Gases, Kinetic Theory and Intermolecular Forces

From Free Particles to Condensed Matter

Three experimental laws combine into the ideal gas equation:

$$PV = nRT$$

with $R = 0.0821$ L atm mol$^{-1}$ K$^{-1}$. Boyle found $P \propto 1/V$ at fixed $T$, Charles found $V \propto T$ at fixed $P$, and Avogadro found $V \propto n$. For a fixed sample the combined law $P_1V_1/T_1 = P_2V_2/T_2$ handles most problems, and temperature must always be in kelvin.

The kinetic molecular model explains why. Gas particles are treated as points in constant random motion, with negligible volume, no intermolecular forces, and perfectly elastic collisions. Average kinetic energy depends only on temperature, which gives the root-mean-square speed

$$u_{\text{rms}} = \sqrt{\frac{3RT}{M}}$$

so at a given temperature light molecules move faster, hence Graham's law of effusion, $r_1/r_2 = \sqrt{M_2/M_1}$. The Maxwell distribution broadens and flattens as temperature rises.

Real gases deviate at high pressure (molecular volume is no longer negligible) and low temperature (attractions matter, and eventually the gas liquefies). Van der Waals corrects both:

$$\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$$

Intermolecular forces are what the ideal model throws away, and they set boiling points. In rising order: London dispersion forces, present in everything and growing with molecular size and polarisability; dipole-dipole attractions between permanent dipoles; and hydrogen bonding, a strong special case when H is bonded to N, O or F. Water's anomalously high boiling point is hydrogen bonding, and the same open hydrogen-bonded lattice makes ice less dense than liquid water.

Solids are classified by what holds them: metallic, ionic, molecular (weak intermolecular forces, low melting), and covalent-network such as diamond, where the whole crystal is one molecule.

Common pitfall: using degrees Celsius in a gas law. Every relation here is proportional to absolute temperature; doubling from 20 °C to 40 °C is a rise of only about 7 %, not 100 %.