Important:  In order to thoroughly understand the material in this unit, you are expected to be familiar with the following topics which were covered in the separate unit Introduction to Acid-Base Chemistry:

• The Arrhenius concept of acids and bases - Review now
• The Brønsted-Lowry concept, conjugate acids and bases - Review now
• Definition of pH and the pH scale - Review now
• Strong vs. weak acids and bases - Review now

You should also have some familiarity with the principles of chemical equilibrium, and know how to write and use equilibrium constants.

Proton transfer reactions

As you should recall from your earlier introduction to acids and bases, the +1 electric charge of the tiny proton (a bare hydrogen nucleus) is contained in such a miniscule volume of space that the resulting charge density is far too large to enable its independent existence in solution; it will always attach to, and essentially bury itself in, the non-bonding orbitals of a solvent. Thus in aqueous solution, what we commonly represent as the "hydrogen ion" H⁺ is more accurately described as the hydronium ion H₃O⁺.

Acid-base chemistry is a transactional process in which protons are exchanged between two chemical species. A molecule or ion that loses or "donates" a proton is acting as an acid; a species that receives or "accepts" that proton plays the role of a base.

Defining an acid as a substance that simply "dissociates" into its component parts

HA → H⁺ + AB^–

fails to capture the concept outlined in the above box.  It is therefore important that you become comfortable with the Brønsted-Lowry model of acid-base behavior that represents the generalized acid-base reaction as a proton transfer process:

Acid-base equilibrium (UCBerkeley, 4 min)

The actual electric charges of the reaction products will of course depend on the particular nature of the species A, but the base will always have one more negative charge than the acid HA.

The acid-base processes we describe in this set of lessons take place in aqueous solution, so we will always assume that H₂O, which can act either as a proton donor or proton acceptor, plays an active role.

Thus a substance that acts as an acid in aqueous solution donates a proton to H₂O, yielding the hydronium ion H₃O⁺.

HA + H₂O → H₃O⁺ + A^–

Similarly, a base will remove a proton from H₂O, yielding the hydroxide ion OH^–.

B + H₂O → BH⁺ + OH^–

Autoprotolysis

A small number of acids are themselves amphiprotic, meaning that one molecule can transfer a proton to another molecule of the same kind. The best-known example of this process, known as autoprotolysis, is of course that of water:

2 H₂O(l) → OH^– + H₃O⁺(1-1)

Pure liquid ammonia and sulfuric acid are other well-known examples:

2 NH₃(l) → NH₂^– + NH₄⁺

2 H₂SO₄(l) → HSO₄^– + H₃SO₄⁺

Autoprotolysis reactions such as these typically proceed to only a very small extent.

The ion product of water

The equilibrium constant for Eq 1 is commonly written as

K_w = [H⁺][OH^–] (1-2)

For more on K_w, see this Wikipedia article, or this somewhat more detailed LSBU (UK) page.

and is usually known as the dissociation constant or ion product of water. For most purposes in elementary courses, we can use the value 1.0 × 10^–14 for K_w. But you should understand that this value is correct only for pure water at 25° C and 1 atm pressure. K_w varies considerably with temperature, pressure, and the ionic content of water: 

It should be obvious that anyone working in the fields of physiological or oceanographic chemistry cannot routinely use 1.0 × 10^–14 for K_w!

 

The equilibrium constants K_a and K_b

Acids and bases vary greatly in their "strength" — that is, their tendencies to donate or accept protons.

As with any chemical reaction, we can define an equilibrium constant K_c whose numerical value reflects the extent of the reaction — that is, the relative concentrations of the products and reactants when the reaction reaches equilibrium.

The strength of an acid HA in water can be defined by the equilibrium

HA + H2O → H3O+ + A–

(1-3a)

Similarly, the strength of the base A− in water is defined by

A− + H2O → HA + OH–

(1-3b)

Note carefully that reaction (3a) is not the reverse of (3b)!

K_a  and K_b are inversely proportional

How are K_a and K_b related? The answer can found by multiplying the above two expressions for K_a and K_b:

(1-3)

Alternatively, we can simply add the two corresponding reaction equations:

and — as you should recall from your earlier study of equilibrium, the equilibrium constant of the sum of two reactions is the product of the two K's.  So either way you figure it, the K_a for an acid and K_b for its conjugate base are related in a very simple way:

K_a K_b = K_w (1-4)

Clearly, as the strength of a series of acids increases, the strengths of their conjugate bases will decrease, this inverse relation between K_a and K_b is implicit in Eq 4.

pK values of some common acids and bases

In order to convert pK_a into K_a, simply take the negative antilog of pK_a. Thus, for acetic acid with
pK_a = 4.7, K_a = 10^–4.7 = 2.0E–5.

The following table gives the pKa values for a number of commonly-encountered acid-base systems which are listed in order of decreasing acid strength. It's worth taking some time to study this table in some detail.

Common acids and bases

ref	acid		pKa	conjugate base	pKb
1	HClO4	perchloric acid	~ –7	ClO4–  chlorate	~ 21
2	HCl	hydrochloric acid	~ –3	Cl – chloride	~ 17
3	H2SO4	sulfuric acid	~ –3	HSO4– bisulfate	~ 17
4	HNO3	nitric acid	~ –1	NO3– nitrate	15
5	H3O+	hydronium ion	0	H2O water	14.00
6	H2SO3	sulfurous acid	1.8	HSO3– bisulfite	12.2
7	HSO4–	bisulfate ion	1.9	SO42– sulfate	12.1
8	H3PO4	phosphoric acid	2.12	H2PO4– dihydrogen phosphate	11.88
9	[Fe(H2O)6]3+	hexaaquoiron(III)	2.10	[Fe(H2O)5OH]2+	11.90
10	HF	hydrofluoric acid	3.2	F – fluoride	10.8
11	CH3COOH	acetic acid	4.76	CH3COO – acetate	9.3
12	[Al(H2O)6]3+	hexaaquoaluminum(III)	4.9	[Al(H2O)5OH]2+	9.1
13	H2CO3	carbonic acid	6.3	HCO3– bicarbonate	7.7
14	H2S	hydrogen sulfide	7.04	HS –bisulfide	6.96
15	H2PO4–	dihydrogen phosphate	7.2	HPO4– monohydrogen phosphate	6.8
16	HSO3–	bisulfite ion	7.21	SO32– sulfite	6.79
17	HOCl	hypochlorous acid	8.0	OCl – hypochlorite	6.0
18	HCN	hydrocyanic acid	9.2	CN – cyanide	4.8
19	H3BO4	boric acid	9.30	B(OH)4– borate	4.79
20	NH4+	ammonium ion	9.25	NH3 ammonia	4.75
21	Si(OH)4	ortho-silicic acid	9.50	SiO(OH)3– silicate	4.50
22	HCO3–	bicarbonate ion	10.33	CO32– carbonate	3.67
23	HPO42–	monohydrogen phosphate ion	12.32	PO43– phosphate	1.67
24	SiO(OH)3–	silicate ion	12.6	SiO2(OH)22–	1.4
25	H2O	water	14.00	OH – hydroxide	0
26	HS –	bisulfide ion	~ 19	S 2– sulfide	~ –5
27	NH3	ammonia	~ 23	NH2 – amide	~ -9
28	OH –	hydroxide ion	~ 24	O 2– oxide	~ –10

 

 

 

 

 

 

Most courses (and their textbooks) do a good job of explaining the concept of proton transfer, but few of them provide you with a clear basis for understanding and making useful predictions about the direction and extent this transfer will take.  For example, when the gas HCl is added to water, you probably know that the reaction proceeds strongly to the right:

HCl + H₂O → H₃O⁺ + Cl^– (2-1)

...but what principle determines this, and what is to prevent the hydronium ion (also an acid) from pushing one of its own protons back onto a chloride ion, and thus reversing the reaction?

The concept of the proton free energy level was introduced in R.W. Gurney's classic Ionic Processes in Solution (1953). The best detailed exposition can be found in the first (1970) edition of Aquatic Chemistry by W. Stumm and J.L. Morgan.

The answer is that a proton is never really "pushed"; it will spontaneously seek out the lowest potential energy state (that is, the strongest base) it can find. Think how a macroscopic object such as a book will tend to fall to a location where its [gravitational] potential energy is as low as possible. Of course, the potential energy of a proton has nothing to do with gravity, but rather with how tightly it can bind to a base. This kind of potential energy is known as proton free energy (PFE) — but in the context of first-year chemistry, we can also just call it "proton energy".

You are not sure what free energy is? Don’t worry about it for the time being; just think of it as you would any other form of potential energy: something that falls when chemical reactions take place. See here for the lesson on free energy.

For the present, all you need to know is that a proton will tend to "fall" to the lowest potential energy state it can find. In keeping with this idea, you can think of an acid as a proton source, and a base as a proton sink.  Using this terminology, we can depict the generalized process HA + B → AB^– + HB⁺ as in this plot, which depicts the proton on HA "falling" to the lower-PFE "sink" provided by the base B, leaving AB^– and HB⁺ as products.

We will be making a lot of use of schematics like this farther on, so take a moment to familiarize yourself with information it depicts.

Take particular note of the following:

• The ordinate represents the relative free energy of a mole of protons when attached to a given base; we label it simply "proton energy" for brevity.
• The gray vertical line at the center of the diagram separates the acidic form of each conjugate pair from its base form; the terms "proton sources" and "proton sinks" are there to remind us of the role these forms play in acid-base chemistry.
• The red diagonal arrow represents the actual process in which a mole of protons "falls" from the acid HA to the base B. The fact that the arrow slants downward indicates that this process is spontaneous, meaning that the reaction proceeds to completion. In other words, the equilibrium constant for this reaction is very large.
• The green horizontal arrows depict the changes (that is, the formation of the two new conjugate species) that result from the proton transfer as the protons "fall" (in free energy) from the acid HA to the base B.

 

Strong acids in water

When you were first introduced to the Brønsted-Lowry model, you learned that the proper way to represent the "dissociation" of a strong acid like hydrochloric is

HCl + H₂O → H₃O⁺ + Cl^– (2-1)

Thus water is revealed as an active participant in the behavior of an acid, rather than as an inert spectator or solvent. Although analogous acid-base reactions can take place in other solvents such as liquid ammonia, virtually all of the acid-base chemistry we encounter in daily life is based on water.

Since water is acting as the proton acceptor here, it must be  fulfilling the role of the base B in the schematic given above. We can therefore construct a similar plot specifically for HCl (or, in fact, for any strong acid).

Recall that, for strong acids, this reaction is essentially complete. This means that a "solution of hydrochloric acid" is in reality a solution of hydronium and chloride ions; except in solutions much more concentrated than 1 M, the species HCl is only present in trace amounts.

Weak acids in water

When acetic acid CH₃COOH is dissolved in water, the resulting solution displays only very mild acidic properties. For simplicity we will represent acetic acid and the acetate ion CH₃COO^– by HAc and Ac^–, respectively:

HAc + H₂O → H₃O⁺ + Ac^– (2-2)

But although this equation has the same form as (2-1) above, it hides an important qualitative difference: HAc is a weak acid, meaning that the proton transfer takes place to only a tiny extent; the equilibrium strongly favors the left side:

HAc + H₂O H₃O⁺ + Ac^–

The corresponding proton free energy diagram now shows the H₃O⁺-H₂O system at the top, with the proton-transfer arrow pointing upward, indicating that the protons get kicked up to a higher free-energy level in this process.

Where does this additional energy come from? Simply from the random thermal energy in the solution.  As you might expect, this is not a thermodynamically favorable process, so it only happens to a very small extent.  In contrast to the HCl example we described previously in which 99.99+ percent of the HCl molecules end up as hydronium ions, almost all of the acetic acid molecules remain unchanged, and only tiny amounts of H₃O⁺ and Ac^– are produced.

 

The fundamental difference between strong acids and weak acids is clearly evident when we combine the preceding two proton-free energy diagrams into one. Strong acids (strong "conjugate-pairs") are always above the PFE of the H₃O⁺-H₂O system, while the weak conjugate pairs are below it.

The other big difference with acetic acid is that water is acting as a base here, rather than as an acid as it did in the HCl example. So it's not surprising that water, the basis of almost all acid-base chemistry, should appear in the middle of these diagrams.

Water can swing both ways

Substances that can both donate and accept protons are said to be amphiprotic. Many oxides and hydroxides behave in this way, and since H₂O falls into both of these classes, it is not surprising that water is amphiprotic:

These two opposing tendencies can be represented by

In the case of water, the contest is a draw; not only does neither side win, but the fraction of
H₂O molecules that break up in this way is miniscule: In pure water, only about one H₂O molecule out of 10⁹ is “dissociated” at any instant:

H₂O H⁺ + OH^– (2-4)

The actual reaction, of course, is the proton transfer

(2-5)

for which the equilibrium constant

K_w = [H₃O⁺][OH^–] (2-6)

is known as the ion product of water.  The value of K_w at room temperature is 1.008 x 10^–14. We can also express this in logarithmic terms: pK_w = 14.0.

As with any equilibrium constant, the value of K_w is affected by the

• temperature: K_w undergoes a 10-fold increase between 0°C and 60°C
• pressure: K_w is about doubled at 1000 atm
• presence of ionic species in the solution.

Because most practical calculations involving K_w refer to ionic solutions rather than to pure water, the common practice of using 10^–14 as if it were a universal constant is unwise; under the conditions commonly encountered in the laboratory, pK_w can vary from about 11 to almost 15. In seawater,
K_w is 6.3 × 10⁻¹². (See Stephen J. Hawkes: “pK_w is almost never 14.0”, J. Chem. Education 1995: 72(9) 799-802)

Notice that under conditions when K_w differs significantly from
1.0 × 10^–14, the pH of a neutral solution will not be 7.0.

For example, at a pressure of 93 kbar and 527°C, K_w = 10^−3.05, the pH of pure water would be 1.5. Similarly unusual conditions apply to deposits of water in geological formations and in undersea vents.

Similarly, the hydroxide ion is the strongest base that can exist in water.
So what happens if you add a soluble oxide such as Na₂O to water? Since O₂^– is a proton sink to H₂O, it will react with the solvent, leaving OH^– as the strongest base present: Na₂O + H₂O → 2 OH^– + Na⁺ .

Thus all bases stronger than OH^– appear equally strong in water — simply because they are all converted to OH^– .

PFE's are the key to understanding acid-base reactions

In the preceding section, we have employed simple PFE (proton free energy) diagrams to help you better understand the concepts of strong and weak acids, water as an acid or a base, and the leveling effect.

But we can gain a more generalized and comprehensive view of acid-base chemistry with the aid of a PFE diagram that shows a whole series of acid-base systems, arranged in order of decreasing acid strengths.

How acid strengths relate to proton free energies

As we explained near the beginning of this lesson, we ordinarily express the strength of an acid by quoting its "dissociation constant" K_a, which is the equilibrium constant for transfer of a proton to the base H₂O.

HA + H₂O → H₃O⁺ + A^–

We also introduced the concept of PFE as an alternative way of expressing the strength of an acid. If both PFE and K_a express the same thing, you may wonder how these two quantities are related. 

If you have had an introduction to thermodynamics and specifically to free energy G, you might wish to look at the Appendix section of this lesson to review the relationship between the fall in free energy and the value of pK_a.

A thorough answer to this question requires a bit of thermodynamics, which you may not have yet studied, but you don't really need any of the math here. It should suffice to show the results in the form of a plot, which yields a simple straight-line relationship when plotted on a semi-logarithmic scale.

Note that

• The extent of any chemical reaction is proportional to the fall in free energy when it takes place. So negative ΔG values correspond to larger equilibrium constants.
• For acid-base reactions specifically, K_a is inversely proportional to ΔG, and thus to the fall in PFE.
• This means that the proton on a strong acid (K_a > 1) has a large negative PFE; it has a "long way to fall" to the H₂O level.
• Conversely, a proton on a weak acid (K_a <1) can only move up to the H₂O level when supplied with thermal energy from the surroundings.
• A K value of 10⁰ = 1 corresponds to a zero-free energy change, meaning         that the reaction is at equilibrum when [HA] = [A^–].
• Proton-free energy changes exceeding –20 kJ mol^–1 result in K_a values so large that the proton transfer can be considered "complete" for most practical purposes.

Understanding the PFE chart

On the PFE diagram shown below, each acid-base system is represented by a horizontal line connecting an acidic species with its conjugate base.  Each line is placed at a height on the diagram that is proportional to its pK_a, as measured on the logarithmic scale (labeled "pH") at the right. What we have, then, is a list of acid-base systems arranged in order of decreasing acid strength.

Think of this chart as an alternative view of the table of acid strengths displayed near the top of this page, in which some of the information that is "hidden" in the table is now revealed in an easily-grasped way.

Take particular note of the following points:

 

The PFE chart helps you visualize (and understand!) acid-base chemistry

The virtue of a PFE chart is that you can see, at a glance, the relations between conjugate acid-base pairs that enables you to make quick, qualitative determinations of what is going on even in complex mixtures of acids and bases. The speed and convenience this affords enables you to build a much deeper conceptual understanding of acid-base chemistry.

In order to do the same thing by consulting a simple list of acid K_a's or pK_a's, you also need to be familiar with the principles of chemical equilibrium and thermodynamics — and then take the time to work through the arithmetic.

In the preceding sections we have shown how the PFE-chart approach can clarify the distinction between strong and weak acids. We will now go on to describe a variety of other principles that are easily grasped by consulting a PFE chart.

 

Significance of the pH scale

You will have noticed the scale labeled "pH" at the left side of the PFE plot.

First, please understand what this scale does NOT do; it does not tell you what the pH of the solution will be when you add an acid to pure water. This pH will depend on both the strength (K_a or pK_a) of the acid, and on its concentration in the resulting solution; you will learn how to carry out pH calculations in later lessons.

All but the very weakest acids can drive the pH down to near the bottom of the scale if their solutions are sufficiently concentrated. The only thing you can be sure of is that the pH of a solution of a solution of an acid in pure water will never be greater than 7.

pH sets the average proton free energy

When pH was first introduced in 1909, it was defined in terms of the "hydrogen ion concentration" [H⁺].  This has since been amended to the hydrogen ion activity {H⁺}. The latter term refers to the "effective" concentration of these ions — that is to the "availability" of protons (regardless of whether they physically exist as H₃O⁺ units or in other forms) to react with bases. But from the standpoint of thermodynamics, "availability" is just a loose term for "free energy". So we can also say that

pH is a measure of the average PFE in an aqueous solution

The more negative the pH, the higher the proton free energy in the solution. (The inverse relation is a consequence of the negative-logarithm definition of pH.)

The pH has the dual advantages of being both dimensionless and readily observable. More importantly, we can easily alter the pH of a solution by adding some strong acid or base.  And when we do this, we are in effect using the pH as a tool for controlling the average PFE in the solution.

More on this in a moment; first, let us introduce an important relation that you may have seen in an earlier chemistry course, and which we will discuss more in a later lesson in this group.

pH can control the relative abundance of conjugate acid-base pairs

Recalling the equilibrium expression for a weak acid

(3-1)

We can solve this for [H⁺]:

(3-2)

Re-writing this in terms of negative logarithms, this becomes

    (3-3)

or, since pK_a = – log K_a, we invert the ratio to preserve the positive sign:

We will make extensive use of this equation in a later lesson on acid-base buffering and titration.

(3-4)

This extremely important relation tells us that the pH of a solution containing a weak acid-base system controls the relative concentrations of the acid and base forms of that system.

Of special interest is the case in which the pH of a solution of an acid-base system is set to the value of its pK_a. According to the above equation, when pH = pK_a, the log term becomes zero, so that the ratio
[AB^–] / [HA] = 10⁰ = 1, meaning that [HA] = [AB^–].  In other words,

This pH adjustment can be easily made by adding appropriate amounts of a strong acid or strong base to the solution.

When the pH of a solution is set to the value of the pK_a of an acid-base pair, the concentrations of the acid- and base forms will be identical.

This condition can be represented schematically:

This is the basis for the arrangement of the various acid-base systems depicted in the PFE diagram; each conjugate pair is placed at a location on the vertical pH scale that corresponds to the acid's pK_a.

It is instructive to consider how sensitively the concentration ratio [A^–] / [HA] depends on the pH.

 

 

We can explore the relation between the pH and the distribution of conjugate species between the acid and base forms by carrying out calculations similar to those in Problem Example 2 for an acid-base system at different pH values.

The orange bars and blue numbers show the mole fractions of the acid and base forms at pH values of the pK_a and the pK_a ± 1.  The green numbers give the percent dissociation of the acid at each pH.

 

The apparent symmetry between the two extreme pH values suggests a simple relation between the log of the ratio
[AB^–] / [HA] and the pH. This is seen very clearly when this ratio is plotted as a function of pH.

 

 

 

 

It is important to note that the [AB^–] / [HA] ratios (calculated here from Eq 3-4) yield only a qualitative picture and do not take into account the concentrations of the different acid-base systems or proton transfers between them.  The latter processes would cause the lowest proton-vacant levels (that is, the strongest bases) to be completely filled from the bottom up, depending on the number of protons available. An exact calculation would require solving a set of simultaneous equations and is beyond the scope of this lesson. 

Make sure you thoroughly understand the following essential concepts that have been presented above.

• Write a chemical equation for the autoprotolysis of a species such as HCO₃^–; write an equation for the equilibrium constant for this reaction.
• Write equations that express the chemical reactions and equilibrium constant expressions that define the strength of an acid or base in terms of proton transfer processes  (Brønsted-Lowry concept).
• Write expressions defining the equilibrium constants K_a and K_b.
• Given the value of any two of K_a, K_b, and K_w, evaluate the third one. (This requires that you can show that K_a × K_b = K_w.)
• Explain the significance of the value of pK_a for a given monoprotic acid-base system —  that is, what it tells us about that particular acid and its conjugate base.
• Sketch out a simple proton free energy (PFE) diagrams illustrating the distinction between monoprotic strong and weak acids.
• State the meaning of the leveling effect,  and construct a simple PFE diagram that illustrates its action.
• Describe, in your own words, the meaning of PFE, how it relates to pH, and in its significance in terms of the relative concentrations of a conjugate acid-base pair in a solution.

---

A. What do we mean by the "strength" of an acid ?

The Brønsted-Lowry model defines the strength of an acid HA as its tendency to donate protons to a base B.  But bases vary in their abilities to accept protons, so the tendency for the complete transaction

HA + B → BH⁺ + A^–(A-1)

to occur depends on the two processes

HA → H⁺ + A^– K₁ (A-2)

B + H⁺ → BH⁺ K₂(A-3)

K_a's are always expressed on a scale relative to the K_b of the solvent

Because it is impossible to study either of these steps independently, what we might call the "absolute" strength of an acid (Eq A-2) cannot be measured. For this reason, we define a "standard" base, usually the solvent, and most commonly, water. Thus we replace Eq A-3 with

H₂O + H⁺ → H₃O⁺K₃(A-4)

so that Eq A^–1 now expresses the strength of HA relative to the base H₂O:

HA + H₂O → H₃O⁺ + A^– (A-5)

which, for convenience, we write in its abbreviated form

HA → H⁺ + A^– (A-5)

whose equilibrium constant is

(A-6)

This value of K_a is, of course, numerically identical to that for Eq A-5.

B. pH and proton free energy

This section may be of interest to readers who have studied elementary thermodynamics, and have some familiarity with Gibbs free energy. it is not required for understanding and using the concept of PFE.

From elementary thermodynamics, the driving force of a chemical reaction is given by the standard free energy change:

ΔG° = –RT ln K (B-1)

Solving this equation for K yields

(B-2)

For a weak acid we know that

(B-3)

Substituting Eq B-2 into this expression, we get

(B-4)

Next, we take the negative log of each term, recalling that ln x = 2.3 log x:

(B-5)

or

(B-6)

For the special case in which the concentrations of the conjugate pairs [H⁺] and [HA] are equal, pH = pK_a, and the rightmost term above disappears, leaving

(B-7)

Thus the pH is a direct measure of the average proton free energy in a solution.