Question

Calcium Lactate is a salt of weak acid and represented as \(\mathrm{Ca}(\mathrm{Lac})_{2} .\) A saturated solution of \(\mathrm{Ca}(\mathrm{Lac})_{2}\) contains \(0.125 \mathrm{~mole}\) of salt in \(0.50 \mathrm{~L}\) solution. The \(\mathrm{pOH}\) of this is 5.60. Assuming complete dissociation of salt, calculate \(K_{\mathrm{a}}\) of lactate acid. \((\log 2.5=0.4)\) (a) \(1.25 \times 10^{-11}\) (b) \(8.0 \times 10^{-4}\) (c) \(3.2 \times 10^{-17}\) (d) \(4 \times 10^{-5}\)

Short answer

The \(K_{a}\) of lactate acid is \(1.25 \times 10^{-11}\).

Step-by-step solution

Calculate the Concentration of Lactate Ion

Since calcium lactate (\(\mathrm{Ca(Lac)}_2\)) completely dissociates in solution, it forms one calcium ion and two lactate ions. This means the concentration of lactate ion is twice the concentration of calcium lactate. Calculate the concentration of lactate anions by multiplying the moles of salt (\(0.125\) mole) by 2 and dividing by the volume of the solution (\(0.50\) L).

Convert pOH to Hydroxide Ion Concentration

The pOH is given as 5.60. Convert this to hydroxide ion concentration (\([OH^-]\)) using the formula \(pOH = -\log[OH^-]\). Take the antilog (inverse log) of the negative pOH to find the hydroxide ion concentration.

Calculate the Hydrogen Ion Concentration

The water dissociation constant (\(K_w\)) is \(1 \times 10^{-14}\) at 25°C. Use the relation \(K_w = [H^+][OH^-]\) to find the concentration of hydrogen ions (\([H^+]\)).

Calculate the Ka of Lactic Acid

Since lactate (Lac) is the conjugate base of lactic acid, the acid dissociation constant (\(K_a\)) for lactic acid can be found using the relationship \(K_a = \frac{K_w}{K_b}\). The \(K_a\) value considering the concentration of lactate ions and hydrogen ions.

Solve for the Numerical Value of Ka

The base dissociation constant (\(K_b\)) of lactate can be calculated using \(K_b = \frac{[OH^{-}][H^{+}]}{[Lac^{-}]}\). Then use this to solve for \(K_a\).

Key concepts

Lactate Ion Concentration

Understanding the concentration of lactate ions in a solution is a key concept in acid-base chemistry and is especially important when dealing with salts of weak acids, such as calcium lactate. When a salt like calcium lactate dissociates in water, it breaks down into its constituent ions. For every mole of calcium lactate, you get one mole of calcium ions and two moles of lactate ions because the chemical formula of calcium lactate is \(\mathrm{Ca(Lac)}_2\).

To calculate the lactate ion concentration after dissociation, you simply take the initial concentration of the calcium lactate and multiply it by two, because each formula unit produces two lactate ions upon dissociation. In a solution where the initial concentration of calcium lactate is given, this step is essential to finding the molar concentration of lactate ions, which is further used to determine the acid dissociation constant (\(K_a\)) of lactic acid.

pOH Conversion

The pOH of a solution is a measure of the alkalinity and is related to the concentration of hydroxide ions (\([OH^-]\)). The formula to convert pOH to the hydroxide ion concentration is \(pOH = -\log[OH^-]\). If you know the pOH of a solution, you can determine the hydroxide ion concentration by taking the antilog of the negative pOH value. This concept is critical as it links a measurable quantity, pOH, to the actual concentration of hydroxide ions in solution, a key factor in the acid-base balance of the system.

Hydroxide Ion Concentration

The hydroxide ion concentration in a solution is a direct indicator of its basicity. From the pOH, as previously discussed, you can calculate the hydroxide ion concentration. Knowing the concentration of hydroxide ions is essential as it allows you to compute the concentration of hydrogen ions via the water dissociation constant, if the temperature is constant. This concentration is a cornerstone measurement for various calculations in solving acid-base equilibrium problems, and it is particularly instrumental when you are working with solutions of salts from weak acids and bases where partial dissociation occurs.

Water Dissociation Constant

The water dissociation constant (\(K_w\)) is a fundamental concept in acid-base chemistry. It signifies the level at which water naturally dissociates into hydrogen and hydroxide ions. The constant value is \(1 \times 10^{-14}\) at 25°C and is the product of the concentrations of these two ions in pure water. In any aqueous solution at this temperature, the product of the hydrogen ion concentration and the hydroxide ion concentration must equal to \(K_w\). This relation is crucial for finding either ion's concentration when the other is known and is directly applied in problems involving acid and base dissociation constants.

Base Dissociation Constant

The base dissociation constant (\(K_b\)) is the equilibrium constant for the dissociation of a base into its conjugate acid and hydroxide ion in water. It is used to measure the strength of a base; the larger the value, the stronger the base. For weak bases, like lactate, this constant allows us to calculate how much of the base exists in its ionized form at equilibrium.

When dealing with salts of weak acids, such as lactate, the \(K_b\) can be determined using the hydroxide ion concentration and the concentration of the base, which in this case is the lactate ion. Finding the \(K_b\) for lactate enables us to subsequently calculate the \(K_a\) of its conjugate acid, lactic acid, through the relationship \(K_a = \frac{K_w}{K_b}\), where \(K_w\) is the water dissociation constant. The knowledge of \(K_a\) and \(K_b\) helps to predict and explain the behavior of acids and bases in various chemical reactions.