Use the Gibbs adsorption isotherm (another name for eqn \(19.50\) ), to show that the volume adsorbed per unit area of solid, \(\mathrm{Va} / \sigma\), is related to the pressure of the gas by \(\mathrm{V},=-(\sigma / \mathrm{RT})(\mathrm{d} \mu / \mathrm{d} \ln \mathrm{p})\), where \(\mu\) is the chemical potential of the adsorbed gas.

Understanding the concept of chemical potential is essential when studying the Gibbs adsorption isotherm. Let's explain this concept in simple terms: chemical potential, denoted as \(\mu\), is essentially a measure of the 'chemical energy' per particle that a substance has. It plays a crucial role in determining the direction of chemical reactions and the movement of molecules.

In the context of adsorption, \(\mu\) represents the potential for the adsorbate gas molecules to adsorb onto a surface. When the chemical potential of a gas phase is higher than on the surface, adsorption is thermodynamically favorable; molecules tend to migrate to the surface.

To link \(\mu\) and pressure \(p\), we consider how potential changes when the system's pressure changes. At a constant temperature, an increase in pressure can lead to more molecules impacting the surface and potentially being adsorbed. This relationship is expressed mathematically and becomes especially relevant when we incorporate it into the Gibbs adsorption isotherm to understand adsorption dynamics.

The term surface excess, denoted by \(\Gamma\), refers to the increased concentration of a substance at the interface or surface compared to the bulk phase. It's like having an additional layer of molecules at the surface that isn't present throughout the bulk material.

This 'excess' arises because molecules at the surface can have different properties than those in the bulk, often due to differing interactions. In the realm of adsorption, this concept helps quantify the amount of a substance that has accumulated on the surface as opposed to being uniformly distributed.

To clarify with an example: picture spilling oil on water. The oil tends to stay together and spreads across the water surface, creating a layer – this is the surface excess. In the adsorption process, measuring this excess can provide insights into how much material is adsorbed and thus the effectiveness of the adsorbent surface.

When dealing with adsorption, it's helpful to measure how much volume of a gas is adsorbed onto a specified area of an adsorbent surface, which we refer to as volume adsorbed per unit area. The equation \(\mathrm{Va} / \sigma\) symbolizes this measurement, with \(\mathrm{Va}\) being the volume adsorbed and \(\sigma\) the area.

We can imagine this as measuring the 'thickness' of the gas that accumulates on the surface. If we envision a sponge (as our adsorbent), the amount of water (or gas) it can hold can be thought of in terms of how deeply the water penetrates the surface of the sponge and spreads over a given area. This measurement is not just an abstract number; it has physical implications for the design of catalytic surfaces and the efficiency of adsorption in processes such as gas purification.

The pressure of the gas, often represented by \(p\), is a fundamental variable in the study of adsorption. It's intuitively understood as the force exerted by the gas molecules that collide with the walls of its container, or, in this context, the adsorbent surface.

As pressure increases, molecules are more frequently and forcefully striking a surface, which can lead to increased chances of adsorption. The measurement of pressure is thus closely linked to how densely packed the molecules are in a given volume – a crucial factor in the rate and extent of adsorption occurring on a surface.

By incorporating the pressure into the Gibbs adsorption isotherm, we are able to predict and calculate the volume of gas that will be adsorbed on a surface at various pressures. This calculation is vital for many industrial applications, from the design of gas masks to the creation of efficient storage tanks for gases.