Question

Phenylacetic acid $\left({\mathrm{C}}_6{\mathrm{H}}_5{\mathrm{CH}}_2\mathrm{COOH}\right)$ is one of the substances that accumulates in the blood of people with phenylketonuria, an inherited disorder that can cause mental retardation or even death. A $0.085\mathrm{M}$ solution of ${\mathrm{C}}_6{\mathrm{H}}_5{\mathrm{CH}}_2\mathrm{COOH}$ has a $\mathrm{pH}$ of $2.68.$ Calculate the $K_a$ value for this acid.

Short answer

The $K_{a}$ value for phenylacetic acid is approximately $5.1\times {10}^{-5}$.

Step-by-step solution

Convert pH to H3O+ concentration

Using the formula for pH, which is: pH = -log10[H3O+] we will first calculate the concentration of hydronium ions (H3O+): 2.68 = -log10[H3O+]

Solve for [H3O+]

Now, we will isolate the [H3O+] by using the inverse of the log function, which is 10^: [H3O+] = 10^(-pH) = 10^(-2.68) [H3O+] ≈ 2.09 × 10^(-3) M

Write the equilibrium expression for Ka

The dissociation of phenylacetic acid (C6H5CH₂COOH) can be represented as: C6H5CH₂COOH ⇄ C6H5CH₂COO⁻ + H3O⁺ The equilibrium constant expression for Ka can be written as: Ka = ([C6H5CH₂COO⁻][H3O⁺])/([C6H5CH₂COOH])

Set up the ICE table

ICE stands for Initial, Change, and Equilibrium concentrations. The table will look like this: | C6H5CH₂COOH | C6H5CH₂COO⁻ | H3O⁺ ------------------------------------------------ Initial | 0.085 M | 0 M | 0 M Change | -x M | +x M | +x M Equilibrium| 0.085-x M | x M | x M As we know the concentration of H3O+ at equilibrium which is 2.09 × 10^(-3) M, so x = 2.09 × 10^(-3) M Now we will substitute the values in the Ka expression: Ka = ([C6H5CH₂COO⁻][H3O⁺])/([C6H5CH₂COOH])

Calculate Ka

Substituting the equilibrium concentrations into the Ka expression: Ka = ((2.09 × 10^(-3))(2.09 × 10^(-3)))/ (0.085 - 2.09 × 10^(-3)) And solving for Ka: Ka ≈ 5.1 × 10^(-5) The Ka value for this acid is approximately 5.1 × 10^(-5).

Key concepts

Equilibrium Expressions

When dealing with weak acids, like phenylacetic acid, we use the concept of equilibrium expressions to understand how these acids dissociate in water. This dissociation forms ions that are present in solution. At equilibrium, the concentration of these ions can be related to the acid dissociation constant, or Ka.
The general form of the equilibrium expression for an acid is based on the reaction:
Acid \rightleftharpoons Conjugate\ Base + H^+.
For phenylacetic acid ${\text{(C}}_{6}ext{H}_5{\text{CH}}_2\text{COOH})$:
The dissociation results in the formation of ${\text{C}}_6{\text{H}}_5{\text{CH}}_2{\text{COO}}^{-}$ and \text{H}_{3}\text{O}^+\.
The equilibrium expression becomes:
$$K_a=\frac{[{\text{C}}_6{\text{H}}_5{\text{CH}}_2{\text{COO}}^{-}][{\text{H}}_3{\text{O}}^{+}]}{[{\text{C}}_6{\text{H}}_5{\text{CH}}_2\text{COOH}]}$$.
This expression tells you that the Ka value is a ratio of the products' concentrations to the reactants' concentration at equilibrium.

• If the Ka is large, it indicates a strong acid (more dissociation).
• A small Ka indicates a weak acid (less dissociation).

Phenylacetic Acid

Phenylacetic acid, ${\text{(C}}_6{\text{H}}_5{\text{CH}}_2\text{COOH})$, is a weak organic acid. This means that not all of its molecules dissociate into ions in solution. In contexts like biochemical processes and medical conditions, understanding its behavior is important.
Phenylacetic acid is relevant in specific medical conditions such as phenylketonuria, where its accumulation can have significant health effects. Its weak acid nature is characterized by a partial ionization in water, making calculations of its dissociation constant essential for understanding its behavior in different environments.
In chemical equations, the reversible arrow $)$ indicates that the dissociation reaches a state of balance, or equilibrium, between its ionized and non-ionized forms.

• This reversible nature enables calculations of pH and Ka.
• Planning safe and effective treatments often requires knowing this relationship.

pH Calculations

pH is a crucial concept in chemistry and biology to describe the acidity of a solution. It's linked closely with the concentration of hydronium ions, ${\text{H}}_3{\text{O}}^{+}$.
You calculate pH using the formula:
$$pH=-{\log }_{10}[{\text{H}}_3{\text{O}}^{+}]$$.
In the case of phenylacetic acid, the given pH was 2.68. From this, we inferred the ${\text{H}}_3{\text{O}}^{+}$ concentration using the formula and found it to be approximately $2.09\times {10}^{-3}$ M.
Understanding how changes in pH affect the concentration of ions helps predict how various substances will behave in different environments.

• A low pH indicates a higher concentration of ${\text{H}}_3{\text{O}}^{+}$ ions, making the solution more acidic.
• Conversely, a high pH corresponds to a less acidic or more basic solution.