Problem 115 - A Complete Resource Book in Chemistry for JEE Main

Question

The dissociation energies of \(\mathrm{CH}_{4}\) and \(\mathrm{C}_{2} \mathrm{H}_{6}\) to convert them into gaseous atoms are 360 and \(620 \mathrm{kcal} \mathrm{mol}\) respectively. The bond energy of \(\mathrm{C}-\mathrm{C}\) bond is (a) \(280 \mathrm{kcal} \mathrm{mol}^{-1}\) (b) \(240 \mathrm{kcal} \mathrm{mol}^{-1}\) (c) \(160 \mathrm{kcal} \mathrm{mol}^{-1}\) (d) \(80 \mathrm{kcal} \mathrm{mol}^{-1}\)

Step-by-Step Solution

Answer

The bond energy of \(\mathrm{C}-\mathrm{C}\) is 80 kcal/mol, option (d).

Step 1: Understand the Problem

We need to find the energy of a single C-C bond in ethane (\(\mathrm{C}_2\mathrm{H}_6\)). Given the dissociation energies of methane (\(\mathrm{CH}_4\)) and ethane, we will calculate by using bond dissociation energies.

Step 2: Determine the Bond Energies in Methane

Methane (\(\mathrm{CH}_4\)) consists of four \(\mathrm{C}-\mathrm{H}\) bonds. Given its total dissociation energy is 360 kcal/mol, the dissociation energy for one \(\mathrm{C}-\mathrm{H}\) bond is the total energy divided by the number of bonds:\[ \text{Energy per } \mathrm{C}-\mathrm{H} = \frac{360}{4} = 90 \, \text{kcal/mol} \]

Step 3: Write the Dissociation Reaction for Ethane

For ethane \(\mathrm{C}_2\mathrm{H}_6\), the dissociation into gaseous atoms involves breaking 6 \(\mathrm{C}-\mathrm{H}\) bonds and 1 \(\mathrm{C}-\mathrm{C}\) bond. The chemical equation can be written as:\[ \mathrm{C}_2\mathrm{H}_6 \rightarrow 2\mathrm{C} + 6\mathrm{H} \]

Step 4: Calculate the Total Energy for \(\mathrm{C}_2\mathrm{H}_6\)

The supplied total dissociation energy to convert \(\mathrm{C}_2\mathrm{H}_6\) into gaseous atoms is 620 kcal/mol. This energy is used to break 6 \(\mathrm{C}-\mathrm{H}\) bonds and 1 \(\mathrm{C}-\mathrm{C}\) bond.

Step 5: Calculate the \(\mathrm{C}-\mathrm{C}\) Bond Energy

Using the known energy for \(\mathrm{C}-\mathrm{H}\) bonds from Step 2, we express the total energy as:\[ 620 = 6 \times 90 + \text{Energy of } \mathrm{C}-\mathrm{C}\]\[ \text{Energy of } \mathrm{C}-\mathrm{C} = 620 - 540 = 80 \, \text{kcal/mol} \]

Step 6: Choose the Correct Answer

The calculated energy of the \(\mathrm{C}-\mathrm{C}\) bond is 80 kcal/mol, which corresponds to option (d).

Key Concepts

Dissociation Energy

Dissociation energy refers to the amount of energy required to break a specific bond in a molecule, leading to the separation of atoms. Each molecule contains different types of bonds, each with its unique energy requirement to break.

• The total dissociation energy of a molecule is the sum of the energies needed to break all its bonds.
• It is crucial in understanding the stability of molecules; the higher the dissociation energy, the more stable the molecule.

For example, methane (\(\mathrm{CH}_4\)) has a total dissociation energy of 360 kcal/mol, meaning that energy is needed to break it into individual gaseous atoms. By knowing the dissociation energy, we can calculate the energy needed to break individual types of bonds within a molecule.
This concept is pivotal in chemical thermodynamics, as it helps to predict how molecules will behave under different conditions.

Methane

Methane is a simple hydrocarbon represented by the chemical formula \(\mathrm{CH}_4\). It is composed of one carbon atom covalently bonded to four hydrogen atoms. These bonds are known as \(\mathrm{C}-\mathrm{H}\) bonds, and they require energy to break.

• The dissociation energy for methane tells us that each \(\mathrm{C}-\mathrm{H}\) bond has an energy of 90 kcal/mol, since the total energy is divided by the number of bonds.
• This is how much energy is needed to break one \(\mathrm{C}-\mathrm{H}\) bond in the molecule.

Methane is often used as a reference fuel due to its simple structure and relative stability. Understanding the properties of methane and its bond energies can help in chemical synthesis, energy production, and even environmental science.

Ethane

Ethane is another simple hydrocarbon, which is represented by \(\mathrm{C}_2\mathrm{H}_6\). It consists of two carbon atoms connected by a covalent \(\mathrm{C}-\mathrm{C}\) bond, with each carbon atom also bonded to three hydrogen atoms.

• Calculating the dissociation energy involves breaking 6 \(\mathrm{C}-\mathrm{H}\) bonds and 1 \(\mathrm{C}-\mathrm{C}\) bond.
• The total dissociation energy for ethane is 620 kcal/mol, which includes the energy for all seven bonds.

By subtracting the energy of the \(\mathrm{C}-\mathrm{H}\) bonds from the total, we determine that the \(\mathrm{C}-\mathrm{C}\) bond requires 80 kcal/mol to break. This value highlights how much strength is needed to rupture the bond between the two carbon atoms, which is vital in chemical reactions and processes.

Chemical Bonds

Chemical bonds are forces that hold atoms together in molecules. There are several types of bonds, but the focus is often on covalent bonds, where atoms share electrons.

• Covalent bonds can vary in strength depending on how they hold the atoms together.
• Bond strength is often analyzed through dissociation energy, helping to clarify how robust the bond is.

The analysis of chemical bonds, such as in methane and ethane, showcases the energy needed to disrupt these connections. Knowing the bond energies allows chemists to predict or manipulate chemical reactions.
For instance, calculating the dissociation energy in ethane helps identify how energy-intensive a reaction is, critical for both research and industrial applications.
This detailed understanding is not only theoretical but also practical in areas like synthetic chemistry and material science.