Question

Malonic acid \(\left(\mathrm{HO}_{2} \mathrm{CCH}_{2} \mathrm{CO}_{2} \mathrm{H}\right)\) is a diprotic acid. In the titration of malonic acid with NaOH, stoichiometric points occur at \(\mathrm{pH}=3.9\) and \(8.8 .\) A 25.00 -mL sample of malonic acid of unknown concentration is titrated with 0.0984 \(M \mathrm{NaOH},\) requiring \(31.50 \mathrm{mL}\) of the NaOH solution to reach the phenolphthalein end point. Calculate the concentration of the initial malonic acid solution. (See Exercise \(113 .\) )

Short answer

The concentration of the initial malonic acid solution is approximately 0.0619 M.

Step-by-step solution

Determine the moles of NaOH used to reach the second stoichiometric point

Since the problem states that the phenolphthalein endpoint occurs at the second stoichiometric point (pH = 8.8), we consider the volume of NaOH used up to that point. We can use the equation: Moles of NaOH = Molarity of NaOH × Volume of NaOH (in L) We are given that the molarity of NaOH is 0.0984 M and the volume used is 31.50 mL, which can be converted to liters as follows: 31.50 mL × (1 L / 1000 mL) = 0.03150 L. Substituting the given values, we get: Moles of NaOH = 0.0984 M × 0.03150 L = 0.0030972 mol

Calculate the moles of malonic acid

Since malonic acid is a diprotic acid, at the second stoichiometric point (pH 8.8), we can write that: Moles of Malonic Acid = Moles of NaOH / 2 Therefore, moles of malonic acid = 0.0030972 mol / 2 = 0.0015486 mol

Calculate the concentration of initial malonic acid solution

To find the concentration of the initial malonic acid solution, we need to divide the moles of malonic acid by the volume of the malonic acid solution (in L). Given, the volume of the malonic acid solution is 25.00 mL, which can be converted to liters as follows: 25.00 mL × (1 L / 1000 mL) = 0.02500 L Now, we can find the concentration of malonic acid using the equation: Concentration of malonic acid = Moles of malonic acid / Volume of malonic acid (in L) Thus, concentration of malonic acid = 0.0015486 mol / 0.02500 L = 0.061944 M So, the concentration of the initial malonic acid solution is approximately 0.0619 M.

Key concepts

Diprotic Acid

A diprotic acid is a type of acid that can donate two protons or hydrogen ions ( H⁺ ) in an aqueous solution. Malonic acid is a common example of this kind of acid, with the molecular formula HO₂CCH₂CO₂H . When it dissolves in water, it dissociates in two steps. In the first step, it donates one proton, and in the second step, the other. This makes the titration of a diprotic acid a bit more complex than a monoprotic acid, which can release only one proton.
The unique property of diprotic acids is that they can have two distinct pK_a decomposition constants, referring to the equilibrium constants for each ionization step. Consequently, this results in two potential stoichiometric or equivalence points in a titration, each representing the complete reaction of one of the protons with a base like NaOH. Understanding this concept is crucial in titrations involving diprotic acids.

Stoichiometric Points

In a titration, stoichiometric points are where the amount of titrant added is exactly enough to react with the reactant in the solution being titrated. For a diprotic acid like malonic acid, there are typically two stoichiometric points.

• The first stoichiometric point refers to the point where one molecule of acid has reacted with one molecule of base, neutralizing the first proton.
• The second stoichiometric point occurs when the second molecule of base completely neutralizes the second proton.

The relevance of these stoichiometric points is easily illustrated in the context of a pH change. For malonic acid, these points occur at pH 3.9 and pH 8.8. Knowing the pH at these specific points allows us to precisely determine how much base is required to reach each point. Stoichiometric points are essential when calculating the concentration of an unknown solution.

Phenolphthalein Endpoint

Phenolphthalein is a common pH indicator used in titrations, known for its distinctive color change. It transitions from colorless in acidic environments to pink in basic solutions, and is especially helpful in determining the endpoint of a titration.
In the titration of malonic acid, the phenolphthalein endpoint corresponds to the second stoichiometric point at a pH of 8.8. This is because phenolphthalein changes color around pH 8.2 to 10.0, making it suitable to detect the completion of the second neutralization step.

• Phenolphthalein is not ideal for the first stoichiometric point due to its high pH range of transition, which is above the first equivalent point.

Therefore, it's crucial to select the appropriate indicator based on the equivalence points' pH when working with polyprotic acids like malonic acid.

Concentration Calculation

Calculating the concentration of malonic acid from a titration involves a methodical approach. The key is using the stoichiometric relationship between the titrant and the analyte.
Firstly, determine the moles of NaOH used to reach the endpoint of the titration. This involves multiplying the volume of NaOH used, converted to liters, by its molarity:\[\text{Moles of NaOH} = \text{Molarity} \times \text{Volume (L)}\]Given that the titration reaches the phenolphthalein endpoint at the second stoichiometric point, calculate the moles of malonic acid by dividing the moles of NaOH by two (since malonic acid is diprotic).
Finally, find the concentration of the initial malonic acid solution by dividing the moles of malonic acid by its volume (also in liters):\[\text{Concentration} = \frac{\text{Moles of Malonic Acid}}{\text{Volume of Malonic Acid (L)}}\]This step-by-step approach ensures accurate determination of concentration in titration experiments.