Question

What is the \(K_{\mathrm{a}}\) of the amino acid serine if it is \(86.3 \%\) dissociated at \(\mathrm{pH}=9.95 ?\)

Short answer

\(K_a = 7.05 \times 10^{-10}\) for serine at 86.3% dissociation and pH 9.95.

Step-by-step solution

Understand the problem

We are given that serine is 86.3% dissociated at pH 9.95, and we need to calculate its acid dissociation constant, \(K_{a}\). This involves understanding the relationship between pH, dissociation percentage, and \(K_a\).

Use pH to find \([H^+]\)

The pH of the solution is 9.95. The concentration of hydrogen ions \([H^+]\) can be calculated using the pH formula: \[pH = -\log([H^+])\]Rearrange to solve for \([H^+]\):\[[H^+] = 10^{-9.95}\]

Calculate \([H^+]\) value

Calculate \([H^+]\) using the formula:\[[H^+] = 10^{-9.95} \approx 1.12 \times 10^{-10} \text{ M}\]

Relate dissociation percentage to \([A^-]\)

If serine is 86.3% dissociated, \([A^-]\) is 0.863 times the initial concentration \(C\). So,\[[A^-] = 0.863C\]

Express \([HA]\) in terms of \([A^-]\)

\([HA]\) is the undissociated concentration, which is the remainder of the initial concentration. Thus, \[[HA] = C - [A^-] = C - 0.863C = 0.137C\]

Write equilibrium expression for \(K_a\)

The expression for \(K_a\) is:\[K_a = \frac{[H^+][A^-]}{[HA]}\]Substitute \([H^+]\), \([A^-]=0.863C\), and \([HA]=0.137C\):\[K_a = \frac{[H^+] \cdot 0.863C}{0.137C}\]

Simplify \(K_a\) expression

Since \(C\) cancels out, simplify to get:\[K_a = \frac{[H^+] \cdot 0.863}{0.137}\]Substitute \([H^+] \approx 1.12 \times 10^{-10}\):\[K_a = \frac{1.12 \times 10^{-10} \times 0.863}{0.137}\]

Calculate \(K_a\) value

Calculate using the simplified \(K_a\) expression:\[K_a \approx \frac{1.12 \times 10^{-10} \times 0.863}{0.137} \approx 7.05 \times 10^{-10}\]

Conclusion

The calculated acid dissociation constant \(K_a\) for serine when it is 86.3% dissociated at pH 9.95 is \(7.05 \times 10^{-10}\).

Key concepts

Amino Acid Chemistry

Amino acids are the building blocks of proteins. They consist of an amino group, a carboxylic acid group, and a unique side chain that defines their characteristics.
Each amino acid can act as both an acid and a base, making them amphiprotic. This means they can donate or accept protons, depending on the solution's pH.
In aqueous solutions, amino acids can exist in different ionization states. At different pH levels, the carboxylic acid and amino groups can either be protonated or deprotonated. Serine, the amino acid discussed here, has a side chain with a hydroxyl group, making it somewhat polar and involved in specific reactions.
Understanding the chemistry of amino acids like serine is crucial when calculating dissociation constants (K_a) because it influences how the amino acid behaves under various conditions.

pH Calculations

The concept of pH is essential in chemistry as it measures the acidity or basicity of a solution. It is calculated using the formula:
\[pH = -\log([H^+])\] where \([H^+]\) represents the concentration of hydrogen ions in the solution.
A low pH value indicates a high concentration of hydrogen ions, signifying an acidic solution, while a high pH value indicates a basic (alkaline) solution.
In the context of this exercise, we find \([H^+]\) by rearranging the formula. Once we know the pH (9.95 in this case), we can calculate the hydrogen ion concentration:\([H^+] = 10^{-9.95}\) which is approximately \(1.12 \times 10^{-10}\). This value helps us understand the acidity of the solution, crucial for further calculations involving \([K_a]\).

Dissociation Percentage

When dealing with acids, dissociation percentage is a measure of how much of the acid has dissociated into ions in a solution. For example, if an acid is 86.3% dissociated, it means that 86.3% of its molecules have separated into ions.
This percentage helps us understand the strength of the acid in a particular pH environment.The dissociation percentage is also vital for calculating the acid dissociation constant \([K_a]\). For our amino acid serine, being 86.3% dissociated at pH 9.95 allows us to express the concentrations of ions involved in equilibrium expressions precisely.
In mathematical terms, an 86.3% dissociation means that, for an initial concentration \(C\), \([A^-] = 0.863C\) and the remaining \([HA] = 0.137C\), which is essential for plugging into the \(K_a\) formula.

Equilibrium Expressions

In chemistry, equilibrium expressions are used to describe the state of a reaction at equilibrium. For a dissociation reaction involving an acid like serine, the equilibrium expression for the acid dissociation constant \(K_a\) is given by:
\[ K_a = \frac{[H^+][A^-]}{[HA]} \]This equation shows the relationship between the concentrations of various species in a reaction at equilibrium.
For serine, which is 86.3% dissociated, we use \([H^+]\), \([A^-] = 0.863C\), and \([HA] = 0.137C\) to form the equation. The initial concentration \(C\) conveniently cancels out, simplifying calculations to solve for \(K_a\).
Equilibrium expressions are crucial tools in chemistry that enable us to calculate the exact concentrations required to maintain balance in reactions. Understanding how to manipulate these equilibria is key to solving practical problems like determining \(K_a\) values.