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 \(620 \mathrm{kcal}\) 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 the C-C bond is 80 kcal/mol (Option d).

Step 1: Understand the Given Information

We are given the dissociation energies for \(\mathrm{CH}_4\) and \(\mathrm{C}_2\mathrm{H}_6\). For \(\mathrm{CH}_4\), the energy required to break all \(\mathrm{C}-\mathrm{H}\) bonds is \(360\ \mathrm{kcal}\ \mathrm{mol}^{-1}\). For \(\mathrm{C}_2\mathrm{H}_6\), which consists of a \(\mathrm{C}-\mathrm{C}\) bond in addition to the \(\mathrm{C}-\mathrm{H}\) bonds, the energy required is \(620\ \mathrm{kcal}\ \mathrm{mol}^{-1}\).

Step 2: Calculate Total Bond Energy for C2H6

The dissociation energy of \(\mathrm{C}_2\mathrm{H}_6\) is the energy required to break all bonds and convert it to gaseous atoms. If we break all the \(\mathrm{C}-\mathrm{H}\) bonds (6 in total) and the \(\mathrm{C}-\mathrm{C}\) bond, we use the energy of \(620\ \mathrm{kcal}\ \mathrm{mol}^{-1}\).

Step 3: Calculate Energy from CH Bond Dissociation in C2H6

Each \(\mathrm{C}-\mathrm{H}\) bond energy can be assumed from \(\mathrm{CH}_4\). The energy required to dissociate one \(\mathrm{C}-\mathrm{H}\) bond in \(\mathrm{CH}_4\) is \(\frac{360}{4} = 90\ \mathrm{kcal}\ \mathrm{mol}^{-1}\). Thus, six \(\mathrm{C}-\mathrm{H}\) bonds in \(\mathrm{C}_2\mathrm{H}_6\) will require \(6 \times 90 = 540\ \mathrm{kcal}\ \mathrm{mol}^{-1}\).

Step 4: Find C-C Bond Energy

Now, subtract the energy calculated for the \(\mathrm{C}-\mathrm{H}\) bonds in \(\mathrm{C}_2\mathrm{H}_6\) from the total dissociation energy: \[620\ \mathrm{kcal}\ \mathrm{mol}^{-1} - 540\ \mathrm{kcal}\ \mathrm{mol}^{-1} = 80\ \mathrm{kcal}\ \mathrm{mol}^{-1}.\]

Step 5: Select the Correct Answer

The bond energy of the \(\mathrm{C}-\mathrm{C}\) bond is \(80\ \mathrm{kcal}\ \mathrm{mol}^{-1}\). Therefore, the correct option is (d) \(80\ \mathrm{kcal}\ \mathrm{mol}^{-1}\).

Key Concepts

Dissociation Energy

Dissociation energy is the amount of energy required to break all the bonds in a molecule and separate it into individual atoms. This energy is essential to understand because it indicates the strength of the bonds within a molecule.
For example, in methane (\(\mathrm{CH}_4\)), the dissociation energy is \(360 ext{ kcal mol}^{-1}\), which represents the total energy needed to break four carbon-hydrogen (\(\mathrm{C}-\mathrm{H}\)) bonds. Meanwhile, in ethane (\(\mathrm{C}_2\mathrm{H}_6\)), an additional carbon-carbon (\(\mathrm{C}-\mathrm{C}\)) bond exists. The total dissociation energy here is \(620 ext{ kcal mol}^{-1}\).
Calculating dissociation energy involves knowing the molecular structure and bond energies. With this data, one can predict how much energy it takes to split the molecule into gaseous atoms, an essential calculation for processes like combustion and chemical reactions.

C-H Bond

The carbon-hydrogen (\(\mathrm{C}-\mathrm{H}\)) bond is a fundamental type of bond found in many organic molecules. Understanding its energy helps in deciphering the strength of these bonds and their role in a compound’s stability.
In the case of methane (\(\mathrm{CH}_4\)), there are four \(\mathrm{C}-\mathrm{H}\) bonds, and the total dissociation energy is \(360 ext{ kcal mol}^{-1}\). This implies that each \(\mathrm{C}-\mathrm{H}\) bond approximately requires \(90 ext{ kcal mol}^{-1}\) to break. Knowing this value is particularly useful because many hydrocarbons have similar bond types that behave largely in parallel fashion.
So, if you encounter a similar molecule, you can use this information to calculate how much energy would be needed to influence its structure or reactions.

C-C Bond

The carbon-carbon (\(\mathrm{C}-\mathrm{C}\)) bond is another critical component when analyzing organic molecules. It is prevalent in alkanes, alkenes, and alkynes, playing a pivotal role in determining molecular shape and stability.
For ethane (\(\mathrm{C}_2\mathrm{H}_6\)), the presence of a single \(\mathrm{C}-\mathrm{C}\) bond coupled with six \(\mathrm{C}-\mathrm{H}\) bonds totals to a dissociation energy of \(620 ext{ kcal mol}^{-1}\).
By knowing the energy to dissociate the \(\mathrm{C}-\mathrm{H}\) bonds, one can subtract to find the energy needed specifically for the \(\mathrm{C}-\mathrm{C}\) bond, which is \(80 ext{ kcal mol}^{-1}\) in this instance.
This information is vital for chemists who need to understand and manipulate molecular structures in organic synthesis and other applications.

Gaseous Atoms Conversion

Converting molecules into gaseous atoms involves breaking all the bonds within a compound, a significant consideration when interpreting dissociation energies. This conversion is crucial, especially in high-temperature processes like combustion, where molecules are broken down into their elemental forms.
Understanding this conversion starts with knowing each bond's energy within the molecule. By calculating the energy needed to disrupt each bond, we can predict how much energy is required to vaporize a compound fully.
In chemical labs, this calculation assists researchers in developing efficient methods for reactions and assessing the energy needed for various processes.
Overall, knowing how much energy is needed for a gaseous atoms conversion is essential in fields such as energetic material research, combustion analysis, and chemical synthesis.