Acid-Base Systems

Solve the general equation for acid-base systems and their titrations.

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Try This Example

Va

Vb

pH

Parameters

Ca_i, pKa₁, pKa₂,.. pKa_n

Cb_i, pKb₁, pKb₂,.. pKb_m

Enter one set of comma-delimited parameters per line.
For strong acids and strong bases, enter C's, only.

Instructions

• This tool solves different types of acid-base problems, including pH calculations involving solutions, mixtures, and their titrations.
• You may also use the tool for cross-mapping the Va, Vb, and pH input fields. To do this, please complete two out of the three fields so the tool can solve for the one left empty.
• To empty a field, just double click it.
• To use the tool, you must include all appropriate parameters; i.e., nominal concentrations, C's, and pK values, where pK = -log(K) with the K's being stepwise equilibrium constants.

What is computed?

• This tool solves the general equation for acid-base systems where
  • Ca and Cb are the acids and bases nominal (analytical, total) concentrations in millimoles per milliliters (mmoles/mL, M).
  • Va and Vb are their nominal volumes in milliliters (mL), usually reported to the nearest 0.1 mL.
  • pH = -log([H⁺])
  • pKa = -log(Ka)
  • pKb = -log(Kb)

• Goals
  The tool was developed as general purpose calculator for addressing different types of acid-base problems and is not limited to the calculation of pH values from titration data. The current version assumes standard conditions of 25° C and zero ionic strength.

  The tool can also be used for cross-mapping

  • pH to Vb and vice versa
  • pH to Va and vice versa
  • Vb to Va and vice versa

  In general, cross-mapping several variables of a given function helps one to have a deeper understanding of the problem at hand even when in the process precision errors are introduced.

  Fortunately for the types of calculations we are dealing with, the error introduced is comparable to the one allowed in volumetric measurements obtained from titration data; i.e, to about one decimal place.

General Acid-Base Equation

• During the titration of an acid of concentration Ca and volume Va one adds a gradually increasing volume Vb of a base with concentration Cb while recording a signal, such as the pH or redox potential E, where pH = -log([H⁺]) (de Levie, 1993).

  If the titration involves aqueous solutions at 25° C with negligible ionic strength, we may replace activities with concentrations and assume that the water autodissociation constant is about Kw = 10^-14.00 = [H⁺][OH^-]. Thus taking logarithms, we may write pKw = pH + pOH = 14.00.

• For titrations involving acid-base solutions and their mixtures, the entire titration curve, or specific points from it, can be computed with a function of the general form

  (1)

  Similar expressions can be found in the literature (de Levie, 1993). By properly setting its terms, (1) can be applied to a diverse types of acid-base problems involving multiple acids, bases, and their salts, but not necessarily involving titrations. This is why we refer to (1) as a general equation for acid-base systems.

  In (1) Δ = [H⁺] - [OH^-] and

  (2)

  (3)

  (4)

  (5)

  where

  • in (2), F_adj is the dissociation function of acid j
  • in (3), α_j,i is the fraction (alpha fraction) of specie i from acid j
  • in (4), F_baj is the association function of base j
  • in (5), α_j,i is the fraction (alpha fraction) of specie i from base j

  and where the K_i's are stepwise equilibrium constants. Most reference texts on acid-base equilibria (Freiser, 1992; Guenther, 1975, Harvey, 2000; Lower, 1975) provide these or their logarithmic version, pK_i = -log(K_i).

  As with pH values, pK_i's should be stated to the same number of decimal places as significant figures are in the K_i's (Dill, 2008). For most titrimetric work, volumes are frequently stated to at least one decimal place while pH and pK values are stated to two decimal places.

pH Calculations Involving Titrations

• As mentioned, our tool solves different types of pH calculations problems involving, though not limited to, the titration of acid-base solutions and their mixtures, regardless of the dissociation strength of the species or number of hydrogen ions involved. As with bases, acids can be monoprotic or polyprotics, strong or weak.

  For instance, acids can be strong and of the HX type, like

  • HCl.

  or weak and of the H_nA type, like

  • H₃PO₄ (pKa₁ = 2.12, pKa₂ = 7.21, pKa₃ = 12.32) where
    F_ad = α₁ + 2α₂ + 3α₃
  • H₂SO₃ (pKa₁ = 1.90, pKa₂ = 7.21) where
    F_ad = α₁ + 2α₂
  • HF (pKa₁ = 3.17) where
    F_ad = α₁

  where the pKa_i's are given as documented in the literature (Ripin & Evans, 2005).

pH Calculations for Special Cases: Amino Acids and Sulfuric Acid

• Amino Acids
  One special case are amino acids. These building blocks of proteins contain amino groups (-NH₂) that accept protons, and carboxyl groups (-COOH) that lose protons.

  If both events occur simultaneously the net effect is a double ion or zwitterion. The pH at which the amino acid molecule carries no charge and thus does not migrate in an electric field is called the isoelectric point, pI, of the amino acid.

  If the neutral molecule is represented as HA^+/-, then the dissociation of HA^+/-in water can be described as forming A^- while its hydrolysis as producing H₂A⁺.

  By adjusting the conditions we can have a solution mainly of H₂A⁺. If this solution is titrated with a strong base, its titration curve will be similar to that of a polyprotic weak acid with two stepwise equilibrium constants, where for neutral side chain amino acids (Hunt & Spinney, 2006), pI = pH = (pKa₁ + pKa₂)/2.

  Such a titration curve has been described (Bacher, 2016) for the titration of 25.0 mL of 0.10 M glycine in the H₂A⁺ form with 0.10 M NaOH, where for

  • Vb = 0.0 mL, pH < 2.00
  • Vb = 12.50 mL, pH = pKa₁.
  • Vb = 25.0 mL, pH = pI.
  • Vb = 37.50 mL, pH = pKa₂.

  and where pKa₁ = 2.35 and pKa₂ = 9.78 (Blatchly, 2004).

  Those results can be easily recomputed with our tool, with a relative error of less than 4% for the pKa₁ determination and close to 0% for the pKa₂. Increasing the concentrations of the titrated and titrant solutions reduces this error.

• Sulfuric Acid (A Strong Polyprotic Acid)
  Another special case is sulfuric acid, H₂SO₄, a polyprotic acid of the H₂A type (pKa₁ = -3.00, pKa₂ = 1.99). Because its first dissociation constant is very large, its titration is very similar to that of a strong monoprotic acid like hydrochloric acid, HCl. However, its neutralization reaction is not 1:1, but 1:2 with an end point slightly above pH = 7.00.

  For instance, consider the titration of 50.0 mL 0.10 M H₂SO₄ with 0.10 M NaOH at 25° C and approximating activities with concentrations; i.e., no corrections for ionic strengths are considered. Assuming some Vb values, we may compute the corresponding equilibrium pH's with our tool. An example is given below.

  
  

  Table 1. Simulated data for 50.0 mL 0.10 M H₂SO₄ titrated with 0.10 M NaOH

  Vb, mL	0.0	50.0	100.0	150.0
  pH	0.96	1.74	7.32	12.40

  
  

  Table 1 results were obtained with the tool by leaving the pH field empty (since it is the missing term to be computed), completing all other form fields including the pKa's field, and then submitting the form.

  The table shows that at the begining of the titration Vb = 0.0 mL so the acid initial pH is about 0.96. In this example, Cb = Ca = 0.10 M so 100.0 mL of 0.10 M NaOH are needed to reach the titration end point which occurs around pH = 7.32.

  These results should agree with those obtained through other tools available online. Any discrepancy can be ascribed to the type of algorithm(s) utilized (i.e., residual analysis, successive approximation methods like Newton-Raphson, activity coefficient model corrections, pK reference values used, significant figures usage,..).

  Our tool implements a residual analysis algorithm which makes unnecessary the use of inversion techniques, succesive approximations, Newton-Raphson, or the evaluation of derivatives or high-degree polynomials.

pH Calculations not Involving Titrations

• The general equation (1) can be solved to compute the pH of systems not undergoing titrations. You just need to assume some experimental conditions and rearrange (1) as needed.

  Table 2 lists some settings to try. Our tool does any rearrangements in the background for you. For specific examples, see the Suggested Exercises section.

  
  

  Table 2. Tool Settings for pH Calculations not Involving Titrations

  Aqueous System	Volumes	Concentrations	pK's
  Acid(s) or Weak Base Salt(s)	Va > 0, Vb = 0	Ca > 0, Cb = 0	pKa's, as needed.
  Base(s) or Weak Acid Salt(s)	Va = 0, Vb > 0	Ca = 0, Cb > 0	pKb's, as needed.
  Acid-Base Salt Pair(s) or Acid-Base Conjugate Pair(s)	Va = Vb > 0	Ca = Cb > 0	pKa and PKb of each pair(s).

  
  

  If the order of the dissociation constants is pKa1 < pKa2

Who can use this tool?

• Lab techs as well as chemistry teachers and their students.

Suggested Exercises

• Warder Titration: 50 mL solution of 0.1 M NaOH and 0.1 M Na₂CO₃ is titrated with 0.10 M HCl. (H₂CO₃ pKa₁ = 6.35 and pKa₂ = 10.33). Compute the pH of the mixture after adding 0.0, 25.0, 50.0, 75.0, and 100.0 mL of HCl.
• Weak Acid + Strong Base Titration: 50.0 mL of 0.10 M weak acid solution, HA (pKa = 5.00), is titrated with 0.10 M NaOH. Compute the pH after adding 0.0, 25.0, 37.5, 50.0, and 75.0 mL of NaOH.
• Highly Diluted Solution: Calculate the pH of 10^-7 M HCl.
• Highly Diluted Solution: Calculate the pH of 10^-7 M NaOH.
• Weak Polyprotic Acid + Strong Base Titration: How much volume of 0.010 M NaOH must be added to 25.0 mL of 0.010 M H₂CO₃ (pKa₁ = 6.35, pKa₂ = 10.33) to obtain pH values equal to the acid pKa's? Compute these volumes to at least 0.1 mL.
• Weak Acid Solution: Calculate the pH of 10^-6 M hypochlorous acid (HOCl, pKa = 8.0).
• Weak Acids Mixture: Calculate the pH of a solution containing 0.1M acetic acid (pKa = 4.75) and 0.001 M chloroacetic acid (pKa = 2.85).
• Acid Salt: Calculate the pH of 0.0010 M sodium sulfate (SO₄^2-, pKb₁ = 17.00, pKb₂ = 12.01).
• Base Salt: Calculate the pH of 0.10 M ammonium chloride (NH₄⁺ pKa = 9.25).
• Salt Pair: Calculate the pH of 0.10 M ammonium acetate (ammonium ion pKa = 9.25, acetate ion pKb = 9.25).
• Amphoteric Salt: Calculate the pH of 0.10 M diammonium phosphate, also known as DPA (NH₄)₂HPO₄ (NH₄⁺ pKa = 9.25; H₃PO₄ pKa₁ = 2.12, pKa₂ = 7.21, pKa₃ = 12.32).
• Conjugate Pair: Calculate the pH of a solution containing 0.010 moles of ammonium chloride (NH₄⁺ pKa = 9.25) and 0.020 moles of ammonia (NH₃ pKb = 4.75) in 100 ml of solution.
• Mixture Preparation: Calculate the pH of 0.01M acetic acid (pKa = 4.75) solution and one that is 0.01M sodium acetate (pKb = 9.25). What would be the pH of the mixture if both solutions are mixed in equal amounts?
• Mixture Preparation: Calculate the pH of the mixture obtained by adding 30 mL of 0.20 M HCl to 20 mL of 0.15 M acetic acid (pKa = 4.75).
  Hint: Va = 30 mL + 20 mL = 50 mL.
• Mixture Preparation: Calculate the pH of the mixture obtained by adding 20 mL of 0.30 M NaH₂PO₄ to 20 mL of 0.30 M NaHS.
  Hint: Both are acid salts, but once mixed an acid-base reaction between both takes place. In addition, H₃PO₄ pKa₁ = 2.12, pKa₂ = 7.21, pKa₃ = 12.32 and H₂S pKa₁ = 7.00, pKa₂ = 19.00.

References

• Bacher, A. D. (2016). Titrimetric Determination of the Concentration and Acid Dissociation Constants of an Unknown Amino Acid Department of Chemistry and Biochemistry. University of California, Los Angeles.
• Blatchly, R. A. (2004). Amino Acids with Acidity Values. Chemistry Department Keene State College.
• de Levie, R. (1993). Explicit Expressions of the General Form of the Titration Curve in Terms of Concentration. Journal of Chemical Education, 70, 209-217.
• Dill, D. (2008). Significant Figures in Numerical Calculations. In Notes on General Chemistry. Department of Chemistry. Boston University.
• Freiser, H. (1992). Concepts & Calculations in Analytical Chemistry. CRC Press, Boca Raton, FL.
• Guenther, W. B. (1975). Chemical Equilibrium. Plenum Press, New York.
• Harvey, D. (2000). Modern Analytical Chemistry. McGraw-Hill, New York
• Coordination properties of cyclam (1,4,8,11-tetraazacyclotetradecane) endowed with two methylphosphonic acid pendant arms in the 1,4-positions
• Hunt, I., & Spinney, R. (2006). Isoelectric point, pI. Department of Chemistry, University of Calgary. In Francis Carey's Organic Chemistry, McGraw-Hill.
• Lower, S. K. (1975). Acid-Base Equilibria and Calculations. Simon Fraser University. York.
• Ripin, D. H., & Evans, D. A. (2005). pKa's of Inorganic and Oxo-Acids. Department of Chemistry and Chemical Biology, Harvard University.
• Wikipedia (2019). Acid Dissociation Constant.
• Aqueous Equilibrium Constants

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