Question

Monochloroacetic acid, \(\mathrm{HC}_{2} \mathrm{H}_{2} \mathrm{ClO}_{2},\) is a skin irritant that is used in "chemical peels" intended to remove the top layer of dead skin from the face and ultimately improve the complexion. The value of \(K_{\mathrm{a}}\) for monochloroacetic acid is \(1.35 \times 10^{-3}\) Calculate the pH of a \(0.10-M\) solution of monochloroacetic acid.

Short answer

The pH of a \(0.10\)M solution of monochloroacetic acid is approximately \(1.44\).

Step-by-step solution

Write the dissociation equation for monochloroacetic acid

The dissociation reaction of monochloroacetic acid in water can be written as: \[\mathrm{HC}_{2}\mathrm{H}_{2}\mathrm{ClO}_{2} \rightleftharpoons \mathrm{H^{+}}+\mathrm{C}_{2}\mathrm{H}_{2}\mathrm{ClO}_{2}^{-}\]

Set up the ICE table

Let's denote monochloroacetic acid as HX, and set up the ICE table with the initial concentration of each species: | Species | Initial (M) | Change (M) | Equilibrium (M) | |:---------:|:-----------:|:----------:|:--------------:| | \(\mathrm{HX}\) | \(0.10\) | \(-x\) | \(0.10 - x\) | | \(\mathrm{H^{+}}\) | \(0\) | \(+x\) | \(x\) | |\(\mathrm{X^{-}}\) | \(0\) | \(+x\) | \(x\) |

Use the \(K_a\) expression and solve for x

The expression for \(K_a\) for the dissociation of monochloroacetic acid is: \[K_a = \frac{[\mathrm{H^{+}}][\mathrm{X^{-}}]}{[\mathrm{HX}]}\] Now, we'll plug in the equilibrium concentrations from the ICE table and solve for x: \[1.35\times10^{-3}=\frac{(x)(x)}{0.10-x}\]

Simplify the equation and solve for x

Since \(K_a\) is small, we make the assumption that the change in the concentration (x) is small compared to the initial concentration (\(0.10\)M). Thus, in the equation we can assume that \((0.10 - x) \approx 0.10\). The equation becomes: \[1.35\times10^{-3}=\frac{x^2}{0.10}\] Solve for x: \[x=\sqrt{1.35\times10^{-3}\times0.10}\] \[x=3.66\times10^{-2}\]

Calculate the pH

The concentration of \(\mathrm{H^{+}}\) ions at equilibrium is equal to x. Now we can calculate the pH using the pH formula: \[pH=-\log([\mathrm{H^{+}}])=-\log(3.66\times10^{-2})\] \[pH\approx1.44\] The pH of a \(0.10\)M solution of monochloroacetic acid is approximately \(1.44\).

Key concepts

Acid Dissociation Constant

The acid dissociation constant, commonly symbolized as \( K_a \), is a critical parameter that helps us understand how an acid behaves in solution. It measures the extent of dissociation of an acid molecule into its ions. The larger the \( K_a \) value, the stronger the acid, meaning it dissociates more completely in water.

In our problem, we're dealing with monochloroacetic acid, for which \( K_a \) is given as \( 1.35 \times 10^{-3} \). This value is relatively small, indicating that monochloroacetic acid is a weak acid.

To calculate the pH of a solution of this acid, we use \( K_a \) in combination with an equilibrium expression derived from the dissociation reaction. This expression highlights the relationship between the concentrations of the acid and its ions in equilibrium. Understanding and applying \( K_a \) is essential for predicting the behavior of acidic solutions and computing their pH.

Monochloroacetic Acid

Monochloroacetic acid, known by its chemical formula \( \mathrm{HC_2H_2ClO_2} \), is a type of carboxylic acid and is known for its usefulness and risks. As a skin irritant, it plays a role in cosmetic procedures such as chemical peels. Despite its industrial use, handling it requires care due to its corrosive nature.

In this exercise, we're interested in its behavior in an aqueous solution. When monochloroacetic acid is dissolved in water, it partially dissociates into hydrogen ions (\( \mathrm{H^+} \)) and residual ions (\( \mathrm{C_2H_2ClO_2^-} \)).

• Its dissociation makes it a candidate for pH calculation, owing to its acid properties.
• Being a weak acid, it doesn't dissociate completely, which is why we observe a balance between dissociated and undissociated forms in solution.

Calculating specific measures like pH involves evaluating how much of it actually dissociates, using the \( K_a \). This reveals the inherent strength of the acid in influencing the solution's acidity.

ICE Table

The ICE table is a useful tool for organizing and calculating the concentrations of species at initial, change, and equilibrium stages in chemical reactions. It stands for Initial, Change, and Equilibrium - the three stages that describe the quantitative progression of a reaction over time.

To set up an ICE table for an acid like monochloroacetic acid, you list the species involved in its dissociation, typically the undissociated acid, the hydrogen ion, and the conjugate base.

Here's how each step works:

• **Initial (I):** Record the starting concentrations before any reaction occurs.
• **Change (C):** Determine the change in concentration, usually represented by \( x \), as the system moves towards equilibrium.
• **Equilibrium (E):** Represent the final concentrations after the reaction has reached equilibrium, often a function of the initial concentrations and \( x \).

This structured approach makes it easier to form an equation based on the dissociation reaction, utilizing the \( K_a \) to relate these concentrations. By solving this equation, you determine \( x \), which then allows you to calculate properties such as pH, reflecting the solution's acidity.