I understand pKa of an acid and pKb of a base, but what meant by pKa of a base? Should it be 14 - pKa? or the pKb of the conjugate base? or something else? In any case, this unfamiliar usage needs to be defined and sourced.

Given the above two points of confusion, I have been unable to decide whether I think the specific reasoning given by 1.129.106.124 above is valid. However I do have difficulty believing that the critical pKa values are -2 for acids and +12 for bases, because I would expect to find symmetry about pH = 7. For acids I agree with pKa = -2, and for bases I would expect pKb = -2 and pKa = 14 - pKb = +16 as the line between strong and weak bases.

Can someone sort out these points of confusion, preferably with sources? Dirac66 (talk) 02:28, 10 September 2019 (UTC)

The buffer capacity plot at the right shows why, with a solution of a base at pH > ca. 11, two equilibria occur simultaneously. I have labelled them (1) and (2) above. So, with a solution of such a base at pH > ca.11 both (1) and (2) have to be taken into account when calculating the concentrations of the species. That's why the statement on line 1 above is not valid. Petergans (talk) 09:38, 10 September 2019 (UTC)

It seems that these calculations of values of the dissociation degree cannot be done accurately without using the different than 1 activity coefficients.--109.166.139.1 (talk) 02:20, 10 May 2020 (UTC)

Relation of thermodynamic constant and concentration quotient

The expression

$K_{\mathrm {a} }={\frac {K^{\ominus }}{\Gamma }}=\mathrm {\frac {[A^{-}][H^{+}]}{[HA]}}$

requires that $\Gamma$ has a value of one. This is, in effect, a definition of the thermodynamic standard state, in which the activity coefficient quotient has a numerical value of one. When $K_{\mathrm {a} }$ is derived from data obtained using media of different ionic strengths its value changes and so does the defined standard state. This is universal practice. It allows for extrapolation of the value to zero ionic strength. See Rossotti and Rossotti, "The Determination of Stability Constants"(1961), Chapter 2, Activity and concentration quotients.

It is true that if a fixed standard state is used $\Gamma$ will be ionic-strength dependent. A citation will be needed if this alternative is to be included. Petergans (talk) 09:28, 18 September 2019 (UTC)

An activity quotient with a value of one and the same numerical value for the two constants doesn't seem to solve the aspect of dimensionality. The dimensional nature of the concentrations constant cannot be justified by dividing a dimensionless thermodynamic constant by something with numerical value of 1 such as the activity quotient, the dimension of the activity quotient (and of each coefficient) must be specified.--109.166.129.52 (talk) 12:30, 18 September 2019 (UTC)