Problem 167 - Chemistry

Question

A gaseous hydrocarbon (containing C and H atoms) in a container of volume \(20.2 \mathrm{~L}\) at \(350 \mathrm{~K}\) and 6.63 atm reacts with an excess of oxygen to form \(205.1 \mathrm{~g}\) of \(\mathrm{CO}_{2}\) and \(168.0 \mathrm{~g}\) of \(\mathrm{H}_{2} \mathrm{O} .\) What is the molecular formula of the hydrocarbon?

Step-by-Step Solution

Answer

The molecular formula of the hydrocarbon is \(C_2H_6\).

Step 1: Determine the moles of \(CO_2\)

First, the number of moles of carbon dioxide can be determined. This is done by dividing the mass of the carbon dioxide produced by its molar mass. The molar mass of \(CO_2\) is approximately 44 g/mol. Therefore, using the formula \(n=\frac{m}{M}\), where \(n\) is the number of moles, \(m\) is mass and \(M\) is molar mass, we get \(n_{_{CO2}}=\frac{205.1 g}{44 g/mol}=4.66 mol\).

Step 2: Determine number of carbon atoms

Knowing that the number of moles of carbon in \(CO_2\) is 4.66 mol and each molecule of \(CO_2\) contains one atom of carbon, therefore, there are 4.66 moles of carbon in the unknown hydrocarbon.

Step 3: Determine moles of \(H_2O\)

Next, determine the number of moles of hydrogen present in the water produced. Divide the mass of the \(H_2O\) produced by its molar mass to get \(n_{_{H2O}}=\frac{168.0 g}{18 g/mol} = 9.33 mol\). Each \(H_2O\) molecule contains two hydrogen atoms, so the total number of moles of hydrogen in the hydrocarbon is \(2 \times 9.33 mol = 18.66 mol\).

Step 4: Find the volume of the Hydrocarbon

Using the Ideal Gas Law \(PV = nRT\), determine the number of moles of the hydrocarbon, where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the gas constant and \(T\) is the temperature. It results in \(n_{_{HC}}=\frac{PV}{RT}=\frac{6.63 atm \times 20.2 L}{0.0821 L atm mol^{-1} K^{-1} \times 350 K}=1.83 mol\).

Step 5: Determine Molecular Formula

Finally, using the number of moles of carbon and hydrogen, determined as 4.66 mol and 18.66 mol respectively, and the total number of moles of the hydrocarbon, calculated as 1.83 mol, we can determine the molecular formula. Since there are approximately 3 times as many moles of hydrogen as carbon, the molecular formula of the hydrocarbon is \(C_{1.83}H_{5.54}\), rounding the subscripts to the nearest whole number gives the empirical formula as \(C_2H_6\). This is also the molecular formula, since the hydrocarbon contains 1.83 moles (approximately equal to 2 moles).

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a powerful equation that helps us understand the behavior of gases under various conditions. It's expressed as \[ PV = nRT \]
where:

• \( P \) is the pressure of the gas,
• \( V \) is the volume,
• \( n \) is the number of moles of the gas,
• \( R \) is the universal gas constant \(0.0821 \, \mathrm{L} \, \mathrm{atm} \, \mathrm{mol}^{-1} \, \mathrm{K}^{-1}\), and
• \( T \) is the temperature in Kelvin.

By manipulating this equation, we can find unknown variables when others are known.
In our problem, we used this equation to find the number of moles of the gaseous hydrocarbon. It's important to ensure all units align, particularly the temperature in Kelvin, common mistakes occur here. Hence, the Ideal Gas Law is crucial in understanding reactions involving gases, such as combustion reactions.

Hydrocarbon Combustion

Combustion is a chemical process that involves the burning of substances in oxygen. When hydrocarbons burn, they typically produce carbon dioxide \(\text{(CO}_2\text{)}\) and water \(\text{(H}_2\text{O)}\). This reaction is key to determining the composition of hydrocarbons.
In our given exercise, the hydrocarbon combusts in excess oxygen. The products formed provide crucial data to backtrack and identify the molecular formula of the original fuel.

• The mass of \(\text{CO}_2\) gives the number of moles of carbon in the original molecule.
• The mass of \(\text{H}_2\text{O}\) helps to determine the number of hydrogen atoms.

During combustion exercises, these measurements allow us to analyze the provided data to find results such as empirical or molecular formulas.
It is incredible how combustion can lead us directly to understand molecular compositions.

Mole Calculation

Mole calculations are fundamental in chemistry to quantify substances in reactions. The term 'mole' acts as a way to count atoms and molecules, similar to how 'dozen' counts eggs. Using the formula \( n = \frac{m}{M} \), the number of moles \( n \) can be calculated by dividing the mass \( m \) of a substance by its molar mass \( M \ex{mol}\).
In the exercise, mole calculations helped determine:

• The moles of \(\text{CO}_2\), reflecting carbon count in the hydrocarbon.
• The moles of \(\text{H}_2\text{O}\), which pinpoint the hydrogen atoms in a molecule.

Such calculations are a bridge to understanding chemical composition and stoichiometric relationships. By efficiently calculating the moles of a substance, one can determine atomic or molecular relationships within a compound, which is a basis for deriving more extensive chemical insights.

Empirical Formula

The empirical formula is the simplest ratio of atoms within a compound. It provides a basic representation of a compound's composition.
The molecular formula, on the other hand, shows the exact number of each type of atom in a molecule, and it can be a multiple of the empirical formula.
For example, from the exercise, following hydrocarbon combustion and subsequent calculations, we derived an empirical formula of \( C_2H_6 \).
Achieving this involves:

• Figuring out the number of moles of each element present,
• Calculating the simplest whole number ratio among these moles, and
• Ensuring that these ratios represent the fundamental structure of the molecule.

Using empirical formulas provides basic insights into what elements a substance comprises and the ratio in which these elements combine. Going from empirical to molecular formula involves determining the actual structure, often using molar masses from experimental data,