Fractional Composition Diagrams

Description

A fractional composition (α) diagram for an acid-base system shows the relative amount of each form of the acid or base at a given pH. For a monoprotic acid, only two forms exist (HA and A−) so there are two alpha expressions:

α(HA) = [HA] / ( [HA] + [A−] )

α(A−) = [A−] / ( [HA] + [A−] )

For a diprotic acid there are now three forms (H2A, HA−, and A2−) and three alpha expressions:

α(H2A) = [H2A] / ( [H2A] + [HA−] + [A2−] )

α(HA−) = [HA−] / ( [H2A] + [HA−] + [A2−] )

α(A2−) = [A2−] / ( [H2A] + [HA−] + [A2−] )

(Note that at any pH the sum of all the α values must equal 1.)

Values of α at any pH can be easily calculated from the equilibrium constant(s). For a monoprotic system:

α(HA) = [H+]  /  ( [H+] + Ka1 )

α(A−) = Ka1  /  ( [H+] + Ka1 )

For a diprotic system:

α(H2A) = [H+]2  /  ( [H+]2 + Ka1[H+] + Ka1Ka2 )

α(HA−) = Ka1[H+]  /  ( [H+]2 + Ka1[H+] + Ka1Ka2 )

α(A2−) = Ka1Ka2  /  ( [H+]2 + Ka1[H+] + Ka1Ka2 )

Note that the numerator in each successive alpha expression is simply the next term in the denominator. A general form of the denominator (D) for an n-protic system can be written as:

D = [H+]n + Ka1[H+]n−1 + Ka1Ka2[H+]n−2 + Ka1Ka2Ka3[H+]n−3 +  ···  +  Ka1Ka2Ka3…Kan

References:

1. Harris, D. C.; Lucy, C. A. Quantitative Chemical Analysis; W. H. Freeman, 2019; pp 230-232.

Exercises and questions

Question 1: Monoprotic acid analysis.

Acetic acid (CH₃COOH) has a pKa of 4.76. Using the simulator:

• Set the acid type to "Monoprotic (HA)" and pKa1 to 4.76.

• Set the pH range from 2 to 8.

Answer the following: 

1. At what pH are the concentrations of HA and A⁻ equal? Explain why this occurs. 

2. At pH 3.76, calculate the ratio of [HA]/[A⁻]. Verify your answer using the simulator. 

3. If the pH is increased by 2 units from the equivalence point, what happens to the fraction of A⁻? Why?

Question 2: Buffer region investigation for a weak acid.

• • • Set the acid type to "Monoprotic (HA)" and pKa1 to 5.5

    • Set the pH range from 3.5 to 7.5

Answer the following:

1. In what pH range would this acid be most effective as a buffer? Justify your answer using the diagram. 

2. What are the fractions of HA and A⁻ at pH 5.0? At pH 6.0? 

3. If you needed to prepare a buffer at pH 5.5, what ratio of acid to conjugate base would you need? Explain how the diagram helps determine this.

Question 3: Diprotic Acid Behavior

Carbonic acid (H₂CO₃) has pKa values of 6.35 and 10.33:

• • • Set the acid type to "Diprotic (H₂A)"

    • Set pKa1 to 6.35 and pKa2 to 10.33

    • Set the pH range from 5 to 12

Answer the following:

1. At what pH does the intermediate species (HCO₃⁻) reach its maximum concentration? What is this maximum fraction? 

2. Identify the two pH values where all three species (H₂CO₃, HCO₃⁻, and CO₃²⁻) exist in significant amounts. 

3. At what pH range would you expect bicarbonate (HCO₃⁻) to be the predominant species (>80%)? Why is this important in biological systems?

Question 4: Triprotic Acid Analysis

Phosphoric acid (H₃PO₄) has pKa values of 2.15, 7.20, and 12.35:

• • • Set the acid type to "Triprotic (H₃A)"

    • Set pKa1 to 2.15, pKa2 to 7.20, and pKa3 to 12.35

    • Set the pH range from 0 to 14

Answer the following:

1. Which phosphate species predominates at physiological pH (7.4)? What fraction of each species exists at this pH?

2. Identify the pH values where adjacent species have equal concentrations.

3. If the pH increases from 7 to 9, explain how the distribution of the four phosphate species changes and why this is significant.

Question 5: Comparative Analysis

Set up each acid below in the simulator and compare their behaviors:

System 1: Weak acid with pKa = 4.0 

System 2: Weak acid with pKa = 7.0

Answer the following: 

1. How does the position of the curves shift between these two systems? Why does this occur? 

2. At pH 5.5, compare the fraction of dissociated species (A⁻) for both acids. Which acid is more dissociated at this pH and why? 

3. For each acid, at what pH is exactly 90% of the acid in the deprotonated form (A⁻)? Explain the relationship between this pH value and the pKa.

4. Download the CSV data for both systems and calculate the difference in fraction of HA between the two acids at various pH values. At what pH is this difference greatest?