Animated Solution for Chemistry - Chemical Thermodynamics: Consider the reaction, $4{\text{NO}}_2(g)+{\text{O}}_2(g)\to 2{\text{N}}_2{\text{O}}_5(g)$, ${\Delta }_{r}H=-111\text{ kJ}$. If ${\text{N}}_2{\text{O}}_5(s)$ is formed instead of ${\text{N}}_2{\text{O}}_5(g)$ in the above reaction, the ${\Delta }_{r}H$ value will be (Given, $\Delta H$ of sublimation for ${\text{N}}_2{\text{O}}_5$ is $54{\text{ kJ mol}}^{-1}$)

Imagine you are standing at the base of a mountain and you want to reach the peak. You could take a direct, steep helicopter ride to the top, or you could hike up a winding trail, stop at a camp, and then continue to the summit. Regardless of the path you choose, your final altitude change is exactly the same.

This is the beautiful essence of Hess's Law in thermodynamics. Enthalpy ($H$) is a state function, meaning its change depends only on the initial and final states, not on the path taken to get there.

In our problem, we are given a direct path: This process releases energy, specifically ${\Delta }_{r}H=-111\text{ kJ}$.

But what if we wanted to form the solid version, $2{\text{N}}_2{\text{O}}_5(s)$, instead? We can construct a hypothetical two-step path to reach our gaseous product: Step 1: Form the solid product from the reactants. Let's call its enthalpy change $\Delta H_{\text{target}}$. Step 2: Sublimate the solid product into a gas. The enthalpy of sublimation is given as $\Delta H_{\text{sub}}=+54\text{ kJ}$.

According to Hess's Law, the sum of the enthalpy changes of our two-step path must equal the enthalpy change of the direct path:

Let's substitute the values we know:

Solving for our unknown:

This gives us our final answer, matching option (a).

A Lesson in Exam Strategy

Now, let's address the elephant in the room. If you are a highly observant student, you might have noticed a subtle trap. The problem states that the enthalpy of sublimation is $54{\text{ kJ mol}}^{-1}$. However, our balanced chemical equation produces two moles of ${\text{N}}_2{\text{O}}_5$.

Technically, the total sublimation energy for the reaction should be:

If we used this mathematically rigorous value, our calculation would be:

Interestingly, $-219\text{ kJ}$ is also an option (d)! So why is the official answer $-165\text{ kJ}$? In the official AIEEE 2011 answer key, the examiners treated the $54\text{ kJ}$ as the sublimation energy for the entire process rather than strictly per mole of product formed.

This is a crucial lesson in competitive exams: sometimes, questions contain slight ambiguities. If you encounter a situation like this, it is often safer to follow the most direct interpretation of the provided numbers unless the context heavily demands stoichiometric scaling. Always keep an eye on the options and the typical conventions of the exam body!