Problem 48 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

Given the bond dissociation enthalpy of \(\mathrm{CH}_{3}-\mathrm{H}\) bond as \(103 \mathrm{kcal} / \mathrm{mol}\) and the enthalpy of formation of \(\mathrm{CH}_{4}(\mathrm{~g})\) as \(-18\) kcal/mol, find the enthalpy of formation of methyl radical. The dissociation energy of \(\mathrm{H}_{2}(\mathrm{~g})\) into \(\mathrm{H}\) (atoms) is 103 kcal/mol. (a) \(-33.5 \mathrm{kcal} / \mathrm{mol}\) (b) \(33.5 \mathrm{kcal} / \mathrm{mol}\) (c) \(18 \mathrm{kcal} / \mathrm{mol}\) (d) \(-9 \mathrm{kcal} / \mathrm{mol}\)

Step-by-Step Solution

Answer

The enthalpy of formation of the methyl radical (CH3·) is +33.5 kcal/mol.

Step 1: Write down the given data

The bond dissociation enthalpy of the CH3-H bond is given as 103 kcal/mol, the enthalpy of formation of CH4(g) is -18 kcal/mol, and the dissociation energy of H2(g) into H (atoms) is 103 kcal/mol.

Step 2: Determine enthalpy change for breaking a CH3-H bond

The breaking of a CH3-H bond requires energy equal to its bond dissociation enthalpy. Therefore, the enthalpy change for breaking a CH3-H bond is +103 kcal/mol because it is an endothermic process.

Step 3: Write the enthalpy change for the formation of CH4 from C, H2 and H

The enthalpy of formation of CH4(g) is given as -18 kcal/mol. This process can be visualized as forming CH4 from elemental carbon and hydrogen gases. The reaction equation can be written as: C(graphite) + 2H2(g) -> CH4(g). The enthalpy change for this reaction is the enthalpy of formation of CH4, which is -18 kcal/mol.

Step 4: Calculate the enthalpy of formation for methyl radical (CH3·)

The formation of a methyl radical from CH4 involves the removal of one hydrogen atom, which is equivalent to breaking one CH3-H bond. The enthalpy of formation of CH4 is -18 kcal/mol, and the energy required to break the CH3-H bond is +103 kcal/mol. Hence, the enthalpy of formation of the methyl radical is the sum of these two quantities: -18 kcal/mol + 103 kcal/mol = 85 kcal/mol.

Step 5: Correct the result for the energy of dissociation of H2 into H atoms

Since the methyl radical was formed from CH4 by removing a hydrogen atom, we must also consider the energy required to provide the H atom. The dissociation of hydrogen molecule H2 into two H atoms requires 103 kcal/mol, but this energy yields two atoms of H, so the energy per atom is half of 103 kcal/mol, which is 51.5 kcal/mol. Since this process is also endothermic, we subtract this value from the previously calculated enthalpy of formation of CH3·: 85 kcal/mol - 51.5 kcal/mol = 33.5 kcal/mol. Therefore, the enthalpy of formation of the methyl radical CH3· is +33.5 kcal/mol.

Key Concepts

Bond Dissociation Enthalpy

Bond dissociation enthalpy (BDE) is the amount of energy required to break a specific chemical bond in one mole of a gaseous substance. It is a critical concept when evaluating the strength of bonds within a molecule and predicting the reactivity of molecules during chemical reactions. For instance, the BDE of the \textbf{CH}\(_3\)-\textbf{H} bond is given as 103 kcal/mol, indicating that to break this bond and separate the methyl group (CH\(_3\)) from the hydrogen atom (H), 103 kcal/mol of energy is needed.

BDE is directly proportional to the strength of the chemical bond; a higher BDE signifies a stronger bond that requires more energy to break. BDE is an essential component in calculating reaction enthalpies, especially in reactions where bond formation and breaking occurs, such as in combustion or in the formation of radicals.

The BDE can be used to calculate the enthalpy changes in reactions by adding the energies for bonds broken and subtracting the energies of bonds formed. This is based on the principle that breaking bonds is an endothermic process (absorbs energy), while forming bonds is an exothermic process (releases energy).

Methyl Radical

Methyl radical, often denoted as CH\(_3$$\bullet\), is a highly reactive species that contains an unpaired electron. It results from the homolytic cleavage of a \textbf{CH}\(_3\)-\textbf{H} bond, in which each atom retains one electron from the shared pair. The radical exhibits a trigonal planar geometry, and due to its unpaired electron, it is extremely reactive and readily participates in subsequent chemical reactions.

The formation of a methyl radical typically involves an endothermic process, such as the breaking of a \textbf{CH}\(_3\)-\textbf{H} bond, which requires energy input. The enthalpy of formation of the methyl radical is crucial when considering reaction mechanisms, especially in organic chemistry, where radicals can lead to chain reactions resulting in complex products.

Calculating the enthalpy of formation of a methyl radical can be tricky as it involves not only the bond dissociation enthalpy of the \textbf{CH}\(_3\)-\textbf{H} bond but also the enthalpy of formation of the original compound (e.g. CH\(_4\)) and any other bond dissociation enthalpies involved in providing the necessary atoms for the radical's formation.

Endothermic Process

An endothermic process is a type of thermodynamic reaction or process that absorbs heat from its surroundings. In chemical reactions, endothermicity indicates that the system gains heat and the reaction proceeds with the consumption of energy. This is contrasted with an exothermic process, where the system releases heat and the reaction provides energy.

The breaking of chemical bonds is a classic example of an endothermic process. In the case of forming a methyl radical from methane (\textbf{CH}\(_4\)), energy must be supplied to overcome the bond dissociation enthalpy of the \textbf{CH}\(_3\)-\textbf{H} bond. Additionally, the dissociation of diatomic hydrogen (H\(_2\)) into individual hydrogen atoms is another endothermic step that must be considered when calculating the complete enthalpy of formation for the methyl radical.

Understanding endothermic processes is essential for predicting how reactions will proceed under different conditions since these reactions require an input of energy to initiate. In educational settings, ensuring that students grasp this concept assists them in solving enthalpic calculations and recognizing the energy flow within a chemical system.