Problem 85 - A Complete Resource Book in Chemistry for JEE Main

Question

A gaseous hydrocarbon has \(85 \%\) carbon and vapour density of 28 . The possible formula of the hydrocarbon will be (a) \(\mathrm{C}_{4} \mathrm{H}_{\mathrm{g}}\) (b) \(\mathrm{C}_{2} \mathrm{H}_{4}\) (c) \(\mathrm{C}_{2} \mathrm{H}_{2}\) (d) \(\mathrm{C}_{3} \mathrm{H}_{6}\)

Step-by-Step Solution

Answer

The possible formula of the hydrocarbon is \(\mathrm{C}_{4} \mathrm{H}_{8}\).

Step 1: Identify Molecular Formula Using Vapour Density

The molecular weight of a compound is twice its vapor density. Therefore, we can calculate the molecular weight as follows:\[ 2 \times 28 = 56 \text{ g/mol} \]

Step 2: Determine Carbon Contribution

Since the hydrocarbon contains 85% carbon, calculate the contribution of carbon in one mole of the hydrocarbon:\[ 0.85 \times 56 = 47.6 \text{ g (carbon per mole of hydrocarbon)} \]

Step 3: Check Carbon Atom Count

The molar mass of carbon is 12 g/mol. Determine how many carbon atoms are present in 47.6 g:\[ \frac{47.6}{12} \approx 3.97 \]This is approximately 4 carbon atoms.

Step 4: Calculate Contribution of Hydrogen

The remaining percentage from 100% will be hydrogen, which is 15%. Calculate the contribution of hydrogen:\[ \text{Remaining mass for hydrogen} = 0.15 \times 56 = 8.4 \text{ g of H per mole of hydrocarbon} \]

Step 5: Determine Hydrogen Atom Count

The molar mass of hydrogen is 1 g/mol. Thus, the number of hydrogen atoms is:\[ \frac{8.4}{1} = 8.4 \]This rounds to 8 hydrogen atoms.

Step 6: Formulate Possible Hydrocarbon

Thus, the formula for the hydrocarbon is approximately \(\mathrm{C}_4\mathrm{H}_8\). Check the options provided.

Step 7: Match with Given Options

From the options, \(\mathrm{C}_{4} \mathrm{H}_{8}\) matches the calculated molecular formula. This means the correct answer is option (a) \(\mathrm{C}_{4} \mathrm{H}_{8}\).

Key Concepts

Molecular Formula Determination

To determine the molecular formula of a gaseous hydrocarbon, we need to use given data like vapor density and element percentages. The first step is linking vapor density to molecular weight. The molecular weight (or molar mass) of a compound is twice its vapor density. This relationship comes from the principle that vapor density is a measure for gases comparing their 'molecular mass' to that of hydrogen. Since the vapor density is given as 28, the molecular weight is calculated as follows:

\[ 2 \times 28 = 56 \text{ g/mol} \]

This calculation gives the total weight of one mole of the hydrocarbon. In this context, the molecular formula is the sum of the weights of the constituent atoms that make up the molecule. We then proceed by associating the carbon and hydrogen contents to these weights to deduce the exact ratio of atoms within the molecule.

Vapour Density

Vapour density is an essential concept in determining the molecular weight of a gas. It serves as a simplifying metric, essentially comparing the mass of a volume of a gas to that of an equivalent volume of hydrogen under the same conditions. By tying together with the concept of molecular weight, it allows us to calculate a gas’s molecular formula, which is crucial in understanding its chemical identity.

Here's how it works:

• The vapor density is the ratio of the density of the gas to the density of hydrogen, where the density is defined by mass per unit volume.
• To find the molecular weight of the gas, you multiply the vapour density by 2. This equation stems from Avogadro’s Law, which states that equal volumes of gases, at the same temperature and pressure, have the same number of molecules.

This gives a simple but powerful way to connect experimental observations (vapor density) with direct calculations (molecular weight), laying the foundation to answer deeper chemistry questions.

Carbon Content Calculation

Carbon content plays a pivotal role in resolving the molecular structure of hydrocarbons. Given a percentage composition, this information allows chemists to backtrack from percentages to the actual composition of atoms in the molecule. In the example, we see how this is practically applied.

Given that the hydrocarbon contains 85% carbon, the mass contribution of carbon in one mole of the hydrocarbon is calculated by:

\[ 0.85 \times 56 = 47.6 \text{ g (carbon per mole of hydrocarbon)} \]

Here, "85% carbon" gives a valuable clue. We can now consider how many carbon atoms this represents:

\[ \frac{47.6}{12} \approx 3.97 \]

Since you can't have a fraction of an atom in this context, the calculation rounds up to approximately 4 carbon atoms. Thus, understanding the concept of carbon content allows us to determine molecular composition from percentage data effectively, enabling further deduction of the full hydrocarbon formula.