• word_count: 200

Surface tension ($S$) is force per unit length along a liquid surface: $S = F/l$ (N/m = J/m$^2$). It arises from unbalanced intermolecular forces on surface molecules, causing the surface to minimize its area. Surface tension decreases with temperature and reaches zero at the critical temperature.

Excess pressure inside curved surfaces follows the Young-Laplace equation. For a spherical drop: $\Delta P = 2S/R$. For a soap bubble (two surfaces): $\Delta P = 4S/R$. Smaller bubbles have higher internal pressure — when connected, air flows from smaller to larger.

Capillary rise: $h = 2S\cos\theta/(\rho gr)$, where $\theta$ is the contact angle. For wetting liquids ($\theta < 90°$, like water on glass), the meniscus is concave and liquid rises. For non-wetting liquids ($\theta > 90°$, like mercury on glass), the meniscus is convex and liquid depresses. Rise is inversely proportional to tube radius.

Energy considerations: work to form a soap bubble of radius $R$ is $8\pi SR^2$. When $n$ drops merge into one, surface energy decreases (released as heat): $\Delta E = 4\pi Sr^2(n - n^{2/3})$.