Q.Calculate the ratio of the mean free paths of the molecules of two gases having molecular diameters 1 Å and 2 Å. The gases may be considered under identical conditions of temperature, pressure and volume.

Molecular Volume Fraction

Kinetic theory begins with a bold assumption: gas molecules are so small compared to the space they roam in that we can treat them as point particles with essentially zero volume. The molecular volume fraction is the number that checks whether that assumption is actually reasonable — it compares the volume genuinely occupied by the molecules themselves to the total volume the gas fills.

The Intuition

Picture a large empty hall with a handful of tennis balls rolling around freely. The balls take up hardly any of the hall's floor space — most of what you see is empty air. That is exactly the picture kinetic theory paints of an ordinary gas: the molecules are the tennis balls, and the container is the hall. The molecular volume fraction simply asks: what fraction of the hall's volume is actually tennis ball?

The Precise Statement

If a gas has $N$ molecules, each modelled as a sphere of radius $r$ (so each molecule has volume $\frac{4}{3}\pi r^{3}$), and the gas occupies a total volume $V$, the molecular volume fraction is:

This is a pure number (no units) — a ratio of two volumes.

A Worked Example (STP Oxygen)

Take one mole of oxygen gas at STP, where $N = N_A = 6.02 \times 10^{23}$ molecules and the molar volume is $V = 22.4\ \text{L} = 2.24 \times 10^{-2}\ \text{m}^3$. Taking the molecular diameter to be about $3\ \text{Å} = 3 \times 10^{-10}\ \text{m}$ (radius $r = 1.5 \times 10^{-10}\ \text{m}$):

So the molecules themselves occupy only about 0.04% of the space the gas fills — the remaining 99.96% is empty. This tiny number is exactly why kinetic theory can safely ignore the size of individual molecules for an ordinary gas.

Why It Matters: Ideal vs. Real Gas …