## Acids Iteration Pattern

**Input Data**

Please keep in mind that this tool considers invalid any input that is not a positive number.**pks value**

Enter solvent dissociation constant, pks. If the solvent is water pks = pkw. If in addition, activity effects are neglected and the temperature is 25^{o}Celsius, pks = pkw = 14. If working at other conditions, use the corresponding pks value.**Ca and pka values**

Enter Ca and pka values in the textarea, in one line, and separated by commas (try example). For a strong acid, H_{d}X, enter the product of d*Ca, where for instance d = 1 for HCl, d = 2 for H_{2}SO_{4}, and so forth. For a weak acid, d is calculated by the tool. In general, d is the deprotonation degree of an acid.- Submit or reset form as needed.

- This tool uses the following iteration pattern to find the pH of an acid solution regardless of its dissociation strength.
[H

^{+}] = (ks + d*Ca*[H^{+}])^{1/2}(1)Defining

*x*= [H^{+}], we can see that (1) is of the form*x = f(x)*(2)Numerical Dynamics Theory (Devaney, 1989a, 1989b, 1992), states that (2) describes an iteration pattern that converges to attractive fixed points when |df(x)/dx| < 1.

Thus, (1) should converge for any number of mixed acids, and positive input values of Ca, pks, pka's, and regardless of the initial guess value of [H

^{+}].In general for a combination of j = 1, 2, 3,...m acids

[H

^{+}] = (ks + (Σd_{j}*Ca_{j})*[H^{+}])^{1/2}(3)where d

_{j}is the deprotonation degree of acid j. For a strong monoprotic acid such as HCl, d_{j}= 1, whereas for a strong diprotic acid such as H_{2}SO_{4}, d_{j}= 2.By contrast, for a weak acid, d

_{j}is a function of its alpha fractionsd

_{j}= α_{1,j}+ 2*α_{2,j}+ 3*α_{3,j}+ ... n*α_{n,j}(4)where the α's are the dissociation fractions of acid j (Freiser, 1992).

The proposed pattern (1) can be easily simplified: If only one acid is present, j = 1 so

[H

^{+}] = (ks + d*Ca*[H^{+}])^{1/2}(5)If this is a weak acid and monoprotic, d = α

_{1}so[H

^{+}] = (ks + α_{1}*Ca*[H^{+}])^{1/2}(6)Finally, if the solvent is water, pks = pkw, hence ks = kw = 10

^{-pkw}[H

^{+}] = (kw + α_{1}*Ca*[H^{+}])^{1/2}(7) - The guessed value is set to [H
^{+}] = 10^{-pks/2}, corresponding to the hydrogen ion concentration of the pure solvent. So if the solvent is water and pks = pkw = 14, the iterations start at [H^{+}] = 1e-7 and stop when the relative error between iterates is less than 1 ppt (one part per thousand). To improve output readability, [H^{+}] results are converted to pH values.

- Data miners, computational chemists, chemical engineers, chemists, and chemistry teachers and their students.

- What is the pH of a 0.01 M aqueous solution of
- aspartic acid (pka
_{1}= 1.990, pka_{2}= 3.900, pka_{3}= 10.002)? - ethylenediamine diacetic acid (pka
_{1}= 1.66, pka_{2}= 2.37, pka_{3}= 6.53, pka_{4}= 9.59)?

- aspartic acid (pka
- In the previous exercise, sort the dissociated acid species by their α fractions. Which are the more and least abundant species and why?
- Compare the deprotonation degree between carbonic acid (pka
_{1}= 6.37, pka_{2}= 10.32) and a hypothetical acid H_{2}A (pka_{1}= 2.00, pka_{2}= 3.00). Assume that both are 0.01 M. What is the chemical significance of the results? - Show that the absolute value of the relative error in [H
^{+}] is Δ[H^{+}]/[H^{+}] = 2.303*ΔpH.

- Devaney, R. L. (1989a). Chaos, Fractals, and Dynamical Systems: Computer Experiments in Mathematics. Addison Wesley, New York.
- Devaney, R. L. (1989b). An Introduction to Dynamical Systems. Addison Wesley, New York.
- Devaney, R. L. (1992). A First Course in Chaotic Dyamical Systems. Addison Wesley, New York.
- Freiser, H. (1992). Concepts & Calculations in Analytical Chemistry: A Spreadsheet Approach. Chapter 4. CRC Press, Boca Raton.

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