Question

If the bond dissociation energies of \(\mathrm{XY}, \mathrm{X}_{2}\) and \(\mathrm{Y}_{2}\) are in the ratio of \(1: 1: 0.5\) and \(\Delta \mathrm{H}_{\mathrm{f}}\) for the formation of \(\mathrm{XY}\) is \(-200 \mathrm{~kJ} / \mathrm{mole}\). The bond dissociation energy of \(\mathrm{X}_{2}\) will be: (a) \(100 \mathrm{~kJ} / \mathrm{mole}\) (b) \(400 \mathrm{~kJ} / \mathrm{mole}\) (c) \(600 \mathrm{~kJ} / \mathrm{mole}\) (d) \(800 \mathrm{~kJ} / \mathrm{mole}\)

Short answer

The bond dissociation energy of \( \mathrm{X}_2 \) is \( 800 \mathrm{kJ/mol} \).

Step-by-step solution

Understanding Bond Dissociation Energies

The bond dissociation energies (BDE) of \( \mathrm{XY}, \mathrm{X}_2, \mathrm{Y}_2 \) are in the ratio \( 1: 1: 0.5 \). This means if we take the BDE of \( \mathrm{XY} \) as \( x \), then \( \mathrm{X}_2 \) also has a BDE of \( x \) and \( \mathrm{Y}_2 \) has a BDE of \( 0.5x \).

Applying the Given Enthalpy Change

The enthalpy change for the formation of \( \mathrm{XY} \) is \(-200 \mathrm{kJ/mol} \). The reaction can be written as: \[ \frac{1}{2} \mathrm{X}_{2} + \frac{1}{2} \mathrm{Y}_{2} \rightarrow \mathrm{XY} \]This is represented by: \[ \Delta H_f = \mathrm{BDE}(\text{reactants}) - \mathrm{BDE}(\text{products}) \]Since the product \( \mathrm{XY} \) has a BDE of \( x \):\[ -200 = \left( \frac{1}{2}x + \frac{1}{2}(0.5x) \right) - x \]

Simplifying the Reaction Equation

We now simplify and solve the equation for \( x \):\[ -200 = \left( 0.5x + 0.25x \right) - x \]This simplifies to:\[ -200 = 0.75x - x \]\[ -200 = -0.25x \]

Solving for x

To find \( x \), solve for:\[ 200 = 0.25x \]\[ x = \frac{200}{0.25} \]\[ x = 800 \]

Finding the Bond Dissociation Energy of \( \mathrm{X}_2 \)

Since \( x \) is the bond dissociation energy for both \( \mathrm{XY} \) and \( \mathrm{X}_2 \) due to the ratio \( 1:1 \), the bond dissociation energy of \( \mathrm{X}_2 \) is \( 800 \mathrm{kJ/mol} \).

Key concepts

Enthalpy of Formation

The enthalpy of formation is a crucial concept in understanding chemical reactions. It refers to the heat change when one mole of a compound is formed from its elements in their standard states. For example, in the formation of a diatomic molecule like \( \mathrm{XY} \), the enthalpy change is given as \( \Delta H_{\mathrm{f}} \). This value can help us predict whether a reaction is exothermic (releases heat) or endothermic (absorbs heat).
The exercise provides an enthalpy of formation of \(-200 \mathrm{kJ/mol}\) for the formation of \( \mathrm{XY} \). This negative value indicates that the reaction releases heat, making it exothermic. Knowing this helps in calculating energy changes during reactions, crucial for understanding energy requirements and outputs in chemical processes.

Reaction Equation

A reaction equation represents a chemical reaction depicting reactants transforming into products. It provides a concise way to express chemical changes.
In the context of the exercise, the reaction equation for forming \( \mathrm{XY} \) is shown as:

• \( \frac{1}{2} \mathrm{X}_{2} + \frac{1}{2} \mathrm{Y}_{2} \rightarrow \mathrm{XY} \)

This equation indicates that half a mole of \( \mathrm{X}_{2} \) reacts with half a mole of \( \mathrm{Y}_{2} \) to form one mole of \( \mathrm{XY} \). Understanding this stoichiometric relationship is key for balancing chemical equations and determining the exact amount of reactants needed to produce a desired quantity of products.

Chemical Bonding

Chemical bonding is the interaction that holds atoms together to form compounds. There are several types of chemical bonds, but the most relevant here are covalent bonds, where atoms share electrons.
In molecules like \( \mathrm{XY} \), bond dissociation energy helps understand the strength of these bonds. It measures the energy required to break the bond in a molecule, which reflects the bond's strength and stability. A higher bond dissociation energy indicates a stronger bond. In this exercise, knowing the bond dissociation energy allows us to determine how much energy is needed to break or form bonds between atoms during chemical reactions.

Bond Energy Calculation

Bond energy calculation is crucial for evaluating energy changes in chemical reactions. It involves determining the bond dissociation energies of all bonds involved.
For the exercise, the bond dissociation energies of \( \mathrm{XY}, \mathrm{X}_2, \mathrm{Y}_2 \) are given in a ratio of \( 1:1:0.5 \). Calculating the exact bond energy involves setting up an equation based on the enthalpy of formation:

• \(-200 = \left( \frac{1}{2}x + \frac{1}{2}(0.5x) \right) - x \)

This formula balances the energy required to break reactant bonds and the energy released when product bonds form. Solving this gives the bond dissociation energy of \( \mathrm{X}_2 \), which is found to be \( 800 \mathrm{kJ/mol} \) in the solution. Understanding bond energy calculations helps predict the feasibility and energy requirements of chemical reactions.