Problem 91 - Chemistry The Molecular Science

Question

Quinine (molar mass \(=324.41 \mathrm{~g} / \mathrm{mol}\) ) is used as a cardiac depressant. It has this percent composition by mass: \(74.04 \% \mathrm{C} ; 7.46 \% \mathrm{H} ; 8.64 \% \mathrm{~N} ;\) and \(9.86 \%\) O. Use these data to determine (a) the empirical formula and (b) the molecular formula of quinine.

Step-by-Step Solution

Answer

(a) Empirical formula: \( \text{C}_{10}\text{H}_{12}\text{N}\text{O} \); (b) Molecular formula: \( \text{C}_{20}\text{H}_{24}\text{N}_2\text{O}_2 \).

Step 1: Convert Percent Composition to Mass

For empirical formula calculations, assume you have 100 grams of the compound. This means: - Carbon (C): 74.04 grams - Hydrogen (H): 7.46 grams - Nitrogen (N): 8.64 grams - Oxygen (O): 9.86 grams

Step 2: Convert Mass to Moles

Use the molar mass of each element to convert the mass to moles:- Moles of C: \( \frac{74.04 \text{ g}}{12.01 \text{ g/mol}} = 6.16 \text{ mol} \)- Moles of H: \( \frac{7.46 \text{ g}}{1.008 \text{ g/mol}} = 7.40 \text{ mol} \)- Moles of N: \( \frac{8.64 \text{ g}}{14.01 \text{ g/mol}} = 0.617 \text{ mol} \)- Moles of O: \( \frac{9.86 \text{ g}}{16.00 \text{ g/mol}} = 0.616 \text{ mol} \)

Step 3: Determine the Simplest Whole Number Ratio

Divide each of the mole values by the smallest number of moles calculated:- C: \( \frac{6.16}{0.616} \approx 10 \)- H: \( \frac{7.40}{0.616} \approx 12 \)- N: \( \frac{0.617}{0.616} \approx 1 \)- O: \( \frac{0.616}{0.616} = 1 \)The empirical formula is \( \text{C}_{10}\text{H}_{12}\text{N}\text{O} \).

Step 4: Calculate the Empirical Formula Weight

Calculate the molar mass of the empirical formula \( \text{C}_{10}\text{H}_{12}\text{N}\text{O} \):- Carbon: \( 10 \times 12.01 = 120.1 \text{ g/mol} \)- Hydrogen: \( 12 \times 1.008 = 12.096 \text{ g/mol} \)- Nitrogen: \( 1 \times 14.01 = 14.01 \text{ g/mol} \)- Oxygen: \( 1 \times 16.00 = 16.00 \text{ g/mol} \)Total: \( 162.206 \text{ g/mol} \)

Step 5: Determine the Molecular Formula

Divide the given molar mass of quinine by the empirical formula weight:\( \frac{324.41 \text{ g/mol}}{162.206 \text{ g/mol}} \approx 2 \).Multiply all the subscripts in the empirical formula by this factor:The molecular formula is \( \text{C}_{20}\text{H}_{24}\text{N}_2\text{O}_2 \).

Key Concepts

Molecular Formula

The molecular formula of a compound tells us the exact number of atoms of each element present in a single molecule of the compound. To determine the molecular formula, one must know both the empirical formula and the molar mass of the compound. The empirical formula provides the simplest whole number ratio of the elements, while the molar mass allows us to scale the empirical formula to obtain the molecular formula.
To find the molecular formula from the empirical formula, you follow these steps:

• Calculate the empirical formula weight, which is the sum of the atomic masses of all atoms in the empirical formula.
• Divide the molar mass of the compound by the empirical formula weight to obtain a multiplying factor.
• Multiply the subscripts in the empirical formula by this factor to get the molecular formula.

In the case of quinine, the empirical formula is \(\text{C}_{10}\text{H}_{12}\text{N}\text{O}\) with an empirical formula weight of 162.206 g/mol. Given the molar mass of quinine is 324.41 g/mol, the factor is approximately 2, leading to the molecular formula \(\text{C}_{20}\text{H}_{24}\text{N}_2\text{O}_2\).

Percent Composition

Percent composition by mass is an important concept in chemistry that describes the percentage of a specific element's mass in the total mass of a compound. This information is crucial as it helps in determining the empirical formula of the compound, which is the simplest representation of the elements present.
Here's how you can interpret and use percent composition:

• Assume a 100 gram sample of the compound to simplify calculations. This makes the percent of each element equal to the grams of that element in the sample.
• Convert these masses to moles using the element's molar mass, allowing you to determine the mole ratio of the elements.
• The simplest whole number ratio of these moles gives the empirical formula, providing insight into the compound's composition.

In the quinine example, we calculated the following percent compositions: Carbon (74.04%), Hydrogen (7.46%), Nitrogen (8.64%), and Oxygen (9.86%). Using these percentages, you can find the numbers of moles for each element and thus determine the compound's empirical formula.

Molar Mass

Molar mass is a fundamental concept in chemistry, representing the mass of one mole of a substance. It is usually expressed in units of grams per mole (g/mol) and is essential for calculating relationships in chemical reactions and conversions from mass to moles and vice versa.
To calculate the molar mass of a compound:

• Add together the atomic masses of all the atoms in the molecule based on the periodic table. Each element's atomic mass is typically given under the element symbol.
• Multiply the atomic mass of each element by its subscripted number in the formula (showing the number of atoms of that element), then sum all these values to get the compound's molar mass.

Understanding DNA's molar mass helps convert between moles and grams, which is paramount for stoichiometric calculations and determining molecular formulas. In the provided quinine problem, the calculated empirical formula weight (molar mass of the empirical formula) helps to figure out the molecular formula of quinine by comparing it with the chemical's known molar mass.