Question

Consider a gas sample consisting of molecules with radius \(r\). (a) Determine the excluded volume defined by two molecules and (b) calculate the excluded volume per mole \((b)\) for the gas. Compare the excluded volume per mole with the volume actually occupied by a mole of the molecules.

Short answer

Determine the excluded volume and calculate it per mole.

Step-by-step solution

Understanding Excluded Volume between Two Molecules

When two spherical molecules of radius \( r \) are in contact, their centers are separated by a distance of \( 2r \). The excluded volume is the volume in which one molecule cannot be, due to the presence of another. This leads to a sphere of radius \( 2r \) having a volume \( V_{excluded} \) for these two molecules. This volume is calculated using the formula for the volume of a sphere, \( V = \frac{4}{3} \pi (2r)^3 \).

Key concepts

Gas Molecules

Gas molecules are the tiny building blocks that make up gases. They are in constant motion, bouncing around and colliding with each other and the walls of their container. This motion is due to their kinetic energy, which is influenced by temperature. Gas molecules can be thought of as tiny spheres, a simplification that helps in understanding various properties, such as volume and pressure.
The behavior of gas molecules is described by various physical laws, including Boyle's and Charles's laws, as well as the ideal gas law. These relationships help us understand how gases will react to changes in pressure, volume, and temperature. Understanding these fundamentals is essential because it helps predict how gases will behave under different conditions.

Radius of Molecules

The radius of a molecule is an average measurement of its size, often treated as the distance from its core to the outer edge. In spherical molecules, this radius is crucial because it helps in calculating the volume the molecule might occupy if it were independent of its surroundings.
When dealing with a sphere, the radius is used in the formula for volume, which is critical in many calculations involving molecules. For example, in the case of two touching gas molecules, the distance between their centers is twice the radius, i.e., \(2r\). This helps in determining the 'excluded volume', an important aspect in understanding real gas interactions. Knowing the radius also aids in comparing theoretical models of gas molecules to actual behavior, gaining insights into their interactions.

Molecular Volume

Molecular volume refers to the space that a single molecule occupies. When molecules are considered as spheres, the molecular volume can be calculated using the formula for the volume of a sphere, \( V = \frac{4}{3} \pi r^3 \). Here, \(r\) is the radius of the molecule.
This concept becomes vital when analyzing how molecules interact with each other. The excluded volume is a crucial facet here—it represents the volume a second molecule cannot occupy because of the space taken up by the first. The calculated volume for two touching molecules is a sphere with radius \(2r\), leading to the formula \( V_{excluded} = \frac{4}{3} \pi (2r)^3 \). This underscores how molecular volume isn't just about the space a molecule occupies, but extends to how it influences the space around it.

Mole Calculation

The concept of a mole is quintessential in chemistry. A mole refers to Avogadro's number (approximately \(6.02 \times 10^{23}\)) of molecules or atoms. This standard unit allows chemists to count particles in a given substance in a manageable way.
For gas calculations, the mole concept is integral for determining quantities like excluded volume per mole. By calculating the excluded volume of one pair of gas molecules and scaling it up by Avogadro's number, we obtain the excluded volume for a mole of molecules, often denoted by \(b\). This enables chemists to compare it with the actual volume a mole of gas occupies, highlighting the difference between ideal and real gases. Such comparisons are crucial in understanding and correcting behaviors of gases in different conditions.