Problem Statement
Describe the structure of the liquid-vapour interface and estimate its thickness.
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Identify given quantities and set up the problem: The liquid-vapour interface is not infinitely sharp. The density profile $\rho(z)$ transitions smoothly from the liquid density $\rho_l$ to vapour density $\rho_v$:
Step 2 — Apply the relevant physical law or equation: $$\rho(z) = \frac{\rho_l+\rho_v}{2} – \frac{\rho_l-\rho_v}{2}\tanh\frac{2z}{\xi}$$
Step 3 — Solve algebraically for the unknown: where $\xi$ is the interface thickness (correlation length).
Step 4 — Substitute numerical values with units: Near the critical point, $\xi$ diverges as $\xi \propto |T-T_c|^{-\nu}$ (with $\nu\approx0.63$). Far below $T_c$: $\xi \approx$ a few molecular diameters ($\sim1$–$2\ \text{nm}$ for water at room temperature).
Step 5 — Compute and check the result: The van der Waals / Cahn-Hilliard theory predicts: $\xi = \sqrt{\kappa_0/c_2}$ where $\kappa_0$ is the square-gradient coefficient. At room temperature, the water interface has $\xi\approx0.3\ \text{nm}$ (theoretical) but appears thicker due to capillary waves.
Worked Calculation
$$\rho(z) = \frac{\rho_l+\rho_v}{2} – \frac{\rho_l-\rho_v}{2}\tanh\frac{2z}{\xi}$$
$$\text{Numerical result} = \text{given expression substituted with values}$$
$$\boxed{\rho(z) = \frac{\rho_l+\rho_v}{2} – \frac{\rho_l-\rho_v}{2}\tanh\frac{2z}{\xi}}$$
The liquid-vapour interface is not infinitely sharp. The density profile $\rho(z)$ transitions smoothly from the liquid density $\rho_l$ to vapour density $\rho_v$:
$$\rho(z) = \frac{\rho_l+\rho_v}{2} – \frac{\rho_l-\rho_v}{2}\tanh\frac{2z}{\xi}$$
where $\xi$ is the interface thickness (correlation length).
Near the critical point, $\xi$ diverges as $\xi \propto |T-T_c|^{-\nu}$ (with $\nu\approx0.63$). Far below $T_c$: $\xi \approx$ a few molecular diameters ($\sim1$–$2\ \text{nm}$ for water at room temperature).
The van der Waals / Cahn-Hilliard theory predicts: $\xi = \sqrt{\kappa_0/c_2}$ where $\kappa_0$ is the square-gradient coefficient. At room temperature, the water interface has $\xi\approx0.3\ \text{nm}$ (theoretical) but appears thicker due to capillary waves.
Answer
$$\boxed{\rho(z) = \frac{\rho_l+\rho_v}{2} – \frac{\rho_l-\rho_v}{2}\tanh\frac{2z}{\xi}}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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