Key examples — intermolecular forces

Where the chapter’s machinery shows up across the bookshelf.

Example 1: water’s bulk modulus from molecular parameters

Estimating $K \sim \varepsilon/\sigma^3$ for water with $\varepsilon \sim 0.25\,\text{eV}$ (hydrogen-bond energy), $\sigma \sim 0.3\,\text{nm}$ (O–O separation) gives $K \sim 1.5\,\text{GPa}$. Measured value: 2.2 GPa. The factor-of-2 agreement is remarkable for a one-line dimensional argument and represents one of the few cases where a macroscopic acoustic quantity is calculable directly from molecular-scale parameters. See Cavitation Ch 1.1.

Example 2: the tensile-strength puzzle

Spinodal-limit calculations give in-principle tensile strengths of $\sim 100\,\text{MPa}$ for water at room temperature. Carefully prepared samples in laboratory have reached $\sim 30\,\text{MPa}$; ordinary water fails at $\sim 1\,\text{MPa}$ or less. The three-orders-of-magnitude gap motivates the heterogeneous-nucleation story of Cavitation Ch 2.2: real liquids contain pre-existing gas-filled crevices on surfaces that fail far below the homogeneous spinodal.

Example 3: high surface tension of water from hydrogen bonds

Surface tension $\sigma$ is approximately the cohesive-energy density times molecular separation: $\sigma \sim \varepsilon/\sigma^2$. For water at room temperature, $\varepsilon_\text{HB} \approx 0.25\,\text{eV}$ and $\sigma \approx 0.3\,\text{nm}$ give $\sigma \approx 70\,\text{mN/m}$ — matching the measured 72 mN/m. The directionality of hydrogen bonds, captured in the chapter’s HydrogenBondAngle visualisation, is what makes water’s surface tension exceptional among common liquids (most others sit at $\sim 20-30\,\text{mN/m}$). See surface-tension chapter.

Example 4: van der Waals coexistence and Maxwell construction

For temperatures below the critical $T_c$, the van der Waals isotherm has a region $(\partial p/\partial v) > 0$ — unstable. The Maxwell equal-area construction on the isotherm gives the coexistence pressure: a horizontal line drawn so the two areas above and below it (between the line and the isotherm) are equal. The endpoints of this line are the equilibrium densities of liquid and vapour. This is the operative tool for predicting liquid-vapour phase diagrams from molecular-level parameters $a$ and $b$.

Example 5: density anomaly of water and ice

Water has a density maximum at 4°C — it expands on cooling below this temperature. Ice is less dense than liquid water. Both anomalies trace to the tetrahedral hydrogen-bond geometry: ice’s open, four-coordinated structure has lower density than the partially-disordered liquid where some hydrogen bonds are broken and molecules pack more closely. This is uniquely consequential for biology (ice floats on lakes) and for the climate.

Cross-book backlinks

• Cavitation Ch 1.1 — what a liquid is: bulk modulus from molecular parameters.
• Cavitation Ch 1.2 — tensile strength in principle: spinodal limit.
• Cavitation Ch 1.3 — the pathetic measured strength: why real water fails so much earlier.
• Cavitation Ch 2.2 — heterogeneous & crevice nucleation: gas-pocket model.
• Physics surface-tension — chapter: $\sigma$ from cohesive energy.