Question

A solution of formic acid \(\left(\mathrm{HCOOH}, K_{\mathrm{a}}=1.8 \times 10^{-4}\right)\) 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 approximately 0.024 M.

Step-by-step solution

Write down the given information

We have the following given information: - pH of the solution: 2.70 - Ka of formic acid: 1.8 x 10^(-4)

Calculate the pKa

To find the pKa, use the formula pKa = -log10(Ka): pKa = -log10(1.8 x 10^(-4)) pKa ≈ 3.74

Use the Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is given by: pH = pKa + log([A-]/[HA]) We can rewrite the equation to solve for the ratio of conjugate base to formic acid concentrations: log([A-]/[HA]) = pH - pKa Plug in the values for pH and pKa from previous steps: log([A-]/[HA]) = 2.70 - 3.74 log([A-]/[HA]) = -1.04

Calculate the ratio of [A-] to [HA]

To find the ratio of [A-] to [HA], exponentiate both sides of the equation: [A-]/[HA] = 10^(-1.04) [A-]/[HA] ≈ 0.091

Set up the reaction stoichiometry

Here is the dissociation reaction of formic acid: HCOOH ↔ H+ + HCOO- Let the initial concentration of formic acid be 'x': Initially: [HCOOH] = x [H+] = 0 [HCOO-] = 0 Assuming that formic acid loses α moles of H+ ions in the dissociation, at equilibrium: [HCOOH] = x - α [H+] = α [HCOO-] = α Now, using the ratio of [A-] to [HA] from step 4, we know: α / (x - α) = 0.091

Solve for x, the initial concentration of formic acid

To find x, solve the equation: (α / (x - α)) = 0.091 Isolate the variable x: x - α = α / 0.091 x = α (1 + 1/0.091) Since the pH of the solution is given, we can write [H+] in terms of pH: [H+] = 10^(-pH) α = [H+] = 10^(-2.70) α ≈ 0.0020 Now, plug α into the equation for x: x = 0.0020 (1 + 1/0.091) x ≈ 0.0020 (1 + 11) x ≈ 0.0020 (12) x ≈ 0.024 Therefore, the initial concentration of formic acid in the solution is approximately 0.024 M.