Question

A solution of formic acid (HCOOH, \(K_{\mathrm{a}}=1.8 \times 10^{-4} )\) has a \(\mathrm{pH}\) of 2.70 . Calculate the initial concentration of formic acid in this solution.

Short answer

The initial concentration of formic acid in the solution is 0.02215 M.

Step-by-step solution

Write down the dissociation equation for formic acid

The dissociation equation for formic acid is: \( HCOOH \rightleftharpoons H^+ + HCOO^- \)

Write down the Ka expression from the equation

The Ka expression refers to the acid dissociation constant, which for formic acid, is given as \( 1.8 \times 10^{-4} \). This is derived from the concentrations of ions in the reaction: \( K_a = \frac{[H^+][HCOO^-]}{[HCOOH]} \)

Calculate the concentration of hydrogen ions using pH

We are given the pH of the solution (2.70). We can find the concentration of hydrogen ions, [H+], using the following formula: \( pH = -\log[H^+] \) We can rearrange this equation to solve for [H+]: \[ [H^+] = 10^{-pH} \] Now we plug in the pH value: \[ [H^+] = 10^{-2.7} = 1.995 \times 10^{-3} \]

Assume that formic acid's initial concentration and ion concentrations can be represented by variables

Let's assume the initial concentration of formic acid is "C". Since formic acid is a weak acid and it doesn't fully dissociate, we can represent the concentrations of the ions as follows: - [HCOOH] = C - x - [H+] = x - [HCOO-] = x

Substitute the variables into the Ka expression

Now we will substitute these values into the Ka expression: \( 1.8 \times 10^{-4} = \frac{x \cdot x}{C - x} \) We also know that the [H+] = x = 1.995 x 10^{-3} from Step 3, so we plug this value into the equation: \( 1.8 \times 10^{-4} = \frac{(1.995 \times 10^{-3})^2}{C - (1.995 \times 10^{-3})} \)

Solve the equation for C

Finally, we solve the equation for C (the initial concentration of formic acid): \[ C = \frac{(1.995 \times 10^{-3})^2}{1.8 \times 10^{-4}} + 1.995 \times 10^{-3} = 0.02215 \] Thus, the initial concentration of formic acid in the solution is 0.02215 M.

Key concepts

Formic Acid

Formic acid is one of the simplest carboxylic acids, known chemically as HCOOH. Also referred to as methanoic acid, it is a naturally occurring substance that is found in the stinging bites of ants and in many plants. Formic acid is a weak acid, which means it does not completely dissociate into its ions in a solution. This partial ionization is depicted by its equilibrium constant, known as the acid dissociation constant, or \( K_a \). For formic acid, the \( K_a \) value is \( 1.8 \times 10^{-4} \), indicating that only a small fraction of the formic acid molecules ionize in a solution.

• Formic acid is commonly used in leather production and as a preservative.
• Its weak acidic nature makes it less corrosive compared to strong acids.
• It has a pungent odor and is colorless when in its pure form.

Understanding the behavior of formic acid in solutions is crucial, especially in applications that involve its use as a pH adjuster or as a coagulant in the manufacturing of rubber.

pH Calculation

The pH of a solution is a measurement of its acidity or alkalinity, represented on a scale that ranges from 0 to 14. A pH of 7 is neutral, while a pH below 7 indicates an acidic solution, and a pH above 7 indicates an alkaline solution. The pH scale is logarithmic, meaning each unit change represents a tenfold change in the concentration of hydrogen ions \([H^+]\).
To calculate the pH from the hydrogen ion concentration, the formula used is:\[ pH = -\log[H^+] \]Conversely, to find the hydrogen ion concentration from the given pH value, we rearrange the formula as:\[ [H^+] = 10^{-pH} \]In the exercise we are considering, the given pH is 2.70. Thus, we calculate [H+] as:\[ [H^+] = 10^{-2.70} = 1.995 \times 10^{-3} \]

• Lower pH values correspond to higher [H+], which implies a more acidic solution.
• Accurate pH measurement and calculation are essential in chemistry, particularly in processes like titration and chemical synthesis.

This step of finding [H+] is fundamental for solving equilibrium problems, such as determining initial concentrations of acids.

Initial Concentration Calculation

To calculate the initial concentration of a weak acid like formic acid in a solution, we must consider its acid dissociation constant (\( K_a \)) and the pH. When the pH is given, we can determine the concentration of hydrogen ions \([H^+]\) as shown in the previous section. For formic acid, where the equilibrium can be represented as:\[ HCOOH \rightleftharpoons H^+ + HCOO^- \]we let the initial concentration of formic acid be "C". Since formic acid is a weak acid, the dissociation is not complete, and we can represent the ion concentrations as:

• \([HCOOH] = C - x\)
• \([H^+] = x\)
• \([HCOO^-] = x\)

Using the given \( K_a \) expression:\[K_a = \frac{[H^+][HCOO^-]}{[HCOOH]}\]and substituting with \( x = [H^+] = 1.995 \times 10^{-3} \), we have:\[1.8 \times 10^{-4} = \frac{(1.995 \times 10^{-3})^2}{C - 1.995 \times 10^{-3}}\]Solving for C gives:\[C = \frac{(1.995 \times 10^{-3})^2}{1.8 \times 10^{-4}} + 1.995 \times 10^{-3} = 0.02215 \, \text{M}\]This method allows us to find the initial concentration of formic acid needed to achieve a specified pH, which is important in chemical formulations and reactions.