Fractional Compositions and Concentrations: Videos & Practice Problems

Understanding fractional compositions in acid-base chemistry is crucial for determining the concentrations of weak acids and their conjugate bases at specific pH levels. For monoprotic acids, the equilibrium expression is given by the acid dissociation constant, Ka = [H⁺][HA] . At pH = pK_a, the concentrations of the acid and conjugate base are equal. In diprotic systems, multiple forms exist, requiring consideration of both K_a1 and K_a2 for accurate calculations of each species' fraction as pH changes.

Fractional Composition

Fractional compositions are essential for understanding the relationship between the amounts of acid and base in a solution at a specific pH. As pH decreases, the solution becomes more acidic due to an increase in the concentration of the weak acid, while an increase in pH indicates a more basic solution with a higher concentration of the conjugate base.

Weak acids do not fully ionize in solution, establishing an equilibrium between the undissociated acid (HA) and its ions, H⁺ and A^-. The equilibrium expression for this system is represented by the acid dissociation constant, K_a, which is defined as:

$K^a=\frac{\mathrm{[H⁺]}\mathrm{[A^-]}}{\mathrm{[HA]}}$

The formal concentration of the solution is the sum of the concentrations of the weak acid and its conjugate base. The fraction of acid molecules present can be denoted by the variable α. At the lowest pH, the solution consists entirely of the acid form (100% HA) and 0% of the conjugate base. As pH increases, the proportion of the conjugate base rises while that of the weak acid decreases. At the point where pH equals pK_a, the concentrations of the weak acid and conjugate base are equal, each constituting 50% of the solution.

This relationship is encapsulated in the Henderson-Hasselbalch equation:

$\mathrm{pH}=\mathrm{pK}+\log \left(\frac{\mathrm{[A^-]}}{\mathrm{[HA]}}\right)$

When pH equals pK_a, the logarithmic term becomes zero, confirming that the concentrations of the weak acid and conjugate base are equal. As pH continues to rise beyond this point, the concentration of the weak acid approaches zero, while the conjugate base approaches 100%.

To calculate the concentration of the weak acid at any pH, the following formula can be used:

$\mathrm{[HA]}=\frac{\mathrm{[H⁺]}}{}$

The fraction of the acid form can be expressed as:

$\alpha =\frac{\mathrm{[H⁺]}}{+}$

Conversely, the fraction of the conjugate base (A^-) is given by:

These equations allow for the determination of the fractions of both the weak acid and its conjugate base in solution, which together sum to 100%. Understanding these concepts is crucial for analyzing weak acid-base equilibria and their behavior in various chemical contexts.

Fractional Composition

In diprotic systems, the behavior of acids is influenced by their ability to donate two protons (H⁺ ions), leading to the existence of multiple forms based on pH levels. Initially, the system exists in its fully protonated acidic form, which contains both acidic hydrogens. Upon losing the first proton, the system transitions to an intermediate form, and finally, upon losing the second proton, it reaches the basic form. Each of these transitions is associated with distinct acid dissociation constants, denoted as K_a1 and K_a2.

The equilibrium expression for the first dissociation can be represented as:

$$ K_{a1} = \frac{[HA^-][H^+]}{[HA]} $$

Rearranging this expression allows us to calculate the concentration of the intermediate form (HA^-) as:

$$ [HA^-] = \frac{[HA] \cdot K_{a1}}{[H^+]} $$

Upon losing the second proton, the equilibrium expression for the second dissociation is given by:

$$ K_{a2} = \frac{[A^{2-}][H^+]}{[HA^-]} $$

From this, we can derive the concentration of the basic form (A^2-) and substitute the expression for HA^- into the equation, leading to a more complex relationship:

$$ [A^{2-}] = \frac{[HA] \cdot K_{a2} \cdot [H^+]}{[H^+]^2} $$

In a diprotic system, the fractions of the three major forms—acid (HA), intermediate (HA^-), and basic (A^2-)—can be expressed through specific equations that depend on the pH. As pH increases, the fraction of the acid form decreases while the fraction of the basic form increases. At the point where pH equals pK_a1, the concentrations of the acid and intermediate forms are equal. Similarly, at pH equal to pK_a2, the concentrations of the intermediate and basic forms are equal.

This relationship illustrates that the fractional composition of each form is contingent upon the pH of the solution. Higher pH values favor the predominance of basic forms, while lower pH values favor acidic forms. Understanding these dynamics is crucial for predicting the behavior of diprotic acids in various chemical environments.

Fractional Composition Calculations 1

In this scenario, we are analyzing a dibasic compound, which is a diprotic species, to determine the fraction of its acidic form at a specific pH of 9.0. The key to solving this problem lies in understanding the relationship between pH, the concentration of hydrogen ions, and the acid dissociation constants.

First, we calculate the concentration of hydrogen ions, denoted as [H⁺], using the formula:

[H⁺] = 10^-pH

For a pH of 9.0, this results in:

[H⁺] = 10^-9.00 = 1.0 x 10^-9 M.

Next, we need to find the acid dissociation constants (K_a1 and K_a2) from the given base dissociation constants (pK_b1 and pK_b2). The relationships between these constants are:

pK_a1 + pK_b2 = 14

pK_a2 + pK_b1 = 14

Substituting the provided values:

pK_a1 = 14 - pK_b2 = 14 - 8 = 6

pK_a2 = 14 - pK_b1 = 14 - 5 = 9

Now, we can convert these pK values to K values using the formula:

K_a1 = 10^-pK_a1 = 10^-6 M

K_a2 = 10^-pK_a2 = 10^-9 M

With these values, we can now calculate the fraction of the acidic form using the formula:

Fraction of acidic form = \(\frac{[H^+]^2}{[H^+]^2 + [H^+] \cdot K_{a1} + K_{a1} \cdot K_{a2}}\)

Substituting the known values:

Fraction of acidic form = \(\frac{(1.0 \times 10^{-9})^2}{(1.0 \times 10^{-9})^2 + (1.0 \times 10^{-9}) \cdot (1.0 \times 10^{-6}) + (1.0 \times 10^{-6}) \cdot (1.0 \times 10^{-9})}\)

This simplifies to:

Fraction of acidic form = \(\frac{1.0 \times 10^{-18}}{1.0 \times 10^{-18} + 2.0 \times 10^{-15}}\)

Calculating the denominator gives us approximately 2.001 x 10^-15, leading to:

Fraction of acidic form ≈ 0.00005.

This result indicates that at a pH of 9, which is significantly higher than the pK_a1 of 6, the concentration of the acidic form is quite low. This aligns with the understanding that as pH increases beyond the pK_a value, the proportion of the acidic form decreases substantially. In this case, the pH being 3 units higher than pK_a1 results in a marked reduction in the acidic form's concentration.

For further practice, consider applying these principles to determine the fraction of tyrosine, utilizing the relevant equations for diprotic species.

Fractional Composition Calculations 1

Tyrosine, an amino acid with two pKa values, behaves as a diprotic acid, meaning it can exist in three forms: acidic, intermediate, and basic. At a pH of 10.00, which is above its second pKa of 8.67, the predominant form of tyrosine will be its basic form. Understanding the distribution of these forms at a given pH involves using specific equations that relate the concentrations of hydrogen ions and the dissociation constants (Ka) of the acid.

The concentration of hydrogen ions, [H⁺], can be calculated from the pH using the formula:

${10}^{-}$

For pH 10.00, this results in [H⁺] = 10^-10 M. The first dissociation constant, Ka₁, is derived from its pKa₁ of 2.37, giving:

${10}^{-}$

Similarly, for the second dissociation constant, Ka₂, with a pKa₂ of 8.67, we have:

${10}^{-}$

To find the fractions of each form, the following equations are used:

Fraction of acidic form:

$\frac{\mathrm{[H⁺]}}{\mathrm{[H⁺]²\; +\; [H⁺]K_a1\; +\; K_a1K_a2\}}}$

Fraction of intermediate form:

$\frac{\mathrm{K_a1[H⁺]}}{\mathrm{[H⁺]²\; +\; [H⁺]K_a1\; +\; K_a1K_a2\}}}$

Fraction of basic form:

$\frac{\mathrm{K_a1K_a2\}}}{\mathrm{[H⁺]²\; +\; [H⁺]K_a1\; +\; K_a1K_a2\}}}$

By substituting the calculated values into these equations, the fractions of each form can be determined. For instance, the fraction in the acidic form is found to be approximately 1.05 x 10^-19, indicating a negligible presence of the acidic form at this pH. The intermediate form fraction is about 0.047, while the basic form fraction is approximately 0.955, confirming that the basic form is the most prevalent at a pH of 10.00.

When expressed as percentages, this translates to about 95.5% of tyrosine existing in its basic form at this pH, with very small amounts of the acidic and intermediate forms present. This analysis illustrates the significant impact of pH on the protonation state of amino acids, which is crucial for understanding their behavior in biological systems.

Calculate the fraction of the intermediate for sulfurous acid, H ₂SO₃, at pH = 11.00? pK_a1 = 1.80, pK_a2 = 7.19. 

• A. 6.31x10^-11
• B. 9.77x10^-14
• C. 0.000155
• D. 0.00977
• E. 1.00

Answer: (C)

Fractional Concentrations

In monoprotic acid-base systems, the relationship between the concentrations of the acid (HA) and its conjugate base (A⁻) can be expressed using specific equations. The fraction of acid molecules, denoted as \( \alpha \), is calculated using the formula:\[\alpha = \frac{[H^+]}{[H^+] + K_a}\]where \( [H^+] \) is the concentration of hydrogen ions and \( K_a \) is the acid dissociation constant. Conversely, the fraction of the conjugate base is given by:\[\alpha_{A^-} = \frac{K_a}{[H^+] + K_a}\]As the pH of the solution increases, the concentration of acid molecules decreases while the concentration of conjugate base molecules increases. Notably, at the point where the pH equals the pK_a (the negative logarithm of \( K_a \)), the concentrations of the acid and conjugate base are equal.To determine the actual concentrations of the acid and conjugate base in a solution, we can modify our equations. The concentration of the acid can be expressed as:\[[HA] = \alpha \times [HA]_{formal}\]where \( [HA]_{formal} \) is the formal concentration of the weak acid solution. Similarly, the concentration of the conjugate base is calculated as:\[[A^-] = \alpha_{A^-} \times [HA]_{formal} = \frac{K_a \times [HA]_{formal}}{[H^+] + K_a}\]By comparing the pH of the solution to the pK_a, one can determine which species predominates. These equations allow for the calculation of the exact fractions of the weak acid and conjugate base, leading to the determination of their respective concentrations in the solution. Understanding these relationships is crucial for analyzing monoprotic systems before moving on to more complex diprotic systems.

Fractional Concentrations

In diprotic systems, compounds can exist in three distinct forms: the acidic form (H₂A), the intermediate form (HA^-), and the basic form (A^2-). The dissociation of these forms is characterized by two acid dissociation constants, K_a1 and K_a2. When the first acidic hydrogen is removed, the reaction produces hydronium ions (H₃O⁺) and the intermediate form, with K_a1 governing this equilibrium. The expression for this equilibrium can be represented as:

$$ K_{a1} = \frac{[H_3O^+][HA^-]}{[H_2A]} $$

Rearranging this allows us to isolate the concentration of the intermediate form (HA^-), which can be expressed as:

$$ [HA^-] = \frac{[H_2A] \cdot K_{a1}}{[H_3O^+]} $$

From the intermediate form, the second acidic hydrogen can be lost, resulting in more hydronium ions and the basic form (A^2-). By substituting the expression for the intermediate form into the equilibrium for the second dissociation, we can derive the concentration of the basic form as:

$$ [A^{2-}] = \frac{[H_2A] \cdot K_{a1} \cdot K_{a2}}{[H_3O^+]^2} $$

The formal concentration (C) of the diprotic acid system can be expressed in terms of the concentrations of the three forms. The mass balance equation states:

$$ C = [H_2A] + [HA^-] + [A^{2-}] $$

By substituting the expressions for the concentrations of the intermediate and basic forms into this equation, we can calculate the new concentrations of each form. It is important to note that all three forms share the same denominator in their respective expressions.

When analyzing the relationship between pH and the concentrations of these forms, a fraction versus pH chart can be utilized. The intersection points on this chart correspond to specific pH values: at pH = pK_a1, the concentrations of the acidic and intermediate forms are equal, while at pH = pK_a2, the concentrations of the intermediate and basic forms are equal. As pH increases, the proportion of the basic form increases, although even at high pH levels, a small amount of the acidic form remains present. This illustrates that all three forms coexist in varying proportions across different pH levels.

Fractional Concentrations Calculations 1

In this scenario, we are analyzing a diprotic compound, referred to as compound B, which has a pK_b1 value of 4.00 and a pK_b2 value of 6.00. The goal is to determine the concentration of the intermediate form of this diprotic acid when its total concentration is 0.150 M and the pH is 8.00.

To find the concentration of the intermediate form, we can use the relationship:

Concentration of intermediate form = Fraction of intermediate form × Formal concentration of diprotic acid.

Expanding this, we have:

Fraction of intermediate form = \(\frac{K_{a1} \cdot [H^+] \cdot [C]}{[H^+]^2 + [H^+] \cdot K_{a1} + K_{a1} \cdot K_{a2}}\)

Where:

• [H⁺] is the hydrogen ion concentration, which can be calculated using the formula [H⁺] = \(10^{-\text{pH}}\). For a pH of 8.00, [H⁺] = \(10^{-8}\) M.
• K_a1 and K_a2 are the acid dissociation constants, which can be derived from the pK values.

To find K_a1 and K_a2, we use the relationships:

• pK_a1 = 14 - pK_b2 = 14 - 6 = 8, thus K_a1 = \(10^{-8}\).
• pK_a2 = 14 - pK_b1 = 14 - 4 = 10, thus K_a2 = \(10^{-10}\).

Now substituting these values into the equation, we have:

Fraction of intermediate form = \(\frac{(10^{-8}) \cdot (10^{-8}) \cdot (0.150)}{(10^{-8})^2 + (10^{-8}) \cdot (10^{-8}) + (10^{-8}) \cdot (10^{-10})}\)

Calculating the numerator:

Numerator = \(1.5 \times 10^{-17}\)

Calculating the denominator:

Denominator = \(10^{-16} + 10^{-16} + 10^{-18} = 2.01 \times 10^{-16}\)

Thus, the concentration of the intermediate form is:

Concentration of intermediate form = \(\frac{1.5 \times 10^{-17}}{2.01 \times 10^{-16}} \approx 0.075 \, \text{M}\)

This indicates that at a pH of 8.00, the concentration of the intermediate form is 0.075 M. It is important to note that as the pH increases, the concentration of the intermediate form will decrease, while the basic form will become more predominant. In this case, since the pH is above the pK_a2 (which is 6), the majority of the diprotic acid exists in its basic form, with lesser amounts in the intermediate and acidic forms.

Fractional Concentrations Calculations 1

To calculate the concentration of the acidic form of 0.230 M tartaric acid at a pH of 6.00, we first need to understand the dissociation of diprotic acids. Tartaric acid has two dissociation constants, represented as pKa₁ = 3.00 and pKa₂ = 4.34. The concentration of the diprotic acid can be expressed as the fraction of the acidic form multiplied by the formal concentration of the acid.

The formula for the concentration of the acidic form is given by:

\[C_{acidic} = \frac{[H^+]^2 \cdot C_{formal}}{[H^+]^2 + [H^+] \cdot K_{a1} + K_{a1} \cdot K_{a2}}\]

Where:

• [H⁺] = 10^-pH = 10^-6
• K_a1 = 10^-pKa₁ = 10^-3
• K_a2 = 10^-pKa₂ = 10^-4.34

Substituting these values into the equation, we calculate:

\[C_{acidic} = \frac{(10^{-6})^2 \cdot 0.230}{(10^{-6})^2 + (10^{-6}) \cdot 10^{-3} + 10^{-3} \cdot 10^{-4.34}}\]

Calculating the numerator:

\[(10^{-6})^2 \cdot 0.230 = 2.30 \times 10^{-13}\]

Calculating the denominator:

\[(10^{-6})^2 + (10^{-6}) \cdot 10^{-3} + 10^{-3} \cdot 10^{-4.34} \approx 4.67098 \times 10^{-8}\]

Thus, the concentration of the acidic form is:

\[C_{acidic} = \frac{2.30 \times 10^{-13}}{4.67098 \times 10^{-8}} \approx 4.92 \times 10^{-6} \text{ M}\]

This result indicates that at a pH of 6.00, the concentration of the acidic form of tartaric acid is very low, reflecting the fact that the majority of the species in solution exists in the basic form. As the pH increases beyond pKa₂, the distribution of the species shifts, resulting in a higher concentration of the basic form and a minimal presence of the acidic form. At pH 6, the acidic form is significantly diminished, while the intermediate and basic forms are more prevalent.

Understanding this distribution is crucial for predicting the behavior of diprotic acids in various pH environments, as the pH provides insight into the relative concentrations of the different forms present in solution.

Here’s what students ask on this topic:

What is the relationship between pH and pKa in determining the fractional composition of weak acids and their conjugate bases?

The relationship between pH and pKa is crucial in determining the fractional composition of weak acids and their conjugate bases. When pH equals pKa, the concentrations of the weak acid (HA) and its conjugate base (A⁻) are equal. This is derived from the Henderson-Hasselbalch equation: ${\mathrm{pH}}_{}={\mathrm{pKa}}_{}+\log \left(\frac{{\mathrm{[A⁻]}}_{}}{{\mathrm{[HA]}}_{}}\right)$. At this point, the log term becomes zero, making pH equal to pKa. As pH increases above pKa, the fraction of the conjugate base increases, and the fraction of the weak acid decreases.

How do you calculate the fractional composition of a diprotic acid system?

In a diprotic acid system, you need to consider both dissociation constants, Ka1 and Ka2. The fractional compositions of the acidic form (H₂A), intermediate form (HA⁻), and basic form (A²⁻) are calculated using specific equations. For the acidic form: ${\alpha }_{=}$. For the intermediate form: ${\alpha }_{=}$. For the basic form: ${\alpha }_{=}$. These equations help determine the fraction of each form at any given pH.

What is the significance of the Henderson-Hasselbalch equation in fractional composition calculations?

The Henderson-Hasselbalch equation is significant in fractional composition calculations because it relates pH, pKa, and the ratio of the concentrations of the conjugate base and weak acid. The equation is: ${\mathrm{pH}}_{}={\mathrm{pKa}}_{}+\log \left(\frac{{\mathrm{[A⁻]}}_{}}{{\mathrm{[HA]}}_{}}\right)$. This equation allows you to determine the pH of a solution if the concentrations of the acid and its conjugate base are known, or vice versa. It is particularly useful for buffer solutions, where the pH is maintained within a narrow range. When pH equals pKa, the concentrations of the acid and conjugate base are equal, which is a key point in understanding fractional compositions.

How do you determine the concentration of a weak acid and its conjugate base at a given pH?

To determine the concentration of a weak acid (HA) and its conjugate base (A⁻) at a given pH, you can use the fractional composition equations. For the weak acid: ${\mathrm{[HA]}}_{}=\frac{{\mathrm{[H⁺]}}_{}{F}_{}}{{\mathrm{[H⁺]}}_{}+{\mathrm{Ka}}_{}}$. For the conjugate base: ${\mathrm{[A⁻]}}_{}=\frac{{\mathrm{Ka}}_{}{F}_{}}{{\mathrm{[H⁺]}}_{}+{\mathrm{Ka}}_{}}$. Here, F is the formal concentration of the solution. These equations allow you to calculate the exact concentrations of the weak acid and its conjugate base at any given pH, helping you understand the solution's composition.

What are the key differences between monoprotic and diprotic acid systems in terms of fractional compositions?

The key differences between monoprotic and diprotic acid systems in terms of fractional compositions lie in the number of dissociation steps and the complexity of the equations. Monoprotic acids have one dissociation constant (Ka) and two forms: the acid (HA) and its conjugate base (A⁻). The fractional compositions are simpler and can be calculated using: ${\alpha }_{=}$ for the acid and ${\alpha }_{=}$ for the conjugate base. Diprotic acids have two dissociation constants (Ka1 and Ka2) and three forms: the acid (H₂A), the intermediate (HA⁻), and the base (A²⁻). The fractional compositions are more complex and require additional equations to account for the intermediate form. This makes diprotic systems more intricate to analyze compared to monoprotic systems.