Chem_{1} General Chemistry Virtual Textbook → Acid-base calculations → Systematic treatment
Systematic Treatment of Acid-Base systems
Exact calculations and their approximations
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The material in this unit is mainly for more advanced courses in analytical and aquatic chemistry. It is not ordinarily presented (or needed) in first-year General Chemistry courses. But if you're curious...
The methods for dealing with acid-base equilibria that we developed in the earlier units of this series are widely used in ordinary practice. Although many of these involve approximations of various kinds, the results are usually good enough for most purposes.
Sometimes, however — for example, in problems involving very dilute solutions, the approximations break down, often because they ignore the small quantities of H^{+} and OH^{–} ions always present in pure water.
In this unit, we look at exact, or "comprehensive" treatment of some of the more common kinds of acid-base equilibria problems.
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 H_{3}O^{+}, 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:
1. The dissociation equilibrium of water must always be satisfied:
[H^{+}][OH^{–}] = K_{w} (1-1)
2. The undissociated acid and its conjugate base must be in mass balance. The actual concentrations of the acid and its conjugate base can depend on a number of factors, but their sum must be constant, and equal to the “nominal concentration”, which we designate here as C_{a}. For a solution of HCl, this equation would be
[HCl] + [Cl^{–}] = C_{a} (1-2)
Because a strong acid is by definition completely dissociated, we can neglect the first term and write the mass balance condition as
[Cl^{–}] = C_{a} (1-3)
3. In any ionic solution, the sum of the positive and negative electric charges must be zero; in other words, all solutions are electrically neutral. This is known as the electroneutrality principle.
[H^{+}] = [OH^{–}] + [Cl^{–}] (1-4)
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^{–}].
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 HClO_{4} 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 HClO_{4} molecules in the solution. What has happened is that about 20% of the H_{3}O^{+} and ClO_{4}^{–} 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.
These very high activity coefficients also explain another phenomenon: why you can detect the odor of HCl over a concentrated hydrochloric acid solution even though this acid is supposedly "100% dissociated".
At these high concentrations, a pair of "dissociated" ions H^{+} and Cl^{–} will occasionally find themselve so close together that they may momentarily act as an HCl unit; some of these may escape as HCl(g) before thermal motions break them up again. Under these conditons, “dissociation” begins to lose its meaning so that in effect, dissociation is no longer complete. Although the concentration of HCl(aq) will always be very small, its own activity coefficient can be as great as 2000, which means that its escaping tendency from the solution is extremely high, so that the presence of even a tiny amount is very noticeable.
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−.
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^{–}] = K_{w}(2-1) (2-2) |
Mass balance | C_{a} = [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)
The latter equation is simplified by multiplying out and replacing [H^{+}][OH^{–}] with K_{w}. We then get rid of the [OH^{–}] term by replacing it with K_{w}/[H^{+}]
[H^{+}] C_{b} + [H^{+}]^{2} – [H^{+}][OH^{–}] = K_{a} C_{a} – K_{a} [H^{+}] + K_{a} [OH^{–}]
[H^{+}]^{2} C_{b} + [H^{+}]^{3} – [H^{+}] K_{w} = K_{a} C_{a} – K_{a} [H^{+}] + ((K_{a} K_{w}) / [H^{+}])
Rearranged into standard polynomial form, this becomes
[H^{+}]^{3} + K_{a}[H^{+}]^{2} – (K_{w} + C_{a}K_{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.
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_{a}C_{a} = 0 (2-7)
and thus, from the quadratic formula,
(2-8)
Calculate the pH of a 0.0010 M solution of acetic acid, K_{a} = 1.74E–5.
Solution: First approximation: [H^{+}] = (1.0E–3 × 1.74E–5)^{½} = (1.74E–8)^{½} = 1.3E–4 M. Applying the "5% test", 1.3E–4 / 1.0E–3 = 0.13, which exceeds 0.05, so we must solve the quadratic (2-7) with factors {1, 1.74E–5, –1.74E–8}. Entering these into a quadratic solver, we obtain the roots +1.2E–4 and –1.4E-4. Taking the positive root, we have pH = – log 1.2E–4 = 3.9
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_{a}C_{a})^{½} (2-10)
Calculate the pH and percent ionization of 0.10 M acetic acid "HAc" (CH_{3}COOH),
K_{a} = 1.74E–5.
Solution: It is usually best to start by using Eq 2-9 as a first approximation:
[H^{+}] = (.10 × 1.74E–5)^{½} = (1.74E–6)^{½} = 1.3E–3 M. This approximation is generally considered valid if [H^{+}] is less than 5% of C_{a}; in this case, [H^{+}]/C_{a} = 0.013, which is smaller than 0.05 and thus within the limit. This same quantity also corresponds to the ionization fraction, so the percent ionization is 1.3%.
The pH of the solution is –log 1.2E-3 = 2.9
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.
Boric acid, B(OH)_{3} ("H_{3}BO_{3}") is a weak acid found in the ocean and in some natural waters. As with many boron compounds, there is some question about its true nature, but for most practical purposes it can be considered to be monoprotic with K_{a} = 7.3E–10:
Bi(OH)_{3} + 2 H_{2}O → Bi(OH)_{4}^{–} + H_{3}O^{+}
Find the [H^{+}] and pH of a .00050 M solution of boric acid in pure water.
Solution: Because this acid is quite weak and its concentration low, we will use the quadratic form Eq 2-7. With the coefficients {1, 7.3E–10, –3.75E–13, –7.3E–24}, this yields the positive root 6.12E–7, corresponding to pH = 6.21. Notice that this is only six times the concentration of H^{+} present in pure water!
It is instructive to compare this result with what the quadratic approximation would yield. The using this quadratic solver, the coefficients {1, 7.3E–10, –3.65E–13} yield [H^{+}] = 6.04E–7, pH = 6.22.
The weak bases most commonly encountered are:
anions A^{–} of weak acids: | A^{–} + H_{2}O → HA + OH^{–} CO_{3}^{2–} + H_{2}O → HCO_{3}^{–} + OH^{–} |
ammonia | NH_{3} + H_{2}O → NH_{4}^{+} + OH^{–} |
amines, e.g. methylamine | CH_{3}NH_{2} + H_{2}O → CH_{3}NH_{3}^{+}+ H_{2}O |
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, H_{2}O, H^{+}, OH^{–} |
Equilibria | [H^{+}][OH^{–}] = K_{w}(2-13) (2-14) |
Mass balance | C_{b} = [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.
Calculate the pH and the concentrations of all species in a 0.01 M solution of methylamine,
CH_{3}NH_{2} (Kb = 4.2E–4).
Solution: We begin by using the simplest approximation Eq 2-14:
[OH^{–}] = (Kb C_{b})^{½}- = 4.2E-4 x 10^{–2} = 2.1E–3
To see if this approximation is justified, we apply a criterion similar to what we used for a weak acid: [OH^{–}] must not exceed 5% of C_{b}. In this case,
[OH^{–}] / C_{b} = (2.1E-3) / 10^{–2} = 0.21, so we must use the quadratic form Eq 2-12, with coefficients {1, 4.2E–4, –4.2E–6}. This yields the positive root 1.9E–3 which corresponds to [OH^{–}]. [H^{+}] = K_{w}/[OH^{–}] = 1E–14 / 1.9E–3 = 5.3E–12,
pH = –log 5.3E–12
= 11.3. From the charge balance equation, solve for
[CH_{3}NH_{2}] = [OH^{–}] – [H^{+}] ≈ [OH^{–}] = 5.3E–12 M. For the concentration of the acid form (methylaminium ion CH_{3}NH_{3}^{+}), use the mass balance equation:
[CH_{3}NH_{3}^{+}] = C_{b} – [CH_{3}NH_{2}] = 0.01 – .0019 =0.0081 M.
Several methods have been published for calculating the hydrogen ion concentration in solutions containing an arbitrary number of acids and bases. These generally involve iterative calculations carried out by a computer. See, for example, J. Chem. Education 67(6) 501-503 (1990) and 67(12) 1036-1037 (1990).
Many practical problems relating to environmental and physiological chemistry involve solutions containing more than one acid.
In this section, we will restrict ourselves to a much simpler case of two acids, with a view toward showing the general method of approaching such problems by starting with charge- and mass-balance equations and making simplifying assumptions when justified.
In general, the hydrogen ions produced by the stronger acid will tend to suppress dissociation of the weaker one, and both will tend to suppress the dissociation of water, thus reducing the sources of H^{+} that must be dealt with.
Consider a mixture of two weak acids HX and HY; their respective nominal concentrations and equilibrium constants are denoted by C_{x} , C_{y} , K_{x} and K_{y} ,
Starting with the charge balance expression
[H^{+}] = [X^{–}] + [Y^{–}] + [OH^{–}](3-1)
We use the equilibrium constants to replace the conjugate base concentrations with expressions of the form
(3-2)
to yield
(3-3)
If neither acid is very strong or very dilute, we can replace equilibrium concentrations with nominal concentrations:
(3-4)
Estimate the pH of a solution that is 0.10M in acetic acid (K_{a} = 1.8E–5) and .01M in formic acid (K_{a} = 1.7E–4).
Solution: Because K_{w} is negligible compared to the C_{a}K_{a} products, we can simplify (3-4):
[H^{+}] = √(1.8E–6 + 1.7E-6) = .0019
Which corresponds to a pH of – log .0019 = 2.7
Note that the pH of each acid separately at its specified concentration would be around 2.8.
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 H_{3}PO_{4}, 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.
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^{–}] = K_{w}(4-1) (4-2) (4-3) |
Mass balance | C_{a} = [H_{2}A] + [HA^{–}] + [A^{2–}](4-4) |
Charge balance | [H^{+}] = [OH^{–}] + [HA^{–}] + 2 [A^{2–}](4-5) (It takes 2 moles of H^{+} to balance the charge of 1 mole of A^{2–}) |
Solving these five equations simultaneously (don't bother!) for K_{1} 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 K_{2} << [H^{+}], then a further simplification can be made that removes K_{2} 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^{+}] ≈ (K_{a} C_{a})^{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.
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, H_{2}O, H^{+}, OH^{–} |
Equilibria | [H^{+}][OH^{–}] = K_{w}(5-1) (5-2 ) |
Mass balance | C_{a} + C_{b} = [HA] + [A^{–}](5-3) C_{b} = [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_{a}K_{a}) [H^{+}] – K_{a}K_{w} = 0 (5-8a )
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:
H^{2} + (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.
Chlorous acid HClO_{2} has a pK_{a} of 1.94. Calculate the pH of a solution made by adding .01 M/L of sodium hydroxide to a .02 M/L solution of chloric acid.
Solution: In the resulting solution, C_{a} = C_{b} = .01M. On the plots shown above, the intersection of the log C_{a} = –2 line with the plot for pK_{a} = 2 falls near the left boundary of the colored area, so we will use the quadratic form (5-10).
Substitution in Eq. 5-10 yields H^{+} + .02 H^{+} – (10^{–1.9} x 10^{–2}) = 0. Substituting
the coefficients {1 .02 –1E–3.9} into an online quadratic solver yields a positive root .0047 = [H^{+}], corresponding to pH = 2.3.
Note: Using the H-H equation (5-11) would give pH = pK_{a} = 1.9.