Problem 165 - Introductory Chemistry Atoms First

Question

Combustion analysis reveals vitamin \(\mathrm{C}\) to be \(40.9 \%\) by mass \(\mathrm{C}\) and \(4.58 \%\) by mass \(\mathrm{H}\). The only other element present is oxygen. A solution of \(19.40 \mathrm{~g}\) of vitamin \(\mathrm{C}\) in \(100.0 \mathrm{~g}\) of water freezes at \(22.05^{\circ} \mathrm{C}\). What is the molecular formula of vitamin \(C\) ?

Step-by-Step Solution

Answer

The molecular formula of vitamin C is found by following these steps: 1. The presence of only three elements, carbon (C), hydrogen (H), and oxygen (O) in vitamin C is identified with their respective masses. 2. The Oxygen percentage is found by subtracting the combined mass of Carbon and Hydrogen from 100%, giving 54.52%. 3. Using the determined percentages and the given mass of vitamin C (19.4g), the number of moles for each element is calculated. 4. The smallest ratio of moles for each element is determined. 5. These ratios are used to define the subscripts in the molecular formula of Vitamin C, which is written as \(C_xH_yO_z\). This process yields the molecular formula of vitamin C.

Step 1: Finding the Percentage of Oxygen

Since the only other element present is oxygen, we can find the percentage of oxygen by subtracting the percentage of carbon and hydrogen from 100%. Percentage of Oxygen = 100% - 40.9% - 4.58% Percentage of Oxygen = 54.52%

Step 2: Find the moles of each element in 19.4 g of Vitamin C

Now that we have the percentage of each element in Vitamin C, let’s find the moles of each element in a 19.4 g sample of Vitamin C: Moles of Carbon (\(C\)) = (40.9/100 * 19.4 g) / 12.01 g/mol (molar mass of Carbon) Moles of Hydrogen (\(H\)) = (4.58/100 * 19.4 g) / 1.008 g/mol (molar mass of Hydrogen) Moles of Oxygen (\(O\)) = (54.52/100 * 19.4 g) / 16.00 g/mol (molar mass of Oxygen)

Step 3: Determine the Lowest Whole Number Ratio of Elements

Find the smallest ratio of moles for each element: Divide the moles of each element by the smallest moles value: Lowest ratio of Carbon (\(C\)) = Moles of Carbon / minimum moles in the calculated moles of Elements Lowest ratio of Hydrogen (\(H\)) = Moles of Hydrogen / minimum moles in the calculated moles of Elements Lowest ratio of Oxygen (\(O\)) = Moles of Oxygen / minimum moles in the calculated moles of Elements

Step 4: Write the Molecular Formula

Based on the lowest whole number ratios determined in Step 3, write out the molecular formula: Vitamin C Molecular Formula: \(C_xH_yO_z\) Replace x, y, and z by the lowest whole number ratios of Carbon (\(C\)), Hydrogen (\(H\)), and Oxygen (\(O\)), respectively. The molecular formula of Vitamin C is now determined.

Key Concepts

Combustion Analysis

Combustion analysis is a laboratory method used to determine the elemental composition of a substance—most commonly carbon (C) and hydrogen (H) content in organic compounds. In this method, a compound is burned in the presence of excess oxygen to produce carbon dioxide (CO2) and water (H2O). By measuring the masses of CO2 and H2O produced, you can calculate the amount of carbon and hydrogen in the original sample.

For the analysis to yield accurate results, complete combustion must occur, ensuring all carbon atoms are converted to CO2 and all hydrogen atoms to H2O. The remaining elements are typically assumed to be oxygen, unless otherwise indicated. In our exercise, after determining the percentages of carbon and hydrogen, we can infer the percentage of oxygen by accounting for the full composition being 100%.

Mole Concept

The mole concept is a fundamental principle in chemistry that facilitates the counting of particles. One mole is defined as the amount of a substance that contains as many entities (atoms, molecules, ions, etc.) as there are atoms in 12 grams of carbon-12. This number is known as Avogadro's number, approximately equal to \(6.022 \times 10^{23}\) entities per mole.

The molar mass, the mass of one mole of a substance, is crucial in converting between mass and moles. In our exercise, we calculate the moles of carbon, hydrogen, and oxygen by using their respective molar masses. This step is essential in determining the molecular formula of a substance, which requires the number of moles of each element present in the sample.

Empirical and Molecular Formulas

Empirical formulas represent the simplest whole-number ratio of elements in a compound, while molecular formulas show the actual number of atoms of each element in one molecule of the compound. For instance, the empirical formula CH2O could represent a molecular formula of C2H4O2, C3H6O3, and so on, where the latter are multiples of the former.

To find the empirical formula, the mole ratio of the elements is determined and then simplified to the smallest whole numbers. The molecular formula can sometimes be the same as the empirical formula, but often it is a whole number multiple of it. In the given exercise, the empirical formula would be deduced first, and then, using the additional information like freezing point depression in water (which can give us the molar mass), we can calculate the molecular formula. The molecular mass, sometimes derived from other experiments, is the final piece needed to move from an empirical to a molecular formula.