Article pubs.acs.org/jchemeduc
Using the Logarithmic Concentration Diagram, Log C, To Teach Acid−Base Equilibrium Jeffrey Kovac* Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600, United States ABSTRACT: Acid−base equilibrium is one of the most important and most challenging topics in a typical general chemistry course. This article introduces an alternative to the algebraic approach generally used in textbooks, the graphical log C method. Log C diagrams provide conceptual insight into the behavior of aqueous acid−base systems and allow students to solve complicated problems using simple geometry and algebra. Examples of the use of log C diagrams are given and the advantages and limitations of the technique discussed.
KEYWORDS: First-Year Undergraduate/General, Physical Chemistry, Problem Solving/Decision Making, Acids/Bases, Equilibrium
A
only tractable for very simple, usually monoprotic, systems. More complicated systems must be solved numerically providing no conceptual insight into the equilibrium behavior in the solution. There is an alternative graphical approach that provides a conceptual understanding of aqueous equilibria and allows the solution of difficult problems: the logarithmic concentration or log C diagram. The log C diagram was proposed by the Swedish chemist Niels Bjerrum in 1914. It is fairly widely used by those who study natural water systems but is relatively unknown in the chemistry community. Few general chemistry textbooks introduce this approach, but there are two more-specialized books published in the 1960s that provide a detailed exposition of the method.1 Log C diagrams are better known in the analytical chemistry community, but they tend to be used as illustrations and not as a calculational method, although the recent analytical chemistry text by Christie G. Enke is an exception.2 The primary advantage of the log C method is that once it is learned, it can easily be applied to both simple and complex systems including open systems such as natural bodies of water in contact with the atmosphere. The log C diagram shows the concentrations of all the species in the system so it is easy to see which are important and which can be neglected. Most problems can be solved using simple geometry, with no complicated algebra. Another advantage of the log C method is that the diagram can be used to construct titration curves, a process that is algebraically tedious.3 This article outlines the basics of the log C approach and discusses its strengths and limitations.4 For simplicity, I have confined the discussion to monoprotic acids. Further details
cid−base equilibrium is one of the most important and most difficult topics in a typical general chemistry course. Aqueous solutions containing weak acids and bases are encountered in many contexts, so a thorough qualitative and quantitative understanding of these systems is essential for many science majors. Typically, acid−base equilibrium problems are solved using an algebraic approach beginning with the fundamental equilibrium constant expression for the weak acid HA with initial (or analytical) concentration Co: HA ⇌ H+ + A− +
(1)
−
K a = [H ][A ]/[HA]
(2)
In solution, there are two conservation laws, mass balance
Co = [HA] + [A−]
(3)
and charge balance [H+] = [A−] + [OH−]
(4)
−
In the general case, [OH ] must be considered because of the autoionization of water. Because there are four variables and only three equations, it is necessary to use the equilibrium expression for the autoionization of water K w = [H+][OH−]
(5)
The general solution of eqs 2−5, when the concentrations of all four species are significant, is difficult. However, in most cases [OH−] ≪ [A−], which simplifies the algebraic problem to the solution of a quadratic equation. One pedagogical problem with the algebraic approach is that the reason that [OH−] is negligible may not be obvious to the beginning student so it becomes something to be memorized and applied blindly. A second problem is that the algebra is © 2012 American Chemical Society and Division of Chemical Education, Inc.
Published: April 18, 2012 905
dx.doi.org/10.1021/ed200675r | J. Chem. Educ. 2012, 89, 905−909
Journal of Chemical Education
Article
and extensions to more complex systems can be found in the works cited in ref 1 as well as ref 5.
■
DRAWING THE Log C DIAGRAM The log C diagram displays the concentrations of all the species in solution for particular values of the analytical concentration of the acid or base, Co, and the equilibrium constant, Ka. The shape of the graph is always the same; the positions of the various lines, however, will move depending on the values of Co and Ka, which define the system point. The method is introduced through a simple example. In an aqueous system, the autoionization of water, described by eq 5. is always present. Taking the common log of both sides of eq 5, log K w = log[H+] + log[OH−]
(6)
with the usual definition, pA = −log A, eq 7 is obtained pK w = pH + pOH
Figure 2. Log C diagram for 0.01 M acetic acid in water, including the autoionization of water. The system point is at (pKa = 4.75, log Co = −2).
(7)
The log C diagram plots the concentrations of all the species in the solution as a function of pH. For [H+] and [OH−] at 25 °C, the graph consists of two straight lines intersecting at the point pH = 7, log C = −7 as shown in Figure 1. Because of the
concentration of 0.01 M. Figure 2 also includes the lines for [H+] and [OH−]. The power of the log C diagram is that the shape is always the same. Once the system point is known, the diagram can be constructed immediately. The horizontal lines are located by the value of log C o and the two lines of slope ±1 are drawn at 45° from the system point. All the straight lines end at a point ±1 log units from the system point and are joined by curved line segments that intersect 0.3 log units below the system point. For most purposes, the curved portions can be hand sketched. (A French curve is helpful.) The log C diagram contains all the information in the relevant equilibrium constant expressions (Ka and Kw) and the mass balance. The only other constraint is the charge balance, which can be used to find the pH of the solution. For this system, the charge balance is given by eq 4. To apply this constraint, the [H+] line is followed beginning at pH = 0 until it crosses the line representing either [A−] or [OH−]. As can be seen in Figure 2, the intersection with the [A−] line, usually called the dominant intersection, occurs between pH 3 and 4. It is clear that the [OH−] concentration is many orders of magnitude smaller and can be safely neglected. At any pH, the various concentrations can be determined by drawing a vertical line and noting where the concentrations intersect that line. The pH can be read from the graph to a precision of about one decimal place, but it can also be easily calculated as can be seen in Figure 3, which shows the region around the intersection. For simplicity, the curved lines representing [HA] and [A−] near the system point have been replaced by the extrapolated straight lines. It is clear from Figure 3 that the [H+] and [A−] lines form an isosceles triangle with the [HA] line as its base. The vertices are at pH = 2 and pH = 4.75, the system point, so the dominant intersection is the midpoint, which in this case is (4.75 + 2.00)/ 2 = 3.375. This is exactly the value of the pH that is obtained using the algebraic approach. One of the limitations of the log C approach is that the simple calculation only works when the dominant intersection occurs in the linear region. If the dominant intersection is close enough to the system point that the lines are curved, one must either find the pH graphically or use an algebraic method. If a precision of ±0.1 pH unit is sufficient, it is easy to use a geometrical analysis to find the intersection in the curved region.5
Figure 1. Log C diagram for the autoionization of water at 25 °C.
autoionization of water, these lines will be fixed on all log C diagrams, although the intersection point will depend on the temperature because Kw is temperature dependent. Using the mass and charge balance conditions, eq 2 can be rewritten in two different ways [HA] = [H+]Co/(K a + [H+])
(8)
and [A−] = K aCo/(K a + [H+])
(9)
Using these two equations, log C can be plotted versus pH for both [HA] and [A−]. The overall shape of the plots can be seen by looking at the limiting behaviors. When Ka ≫ [H+] or pH ≫ pKa, the graph of log [HA] versus pH will be linear with a slope of −1, whereas the graph of log [A−] will be constant and equal to log Co. In the other limit, pH ≪ pKa, log [HA] is constant and log [A−] is linear with a slope of +1. The two 45° lines, if extended, would intersect at what is called the system point (pH, log Co). Within ±1 pH unit of the system point, the lines are curved and intersect at a point where pH = pKa and log [HA] = log [A−] = log Co − log 2. This is illustrated in Figure 2 for an acetic acid solution (pKa = 4.75) with a 906
dx.doi.org/10.1021/ed200675r | J. Chem. Educ. 2012, 89, 905−909
Journal of Chemical Education
Article
change is in the proton condition because the two species present at the zero level are A− and H2O. This leads to the following proton condition [OH−] = [H+] + [HA]
Returning to Figure 2, the dominant intersection is found by following the [OH−] line from high pH until it intersects the [HA] line at a pH near 9. The simple geometric calculation gives the exact answer as pH = (12.0 + 4.75)/2 = 8.38.
■
“BACK OF THE ENVELOPE” CALCULATIONS In both examples, it was unnecessary to use a precisely drawn log C diagram. Knowing the system point allows one to sketch a rough diagram and then use simple geometry to calculate the pH of solution. This will be illustrated by solving a problem that is difficult to solve using the algebraic method, the pH of a mixture of an acid and a base. Problems similar to this one are rarely considered in a general chemistry course partly because they are a bit esoteric and partly because they are difficult to solve, but the log C method makes them almost trivial. Consider a solution containing 0.05 M acetic acid (HAc) and 0.150 M sodium cyanide (NaCN). The pKa of HCN is 9.31. To solve this algebraically, one would need two equilibrium constant expressions and several mass balance and charge balance equations. With the log C approach, the problem is easy to solve. The proton condition is shown in Figure 5 and displayed as
Figure 3. Expanded view of the dominant intersection taken from Figure 2 showing the geometric calculation of the pH; see text for details.
■
THE PROTON CONDITION Instead of the charge balance equation, an alternative, and simpler, method for finding the dominant intersection is to use what is called the proton condition, which can be used for any system including mixtures. Begin by identifying all species in the solution, including water, before the ionization of the protons. These species might include ions that result from the dissociation of a salt such as sodium acetate. This is called the zero level. Then, identify all species in the solution after ionization including the un-ionized species. This is called the final condition. Species in the zero level are connected with those in the final condition that differ by the loss or gain of one or more protons. This is illustrated for a monoprotic acid in water in Figure 4. The proton condition is
[HCN] + [H+] = [Ac−] + [OH−]
Figure 5. Proton condition for a mixture of sodium cyanide, NaCN, and acetic acid, HAc, in water. Figure 4. Proton condition for a weak acid in water. The zero level is all the species prior to ionization. The final condition is all ionized and un-ionized species. The arrow indicates the gain or loss of a proton.
gain of protons = loss of protons
The relevant lines from the log C diagram, which are determined from the proton condition, are shown in a simple pencil sketch in Figure 6A. The important part of the diagram is expanded in Figure 6B to show that the pH of the mixture can be calculated very simply. From the proton condition and the diagram, it is easy to see that the dominant intersection is where [HCN] = [Ac−]. Knowing the two concentrations and the pKa of HCN, the pH of the mixture can be calculated using simple geometry. The diagram makes clear that the [H+] and [OH−] concentrations are orders of magnitude smaller.
(10)
This is a proton conservation equation. In the case illustrated in Figure 4, the proton condition is [H+] = [A−] + [OH−]
■
To make the gain and loss of protons clearer, H3O+ is identified in Figure 4, but simplified to H+ in the expression for the proton condition.
SOLVING PROBLEMS FOR VARIABLES OTHER THAN pH So far, all the examples have involved calculating the pH given an initial concentration of acid or base. Once students become comfortable with the log C method, it can be used to solve other kinds of problems. For example, given that the pH of a sodium cyanide (NaCN) solution is 11.5, the initial
■
BASES AND BASIC SALTS A standard problem in acid−base equilibrium is to calculate the pH of a basic salt such as sodium acetate. The log C diagram for the conjugate base is the same as that for the acid. The only 907
dx.doi.org/10.1021/ed200675r | J. Chem. Educ. 2012, 89, 905−909
Journal of Chemical Education
Article
which can easily be solved to give log Co = −0.31 or Co = 0.490. This is just one example. Other kinds of problems can be solved using the log C method. In ref 5 the log C method is used to treat open systems including natural water systems.
■
LIMITATIONS OF THE METHOD A major limitation of the log C method is the treatment of buffer solutions. Buffer systems, which are nearly equimolar mixtures of a weak acid and the conjugate base, are very near the system point and in the curved region of the log C diagram. The simple calculations used to solve the problems in this article will not be sufficiently accurate in this region, although the log C diagram shows why the pH is relatively constant and also shows why the buffering action only occurs within one pH unit of the pKa. A carefully drawn diagram can be used to solve buffer problems, but it is probably better to use the Henderson−Hasselbach equation and solve the problem algebraically. As noted earlier, if the system being studied lies in the curved region near the system point, it may be necessary to use the algebraic method to obtain an accurate result for the pH or concentrations. At very low concentrations of weak acid or base where the H+ and OH− concentrations are comparable to the concentrations of HA and A− and the proton condition involves more than two terms, the simple geometric method of obtaining the pH will fail, but it is relatively easy to incorporate the additional term geometrically.5 This is a complication that is unlikely to be covered in the typical general chemistry course.
Figure 6. Pencil sketch of the back of the envelope calculation of the pH of a mixture 0.05 M NaCN and 0.150 M HAc (acetic acid). (A) Sketch of the relevant concentrations and (B) expanded view of the region around the dominant intersection showing the method of calculation. The y axis is log C and the x axis is pH.
■
CONCLUSION The log C method is a simple and powerful method for understanding acid−base equilibria. It can also be applied to both solubility and oxidation−reduction equilibria. It provides a deep conceptual understanding of these systems and allows one to solve complex problems using elementary geometry and algebra. Students certainly need to understand the algebraic approach to equilibrium so they can apply it in situations, such as buffers, where the log C method is not applicable, but for most problems, the log C method is superior. This article has provided an introduction to the use of these diagrams. More details can be found in the references.
■
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected].
Figure 7. Back of the envelope log C diagram to determine the initial concentration of a NaCN solution with pH = 11.5.
Notes
The authors declare no competing financial interest.
■
concentration of NaCN can be calculated. The simplified log C diagram is shown in Figure 7. The proton condition can easily be determined
ACKNOWLEDGMENTS I am grateful to Fred Tabbutt who introduced me to the log C method long ago, then helped me relearn it by inviting me to work with him on Water and Sun.
[HCN] + [H+] = [OH−]
■
Because this is a base hydrolysis problem, the dominant intersection will be the intersection between the [OH−] line and the [HCN] line. As can be seen in Figure 7, the [H+] is insignificant. The relevant lines and intersections are sketched in Figure 7. We know that the dominant intersection will be at pH = 11.5 and that the two intersecting lines have slopes of +1 and −1. Simple geometric considerations allow us to write the equation
REFERENCES
(1) Butler, J. N. Solubility and pH Calculations; Addison-Wesley: Reading, MA, 1964. Butler, J. N. Ionic Equilibrium, A Mathematical Approach; Addison-Wesley: Reading, MA, 1964. (2) Enke, C. G. The Art and Science of Chemical Analysis; J. Wiley: Hoboken, NJ, 2001. (3) The construction of titration curves from log C diagrams is explained in detail in Tabbutt, F. D. J. Chem. Educ. 1966, 43, 245. (4) I learned the log C method from Fred Tabbutt as a freshman at Reed College long ago (Tabbutt, F. D.; , Instant Equilibrium, privately
[(14 + log Co) + pK a]/2 = 11.5 908
dx.doi.org/10.1021/ed200675r | J. Chem. Educ. 2012, 89, 905−909
Journal of Chemical Education
Article
published, 1968) but had mostly forgotten it until recently when I relearned it in the course of collaborating on a textbook that uses environmental systems as a vehicle to teach general chemistry and uses log C diagrams extensively. I have been impressed by the power and simplicity of the approach and have begun to use it in my own teaching in parallel with the algebraic approach. (5) Tabbutt, F.; Kovac, J.; , Water and Sun, in preparation.
909
dx.doi.org/10.1021/ed200675r | J. Chem. Educ. 2012, 89, 905−909
