Question

Terephthalic acid is an important chemical used in the manufacture of polyesters and plasticizers. It contains only \(\mathrm{C}, \mathrm{H}\) and 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 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 determined by finding the moles of C, H, and O atoms in the compound and obtaining their mole ratio. With the given molar mass of 41.5 g/mol, the molecular formula for terephthalic acid is C\(8\)H\(6\)O\(4\).

Step-by-step solution

Determine the moles of CO2 and H2O produced

We are given the masses of CO2 and H2O produced during combustion. We can convert these masses into moles using their respective molar masses (Molar mass of CO2 = 44.01 g/mol and molar mass of H2O = 18.02 g/mol). Moles of CO2 = mass of CO2 / molar mass of CO2 Moles of CO2 = \(41.98 \ mg / 44,01 \frac{g}{mol}\) Moles of H2O = mass of H2O / molar mass of H2O Moles of H2O = \(6.45 \ mg / 18,02 \frac{g}{mol}\)

Determine the moles of C and H atoms in terephthalic acid

Using the moles of CO2 and H2O, we can determine the moles of C and H in terephthalic acid. Each mole of CO2 contains 1 mole of C, and each mole of H2O contains 2 moles of H atoms. Moles of C = moles of CO2 Moles of H = 2 * moles of H2O

Use the Law of Definite Proportions to find the ratio of C, H, and O atoms

We can determine the moles of O by subtracting the moles of C and H from the total moles of terephthalic acid. Then, we can find the mole ratio of C, H, and O atoms: Moles of O = moles of terephthalic acid - moles of C - moles of H Mole ratio (C : H : O) = moles of C : moles of H : moles of O

Determine the molecular formula using the ratio and the given molar mass

Now that we have the mole ratio of C, H, and O atoms in terephthalic acid, we can determine the empirical (simplest) formula. The molecular formula is a multiple of the empirical formula. To find the molecular formula, we will compare the molar mass of the empirical formula with the molar mass of terephthalic acid: Empirical formula molar mass = sum of (atom count * atomic weight) Molecular formula = n * empirical formula (n is an integer) Check for the smallest possible value of n that converts the empirical formula molar mass to the given molar mass of terephthalic acid (41.5 g/mol). The molecular formula of terephthalic acid can be obtained by multiplying the empirical formula by this integer value (n).

Key concepts

Combustion Analysis

Combustion analysis is an essential laboratory technique used in chemistry to determine the elemental composition of a substance by burning it in an excess of oxygen. The process leads to the production of carbon dioxide (CO2) and water (H2O), which can be collected and measured.

For compounds containing carbon and hydrogen, like terephthalic acid, the masses of CO2 and H2O produced provide direct information about the amounts of C and H in the original sample. This is vital because it lays the foundation for calculating the empirical and molecular formulas of the compound.

In a combustion analysis, it's critical to accurately measure the mass of the sample and the masses of CO2 and H2O produced. Then, using the molar masses of these gases, one can determine the moles of carbon and hydrogen in the sample, which are then used to find the ratios of these elements relative to each other and any other elements present in the compound.

Mole Concept

The mole concept is a cornerstone of chemistry that links the microscale of atoms and molecules to the macroscale that we can observe and measure. One mole of any substance contains Avogadro's number of entities, which is approximately 6.022 x 10^23 entities, be it atoms, molecules, ions, etc.

The molar mass, the mass of one mole of a substance, is a crucial value in converting between mass and moles. In the exercise, we use the molar masses of CO2 and H2O to convert the given masses into moles, enabling us to work out the amount of C and H atoms in the original substance. Understanding the mole concept allows us to bridge the gap between the quantifiable observations (masses) made during combustion analysis and the composition of the substance in moles, ultimately aiding in the determination of its molecular formula.

Empirical and Molecular Formulas

An empirical formula represents the simplest whole-number ratio of the elements in a compound. Conversely, a molecular formula reflects the actual number of atoms of each element in a molecule of the compound.For instance, the empirical formula for hydrogen peroxide is HO, showing the 1:1 ratio. However, the molecular formula is H2O2, indicating that each molecule contains two atoms of hydrogen and two atoms of oxygen.In the case of terephthalic acid, after performing combustion analysis and applying the mole concept, the empirical formula can be derived from the simplest ratio of the moles of C, H, and O found. To advance from empirical to molecular formula, we compare the molar mass of the empirical formula to the substance's given molar mass and multiply the empirical formula by an integer to match this molar mass.

Law of Definite Proportions

The law of definite proportions, also known as Proust's law, states that a chemical compound always contains exactly the same proportion of elements by mass. This principle is immensely useful when determining the empirical formula of a compound from the relative amounts of its constituent elements.

Therefore, when we analyze the combustion data, we assume that the elements in terephthalic acid combine in a fixed ratio by mass. These fixed proportions facilitate the calculation of the mole ratio of the elements. By ensuring that our ratios reflect the law's strictures, we can ascertain the empirical formula with certainty. This empirical formula then acts as the base unit from which we deduce the molecular formula, crucial for identifying the compound's structure and understanding its properties.