Problem 106 - Chemistry: The Central Science

Question

(a) By titration, \(15.0 \mathrm{~mL}\) of \(0.1008 \mathrm{M}\) sodium hydroxide is needed to neutralize a \(0.2053-\mathrm{g}\) sample of a weak acid. What is the molar mass of the acid if it is monoprotic? (b) An elemental analysis of the acid indicates that it is composed of \(5.89 \% \mathrm{H}, 70.6 \% \mathrm{C}\), and \(23.5 \% \mathrm{O}\) by mass. What is its molecular formula?

Step-by-Step Solution

Answer

The molar mass of the weak monoprotic acid is 135.74 g/mol. Its molecular formula is \(\ce{C8H8O2}\).

Step 1: Calculate moles of NaOH

Since we know the volume (\(V = 15.0 \mathrm{~mL}\)) and molarity (\(0.1008 \mathrm{M}\)) of the sodium hydroxide used in the titration, we can calculate how many moles of NaOH were used: Moles of NaOH = Molarity × Volume = \(0.1008\;\mathrm{M} \times 0.0150\;\mathrm{L} = 0.001512\;\mathrm{moles}\)

Step 2: Determine moles of the weak acid

As the acid is monoprotic (provides only one H+ ion), it reacts with NaOH in 1:1 ratio. Thus, the moles of the weak acid present in the 0.2053 g sample is also 0.001512 moles.

Step 3: Calculate the molar mass of the acid

Since we know the mass and moles of the acid, we can find its molar mass by dividing the mass by the number of moles: Molar mass of the acid = \(\frac{0.2053 \; \mathrm{g}}{0.001512 \; \mathrm{moles}} = 135.74 \; \mathrm{g/mol}\) (rounded to two decimal points)

Step 4: Find the empirical formula of the acid

Using the mass percentages given, let's assume we have a 100 g sample of the weak acid. In this case, we would have: - 5.89 g of H - 70.6 g of C - 23.5 g of O Next, we need to convert these masses into moles: Moles of H = \(\frac{5.89 \; \mathrm{g}}{1.00784 \; \mathrm{g/mol}} = 5.84 \; \mathrm{moles}\) Moles of C = \(\frac{70.6 \; \mathrm{g}}{12.0107 \; \mathrm{g/mol}} = 5.88 \; \mathrm{moles}\) Moles of O = \(\frac{23.5 \; \mathrm{g}}{15.9994 \; \mathrm{g/mol}} = 1.47 \; \mathrm{moles}\) Now, we need to find the simplest ratio of these moles by dividing by the smallest value: Ratio of moles: H = \(\frac{5.84}{1.47} \approx 4\); C = \(\frac{5.88}{1.47} \approx 4\); O = \(\frac{1.47}{1.47}\approx 1\) So, the empirical formula is \(\ce{C4H4O}\) with a mass of 68.08 g/mol.

Step 5: Find the molecular formula of the acid

Now that we have the empirical formula and its mass, we can use the molar mass we calculated in Step 3 to find the molecular formula. We need to find the multiple of the empirical mass that gives the molar mass of the acid: Multiplier = \(\frac{135.74 \; \mathrm{g/mol}}{68.08 \; \mathrm{g/mol}} \approx 2\) Now, we multiply the empirical formula by this integer: Molecular formula = \(2 \times (\ce{C4H4O}) = \ce{C8H8O2}\) So, the weak acid's molecular formula is \(\ce{C8H8O2}\).

Key Concepts

Understanding Titration

Titration is a laboratory method used to determine the concentration of a solution by reacting it with a solution of known concentration, called a titrant. In a typical titration, the titrant is added to a sample until the reaction reaches an endpoint, which is often indicated by a color change due to an indicator or by reaching a known pH value.

For instance, if a known volume of a base, such as sodium hydroxide (NaOH), is used to neutralize an acid, we can deduce the amount of acid present in a solution. When the neutralization reaction is complete, the number of moles of the base will be equal to the number of moles of the monoprotic acid, assuming a 1:1 mole ratio between the acid and base. By knowing the volume and molarity of the titrant, one can calculate the moles of substance in the sample and, consequently, other important properties, such as its molar mass.

The Role of a Monoprotic Acid in Titration

A monoprotic acid is an acid that can donate only one proton (hydrogen ion) per molecule to a base during a chemical reaction. This characteristic simplifies the stoichiometry of a titration, as the acid will react with a base in a one-to-one molar ratio.

In the given exercise, the monoprotic nature of the acid ensures that each mole of NaOH neutralizes exactly one mole of the acid. Understanding that the reaction is a 1:1 ratio allows us to accurately calculate the molar mass of the acid by measuring the amount of a standard NaOH solution required to neutralize a known mass of the acid.

Determining an Empirical Formula

The empirical formula represents the simplest whole number ratio of the elements in a compound. It is determined by converting the mass percentages of each element into moles and then finding the simplest whole number ratio of these moles.

For example, by assuming a 100 g sample based on the percentage composition given, one can convert the mass of each element to moles and simplify the mole ratio. It's important to use the atomic mass of each element, expressed in g/mol, to make this conversion. A clear understanding of the mole concept is critical here, as it helps to translate the mass of elements into the empirical formula that reflects the actual number ratio of atoms in a sample of the compound.

Calculating the Molecular Formula

Once the empirical formula is known, the molecular formula can often be determined if the molar mass of the compound is also known. The molecular formula is the formula that represents the actual number of atoms of each element in a molecule of the compound and may be a multiple of the empirical formula.

In the case of the weak acid in our exercise, the empirical formula determined was (C4H4O), with a corresponding mass of 68.08 g/mol. The molecular formula can be found by comparing the empirical formula mass with the molar mass calculated from the titration experiment. By dividing the molar mass of the compound by the mass of the empirical formula, one obtains a factor that, when multiplied by the empirical formula, yields the molecular formula. This final step provides insight into the actual structure and composition of the compound in question.