Animated Solution for Chemistry - Chemical Thermodynamics: For a particular reversible reaction, at temperature $T$, $\Delta H$ and $\Delta S$ were found to be both +ve. If $T_e$ is the temperature at equilibrium, the reaction would be spontaneous when

The universe is governed by a cosmic dance between two fundamental forces: the drive towards lower energy and the drive towards maximum chaos. In the realm of chemistry, this epic battle dictates whether a reaction will happen on its own—a concept we call spontaneity.

To truly master chemical thermodynamics, we must look beyond rote memorization and understand the physical story told by the Gibbs Free Energy equation. Let's embark on a journey to decode this beautiful mathematical relationship.

Enthalpy vs. Entropy

Before we dive into the math, let's meet the two main characters of our story.

First, we have Enthalpy ($\Delta H$). Think of enthalpy as the financial cost of a reaction. Nature is inherently lazy; it prefers to release energy rather than absorb it. An exothermic reaction ($\Delta H<0$) is like getting paid to do a job—it's highly favorable. However, in our specific problem, we are told that $\Delta H$ is positive. This means the reaction is endothermic; it requires an input of heat. From an enthalpy perspective, this reaction is unfavorable. It's an uphill battle.

Second, we have Entropy ($\Delta S$). Entropy is the measure of randomness, chaos, or the number of microstates available to a system. The universe loves chaos! A positive entropy change ($\Delta S>0$) means the system is becoming more disordered, which is highly favorable. In our problem, we are given that $\Delta S$ is positive. So, while enthalpy is fighting against the reaction, entropy is cheering it on.

Gibbs Free Energy

How do we resolve this conflict? Who wins when enthalpy says "no" but entropy says "yes"? Enter Josiah Willard Gibbs and his master equation:

Here, Gibbs Free Energy ($\Delta G$) is the ultimate judge. For a reaction to be spontaneous—meaning it will proceed without any continuous external intervention—the system must release free energy. Mathematically, this means $\Delta G$ must be strictly less than zero ($\Delta G<0$).

Notice the role of Temperature ($T$) in this equation. Temperature acts as an amplifier, a volume knob for entropy. The higher the temperature, the more heavily the entropy term ($-T\Delta S$) influences the overall free energy.

Equilibrium

Before a reaction becomes spontaneous, it must cross a threshold. This tipping point is known as equilibrium. At equilibrium, the forward and reverse reactions are perfectly balanced. The system has no preference to move in either direction, which means there is no "free energy" pushing it.

Mathematically, at equilibrium, the change in Gibbs free energy is exactly zero:

Let's call the temperature at which this happens the equilibrium temperature, $T_e$. Substituting this into our master equation gives:

By rearranging this equation, we can find the exact temperature where the battle between enthalpy and entropy is a perfect tie:

This $T_e$ is a critical landmark. It is the exact temperature where the reaction transitions from being non-spontaneous to spontaneous.

Now, let's return to our ultimate goal: finding the condition for the reaction to be spontaneous. We know that spontaneity requires a negative free energy change:

Substituting the Gibbs equation into this inequality, we get:

We want to solve for the temperature $T$. Let's move the entropy term to the other side of the inequality:

Because the problem explicitly states that $\Delta S$ is positive, we can safely divide both sides by $\Delta S$ without having to flip the inequality sign. (Remember your algebra rules: dividing by a negative number flips the inequality, but dividing by a positive number keeps it the same!).

Or, written more naturally:

But wait! We already discovered what $\frac{\Delta H}{\Delta S}$ represents. It is exactly our equilibrium temperature, $T_e$. Substituting $T_e$ back into our inequality yields the final, elegant result:

If we were to plot $\Delta G$ on the y-axis and Temperature ($T$) on the x-axis, the equation $\Delta G=\Delta H-T\Delta S$ represents a straight line.

Because $\Delta H$ is positive, the y-intercept is above the origin. Because $\Delta S$ is positive, the slope of the line (which is $-\Delta S$) is negative. The line starts high up on the y-axis and slopes downwards as temperature increases.

The exact point where this line crosses the x-axis (where $\Delta G=0$) is $T_e$. As we move to the right of $T_e$—meaning we increase the temperature so that $T>T_e$—the line dips below the x-axis into the negative $\Delta G$ territory. This visual perfectly confirms our algebraic derivation: the reaction only becomes spontaneous when the temperature is cranked up high enough to let the favorable entropy overpower the unfavorable enthalpy.

In physical chemistry, equations are not just letters and numbers; they are the language of nature. By understanding the interplay between enthalpy, entropy, and temperature, you gain the power to predict the behavior of the universe itself.