Question

Adipic acid is used to make nylon. If the molar mass is \(147 \mathrm{~g} / \mathrm{mol}\) and the empirical formula is \(\mathrm{C}_{3} \mathrm{H}_{5} \mathrm{O}_{2}\), what is the molecular formula of the compound?

Short answer

The molecular formula is \(\text{C}_6\text{H}_{10}\text{O}_4\).

Step-by-step solution

Find Molar Mass of Empirical Formula

Calculate the molar mass of the empirical formula \(\text{C}_3\text{H}_5\text{O}_2\) by adding the atomic masses from the periodic table: Carbon (C) has an atomic mass of approximately \(12 \text{ g/mol}\), Hydrogen (H) is \(1 \text{ g/mol}\), and Oxygen (O) is \(16 \text{ g/mol}\). The calculation is as follows: \(3(12) + 5(1) + 2(16) = 36 + 5 + 32 = 73 \text{ g/mol}\).

Determine Ratio of Molar Masses

Divide the given molar mass of the compound by the calculated molar mass of the empirical formula. Use the formula \(n = \frac{\text{Molar mass of compound}}{\text{Molar mass of empirical formula}}\). Substitute the known values: \(n = \frac{147}{73} \approx 2\).

Calculate Molecular Formula

To find the molecular formula, multiply each subscript in the empirical formula by the integer obtained in Step 2 (which is 2). The empirical formula \(\text{C}_3\text{H}_5\text{O}_2\) becomes \(\text{C}_{(3 \times 2)}\text{H}_{(5 \times 2)}\text{O}_{(2 \times 2)} = \text{C}_6\text{H}_{10}\text{O}_4\).

Key concepts

Molar Mass

The molar mass of a compound is the mass of one mole of its entities, which could be atoms or molecules. It's an essential concept in chemistry for converting between mass and moles. Molar mass is expressed in grams per mole (g/mol). To find it, you sum up the atomic masses of each atom present in a molecule.

To illustrate, the molar mass of the empirical formula for adipic acid, \(\text{C}_3\text{H}_5\text{O}_2\), is calculated by adding the atomic masses:

• Carbon (C) has an atomic mass of about 12 g/mol.
• Hydrogen (H) has an atomic mass of 1 g/mol.
• Oxygen (O) has an atomic mass of 16 g/mol.

Altogether, three carbons, five hydrogens, and two oxygens add up to \(3(12) + 5(1) + 2(16) = 73 \text{ g/mol}\). Thus, understanding and calculating molar mass helps dive deeper into exploring chemical compounds and their properties. Make sure you use atomic masses accurate to their decimals from the Periodic Table for more precision.

Empirical Formula

The empirical formula of a compound represents the simplest whole-number ratio of the atoms of each element in the compound. It's a crucial building block for understanding the structure of a molecule, as it gives insight into the basic composition without showing the actual number of atoms.

For adipic acid, the empirical formula is \(\text{C}_3\text{H}_5\text{O}_2\), meaning that for every three carbon atoms, there are five hydrogen atoms and two oxygen atoms. This acts as a fundamental stepping stone in determining the molecular formula. To transform the empirical formula into the molecular formula, you must know the molar mass of the compound and compare it to that of the empirical formula.

The molecular formula could be a multiple of the empirical formula, making this conversion dependent upon the compound's determined molar mass. Hence, empirical formulas are a vital part of stoichiometry, as they provide the groundwork for all further chemical calculations.

Atomic Mass

Atomic mass is a measure of the mass of an atom, typically expressed in atomic mass units (amu). It is approximately equivalent to the number of protons and neutrons in the atom, as electrons have negligible mass compared to protons and neutrons.

This concept is fundamental to calculating molar mass. For adipic acid with an empirical formula \(\text{C}_3\text{H}_5\text{O}_2\), understanding the atomic mass of each constituent atom is key. Carbon has an atomic mass close to 12 amu, hydrogen about 1 amu, and oxygen approximately 16 amu. Knowing these values allows chemists to sum them up for each atom in a molecule and compute the overall molar mass.

Additionally, atomic mass is crucial in finding out reactions and ratios in chemical equations. By understanding the atomic masses, determining the molar ratios using the empirical formula and thereby deriving the molecular formula becomes feasible in chemical analyses.