Animated Solution for Chemistry - Chemical Thermodynamics: If the bond dissociation energies of $XY$, $X_2$ and $Y_2$ (all diatomic molecules) are in the ratio of $1:1:0.5$ and $\Delta H_f$ for the formation of $XY$ is $-200{\text{ kJ mol}}^{-1}$. The bond dissociation energy of $X_2$ will be

Imagine you are in a molecular laboratory, tasked with synthesizing exactly one mole of the diatomic gas $XY$. To do this, you must start with the elemental building blocks in their standard states: $X_2$ and $Y_2$ gases.

Visualizing the Reaction

The standard enthalpy of formation, denoted as $\Delta H_f$, is strictly defined for the creation of one mole of a substance. Therefore, our balanced chemical equation must reflect this constraint:

This fractional stoichiometry is crucial. If you were to write $X_2+Y_2\to 2XY$, you would be calculating the enthalpy for two moles, which is a classic trap that leads to silly mistakes!

Hess's Law

To find the enthalpy change of this reaction, we rely on Hess's Law, specifically using bond dissociation energies. The logic is beautifully simple: we must invest energy to break the bonds of the reactants (an endothermic process), and the system will release energy when the new bonds of the products are formed (an exothermic process).

Substituting our specific molecules into this master equation, we get:

The problem provides a fascinating ratio for the bond dissociation energies of $XY$, $X_2$, and $Y_2$, which is $1:1:0.5$. To make our algebra elegant, let's introduce a variable $a$ to represent the bond energy of $XY$.

Let ${\text{BE}}_{XY}=a$ This implies ${\text{BE}}_{X_2}=a$ And ${\text{BE}}_{Y_2}=0.5a$

We are also given that the standard enthalpy of formation $\Delta H_f$ is $-200{\text{ kJ mol}}^{-1}$. Let's substitute all these values into our equation:

Now, we just need to carefully solve for $a$. Let's simplify the terms inside the bracket:

Combining the positive terms gives us $0.75a$. Subtracting $a$ from this yields:

To isolate $a$, we divide both sides by $-0.25$:

Since we defined the bond dissociation energy of $X_2$ as $a$, we have our final answer:

Looking at the given options—$400$, $300$, and $200$—our calculated value of $800$ is nowhere to be found. Therefore, the correct choice is confidently None of these. This problem is a brilliant test of your fundamental understanding of standard states and your ability to trust your algebraic execution!