Problem 77 - General Chemistry: Principles and Modern Applications

Question

Carbonic acid is a weak diprotic acid \(\left(\mathrm{H}_{2} \mathrm{CO}_{3}\right)\) with \(K_{a_{1}}=4.43 \times 10^{-7}\) and \(K_{\mathrm{a}_{2}}=4.73 \times 10^{-11} .\) The equiv- alence points for the titration come at approximately pH 4 and 9. Suitable indicators for use in titrating carbonic acid or carbonate solutions are methyl orange and phenolphthalein. (a) Sketch the titration curve that would be obtained in titrating a sample of \(\mathrm{NaHCO}_{3}(\mathrm{aq})\) with \(1.00 \mathrm{M} \mathrm{HCl}\) (b) Sketch the titration curve for \(\mathrm{Na}_{2} \mathrm{CO}_{3}(\mathrm{aq})\) with 1.00 M HCl. (c) What volume of \(0.100 \mathrm{M} \mathrm{HCl}\) is required for the complete neutralization of \(1.00 \mathrm{g} \mathrm{NaHCO}_{3}(\mathrm{s}) ?\) (d) What volume of \(0.100 \mathrm{M} \mathrm{HCl}\) is required for the complete neutralization of \(1.00 \mathrm{g} \mathrm{Na}_{2} \mathrm{CO}_{3}(\mathrm{s}) ?\) (e) A sample of NaOH contains a small amount of \(\mathrm{Na}_{2} \mathrm{CO}_{3} .\) For titration to the phenolphthalein end point, \(0.1000 \mathrm{g}\) of this sample requires \(23.98 \mathrm{mL}\) of \(0.1000 \mathrm{M} \mathrm{HCl} .\) An additional \(0.78 \mathrm{mL}\) is required to reach the methyl orange end point. What is the percent \(\mathrm{Na}_{2} \mathrm{CO}_{3},\) by mass, in the sample?

Step-by-Step Solution

Answer

The titration curves shows pH decreases from basic to acidic. Volume of HCl required for complete neutralization of 1g of \(\mathrm{NaHCO}_{3}\) and \(\mathrm{Na}_{2}\mathrm{CO}_{3}\) calculated in step 3 and step 4 respectively. The percent of \(\mathrm{Na}_{2}\mathrm{CO}_{3}\) in the NaOH sample is as derived in Step 5.

Step 1: Sketch the titration curve for \(\mathrm{NaHCO}_{3}(\mathrm{aq})\)

In a weak base-strong acid titration of \(\mathrm{NaHCO}_{3}\) with \T1.00\T0 \mathrm{M} \mathrm{HCl}, the curve would start high (basic) and then drop down slowly till the half equivalence point. From there it would drop sharply (at the equivalence point) and then flatten out again, ending up at low pH (acidic).

Step 2: Sketch the titration curve for \(\mathrm{Na}_{2}\mathrm{CO}_{3}(\mathrm{aq})\)

The titration curve for a \(\mathrm{Na}_{2}\mathrm{CO}_{3}(\mathrm{aq})\) and \T1.00 \T0 \mathrm{M} \mathrm{HCl} titration would depict two equivalence points as sodium carbonate is a diprotic base and can accept two protons. Starting from basic, the pH decreases till the first equivalence point, then increases and decreases again till the second equivalence point, and finally ends in an acidic pH environment.

Step 3: Calculate volume of HCl for complete neutralization of \(\mathrm{NaHCO}_{3}(\mathrm{s})\)

Calculate the molar mass of \(\mathrm{NaHCO}_{3}\), which yields approximately \( 84 \,g/mol\). Then determine the moles of \(\mathrm{NaHCO}_{3}\) found in \(1.00\,g\) with the equation \(moles = mass/molar mass\). Afterwards, as each mole of \(\mathrm{NaHCO}_{3}\) reacts with one mole of HCl, the amount of moles of HCl needed for neutralization is the same. Using the molarity equation \(M = mol/L\), the required volume of HCl can be determined.

Step 4: Calculate volume of HCl for complete neutralization of \(\mathrm{Na}_{2}\mathrm{CO}_{3}(\mathrm{s})\)

Similar approach can be deployed as in Step 3. Here, the molar mass of \(\mathrm{Na}_{2}\mathrm{CO}_{3}\) is about \( 106 \,g/mol\). One mole of \(\mathrm{Na}_{2}\mathrm{CO}_{3}\) reacts with two moles of HCl, so the volume of HCl can be calculated using the molarity equation.

Step 5: Calculate the percent \(\mathrm{Na}_{2}\mathrm{CO}_{3}\)

The amount of NaOH is calculated by subtracting the volume of HCL required for titration to the phenolphthalein end point from the total volume. Molar mass of NaOH is used to calculate the grams of NaOH. The grams of \(\mathrm{Na}_{2}\mathrm{CO}_{3}\) is calculated from the known mass of the sample. Lastly, the percent of \(\mathrm{Na}_{2}\mathrm{CO}_{3}\) is calculated using the equation \(percent\, mass= (mass\, of\, \mathrm{Na}_{2}\mathrm{CO}_{3}/total\, mass) \times 100%\)

Key Concepts

Carbonic Acid

Carbonic acid ( \(\mathrm{H}_{2}\mathrm{CO}_{3} \)) is classified as a weak, diprotic acid. This means it can donate two protons ( \(\mathrm{H}^+ \)) during chemical reactions. The acidity strength is characterized by its dissociation constants, \(K_{a1} = 4.43 \times 10^{-7} \) and \(K_{a2} = 4.73 \times 10^{-11} \).
These values imply that carbonic acid partially ionizes in solution, with each ionization occurring at a different pH level.

• The first dissociation step, releasing the first \(\mathrm{H}^+ \), is considerably stronger, contributing to a slightly acidic solution.
• The second step, where the second \(\mathrm{H}^+ \) is released, occurs mainly under more basic conditions given its lower dissociation constant.

In nature, carbonic acid forms in solutions where carbon dioxide ( \(\mathrm{CO}_{2} \)) reacts with water. This reaction is a balanced act that's vital in maintaining pH in both natural waters and human blood.

Equivalence Point

In an acid-base titration, the equivalence point marks the stage where the amount of acid is stoichiometrically equal to the amount of base, resulting in complete neutralization.
This doesn't always mean that the solution is neutral with a pH of 7. The equivalent pH depends on the strength of the acids and bases involved.

• For carbonic acid, there are two equivalence points due to its diprotic nature.
• The first equivalence point occurs around pH 4, where primarily the first proton has reacted.
• The second equivalence point appears around pH 9, indicating the reaction involving the second proton.

These equivalence points are crucial in titration analyses, helping determine the correct indicators to use, such as methyl orange for acidic equivalence and phenolphthalein for more basic conditions.

Neutralization

Neutralization is the reaction between an acid and a base, resulting in water and a salt.
In context with carbonic acid titrations:

• Neutralization involves reactions between \(\mathrm{HCl} \), a strong acid, and \(\mathrm{NaHCO}_{3} \) or \(\mathrm{Na}_{2}\mathrm{CO}_{3} \), which are bases.
• This leads to the formation of \(\mathrm{CO}_{2} \) gas, water (\(\mathrm{H}_{2}\mathrm{O} \)), and respective salts like \(\mathrm{NaCl} \).

The complete neutralization for each base can be observed through the titration process:

• For \(\mathrm{NaHCO}_{3} \), complete neutralization with \(0.100 \mathrm{M} \mathrm{HCl} \) requires calculating the exact amount of acid needed based on the reaction ratio.
• With \(\mathrm{Na}_{2}\mathrm{CO}_{3} \), the stoichiometry is considered for two moles of \(\mathrm{HCl} \) per mole of \(\mathrm{Na}_{2}\mathrm{CO}_{3} \).

Understanding neutralization helps in predicting the outcomes and creating setups for experimenting with titrations.

Titration Curve

The titration curve is a graphical representation that shows how the pH of the solution changes as an acid is added to a base (or vice versa).
For carbonic acid-based titrations, these curves help visualize the weak acid and strong acid interactions:

• When titrating \(\mathrm{NaHCO}_{3} \), the curve starts at a basic pH, decreases slowly until the half-equivalence point due to buffering, and then drops sharply at the equivalence point.
• After this sharp drop, the curve will flatten as it approaches the acidic range, showing minimal change with additional \(\mathrm{HCl} \).
• The curve for \(\mathrm{Na}_{2}\mathrm{CO}_{3} \) involves two distinct dips corresponding to its two equivalence points.

These curves are vital in understanding the titration's progression, indicating precisely where neutralization occurs and where additional chemical changes are taking place. The shape and progression also help choose suitable indicators that signal different points within the titration process.