Problem 93 - Chemistry An Atoms First Approach

Question

Cumene is a compound containing only carbon and hydrogen that is used in the production of acetone and phenol in the chemical industry. Combustion of \(47.6 \mathrm{mg}\) cumene produces some \(\mathrm{CO}_{2}\) and \(42.8 \mathrm{mg}\) water. The molar mass of cumene is between 115 and \(125 \mathrm{g} / \mathrm{mol} .\) Determine the empirical and molecular formulas.

Step-by-Step Solution

Answer

The empirical formula for cumene is \(\text{C}_1\text{H}_{44}\) and the molecular formula is \(\text{C}_2\text{H}_{88}\).

Step 1: Calculate the amount of hydrogen in cumene from water produced

Since water is produced when hydrogen in the cumene reacts with oxygen, we can calculate the number of moles of hydrogen in the sample by using the mass of water produced. We know that the mass of water produced is \(42.8 \mathrm{mg}\), so we can convert this mass to moles: \[\text{moles of H}_2\text{O} = \frac{42.8\, \mathrm{mg}}{18.015\, \mathrm{g/mol}} \times \frac{1\, \mathrm{g}}{1000\, \mathrm{mg}} = 0.002375\, \mathrm{mol}.\] Since one water molecule is formed from two hydrogen atoms, the moles of hydrogen in cumene is twice the moles of water: \[\text{moles of H} = 2 \times \text{moles of H}_2\text{O} = 2 \times 0.002375\, \mathrm{mol} = 0.004750\, \mathrm{mol}.\]

Step 2: Calculate the amount of carbon in cumene from CO2 produced

Carbon dioxide is produced when carbon in the cumene reacts with oxygen. We can calculate the mass of CO2 produced by subtracting the mass of water from the initial mass of cumene: \[\text{mass of CO}_2 = 47.6\, \mathrm{mg} - 42.8\, \mathrm{mg} = 4.8\, \mathrm{mg}.\] Now, we can calculate the moles of carbon in the cumene by converting the mass of CO2 to moles of carbon: \[\text{moles of CO}_2 = \frac{4.8\, \mathrm{mg}}{44.01\, \mathrm{g/mol}} \times \frac{1\, \mathrm{g}}{1000\, \mathrm{mg}} = 0.000109\, \mathrm{mol}.\] Since there is one mole of carbon for every mole of CO2, the moles of carbon in cumene are the same as the moles of CO2 produced (0.000109 mol).

Step 3: Determine the empirical formula

To find the empirical formula, we need to find the whole number ratio of moles of carbon and hydrogen. Divide both the moles of H and C by the smallest value to find the ratio: \[\frac{\text{moles of C}}{\text{smallest value}} = \frac{0.000109}{0.000109} = 1\] \[\frac{\text{moles of H}}{\text{smallest value}} = \frac{0.004750}{0.000109} = 43.58 \approx 44\] The empirical formula for cumene is \(\text{C}_1\text{H}_{44}\).

Step 4: Calculate the molecular formula

We are given that the molar mass of cumene is between \(\text{115 to 125 g/mol}\) and the empirical formula mass can be calculated as follows: Empirical formula mass = \(1 \times 12.01\, \mathrm{g/mol} + 44 \times 1.008\, \mathrm{g/mol} = 56.35\, \mathrm{g/mol}\) Now, we can determine the molecular formula by finding the ratio of the molecular mass to the empirical formula mass, and multiplying the empirical formula by this ratio. n = (molecular mass)/(empirical formula mass) Since the molecular mass must be between \(\text{115 and 125 g/mol}\), we can estimate the lowest and highest possible values of n: \[\text{Lowest value of n} = \frac{115\, \mathrm{g/mol}}{56.35\, \mathrm{g/mol}} = 2.04 \approx 2\] \[\text{Highest value of n} = \frac{125\, \mathrm{g/mol}}{56.35\, \mathrm{g/mol}} = 2.22 \approx 2\] Since both values of n are approximately equal to 2, the molecular formula of cumene is \(\text{C}_2\text{H}_{88}\).

Key Concepts

Combustion Analysis

Combustion analysis is a crucial method in chemistry used to determine the elemental composition of compounds, specifically those containing carbon and hydrogen. In such analyses, a sample of the compound is burned in the presence of oxygen. This process triggers a chemical reaction resulting in the formation of carbon dioxide (CO₂) and water (H₂O). From these reaction products, one can work backward to determine the original composition of the compound. This type of analysis is especially useful in organic chemistry, where many compounds are hydrocarbon-based.
To break it down further:

• Burn a known mass of the compound.
• Measure the masses of CO₂ and H₂O produced.
• Calculate the amount of carbon and hydrogen originally present by converting the measured products back into moles of the constituent elements.

This method assumes complete combustion, which means all hydrogen atoms in the sample become part of water molecules and all carbon atoms form carbon dioxide. In the exercise, combustion analysis was integral as it provided the needed data to derive the empirical formula of cumene.

Chemical Formula Determination

The chemical formula of a compound can be derived through a series of mathematical steps starting with combustion analysis, as illustrated in the original exercise. To derive it, it's necessary to calculate the amount of each element in the compound from the mass of combustion products.
First, the molar amounts of elements, such as carbon and hydrogen, are determined from CO₂ and H₂O. In the exercise, we measured the moles of each element by using the molecular weights of CO₂ and H₂O:

• For CO₂, remember that each mole corresponds to one mole of carbon atoms.
• For H₂O, the amount of hydrogen is double the number of moles of water produced, as two hydrogen atoms are present in each molecule of water.

With these mole amounts, the empirical formula can be established by dividing by the smallest mole number to find the simplest whole number ratio of atoms. This allows us to express the relative numbers of each type of atom in the compound, providing the empirical formula.

Organic Chemistry Calculations

In organic chemistry, calculating molecular formulas based on analysis data is foundational to understanding compound structures. Such calculations allow chemists to accurately describe the composition of hydrocarbons, like cumene. Once an empirical formula is determined (as in our previous section), converting it to a molecular formula requires knowledge of the compound’s molar mass.
Molecular formulas are often a multiple of empirical formulas. Given an approximate molar mass range, determine the likely integer multiple (n) by dividing the molar mass of the compound by the mass of the empirical formula. This calculation should yield a number close to a whole integer:

• Multiply all subscripts in the empirical formula by this integer to get the molecular formula.
• Check the calculated molar mass against the provided range to ensure accuracy.

This ensures that our conclusions align with both the compositional and experimental data, resulting in a reliable molecular formula. In our example with cumene, we completed this step to find the molecular formula C₂H₈₈, using the given molar mass range to confirm correctness.