Question

Terephthalic acid is an important chemical used in the manufacture of polyesters and plasticizers. It contains only C, \(\mathrm{H}\), and \(\mathrm{O}\). Combustion of \(19.81 \mathrm{mg}\) terephthalic acid produces \(41.98 \mathrm{mg} \mathrm{CO}_{2}\) and \(6.45 \mathrm{mg} \mathrm{H}_{2} \mathrm{O}\). If \(0.250 \mathrm{~mole}\) of terephthalic acid has a mass of \(41.5 \mathrm{~g}\), determine the molecular formula for terephthalic acid.

Short answer

The molecular formula for terephthalic acid is C₁₀H₁₀O₅.

Step-by-step solution

Find the mass of Carbon, Hydrogen, and Oxygen in the sample.

We know the mass of CO2 and H2O produced as a result of the combustion reaction. One mole of CO2 contains one mole of Carbon, and one mole of H2O contains two moles of Hydrogen. We can use this information to find the mass of C and H in the sample and then, by difference, find the mass of oxygen. Given mass of CO2 = 41.98 mg Molar mass of CO2 = 12.01 g/mol (C) + 2(16.00 g/mol) (O) = 44.01 g/mol Given mass of H2O = 6.45 mg Molar mass of H2O = 2(1.01 g/mol) (H) + 16.00 g/mol (O) = 18.02 g/mol Now the number of moles CO2 = \( \frac{41.98 \,\text{mg}}{44.01 \frac{\text{g}}{\text{mol}}}\times \frac{1\,\text{g}}{1000\,\text{mg}} \) = 0.0010 moles Likewise, Number of moles of H2O = \( \frac{6.45 \,\text{mg}}{18.02 \frac{\text{g}}{\text{mol}}}\times \frac{1\,\text{g}}{1000\,\text{mg}} \) = 0.00036 moles Total mass of the sample = 19.81 mg Mass of C = moles × molar mass = 0.0010 moles × 12.01 g/mol = 12.0 mg Mass of H = moles × molar mass = 0.00036 moles × 2(1.01 g/mol) = 0.73 mg Mass of O (by difference) = total mass - mass of C - mass of H = 19.81 mg - 12.0 mg - 0.73 mg = 7.08 mg

Convert masses to moles.

Now that we have the masses of C, H, and O in the sample, we can convert them to moles. Number of moles of Carbon = \( \frac{12.0 \,\text{mg}}{12.01\, \frac{\text{g}}{\text{mol}}} \times \frac{1\,\text{g}}{1000\,\text{mg}}\) = 1.0 mmoles Number of moles of Hydrogen = \( \frac{0.73 \,\text{mg}} {1.01\,\frac{\text{g}}{\text{mol}}} \times \frac{1\,\text{g}}{1000\,\text{mg}}\) = 0.72 mmoles Number of moles of Oxygen = \( \frac{7.08 \,\text{mg}}{16.00\,\frac{\text{g}}{\text{mol}}} \times \frac{1\,\text{g}}{1000\,\text{mg}}\) = 0.44 mmoles

Find the empirical formula.

To find the empirical formula, we need to find the smallest whole number ratio of the elements. We can do this by dividing each element’s moles by the smallest moles value, which is 0.44 mmoles. C: \( \frac{1.0 \,\text{mmoles}}{0.44 \,\text{mmoles}}\) = 2.3 which is approximately 2 H: \( \frac{0.72 \,\text{mmoles}}{0.44 \,\text{mmoles}}\) = 1.6 which is approximately 2 O: \( \frac{0.44 \,\text{mmoles}}{0.44 \,\text{mmoles}}\) = 1 So the empirical formula is C₂H₂O.

Find the empirical and molecular formula mass.

Given that 0.250 moles of terephthalic acid has a mass of 41.5 g, we can find the molar mass of terephthalic acid: Molar mass of terephthalic acid = \(\frac{41.5 \,\text{g}}{0.250 \,\text{moles}}\) = 166 g/mol Empirical formula mass = 2(12.01 g/mol) + 2(1.01 g/mol) + 16.00 g/mol = 32.04 g/mol

Determine the molecular formula.

Finally, we'll compare the empirical formula mass with the molar mass: Molecular formula = Empirical formula × n where n = \( \frac{\text{Molar mass}}{\text{Empirical formula mass}} \) = \( \frac{166 \,\text{g/mol}}{32.04 \,\text{g/mol}} \) = 5.18 ≈ 5 Since n is approximately 5, we can multiply the empirical formula by 5 to get the molecular formula: Molecular formula: C₂H₂O × 5 = C₁₀H₁₀O₅ The molecular formula for terephthalic acid is C₁₀H₁₀O₅.

Key concepts

Combustion Analysis

Combustion analysis is a significant laboratory technique used to determine the elemental composition of an organic compound. In essence, this process involves burning a compound to break it down into simpler substances, such as carbon dioxide (CO₂) and water (H₂O). By carefully measuring the amounts of CO₂ and H₂O produced during combustion, scientists can deduce the amounts of carbon and hydrogen in the original compound.
For example, in the case of terephthalic acid, we know the mass of CO₂ (41.98 mg) and H₂O (6.45 mg) produced during combustion. Understanding that each molecule of CO₂ represents one atom of carbon and each molecule of H₂O represents two atoms of hydrogen helps in determining their respective amounts in the initial sample. Oxygen, the third element in the compound, can be calculated by subtracting the masses of carbon and hydrogen from the total mass of the sample.
This straightforward procedure allows for accurate elemental analysis, which is essential for finding other necessary chemical formulations, such as the empirical formula.

Empirical Formula

The empirical formula represents the simplest whole-number ratio of the atoms of each element in a compound. It provides a fundamental blueprint for understanding how elements are combined in a compound. To find the empirical formula, we begin by determining the mole ratios of each element present in the compound.
Using the mole calculations from the combustion analysis, we can transition from mass to moles for each element. For instance, if the sample contains 1.0 millimoles of carbon, 0.72 millimoles of hydrogen, and 0.44 millimoles of oxygen, we can establish these in a simplistic ratio: (C: H: O). By normalizing each value to the smallest quantity of moles (in this case, 0.44 millimoles), we arrive at the ratios of 2 to 2 to 1. Therefore, the empirical formula is deduced as C₂H₂O.
This formula tells us that in the simplest terms, there are twice as many carbon and hydrogen atoms for every oxygen atom. It's imperative to note that the empirical formula by itself does not reveal the actual number of atoms, only the ratio.

Molar Mass Calculation

Molar mass, a cornerstone concept in chemistry, refers to the mass of one mole of a substance. It's immensely valuable when determining various chemical properties of a compound. In these calculations, knowing the mass and the number of moles of a compound allows for determining its molar mass.
For example, in the case of terephthalic acid, if 0.250 moles have a mass of 41.5 grams, its molar mass can be calculated by dividing the mass by the number of moles: \(\frac{41.5 \, \text{g}}{0.250 \, \text{moles}}\), resulting in 166 g/mol. This calculation is crucial since it bridges the empirical formula with the molecular formula. It essentially tells us how much one mole of terephthalic acid weighs.
Accurate molar mass determination is essential for effectively establishing the relationship between empirical and molecular formulas, especially when extending these calculations to larger scales in industrial contexts.

Stoichiometry

Stoichiometry is a fundamental principle in chemistry that deals with the quantitative relationships among the reactants and products in a chemical reaction. It is based on the law of conservation of mass, which states that matter cannot be created or destroyed.
In the context of our example, stoichiometry can be applied to establish the molecular formula of terephthalic acid from its empirical formula. Using the molar mass of 166 g/mol as a reference, stoichiometry helps in finding the exact multiplier required to convert the empirical formula, C₂H₂O, into the molecular formula. In this case, by comparing the molar mass of the substance with the empirical formula mass (32.04 g/mol), we find a factor of approximately 5 (\(\frac{166 \, \text{g/mol}}{32.04 \, \text{g/mol}}\)). Hence, the molecular formula becomes C₁₀H₁₀O₅.
This aspect of stoichiometry is vital in scientific research as it aids in predicting the amounts of substances consumed and produced in a given reaction, thereby optimizing resource use.