The usual definition of a “strong” acid or base is one that is completely dissociated in aqueous solution. Hydrochloric acid is a common example of a strong acid. When HCl gas is dissolved in water, the resulting solution contains the ions H₃O⁺, OH^–, and Cl^–, but except in very concentrated solutions, the concentration of HCl is negligible; for all practical purposes, molecules of “hydrochloric acid”, HCl, do not exist in dilute aqueous solutions.

In order to specify the concentrations of the three species present in an aqueous solution of HCl, we need three independent relations between them. These relations are obtained by observing that certain conditions must always hold:

The next step is to combine these three equations into a single expression that relates the hydronium ion concentration to C_a. This is best done by starting with an equation that relates several quantities, such as Eq 6-4 , and substituting the terms that we want to eliminate. Thus we can get rid of the [Cl^–] term by substituting Eq 3 into Eq 4 :

[H⁺] = [OH^–] + C_a (1-5)

The [OH^–] term can be eliminated by the use of Eq 1:

  (1-6)

This equation tells us that the hydronium ion concentration will be the same as the nominal concentration of a strong acid as long as the solution is not very dilute. As the acid concentration falls below about 10^–6 M, however, the second term predominates; [H⁺] approaches √K_w, or 10^–7 M. The hydronium ion concentration can of course never fall below this value; no amount of dilution can make the solution alkaline!

Notice that Eq 6 is a quadratic equation; in regular polynomial form it would be

[H⁺]² – C_a[H⁺] – K_w = 0 (1-7)

Most practical problems involving strong acids are concerned with more concentrated solutions in which the second term of Eq 7 can be dropped, yielding the simple relation [H⁺] = [A^–].

Concentrated solutions of strong acids

In more concentrated solutions, interactions between ions cause their “effective” concentrations, known as their activities, to deviate from their “analytical” concentrations.

Thus in a solution prepared by adding 0.5 mole of the very strong acid HClO₄ to sufficient water to make the volume 1 liter, freezing-point depression measurements indicate that the concentrations of hydronium and perchlorate ions are only about 0.4 M. This does not mean that the acid is only 80% dissociated; there is no evidence of HClO₄ molecules in the solution. What has happened is that about 20% of the H₃O⁺ and ClO₄^– ions have formed ion-pair complexes in which the oppositely-charged species are loosely bound by electrostatic forces. Similarly, in a 0.10 M solution of hydrochloric acid, the activity of H⁺ is 0.81, or only 81% of its concentration. (See the green box below for more on this.)

Activities are important because only these work properly in equilibrium calculations. The relation between the concentration of a species and its activity is expressed by the activity coefficient γ (gamma):

The relation between the concentration C of a species and its activity a is expressed by the activity coefficient γ (gamma):

a = γ C (1-8)

As a solution becomes more dilute, γ approaches unity. At ionic concentrations below about 0.001 M, concentrations can generally be used in place of activities with negligible error.

Recall that pH is defined as the negative logarithm of the hydrogen ion activity, not its concentration.

* Activities of single ions cannot be determined, so activituy coefficients in ionic solutions are always the average, or mean, of those for all ionic species present. This quantity is denoted as γ_±.

At very high concentrations, activities can depart wildly from concentrations. This is a practical consideration when dealing with strong mineral acids which are available at concentrations of 10M or greater. In a 12 M solution of hydrochloric acid, for example, the mean ionic activity coefficient* is 207. This means that under these conditions with [H⁺] = 12, the activity {H⁺} = 2500, corresponding to a pH of about –3.4, instead of –1.1 as might be predicted if concentrations were being used.

Other sources that discuss the material in this section include the
ChemBuddy site and Chapter 5 of J.L. Burgot: Ionic Equilibria in Analytic Chemistry. Images adapted from both of these sources are used here.

Most acids are weak; there are hundreds of thousands of them, whereas there are no more than a few dozen strong acids. We can treat weak acid solutions in exactly the same general way as we did for strong acids. The only difference is that we must now include the equilibrium expression for the acid. We will start with the simple case of the pure acid in water, and then go from there to the more general one in which strong cations are present. In this exposition, we will refer to “hydrogen ions” and [H⁺] for brevity, and will assume that the acid HA dissociates into H⁺ and its conjugate base A−.

Pure acid in water

In addition to the species H⁺, OH^–, and A− which we had in the strong-acid case, we now have the undissociated acid HA; four variables, requiring four equations.

Equilibria	[H+][OH–] = Kw(2-1) (2-2)
Mass balance	Ca = [HA] + [A–](2-3)
Charge balance	[H+] = [OH–] + [HA–](2-4)

In order to eliminate [HA] from Eq 2-2, we solve Eq 2-4 for this term, and substitute the resulting expression into the numerator:

(2-5)

[H⁺]³ + K_a[H⁺]² – (K_w + C_aK_a) [H⁺] – K_a K_w = 0 (2-5)

For most practical applications, we can make approximations that eliminate the need to solve a cubic equation.

Approximations

1. Unless the acid is extremely weak or the solution is very dilute, the concentration of OH^– can be neglected in comparison to that of [H⁺]. If we assume that [OH^–] ≪ [H⁺], then Eq 2-5 can be simplified to

(2-6)

which is a quadratic equation:

[H⁺]² +K_a[H⁺]– K_aC_a = 0 (2-7)

and thus, from the quadratic formula,

(2-8)

 

 

2. If the acid is fairly concentrated (usually more than 10^–3 M), a further simplification can frequently be achieved by making the assumption that
[H⁺] << C_a. This is justified when most of the acid remains in its protonated form [HA], so that relatively little H⁺ is produced. In this event, Eq 2-6 reduces to

(2-9)

or

[H⁺] = (K_aC_a)^½ (2-10)

 

 

3. If the acid is very weak or its concentration is very low, the H⁺ produced by its dissociation may be little greater than that due to the ionization of water. But if the solution is still acidic, it may still be possible to avoid solving a cubic equation by assuming that the term ([H⁺] - [OH^–]) << C_a in Eq 2-5:

(2-11)

This can be rearranged into standard quadratic form

[H⁺]² + K_a [H⁺] – K_a C_a = 0(2-12)

If you have access to the Internet, you might find this cubic equation solver useful.

For dilute solutions of weak acids, an exact treatment may be required. With the aid of a computer or graphic calculator, solving a cubic polynomial is now far less formidable than it used to be. But round-off errors can cause these computerized cubic solvers to blow up; it is generally safer to use a quadratic approximation.

 

 

Weak bases

The weak bases most commonly encountered are:

anions A– of weak acids:	A– + H2O → HA + OH– CO32– + H2O → HCO3– + OH–
ammonia	NH3 + H2O → NH4+ + OH–
amines, e.g. methylamine	CH3NH2 + H2O → CH3NH3++ H2O

Solution of an anion of a weak acid

Note that, in order to maintain electroneutrality, anions must be accompanied by sufficient cations to balance their charges. Thus for a C_b M solution of the salt NaA in water, we have the following conditions:

Species	Na+, A–, HA, H2O, H+, OH–
Equilibria	[H+][OH–] = Kw(2-13) (2-14)
Mass balance	Cb = [Na+] = [HA] + [A–](2-15)
Charge balance	[Na+] + [H+] = [OH–] + [A–](2-16)

Replacing the [Na⁺] term in Eq 2-15 by C_b and combining with K_w and the mass balance, a relation is obtained that is analogous to that of Eq 2-5 for weak acids:

(2-17)

 

The approximations

(2-18)

and

[OH^–] ≈ (K_b C_b)^½(2-19)

can be derived in a similar manner.

 

to yield

(3-3)

If neither acid is very strong or very dilute, we can replace equilibrium concentrations with nominal concentrations:

(3-4)

 

But if 0.001M chloroacetic acid (K_a= .0014) is used in place of formic acid, the above expression becomes

(3-5)

which exceeds the concentration of the stronger acid; because the acetic acid makes a negligible contribution to [H⁺] here, the simple approximation given above (3-3) is clearly invalid. We now use the mass balance expression for the stronger acid

[HX] + [X^–] = C_x(3-6)

to solve for [X^–] which is combined with the equilibrium constant K_x to yield

(3-?)

Solving this for [X^–] gives

(3-8)

The approximation for the weaker acetic acid (HY) is still valid, so we retain it in the substituted electronegativity expression:

(3-9)

which is a cubic equation that can be solved by approximation.

Owing to the large number of species involved, exact solutions of problems involving polyprotic acids can become very complicated. Thus for phosphoric acid H₃PO₄, the three "dissociation" steps yield three conjugate bases:

Fortunately, it is usually possible to make simplifying assumptions in most practical applications. In the section that follows, we will show how this is done for the less-complicated case of a diprotic acid.

Diprotic acids

A diprotic acid HA can donate its protons in two steps, yielding first a monoprotonated species HA^– and then the completely deprotonated form A^2–.

Since there are five unknowns (the concentrations of the acid, of the two conjugate bases and of H⁺  and OH^–), we need five equations to define the relations between these quantities. These are

Equilibria	[H+][OH–] = Kw(4-1) (4-2) (4-3)
Mass balance	Ca = [H2A] + [HA–] + [A2–](4-4)
Charge balance	[H+] = [OH–] + [HA–] + 2 [A2–](4-5) (It takes  2 moles of H+ to balance the charge of 1 mole of A2–)

Solving these five equations simultaneously (don't bother!) for K₁ yields the rather intimidating expression

 (4-6)

which is of little practical use except insofar as it provides the starting point for various simplifying approximations:

 

If the solution is even slightly acidic, then ([H+] – [OH–]) ≈ [H+]	(4-7)
For any of the common diprotic acids, K2 is much smaller than K1. If the solution is sufficiently acidic that K2 << [H+], then a further simplification can be made that removes K2 from the equation. This is the starting point for most practical calculations.	(4-8)
Finally, if the solution is sufficiently concentrated and K1 sufficiently small so that [H+] << Ca, then the equation reduces to the simplest form.	[H+] ≈ (Ka Ca)½

Solutions containing a weak acid together with a salt of the acid are collectively known as buffers. When they are employed to control the pH of a solution (such as in a microbial growth medium), a sodium or potassium salt is commonly used and the concentrations are usually high enough for the Henderson-Hasselbalch equation to yield adequate results.

Exact solution

An alternative treatment of exact buffer calculations can be found on John Denker's pH vs Concentration page. On this same page, he comments on the perils of depending on standard algebraic formulas for solving cubic equations, which can give wrong or even absurd answers owing to roundoff errors.

In this section, we will develop an exact analytical treatment of weak acid-salt solutions, and show how the H^–H equation arises as an approximation.

A typical buffer system is formed by adding a quantity of strong base such as sodium hydroxide to a solution of a weak acid HA. Alternatively, the same system can be made by combining appropriate amounts of a weak acid and its salt NaA. A system of this kind can be treated in much the same way as a weak acid, but now with the parameter C_b in addition to C_a.

Species	Na+, A–, HA, H2O, H+, OH–
Equilibria	[H+][OH–] = Kw(5-1) (5-2 )
Mass balance	Ca  + Cb = [HA] + [A–](5-3) Cb = [Na+](5-4)
Charge balance	[Na+] + [H+] = [OH–] + [A–](5-5)

Substituting Eq 5-4 into 5-5 yields an expression for [A^–]:

[A^–] = C_b + [H⁺] – [OH^–](5-6)

Inserting this into Eq 5-3 and solving for [HA] yields

[HA] = C_b + [H⁺] – [OH^–](5-7)

Finally, we substitute these last two expressions into the equilibrium constant (5-2):

(5-8)

which becomes cubic in [H⁺]  when [OH^–] is replaced by (K_w / [H⁺]).

[H⁺]³ +(C_b +K_a)[H⁺]² – (K_w + C_aK_a) [H⁺] – K_aK_w = 0 (5-8a )

Approximations

In almost all practical cases it is possible to make simplifying assumptions. Thus if the solution is known to be acidic or alkaline, then the [OH^–] or [H⁺] terms in Eq 5-8 can be neglected. In acidic solutions, for example, Eq 5-8 becomes

(5-9)

which can be rearranged into a quadratic in standard polynomial form:

H² + (C_b + C_a) H⁺ – K_a C_a = 0(5-10)

If the concentrations C_a and C_b are sufficiently large, it may be possible to neglect the [H⁺] terms entirely, leading to the commonly-seen Henderson-Hasselbalch equation*.

(5-11)Must know this!

It's important to bear in mind that the H-H equation is an "approximation of an approximation" that is generally valid only for combinations of K_a and concentrations that fall within the colored portion of this plot.

Most buffer solutions tend to be fairly concentrated, with C_a and C_b typically around .01 - 0.1 M.

For more dilute buffers and larger K_a's that bring you near the boundary of the colored area, it is safer to start with 5-9.