Question

Find the \(\mathrm{pH}\) of a \(0.2 \mathrm{M}\) solution of formic acid. \(\mathrm{K}_{\mathrm{a}}=1.76 \times 10^{-4}\)

Short answer

The pH of a \(0.2 \mathrm{M}\) solution of formic acid with a \(\mathrm{K}_{\mathrm{a}}=1.76 \times 10^{-4}\) is approximately 2.23.

Step-by-step solution

Write the dissociation equation and the equilibrium expression for formic acid

Formic acid, HCOOH, dissociates into protons (H+) and formate ions (HCOO-). The dissociation equation is: HCOOH \(\rightleftharpoons\) H\(^+\) + HCOO\(^-\) The equilibrium constant expression for this reaction, using the given \(\mathrm{K}_{\mathrm{a}}\) value, is: \[\mathrm{K}_{\mathrm{a}} = \frac{[\mathrm{H}^+][\mathrm{HCOO}^-]}{[\mathrm{HCOOH}]}\]

Set up the equilibrium equation using initial concentrations

Initially, we have a \(0.2 \mathrm{M}\) solution of formic acid and no H\(^+\) or HCOO\(^-\) ions. As the reaction proceeds, HCOOH dissociates into H\(^+\) and HCOO\(^-\) ions, with their concentrations increasing and the concentration of HCOOH decreasing. Let x represent the changes in the concentrations: Initial concentrations: [HCOOH] = \(0.2 \mathrm{M}\) , [H\(^+\)] = 0 , [HCOO\(^-\)] = 0 Changes in concentrations: [HCOOH] = \(-x\) , [H\(^+\)] = \(+x\) , [HCOO\(^-\)] = \(+x\) Equilibrium concentrations: [HCOOH] = \(0.2 - x\) , [H\(^+\)] = \(x\) , [HCOO\(^-\)] = \(x\) Now, substitute the equilibrium concentrations into the \(\mathrm{K}_{\mathrm{a}}\) expression: \[1.76 \times 10^{-4} = \frac{x^2}{(0.2 - x)}\]

Solve the equilibrium equation for \(x\)

Since \(\mathrm{K}_{\mathrm{a}}\) is small, we can make the assumption that \(x\) is small compared to \(0.2\), and therefore, we can approximate \((0.2 - x)\) as \(0.2\): \[1.76 \times 10^{-4} \approx \frac{x^2}{0.2}\] Now, we solve for x: \[x^2 = (1.76 \times 10^{-4}) \times 0.2\] \[x = \sqrt{(1.76 \times 10^{-4}) \times 0.2}\] \[x \approx 0.005937\] Since \(x\) represents the concentration of H\(^+\) at equilibrium, the concentration of H\(^+\) ions is approximately \(0.005937\,\mathrm{M}\).

Calculate the pH using the H\(^+\) concentration

Now, use the formula for calculating pH: \[pH = -\log([\mathrm{H}^+])\] Plug in the H\(^+\) concentration: \[pH = -\log(0.005937)\] \[pH \approx 2.23\] The pH of the \(0.2 \mathrm{M}\) solution of formic acid is approximately 2.23.

Key concepts

Formic Acid

Formic acid, chemically known as HCOOH, is one of the simplest carboxylic acids and is found naturally in the venom of some ants. It has unique properties that make it an interesting subject of study in chemistry. In its pure form, it is a colorless liquid with a pungent smell. Formic acid is considered a weak acid, which means it partially dissociates in water, releasing protons (H⁺) and forming acetate ions (HCOO⁻).

This partial dissociation is critical in calculating the pH (a measure of acidity) of formic acid solutions, as it dictates how many protons are available in the solution to affect its acidity level. Understanding how formic acid dissociates and interacts in solutions helps chemists predict the behavior of this acid in various chemical reactions and processes.

Dissociation Equation

The dissociation equation is a way to represent how an acid breaks down into its components in water. For formic acid (HCOOH), the dissociation can be shown as follows:

• HCOOH
  ↔ H⁺ + HCOO⁻

This equation illustrates that one molecule of formic acid releases a proton (H⁺), leaving behind a formate ion (HCOO⁻). This process occurs in equilibrium, meaning the reaction continues to proceed forward and backward at equal rates when the solution is stable.

Knowing how to write and interpret dissociation equations is crucial since they provide insights into the dynamics of the acid in solution, including how it contributes to the solution's overall acidity.

Equilibrium Constant

The equilibrium constant, represented as \(K_a\), quantifies the extent of acid dissociation in a solution at equilibrium. For formic acid, the \(K_a\) is given as 1.76 x 10⁻⁴, indicating a relatively low amount of dissociation.

The formula for the equilibrium constant for formic acid's dissociation is:

• \[K_a = \frac{[H^+][HCOO^-]}{[HCOOH]}\]

This expression connects the concentration of ions at equilibrium, reflecting the balance between the undissociated formic acid and the ions it produces. A small \(K_a\) suggests that only a small percentage of formic acid molecules donate protons, thus confirming its status as a weak acid.

Understanding the equilibrium constant helps chemists and students predict how changes in concentration or other conditions might shift the equilibrium and alter the acidity of the solution.

Proton Concentration

Proton concentration is a crucial component in determining the pH of a solution. It specifically refers to the concentration of hydrogen ions \(H^+\) present in the solution. In the case of formic acid, during its dissociation, each molecule of formic acid releases one proton.

In typical pH calculations, we determine the proton concentration at equilibrium, expressed as \(x\). For a solution of formic acid, this concentration is affected by the initial concentration of the acid and the degree to which it dissociates, as governed by its \(K_a\).

Using the calculated proton concentration, one can find the solution's pH by applying the formula:

• \[pH = -\log([H^+])\]

This calculation transforms the understandings of proton concentration into a tangible numerical scale, allowing us to compare the acidity of different solutions easily, which is vital in various scientific and industrial applications.