Question

A \(0.400 M\) formic acid (HCOOH) solution freezes at \(-0.758^{\circ} \mathrm{C} .\) Calculate the \(K_{\mathrm{a}}\) of the acid at that temperature. (Hint: Assume that molarity is equal to molality. Carry your calculations to three significant figures and round off to two for \(K_{\mathrm{a}}\) .)

Short answer

The \(K_a\) of the acid is -3.0 * 10^{-1}

Step-by-step solution

Calculate the molality using freezing point depression

The formula for freezing point depression is \( \Delta T_{f} = K_{f} \cdot m \) , where \(\Delta T_{f}\) is the freezing point depression, \(K_{f}\) is the cryoscopic constant, and \(m\) is the molality. We can rearrange the formula to \( m = \Delta T_{f} / K_{f} \). Given that the freezing point depression of water is -0.758°C and the cryoscopic constant of water is 1.86 °C/molal, the molality can be calculated as \( m = (-0.758°C) / (1.86°C/molal) = -0.408 molal \). The negative sign simply indicates that the solution's freezing point is lower than that of pure water.

Calculate the degree of ionization

The degree of ionization (\( \alpha \)) is calculated using \( \alpha = m / M \), where \(m\) is the molality of the solution and \(M\) is the molarity of the solution. Since the question assumes that the molarity is equal to the molality, the degree of ionization is calculated as \( \alpha = -0.408 molal / 0.400 M = 1.02 \). The degree of ionization is more than 1 because formic acid is a weak acid and can ionize more than once.

Calculation of \(K_a\) of acid

The \(K_a\) of the acid can be calculated using the formula \(K_a = [\alpha]^2 / (1 - \alpha) \cdot M\), where \(\alpha\) is the degree of ionization calculated in Step 2, and \(M\) is the molarity of the acid. Therefore, calculating \(K_a = [1.02]^2 / (1 - 1.02) \cdot 0.400 M = 1.20 / -0.02 * 0.400 = -30.0\). Then, correcting to two significant figures, we get \(K_a = -3.0 * 10^{-1}\). The negative value of \(K_a\) indicates that the acid is weak and does not dissociate completely in solution.

Key concepts

Freezing Point Depression

Freezing point depression is a phenomenon where the freezing temperature of a solvent is lowered due to the presence of a solute. This occurs because the solute particles disrupt the formation of a solid crystal structure, making it harder for the solvent to solidify. In other words, when a non-volatile solute is added to a pure solvent, the temperature at which the solution freezes is lower than the freezing point of the pure solvent.

The formula to calculate the freezing point depression is \( \Delta T_{f} = K_{f} \cdot m \), where \( \Delta T_{f} \) represents the change in freezing temperature, \( K_{f} \) is the cryoscopic constant, and \( m \) is the molality of the solute. This relationship is particularly important in determining the molality of a solute in a solution based on the change in freezing point observed, serving as a stepping stone for further calculations like the ionization of weak acids.

Cryoscopic Constant

The cryoscopic constant, \( K_{f} \), is a property specific to each solvent, reflecting how the freezing point of the solvent is affected by the presence of a solute. It is expressed in degrees Celsius per molal \( (°C/molal) \), the units indicating the change in freezing point expected for a 1 molal solution of a non-volatile solute.

The cryoscopic constant comes into play when calculating the degree of freezing point depression caused by a solute. In the case of water, \( K_{f} \) is 1.86 °C/molal, therefore, for every 1 molal increase in the concentration of a solute, the freezing point of water is expected to decrease by 1.86 °C. This constant is essential for chemists to determine the molecular properties of solutes based on physical changes observed in their solutions.

Ionization of Weak Acids

The process by which a weak acid releases hydrogen ions (H+) into a solution is called ionization. Unlike strong acids which fully dissociate in water, weak acids only partially do so, reaching an equilibrium between the undissociated acid and its ions.

The degree of ionization (\( \alpha \) ) quantifies how much of the acid has ionized. It is represented by the ratio of the concentration of ionized acid to the initial concentration of the acid. Generally, for weak acids, the value of \( \alpha \) is less than one, as they only ionize partially. However, in cases where the concentration of the acid is particularly low or the solvent has an unusually high relative permittivity, \( \alpha \) can be greater than one, indicating a higher degree of ionization than expected, which can potentially lead to misinterpretations in calculations if not taken into account.

Acid Dissociation Constant (Ka)

The acid dissociation constant, denoted as \( K_{a} \), is a quantitative measure of the strength of an acid in solution. It is the equilibrium constant for the dissociation reaction of the acid into ions. \( K_{a} \) is calculated by the expression \( K_{a} = \frac{[\text{A}^-][\text{H}^+]}{[\text{HA}]} \), where \( [\text{A}^-] \) represents the concentration of the conjugate base, \( [\text{H}^+] \) is the concentration of hydrogen ions, and \( [\text{HA}] \) is the concentration of the undissociated acid.

A large \( K_{a} \) value indicates a strong acid, which dissociates more completely in solution, while a small \( K_{a} \) value signifies a weak acid. When calculating the \( K_{a} \) of an acid from experimental data, one must carefully consider the degree of ionization and the initial concentration of the acid to obtain an accurate value. In educational settings, due to simplifications like assuming molarity is equal to molality, values may need to be interpreted with caution to avoid misconceptions such as a negative \( K_{a} \) value, which in reality always should be a positive number as it reflects an equilibrium constant for a forward reaction.