Problem Statement
Solve the fluid mechanics problem: Explain capillary condensation: why vapour condenses in narrow pores at pressures below the bulk saturation pressure. All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see Concepts section) This problem draws on fundamental physical principles.
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
Surface tension $\sigma$ is the energy per unit area (or force per unit length) at a liquid surface. It arises from cohesive intermolecular forces. Capillary rise results from the balance between surface tension pulling liquid up and gravity pulling it down. The Laplace pressure across a curved interface is $\Delta P = 2\sigma/r$ (sphere) or $\sigma/r$ (cylinder).
- $h = 2\sigma\cos\theta/(\rho g r)$ — capillary height
- $\Delta P = 2\sigma/r$ — excess pressure inside a droplet
- $W = \sigma \Delta A$ — work done against surface tension
Step-by-Step Solution
Step 1 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 2 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Worked Calculation
$$P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g h_2$$
$$v = \sqrt{2gh} = \sqrt{2\times9.8\times2} = \sqrt{39.2} \approx 6.26\,\text{m/s}$$
$$\boxed{v_{\text{efflux}} = \sqrt{2gh}}$$
Answer
$$\boxed{v_{\text{efflux}} = \sqrt{2gh}}$$
Physical Interpretation
Capillary action allows plants to draw water from roots to leaves against gravity. The thinner the tube, the higher the rise — but also the smaller the volume transported.
Leave a Reply