Estimate the distance (in nm ) between molecules of water...

Estimate the distance (in $\mathrm{nm}$ ) between molecules of water vapor at $100^{\circ} \mathrm{C}$ and 1.0 atm. Assume ideal behavior. Repeat the calculation for liquid water at $100^{\circ} \mathrm{C},$ given that the density of water is $0.96 \mathrm{g} / \mathrm{cm}^{3}$ at that temperature. Comment on your results. (Assume each water molecule to be a sphere with a diameter of $0.3 \mathrm{nm} .$ ) (Hint: First calculate the number density of water molecules. Next, convert the number density to linear density, that is, the number of molecules in one direction.)

Key Concepts

Liquid Density

Liquid density is defined as the mass per unit volume of a liquid. In problems involving liquid-phase calculations, the density is used to determine the number density by relating the mass of a single molecule or a formula unit to the overall mass per unit volume. This concept is key when comparing molecular spacing in liquids versus gases.

Linear Density and Molecular Spacing

While number density gives the count per unit volume, linear density refers to the number of molecules along a single direction. By taking the cube root of the number density, one can estimate the average distance between molecules, effectively converting a three-dimensional density into a one-dimensional spacing estimate.

Spherical Molecule Approximation

The spherical molecule approximation assumes that molecules can be modeled as spheres of a given diameter. This simplification allows for estimating molecular sizes and spacing in a more straightforward manner, providing a basis for comparing calculated intermolecular distances with the assumed physical size of the molecule.

Ideal Gas Law

The ideal gas law relates the pressure, volume, temperature, and number of moles of a gaseous system, typically expressed as PV = nRT. This law is used to determine the number density of molecules in a gas by converting pressure and temperature measurements into the number of molecules per unit volume, assuming ideal (non-interacting) behavior.

Number Density

Number density is a measure of the number of particles (or molecules) present in a given volume. It is a crucial concept because it provides the starting point for estimating intermolecular distances by establishing how tightly the molecules are packed in either a gas or a liquid state.

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