Animated Solution for Chemistry - Chemical Thermodynamics: Standard entropy of $X_2,Y_2$ and $X{Y}_3$ are $60$, $40$ and $50{\text{ JK}}^{-1}{\text{mol}}^{-1}$, respectively. For the reaction, $\frac{1}{2}{X}_2+\frac{3}{2}{Y}_2⟶X{Y}_3,\Delta H=-30\text{ kJ}$, to be at equilibrium, the temperature will be

The article should be 800-2500 words. I will write a comprehensive, engaging article that explains the concepts of entropy, enthalpy, and Gibbs free energy, and how they interplay to determine the equilibrium state of a chemical reaction.

Imagine you are a chemical engineer tasked with controlling a delicate reaction. You have a mixture of two diatomic gases, $X_2$ and $Y_2$, and you want them to react and form a new compound, $X{Y}_3$. The balanced chemical equation for this process is given as:

Before we even touch a calculator, let's look at the physical reality of what is happening here. We are taking $2$ total moles of gas ($0.5$ moles of $X_2$ and $1.5$ moles of $Y_2$) and compressing them into just $1$ mole of product gas ($X{Y}_3$). What does this tell us? It tells us that the system is becoming more ordered. The particles have fewer ways to arrange themselves. In the language of thermodynamics, the randomness or entropy of the system is decreasing.

We are also given the standard molar entropies for each substance: $S^{\circ }(X_2)=60{\text{ J K}}^{-1}{\text{mol}}^{-1}$ $S^{\circ }(Y_2)=40{\text{ J K}}^{-1}{\text{mol}}^{-1}$ * $S^{\circ }(X{Y}_3)=50{\text{ J K}}^{-1}{\text{mol}}^{-1}$

Furthermore, we are told that the enthalpy change for the reaction, $\Delta H$, is $-30\text{ kJ}$. The negative sign is crucial here. It means the reaction is exothermic; it releases heat into the surroundings.

Our ultimate goal is to find the exact temperature at which this reaction reaches equilibrium. Equilibrium is that magical state where the forward and reverse reactions occur at the exact same rate, and the system experiences no net change.

To find the equilibrium temperature, we need to calculate the total change in entropy for the reaction, denoted as $\Delta S_{\text{reaction}}$. The formula is straightforward: we subtract the total entropy of the reactants from the total entropy of the products.

Let's carefully substitute our given values into this equation. Remember, entropy is an extensive property, meaning it depends on the amount of substance present. Therefore, we must multiply the molar entropy of each substance by its stoichiometric coefficient from the balanced equation.

Substituting the numbers:

Now, let's do the math. Half of $60$ is $30$, and three-halves of $40$ is $60$.

As we predicted earlier, the entropy change is negative. The system has indeed become more ordered.

Now we bring in the heavy artillery: the Gibbs Free Energy equation. This equation is the ultimate arbiter of chemical spontaneity. It relates enthalpy, entropy, and temperature to tell us which way a reaction will go.

Here is the catch, and it is a beautiful one. At equilibrium, the system has reached a state of minimum free energy. There is no driving force pushing it forward or backward. Therefore, at equilibrium, the change in Gibbs free energy, $\Delta G$, is exactly zero.

This allows us to rearrange the equation to solve for the equilibrium temperature, $T$:

Before we plug in our numbers, we must address a classic trap. Our enthalpy, $\Delta H$, is given in kilojoules ($-30\text{ kJ}$), but our entropy, $\Delta S$, is in Joules ($-40{\text{ J K}}^{-1}{\text{mol}}^{-1}$). We cannot divide kilojoules by Joules! We must convert $\Delta H$ to Joules by multiplying by $1000$.

Now, we are ready for the final substitution:

The negative signs cancel out beautifully, leaving us with:

And there we have it. At exactly $750\text{ K}$, the enthalpy term and the entropy term perfectly balance each other out, and the reaction sits in perfect equilibrium.

What happens if we change the temperature? Because both $\Delta H$ and $\Delta S$ are negative, this reaction is driven by enthalpy but hindered by entropy. At temperatures below $750\text{ K}$, the enthalpy term dominates, $\Delta G$ becomes negative, and the reaction is spontaneous. At temperatures above $750\text{ K}$, the entropy term takes over, $\Delta G$ becomes positive, and the reaction becomes non-spontaneous. This interplay between heat and disorder is the very heartbeat of chemical thermodynamics!