Animated Solution for Chemistry - Chemical Thermodynamics: A reaction is non-spontaneous at the freezing point of water but is spontaneous at the boiling point of water, then

Have you ever wondered why ice melts on a hot summer day, but water doesn't spontaneously freeze in the middle of July? Or why a matchstick burns fiercely once struck, but doesn't un-burn itself back into pristine wood? The answer to all these questions lies in the heart of chemical thermodynamics, specifically in a magical quantity known as the Gibbs Free Energy ($\Delta G$).

To understand whether a chemical reaction will happen on its own—a concept we call spontaneity—we rely on the master equation of thermodynamics:

This equation is like a cosmic accounting book. Let's break down the players: $\Delta H$ (Enthalpy Change): This is the heat budget of the reaction. If $\Delta H$ is negative, the reaction releases heat (exothermic), which the universe generally loves. If it's positive, the reaction absorbs heat (endothermic), which is like paying a tax. $\Delta S$ (Entropy Change): This measures the change in randomness or chaos. The universe naturally tends towards higher entropy. A positive $\Delta S$ means things are getting messier, which is highly favored! $T$ (Absolute Temperature):* Measured in Kelvin, temperature acts as a volume knob. It amplifies the effect of the entropy change. The hotter it gets, the more the universe cares about chaos.

For any process to be spontaneous, the golden rule is that $\Delta G$ must be strictly negative ($\Delta G<0$). If $\Delta G$ is positive, the reaction is non-spontaneous and won't proceed without external help.

Let's dive into the specific problem at hand. We are given a mysterious reaction and two crucial clues about its behavior at different temperatures.

The first clue states that at the freezing point of water, the reaction is non-spontaneous. The freezing point of water ($0^{\circ }\text{C}$ or $273\text{ K}$) is a relatively low temperature in the grand scheme of chemical reactions.

What does this tell us? According to our golden rule, if a reaction is non-spontaneous, its Gibbs Free Energy must be positive.

At this low temperature, the system is perfectly happy as it is. The reactants have no natural desire to transform into products. The energy barrier or the lack of sufficient chaos prevents the reaction from moving forward.

Now, let's turn up the heat! The second clue reveals that when we raise the temperature to the boiling point of water ($100^{\circ }\text{C}$ or $373\text{ K}$), the reaction suddenly wakes up and becomes spontaneous.

This is a dramatic shift in behavior. By simply providing more thermal energy, we have coaxed the reaction into happening on its own. Mathematically, this means that at this high temperature, the Gibbs Free Energy has crossed the zero threshold and become negative.

This flip from positive to negative $\Delta G$ is the key to unlocking the signs of both $\Delta H$ and $\Delta S$.

To truly master this concept, we must visualize it. Imagine plotting a graph of $\Delta G$ on the y-axis versus Temperature ($T$) on the x-axis.

Our master equation, $\Delta G=\Delta H-T\Delta S$, is actually the equation of a straight line! Think of it in the familiar $y=mx+c$ format: $y$ is $\Delta G$ $x$ is $T$ The slope ($m$) is $-\Delta S$ The y-intercept ($c$) is $\Delta H$

Let's plot our two clues on this graph. At a low temperature ($T_{\text{freeze}}$), our point is above the x-axis because $\Delta G>0$. At a high temperature ($T_{\text{boil}}$), our point is below the x-axis because $\Delta G<0$.

If we draw a line connecting these two points, we get a straight line that slopes downwards from left to right. As temperature increases, $\Delta G$ steadily decreases.

Now comes the elegant part. We can read the signs of $\Delta H$ and $\Delta S$ directly from our graph!

First, let's look at the slope. Our line is sloping downwards, which means the slope is a negative value. We know that the slope of this line is equal to $-\Delta S$.

If the negative of $\Delta S$ is a negative number, then $\Delta S$ itself must be positive ($\Delta S>0$). This makes perfect physical sense! The reaction only becomes spontaneous at high temperatures because the $T\Delta S$ term grows large enough to drive the reaction. The system is moving towards a state of higher randomness.

Next, let's find the enthalpy change, $\Delta H$. This is the y-intercept of our graph, which represents the value of $\Delta G$ when the temperature is absolute zero ($T=0\text{ K}$).

Look at our downward-sloping line. To pass through a positive value at $T_{\text{freeze}}$ and then drop to a negative value at $T_{\text{boil}}$, the line must have started from a positive value on the y-axis.

Therefore, $\Delta H$ must be positive. The reaction is endothermic. It requires an input of heat to proceed, which is why it refuses to happen at low temperatures where thermal energy is scarce.

We have successfully decoded the thermodynamic DNA of this reaction. For a process to be non-spontaneous at low temperatures but spontaneous at high temperatures, it must be an endothermic process that leads to an increase in entropy.

$\Delta H=+\text{ve}$ $\Delta S=+\text{ve}$

This is a classic "entropy-driven" reaction. At low temperatures, the positive $\Delta H$ dominates, making $\Delta G$ positive. But as the temperature rises, the $T\Delta S$ term (which is subtracted) becomes larger and larger, eventually overpowering the enthalpy term and pulling $\Delta G$ into the negative, spontaneous territory.

Understanding this interplay between heat and chaos is one of the most profound realizations in all of chemistry!