(a) Consider 1.00 mol of an ideal gas at 285 K and 1.00 atm pressure. Imagine that the molecu

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(a) Consider 1.00 mol of an ideal gas at 285 K and 1.00...

(a) Consider $1.00 \mathrm{~mol}$ of an ideal gas at $285 \mathrm{~K}$ and $1.00 \mathrm{~atm}$ pressure. Imagine that the molecules are for the most part evenly spaced at the centers of identical cubes. Using Avogadro's constant and taking the diameter of a molecule to be $3.00 \times 10^{-8} \mathrm{~cm}$, find the length of an edge of such a cube and calculate the ratio of this length to the diameter of a molecule. The edge length is an estimate of the distance between molecules in the gas. (b) Now consider a mole of water having a volume of $18 \mathrm{~cm}^{3}$. Again imagine the molecules to be evenly spaced at the centers of identical cubes and repeat the calculation in $(a)$.

Key Concepts

Comparison of Molecular Distance to Molecular Diameter

After calculating the cube edge length (representing the distance between neighboring molecules), comparing this length to the actual diameter of a molecule provides insight into how far apart the molecules are relative to their size. This ratio is especially important in distinguishing the sparse arrangement of molecules in a gas from the much denser packing in liquids.

Cube Model for Molecular Spacing

This model assigns a cube of equal volume to each molecule such that the molecule is situated at the center of the cube. The edge length of the cube can be found by taking the cube root of the volume per molecule. This simplifies the geometric interpretation of intermolecular distances and helps compare these distances with the actual size of the molecules.

Avogadro's Number

Avogadro’s number is the number of particles (atoms or molecules) in one mole of a substance, approximately 6.022 × 10^23. This constant is crucial for converting between macroscopic amounts of substance and the microscopic scale, where each molecule is assigned an equal share of the total volume, enabling the calculation of volume per molecule.

Volume per Molecule

By dividing the total volume occupied by a mole of substance by Avogadro’s number, one obtains the volume accessible to a single molecule. This concept is essential in determining the spacing between molecules when assuming they are isolated in identical volumes, as in the cube model used to approximate molecular distance.

Ideal Gas and Ideal Gas Law

This concept describes a hypothetical gas whose molecules do not interact except through elastic collisions and occupy negligible space relative to the container. The ideal gas law, PV = nRT, relates the pressure, volume, number of moles, and temperature, allowing one to compute the volume taken by a mole of gas under specified conditions, which is fundamental when estimating how molecules are spaced in a gas phase.

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