Animated Solution for Chemistry - Surface Chemistry: Select the graph that correctly describes the adsorption isotherms at two temperatures $T_1$ and $T_2$ ($T_1>T_2$) for a gas. ($x=$ mass of the gas adsorbed ; $m=$ mass of adsorbent ; $p=$ pressure)

Understanding Adsorption Isotherms

Imagine you have a piece of charcoal and you place it in a closed container filled with a gas. Over time, you'll notice the pressure of the gas drops. Where did the gas go? It didn't vanish; it stuck to the surface of the charcoal. This phenomenon, where molecules of a gas or liquid accumulate on the surface of a solid, is called adsorption.

To truly understand this process, chemists use a powerful visual tool called an adsorption isotherm. An isotherm is simply a graph that plots the extent of adsorption against the pressure of the gas, keeping the temperature strictly constant. Let's dive into the mathematics and thermodynamics behind these fascinating curves.

Why Curves, Not Lines?

When we plot the mass of the gas adsorbed per unit mass of the adsorbent (denoted as $\frac{x}{m}$) against the pressure ($p$), we don't get a straight line. Why? Because the relationship isn't perfectly linear across all pressures.

At moderate pressures, the behavior is beautifully described by the Freundlich Adsorption Isotherm equation:

Here, $k$ and $n$ are constants that depend on the nature of the gas and the solid at a given temperature. The crucial part is the exponent $\frac{1}{n}$, which typically has a value between $0$ and $1$. Because this exponent is a fraction, the mathematical plot of this equation is a curve that gradually flattens out, rather than a straight line shooting off to infinity. This immediately tells us that any graph showing straight lines for an adsorption isotherm over a wide pressure range is fundamentally incorrect.

A Battle of Energy and Chaos

Now, let's introduce temperature into the mix. What happens if we heat the system? To answer this, we must look at the thermodynamics of the adsorption process.

When gas molecules, which are freely zipping around in the container, suddenly stick to a solid surface, their freedom of movement is drastically restricted. In thermodynamic terms, the randomness or entropy of the system decreases ($\Delta S<0$).

For any process to happen spontaneously, the Gibbs free energy change ($\Delta G$) must be negative. We know from the famous equation:

Since $\Delta S$ is negative, the term $-T\Delta S$ becomes positive. To ensure that $\Delta G$ remains negative and the process stays spontaneous, the enthalpy change ($\Delta H$) must be highly negative. A negative $\Delta H$ means the process releases heat; it is exothermic.

Because adsorption is an exothermic equilibrium, we can predict its behavior using Le Chatelier's Principle. This principle states that if you apply a stress to a system at equilibrium, the system will shift to counteract that stress.

If we increase the temperature (adding heat), the system will try to consume that excess heat by shifting the equilibrium in the backward, endothermic direction. This means the adsorbed gas molecules will gain enough thermal energy to break free from the surface and return to the gaseous phase. This reverse process is called desorption.

Therefore, as temperature increases, the extent of adsorption ($\frac{x}{m}$) strictly decreases.

If we look at our graphs and pick a constant pressure $p_0$, the curve corresponding to the higher temperature ($T_1$) must yield a lower value of $\frac{x}{m}$ compared to the curve for the lower temperature ($T_2$). Visually, this means the curve for $T_2$ will always lie above the curve for $T_1$.

Combining these two profound insights—that the isotherms must be curves due to the Freundlich equation, and that the lower temperature curve must sit higher due to Le Chatelier's principle—we arrive flawlessly at the correct graphical representation.