Communication pubs.acs.org/jchemeduc
A Better Ion Fraction for a Neutral Aqueous Solution David W. Ball* Department of Chemistry, Cleveland State University, Cleveland, Ohio 44115, United States ABSTRACT: There is some minor confusion in some literature regarding the number of H+ and OH− ions in a neutral aqueous solution, presumably based on an incorrect application of [H+] = [OH−] = ∼1 × 10−7 M. Here, the method to determine the proper number of ions present in a neutral solution is demonstrated.
KEYWORDS: First-Year Undergraduate/General, High School/Introductory Chemistry, Physical Chemistry, Calculator-Based Learning, Misconceptions/Discrepant Events, Aqueous Solution Chemistry, Equilibrium, Water/Water Chemistry
I
n reading Dunn’s fascinating book Caveman Chemistry,1,2 there is an interesting phrase about the autoionization of water: “water...can be viewed as hydrogen hydroxide (HOH) and some small fraction (1 in 10,000,000) will ionize to form hydrogen (H+) and hydroxide (OH−) ions...” [1, p 221]. The number 10,000,000 should be recognized as 107, so 1 in 10,000,000 represents a fraction of 10−7, suggesting (but not proving, admittedly) that this statistic came from the wellknown 10−7 M concentration of H+ and OH− ions in neutral water. However, 1 in 10,000,000 is not equivalent of a concentration of 10−7 M. One wonders if the incorrect connection of 10−7 M to 1 in 10,000,000 molecules is made elsewhere. In the general chemistry textbook by Brown, Lemay, Bursten, and Murphy, the extent of ionization is given as “only about 2 out of every 109 molecules”,3 whereas in the OWL online Web-based learning system of Cengage Publishing company, Vining states that it is “approximately 2 out of every billion”.4 (A more global search of textbooks and other references was not performed; however, some general chemistry textbooks do not actually cite a specific number.) Although consistent with each other, the citation of actual numbers begs the question of exactly how this number is determined and whether one can be more exact than “only about” or “approximately” two ionizations per billion molecules.
H2O, the density of water at a given temperature, and the autoionization constant of H2O, KW, at the same temperature. Only one approximation need be made (and this can be skipped for general chemistry course): here the activities of H+ and OH−, aH+ and aOH−, are assumed to be numerically equivalent to their concentrations for such dilute solutions;5 that is, a H + = [H +]
As such, rather than using K w = (a H +)(aOH−)
to express the autoionization constant, the concentrations can be used K w = [H +][OH−]
which is the approximation usually made in entry-level courses. The temperature is assumed to be 25 °C, a common reference temperature for thermodynamic considerations. This assumption is important, because the experimental values (which are needed) for the density of water and Kw can vary significantly with temperature (vide infra). At 25 °C, the density of water is 997.0479 g/L.6 This can be combined with the molar mass of H2O, using the most recent atomic masses available from the International Union of Pure and Applied Chemistry, IUPAC,7 to get the number of moles of H2O molecules in 1 L:
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CALCULATIONS Actual data exist to several significant figures that allow the calculate of what fraction of ions exists in neutral water. The calculation is a good exercise in converting units and significant figures and would be suitable for students with a good understanding of general chemistry principles. It can be done in class individually or in groups or assigned as a homework problem. The data that are necessary are the molar mass of © 2012 American Chemical Society and Division of Chemical Education, Inc.
aOH− = [OH−]
molar mass of H2O = 2(1.00794) + 15.9994 = 18.0153 g/mol 997.0479 g ×
1 mol = 55.3445 mol for 1 L H2O 18.0153 g
Published: February 8, 2012 683
dx.doi.org/10.1021/ed200521v | J. Chem. Educ. 2012, 89, 683−684
Journal of Chemical Education
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Now Kw is needed. At 25 °C and atmospheric pressure, the pKw for H2O is 13.995.8 Because pKw = −log(Kw), upon rearranging
CONCLUSION The ultimate lesson presented here is that equilibrium constants are expressed in terms of concentrations or activities. If quantities in terms of amounts are desired, a bit of math is required. With some dimensional analysis, the amounts can be determined to the limit of the precision of experimental data.
K w = inv log( −pK w )
And thus,
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K w = inv log(− 13.995) = 1.01 × 10−14
It is this number that ultimately limits the significant figures of our final answer. Using the expression for Kw above and assuming that [H+] = [OH−] at equilibrium, [H+] is
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected].
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ACKNOWLEDGMENTS Thanks to Barry W. Hicks, a colleague at the U.S. Air Force Academy Department of Chemistry, for unwittingly inspiring this communication.
K w = [H +][OH−] = [H +]2 = 1.01 × 10−14
[H +] = 1.01 × 10−7 M
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This is not far from the textbook answer. For clarity, this is rewritten as
REFERENCES
(1) Dunn, K. M. Caveman Chemistry; Universal Publishers: Parkland, FL, 2003. (2) Matthews, M. S. J. Chem. Educ. 2001, 84, 490. This is a review of Dunn’s book Caveman Chemistry.1 (3) Brown, T. L.; Lemay Jr., H. E.; Bursten, B. E.; Murphy, C. J. Chemistry: The Central Science, 11th ed.; Pearson: Upper Saddle River, NJ, 2009. (4) Autoionization of Water. http://s-owl.cengage.com/ebooks/ vining_owlbook_prototype/ebook/ch16/Sect16-2-a.html (accessed Jan 2012). (5) Hawkes, S. J. J. Chem. Educ. 1995, 72, 799−802. (6) CRC Handbook of Chemistry and Physics, 70th ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 1990. (7) IUPAC Green Book (Quantities, Units and Symbols in Physical Chemistry), available online at http://www.iupac.org (accessed Jan 2012). (8) (a) International Association for the Properties of Water and Steam. Release on the Ionization Constant of H2O, 2007. Available at http://www.iapws.org (accessed Jan 2012). (b) Bandura, A. V.; Lvov, S. N. J. Phys. Chem. Ref. Data 2006, 35, 15. (9) Shoesmith, D. W.; Lee, W. Can. J. Chem. 1976, 54, 3553−3558.
[H +] = 1.01 × 10−7 mol H +/L solution
Ignoring the volume of the H+ ions and using the number of moles of H2O per liter of H2O as determined above, inverting so that the liter units cancel, gives: 1.01 × 10−7
Communication
mol H + 1L × L 55.3445 mol H2O
= 1.82 × 10−9 mol H +/mol H2O
Multiplying the numerator and denominator by 109, the final answer to three significant figures is 1.82 mol H +/109 mol H2O
or 1.82 H+ ions per 1,000,000,000 H2O molecules. Thus, water is ionized to the extent of 1.82 (or about 2) per billion at 25 °C, based on the data cited here. This value is consistent with the approximations cited by Brown et al.3 and Vining,4 but now its derivation has been established. Dunn’s statement in Caveman Chemistry is obviously incorrect.
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TEMPERATURE DEPENDENCE As hinted above, the values of density and pKw vary with temperature, sometimes significantly. Values at the appropriate temperature must be used if this calculation is to be demonstrated at other than 25.0 °C. To aid the instructors who may wish to ask this of their students, Table 1 lists some temperatures and the pKw Table 1. pKw Values, Densities (d), Degree of Ionization (DI) for Liquid H2O and D2O at Various Temperatures T/°C
pKw (H2O)
d (H2O)/ (g L−1)
DI (H+) (ppb)
pKw (D2O)
d (D2O)/ (g L−1)
DI (D+) (ppb)
0 25 50 75 100
14.946 13.995 13.264 12.696 12.252
999.57 997.05 988.07 974.89 958.38
0.61 1.82 4.25 8.29 14.1
14.957 14.184 13.571 13.115
1104.45 1095.70 1081.58 1063.46
0.60 1.55 3.03 5.22
and density values for H2O at those temperatures,6,8a along with the ppb(H+) as calculated by the author. To assess isotope effects, as similar data are available for D2O,6,9 Table 1 also lists pKw and density values for D2O as well. The trend in the ionizations of H2O and D2O can also be discussed in terms of their adherence to Le Châtelier’s principle. 684
dx.doi.org/10.1021/ed200521v | J. Chem. Educ. 2012, 89, 683−684
