Question

Formic acid \(\left(\mathrm{HCO}_{2} \mathrm{H}\right)\) is secreted by ants. Calculate \(\left[\mathrm{H}^{+}\right]\) and the \(\mathrm{pH}\) of a \(0.025 M\) solution of formic acid \(\left(K_{\mathrm{a}}=1.8 \times 10^{-4}\right)\).

Short answer

The concentration of hydrogen ions \([\mathrm{H}^{+}]\) in the 0.025 M formic acid solution is approximately \(2.12 \times 10^{-3} \mathrm{M}\), and the pH of the solution is approximately 2.67.

Step-by-step solution

Write the dissociation equation

The dissociation equation of formic acid \((\mathrm{HCO}_{2} \mathrm{H})\) into hydrogen ions \((\mathrm{H}^{+})\) and formate ions \((\mathrm{HCO}_{2}^{-})\) is as follows: \[\mathrm{HCO}_{2}\mathrm{H} \rightleftarrows \mathrm{H}^{+} + \mathrm{HCO}_{2}^{-}\] 2. Write the expression of Ka in terms of the concentrations of the ions and formic acid

Write the Ka expression

Using the dissociation equation, the equilibrium expression for the acid dissociation constant, Ka, can be written as: \[K_{\mathrm{a}} = \frac{[\mathrm{H}^{+}][\mathrm{HCO}_{2}^{-}]}{[\mathrm{HCO}_{2}\mathrm{H}]}\] 3. Let \(x\) represent the concentration of dissociated hydrogen ions

Use \(x\) to represent the concentration of \(\mathrm{H}^{+}\)

Since the concentration of \(\mathrm{H}^{+}\) is equal to the concentration of \(\mathrm{HCO}_{2}^{-}\), let's use the variable \(x\) to simplify the expression: \[[\mathrm{H}^{+}] = [\mathrm{HCO}_{2}^{-}] = x\] 4. Calculate the equilibrium amounts of each species in the solution

Use initial concentration and \(x\) to calculate equilibrium concentrations

Formic acid initially has a concentration of 0.025 M. During dissociation, its concentration will decrease by \(x\), so the equilibrium concentration becomes \((0.025 - x)\). Using this and step 3, we can rewrite the Ka expression as: \[K_{\mathrm{a}} = \frac{x^2}{(0.025 - x)}\] 5. Calculate the concentration of hydrogen ions (H+)

Solve for x

Insert the given value for Ka (1.8 x 10^-4) and solve for \(x\) (we'll assume x<<0.025 since it will simplify the equation): \[\begin{aligned} 1.8 \times 10^{-4} &= \frac{x^2}{0.025}\\ x^2 &= 1.8 \times 10^{-4} \times 0.025\\ x &= \sqrt{4.5 \times 10^{-6}}\\ x &= 2.12 \times 10^{-3} \end{aligned}\] So, the concentration of hydrogen ions is approximately \(2.12 \times 10^{-3} \mathrm{M}\). 6. Calculate the pH of the solution

Calculate the pH

Now that we know the concentration of \(\mathrm{H}^{+}\), we can calculate the pH using the formula: \[\mathrm{pH}=-\log [\mathrm{H}^{+}]\] \[\mathrm{pH} = -\log(2.12 \times 10^{-3}) \approx 2.67\] In conclusion, the [H+] is approximately \(2.12 \times 10^{-3} \mathrm{M}\), and the pH of the 0.025 M formic acid solution is approximately 2.67.

Key concepts

Acid Dissociation Constant

The acid dissociation constant, denoted as \( K_{a} \), is a quantitative measure of the strength of an acid in solution. It specifically represents the equilibrium constant for the dissociation of an acid into hydrogen ions \( (H^{+}) \) and the conjugate base. Higher \( K_{a} \) values indicate stronger acids, which dissociate more in water.

For a generic acid represented by \( HA \), the dissociation can be depicted as: \[ HA \rightleftarrows H^{+} + A^{-} \] The \( K_{a} \) value is then defined by the ratio of the concentration of the products to the concentration of the reactants at equilibrium: \[ K_{a} = \frac{[H^{+}][A^{-}]}{[HA]} \] Understanding \( K_{a} \) is crucial when predicting the behavior of acids in solutions, calculating the pH, and determining buffer capacities.

Formic Acid Dissociation

Formic acid \( (HCO_{2}H) \) is an example of a weak acid, meaning it only partially dissociates in solution. The dissociation reaction for formic acid is: \[ HCO_{2}H \rightleftarrows H^{+} + HCO_{2}^{-} \] In this reaction, formic acid \( (HCO_{2}H) \) is the reactant, \( H^{+} \) represents the hydrogen ion, and formic ions \( (HCO_{2}^{-}) \) are the conjugate base.

The extent to which formic acid dissociates is governed by its \( K_{a} \) value. When formic acid dissociates, it increases the concentration of hydrogen ions in the solution, which affects the solution's acidity, and consequently the pH value. The knowledge of the acid dissociation constant allows chemists to estimate the degree of dissociation and predict the pH of the solution.

Concentration of Hydrogen Ions

The concentration of hydrogen ions, often represented as \( [H^{+}] \), is a direct measure of a solution's acidity. In the dissociation of formic acid, \( [H^{+}] \) is proportional to the concentration of formate ions \( [HCO_{2}^{-}] \) at equilibrium due to the stoichiometry of the reaction.

Calculating \( [H^{+}] \) is key for finding the pH, which is a scale used to specify the acidity or basicity of an aqueous solution. The pH is the negative logarithm of the hydrogen ion concentration: \[ pH = -\log[H^{+}] \] By knowing the \( [H^{+}] \) of a solution, one can determine how acidic or basic the solution is, which is essential for many chemical analyses and reactions.

Equilibrium Concentration Calculation

Equilibrium concentration calculation involves determining the concentrations of reactants and products when a reaction has reached a state where the forward and reverse reactions occur at the same rate. For the dissociation of formic acid, we assume that the initial concentration of formic acid decreases by an amount \( x \) which is the amount that dissociates into \( H^{+} \) and \( HCO_{2}^{-} \) ions.

Using the initial concentration and the value of \( x \) obtained from the \( K_{a} \) expression, we can find the equilibrium concentrations. The initial minus the change gives the equilibrium concentration of formic acid \( (0.025 - x) \) M, and the change gives the equilibrium concentrations of the hydrogen and formate ions \( (x) \) M each. This process allows chemists to predict how much of an acid will dissociate and to calculate the resultant pH of the solution.