One way to gain a physical understanding of \(b\) in the van der Waals equation is to calculate the "excluded volume." Assume that the distance of closest approach between two similar atoms is the sum of their radii \((2 r) .\) (a) Calculate the volume around each atom into which the center of another atom cannot penetrate. (b) From your result in (a), calculate the excluded volume for 1 mole of the atoms, which is the constant \(b\). How does this volume compare with the sum of the volumes of 1 mole of the atoms?

The concept of excluded volume is pivotal in understanding the physical behavior of gases, especially when we divert from ideal conditions to the realm of real gases. Excluded volume can be thought of as the space around an atom or molecule into which other atoms or molecules cannot enter due to steric hindrance, essentially forming an exclusion zone.

In a simplistic model, if we consider an atom to be a sphere with radius r, the exclusion volume for this single atom is the volume of a sphere with radius r. Interestingly, when we refer to a collection of atoms, like one mole of gas, we need to take into account that each atom contributes to excluding others from their spaces. Therefore, we calculate the excluded volume by multiplying the volume of a single atom's exclusion sphere by Avogadro's number, thus scaling up from an individual atom to a mole.

When delving into Avogadro's number, we touch upon one of the fundamental constants of chemistry. Defined as the number of atoms, molecules, or ions in one mole of a substance, it provides a bridge between the microscopic scale of atoms and the macroscopic world we can measure. Avogadro's number, approximately equal to \(6.022 \times 10^{23}\text{ entities/mole}\), allows us to count particles in a substance by weighing it.

Considering a mole of a substance, we can understand the massive amount of atoms involved and how this number is used to scale properties, such as volume, from a single atom to molar levels. Any calculation that requires consideration of individual particles within a sample must refer to this massive constant.

The molar volume of a gas is the volume that one mole of the substance occupies under specified conditions of temperature and pressure. In the context of ideal gases, it's a straightforward calculation based on the ideal gas law, leading to a value that, at standard temperature and pressure (STP), is roughly \(22.4 \text{ liters/mole}\).

However, this becomes more complex with real gases, which do not always behave ideally, especially under high pressure or low temperature. We must then turn to corrections for non-ideal behavior, such as those proposed by the Van der Waals equation which incorporates terms for both excluded volume and intermolecular forces.

The study of real gases opens up when deviations from the ideal gas law become apparent. Real gases exhibit behavior that cannot be explained by the simple assumptions of the ideal gas model. These discrepancies arise due to interactions between gas particles and the finite volume that these particles occupy.

Real gases, unlike ideal gases, have interactions between molecules; they attract one another when at a distance and repel when very close. Furthermore, the concept of excluded volume is introduced as the volume truly occupied by the gas particles themselves is non-negligible at high pressures or low temperatures. This is why the Van der Waals equation is a useful tool since it corrects for these factors, aligning predictions more closely with real-life observations.

Considering the molecular volume leads us to ponder the actual space occupied by a molecule, taking into account its geometric size. When we say molecular volume, we refer to the volume a single molecule occupies, which can be approximated by modeling the molecule as a sphere and using the formula for the volume of a sphere.

However, the volume each atom 'excludes,' as in the Van der Waals equation, is different from the actual molecular volume. This difference arises because excluded volume considers the space into which the center of another atom cannot penetrate, rather than the space each atom physically occupies. It's a hint of how dimensionality affects the properties and behaviors of particles in gases, and it underscores the nuances between models that describe ideal versus real gases.