Problem 96 - Chemistry and Chemical Reactivity

Question

A noncarbonated soft drink contains an unknown amount of citric acid, \(\mathrm{H}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7} .\) If \(100 .\) mL of the soft drink requires \(33.51 \mathrm{mL}\) of \(0.0102 \mathrm{M} \mathrm{NaOH}\) to neutralize the citric acid completely, what mass of citric acid does the soft drink contain per \(100 .\) mL? The reaction of citric acid and \(\mathrm{NaOH}\) is \mathrm{H}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}(\mathrm{aq})+3 \mathrm{NaOH}(\mathrm{aq}) \longrightarrow \mathrm{Na}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}(\mathrm{aq})+3 \mathrm{H}_{2} \mathrm{O}(\ell)

Step-by-Step Solution

Answer

The soft drink contains approximately 0.0219 grams of citric acid per 100 mL.

Step 1: Write the Balanced Chemical Equation

The balanced chemical equation is already provided:\[\mathrm{H}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}(\mathrm{aq}) + 3 \mathrm{NaOH}(\mathrm{aq}) \longrightarrow \mathrm{Na}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}(\mathrm{aq}) + 3 \mathrm{H}_{2} \mathrm{O}(\ell)\]This reaction indicates that one mole of citric acid reacts with three moles of \(\mathrm{NaOH}\).

Step 2: Determine Moles of NaOH Used

Use the volume and molarity of the \(\mathrm{NaOH}\) solution to find the number of moles of \(\mathrm{NaOH}\) used:\[\text{Moles of } \mathrm{NaOH} = \text{Volume (L)} \times \text{Molarity} = 0.03351 \times 0.0102 \approx 0.000341602 \,\text{moles}\]

Step 3: Calculate Moles of Citric Acid

According to the reaction stoichiometry, 1 mole of citric acid reacts with 3 moles of \(\mathrm{NaOH}\). Therefore, divide the moles of \(\mathrm{NaOH}\) by 3 to find the moles of citric acid:\[\text{Moles of citric acid} = \frac{0.000341602}{3} \approx 0.000113867 \,\text{moles}\]

Step 4: Calculate Mass of Citric Acid

The molar mass of citric acid \(\mathrm{H}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}\) is approximately 192.13 g/mol. Use the moles of citric acid to find its mass:\[\text{Mass of citric acid} = \text{Moles} \times \text{Molar Mass} = 0.000113867 \times 192.13 \approx 0.02187 \, \text{grams}\]

Step 5: Conclusion

The mass of citric acid in 100 mL of the soft drink is approximately 0.0219 grams.

Key Concepts

Balanced Chemical Equation

When dealing with chemical reactions, it is crucial to understand the role of a balanced chemical equation as it provides a representation of the reaction.
The equation for the reaction between citric acid and sodium hydroxide (NaOH) is:\[ \mathrm{H}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}(\mathrm{aq}) + 3 \mathrm{NaOH}(\mathrm{aq}) \longrightarrow \mathrm{Na}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}(\mathrm{aq}) + 3 \mathrm{H}_{2} \mathrm{O}(\ell) \]This equation tells us that:

• One molecule of citric acid reacts with three molecules of NaOH.
• The products are sodium citrate and water.

A balanced equation ensures that the number of atoms for each element is the same on both sides of the equation. This conservation of atoms reflects the conservation of mass and is fundamental for correctly analyzing the reaction and its stoichiometry.

Stoichiometry

Stoichiometry is all about the quantitative relationships between the reactants and products in a chemical reaction.
In our equation, stoichiometry tells us that every one mole of citric acid will completely react with three moles of NaOH.
Here's how it applies in our problem:

• We start with the moles of NaOH calculated using its volume and molarity.
• Dividing the moles of NaOH by 3, we find the amount of citric acid involved because of the 1:3 ratio in the balanced equation.

By understanding the stoichiometry of the reaction, you can determine the quantities needed for complete reactions, which is the key to finding the citric acid's mass in the soft drink.

Molarity

Molarity (M) is a measure of the concentration of a solution in terms of moles of solute per liter of solution.
In the context of titration, it is used to figure out how many moles of a reactant are present in a given volume of solution.
For this exercise:

• The concentration of NaOH is given as 0.0102 M.
• The volume of NaOH used is 33.51 mL, which can be converted to liters by dividing by 1000 (0.03351 L).
• The product of these two gives you the moles of NaOH, which is a critical step in the titration calculation.

Understanding molarity allows you to calculate the amount of a substance in a solution and is vital in all sorts of chemical calculations.

Molar Mass

Molar mass is the mass of one mole of a substance, often expressed in grams per mole (g/mol).
For citric acid, the molar mass is calculated based on its chemical formula. - Citric acid (\(\mathrm{H}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}\)) has a molar mass of approximately 192.13 g/mol.Once you have determined the number of moles of a substance (like citric acid) through stoichiometry:

• Multiply the number of moles by the molar mass to get the mass of the substance.
• In our problem, the moles of citric acid were multiplied by its molar mass to find that the soft drink contains about 0.0219 grams of citric acid per 100 mL.

The concept of molar mass links the microscopic scale of atoms and molecules to the macroscopic scale of grams that we can measure in the lab.