Problem Statement
Derive Young’s equation for the contact angle $\theta$ of a liquid drop on a solid surface.
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: At the three-phase contact line, three surface tensions act: solid-liquid ($\sigma_{SL}$), solid-gas ($\sigma_{SG}$), and liquid-gas ($\sigma_{LG}$). Horizontal force balance at the contact line:
Step 2 — Apply the relevant physical law or equation: $$\sigma_{SG} = \sigma_{SL} + \sigma_{LG}\cos\theta$$
$$\boxed{\cos\theta = \frac{\sigma_{SG}-\sigma_{SL}}{\sigma_{LG}}}$$
Step 3 — Solve algebraically for the unknown: This is Young’s equation.
Step 4 — Substitute numerical values with units:
- $\theta < 90°$: wetting (hydrophilic); the liquid spreads.
- $\theta > 90°$: non-wetting (hydrophobic); mercury on glass ($\theta \approx 140°$).
- $\theta = 0°$: complete wetting; liquid spreads indefinitely.
Worked Calculation
$$\sigma_{SG} = \sigma_{SL} + \sigma_{LG}\cos\theta$$
$$\boxed{\cos\theta = \frac{\sigma_{SG}-\sigma_{SL}}{\sigma_{LG}}}$$
$$\text{Numerical result} = \text{given expression substituted with values}$$
At the three-phase contact line, three surface tensions act: solid-liquid ($\sigma_{SL}$), solid-gas ($\sigma_{SG}$), and liquid-gas ($\sigma_{LG}$). Horizontal force balance at the contact line:
$$\sigma_{SG} = \sigma_{SL} + \sigma_{LG}\cos\theta$$
$$\boxed{\cos\theta = \frac{\sigma_{SG}-\sigma_{SL}}{\sigma_{LG}}}$$
This is Young’s equation.
- $\theta < 90°$: wetting (hydrophilic); the liquid spreads.
- $\theta > 90°$: non-wetting (hydrophobic); mercury on glass ($\theta \approx 140°$).
- $\theta = 0°$: complete wetting; liquid spreads indefinitely.
Answer
$$\boxed{\cos\theta = \frac{\sigma_{SG}-\sigma_{SL}}{\sigma_{LG}}}$$
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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