Animated Solution for Chemistry - Chemical Thermodynamics: For the reaction, $2{\text{NO}}_2(g)\rightleftharpoons {\text{N}}_2{\text{O}}_4(g)$, when $\Delta S=-176.0{\text{ J K}}^{-1}$ and $\Delta H=-57.8{\text{ kJ mol}}^{-1}$, the magnitude of $\Delta G$ at $298\text{ K}$ for the reaction is ......... ${\text{kJ mol}}^{-1}$. (Nearest integer)

Imagine you are watching a microscopic dance where two molecules of nitrogen dioxide ($2{\text{NO}}_2$) collide and fuse to form a single molecule of dinitrogen tetroxide (${\text{N}}_2{\text{O}}_4$). This is a classic dimerization reaction. But how do we know if this reaction will happen on its own? Will it proceed spontaneously, or does it need a push?

To answer this, we must consult the ultimate judge of chemical spontaneity: Gibbs Free Energy ($\Delta G$).

The spontaneity of any process is governed by a delicate balance between two fundamental thermodynamic forces: Enthalpy ($\Delta H$), which represents the heat energy exchanged, and Entropy ($\Delta S$), which represents the degree of randomness or disorder in the system.

These two forces are tied together by the master equation:

If $\Delta G$ is negative, the reaction is spontaneous. If it is positive, the reaction is non-spontaneous. Our goal is to calculate the exact value of $\Delta G$ at a temperature of $298\text{ K}$.

Before we rush into plugging numbers into our equation, we must pause and look closely at the units provided in the question. This is where the examiners set a classic trap for unwary students.

The enthalpy change is given as $\Delta H=-57.8{\text{ kJ mol}}^{-1}$. Notice the prefix "kilo".

However, the entropy change is given as $\Delta S=-176.0{\text{ J K}}^{-1}{\text{mol}}^{-1}$. Notice that this is in standard Joules, not kilojoules!

You cannot directly add or subtract Joules and kilojoules. It is like trying to subtract meters from kilometers without converting them first. We must convert the entropy value into kilojoules by dividing by $1000$:

Now that our units are perfectly aligned, we can safely substitute the values into the Gibbs free energy equation. Let's first evaluate the temperature-entropy term ($T\Delta S$):

Next, we subtract this value from the enthalpy change:

Be careful with the double negative! Subtracting a negative number is the same as adding a positive number:

The question specifically asks for the magnitude of $\Delta G$ rounded to the nearest integer. The magnitude is simply the absolute value, which strips away the negative sign:

Rounding $5.352$ to the nearest integer gives us our final answer: $5$.

As a final thought, notice that $\Delta G$ is negative, confirming that the dimerization of ${\text{NO}}_2$ is indeed spontaneous at room temperature ($298\text{ K}$). However, because both $\Delta H$ and $\Delta S$ are negative, this reaction is temperature-dependent. If we were to heat the system significantly, the $T\Delta S$ term would eventually overpower the $\Delta H$ term, making $\Delta G$ positive and the reaction non-spontaneous!