Capillarity — Revision Notes

⚡ 30-Second Revision

Capillarity: — Rise/fall of liquid in narrow tubes due to surface tension, cohesive, and adhesive forces.
Jurin's Law: — $h = \frac{2T\cos\theta}{r\rho g}$
$T$ — Surface Tension (N/m)
$\theta$ — Angle of Contact

- Acute ($\theta < 90^\circ$): Rise, concave meniscus, wetting (e.g., water/glass) - Obtuse ($\theta > 90^\circ$): Fall, convex meniscus, non-wetting (e.g., mercury/glass) - $90^\circ$: No action, flat meniscus

$r$ — Radius of tube (m) ($h \propto 1/r$)
$\rho$ — Density of liquid (kg/m$^3$) ($h \propto 1/\rho$)
$g$ — Acceleration due to gravity (m/s$^2$) ($h \propto 1/g$)
Inclined Tube: — Vertical height $h$ remains same, length along tube $L = h/\cos\alpha$ (where $\alpha$ is angle with vertical).
Insufficient Length: — Liquid rises to top, meniscus flattens (radius of curvature increases), no overflow.

2-Minute Revision

Capillarity is the phenomenon where liquids rise or fall in narrow tubes, driven by surface tension and the balance of cohesive (liquid-liquid) and adhesive (liquid-solid) forces. The key formula is Jurin's Law: $h = \frac{2T\cos\theta}{r\rho g}$. Here, $h$ is the height of rise/fall, $T$ is surface tension, $\theta$ is the angle of contact, $r$ is the tube radius, $\rho$ is liquid density, and $g$ is gravity.

If the angle of contact ($\theta$) is acute (less than $90^\circ$), adhesive forces are stronger, the liquid wets the surface, forms a concave meniscus, and rises. If $\theta$ is obtuse (greater than $90^\circ$), cohesive forces are stronger, the liquid doesn't wet, forms a convex meniscus, and falls.

Remember the proportionalities: $h$ is inversely proportional to $r$, $\rho$, and $g$, and directly proportional to $T$ and $\cos\theta$. For an inclined tube, the vertical height remains $h$, but the length of the liquid column along the tube increases to $L = h/\cos\alpha$, where $\alpha$ is the angle with the vertical. If a tube is too short, the liquid rises to the top, and the meniscus flattens, preventing overflow.

5-Minute Revision

Capillarity describes the spontaneous movement of a liquid in a narrow tube, either rising (capillary rise) or falling (capillary fall). This behavior is a result of surface tension, which acts like a stretched skin on the liquid surface, and the competition between cohesive forces (attraction within the liquid) and adhesive forces (attraction between liquid and tube material).

The angle of contact ($\theta$), measured inside the liquid at the solid-liquid interface, is crucial. An acute angle ($\theta < 90^\circ$) signifies that adhesive forces dominate, leading to wetting, a concave meniscus, and capillary rise (e.

g., water in glass). An obtuse angle ($\theta > 90^\circ$) means cohesive forces dominate, resulting in non-wetting, a convex meniscus, and capillary fall (e.g., mercury in glass).

Quantitatively, capillary action is governed by Jurin's Law: $h = \frac{2T\cos\theta}{r\rho g}$. Let's break down its components:

$T$: Surface tension of the liquid. Higher $T$ means greater $h$.
$\cos\theta$: Determines direction and magnitude. Positive for rise, negative for fall.
$r$: Radius of the capillary tube. Smaller $r$ leads to larger $h$ ($h \propto 1/r$).
$\rho$: Density of the liquid. Denser liquids show smaller $h$ ($h \propto 1/\rho$).
$g$: Acceleration due to gravity. Weaker $g$ (e.g., in space) leads to larger $h$.

Example: If water rises $5,\text{cm}$ in a tube of radius $0.5,\text{mm}$, how much will it rise in a tube of radius $0.25,\text{mm}$? Since $h \propto 1/r$, if the radius is halved, the height will double. So, $h' = 2 \times 5,\text{cm} = 10,\text{cm}$.

Special Cases:

1
Inclined Tube: — If a tube is inclined at angle $\alpha$ to the vertical, the vertical height $h$ remains the same. The length of the liquid column along the tube, $L$, is $L = h/\cos\alpha$. So, $L$ will be greater than $h$.
2
Insufficient Length: — If the tube's physical length is less than the calculated $h$, the liquid will rise to the top but will not overflow. Instead, the meniscus will flatten (its radius of curvature increases) to adjust the upward surface tension force to exactly balance the weight of the liquid column present.
3
Effect of Temperature/Impurities: — Increasing temperature generally decreases $T$, thus decreasing $h$. Impurities can either increase or decrease $T$, affecting $h$ accordingly.

Mastering these concepts and the formula is key for NEET questions.

Prelims Revision Notes

Capillarity: NEET Revision Notes

1. Definition: The phenomenon of rise or fall of a liquid in a narrow tube (capillary tube) due to surface tension and the balance of cohesive and adhesive forces.

2. Key Forces:

* Cohesive Forces: Attraction between molecules of the *same* liquid. * Adhesive Forces: Attraction between liquid molecules and the *solid* tube material.

3. Angle of Contact ($\theta$):

* Angle formed by the tangent to the liquid surface at the point of contact with the solid, measured *inside* the liquid. * **$\theta < 90^\circ$ (Acute):** Adhesive forces > Cohesive forces. Liquid 'wets' the solid.

Concave meniscus. Capillary Rise. (e.g., Water in clean glass, $\theta \approx 0^\circ$) * **$\theta > 90^\circ$ (Obtuse):** Cohesive forces > Adhesive forces. Liquid does not 'wet' the solid. Convex meniscus.

Capillary Fall (Depression). (e.g., Mercury in glass, $\theta \approx 140^\circ$) * **$\theta = 90^\circ$:** Adhesive $\approx$ Cohesive. Flat meniscus. No capillary action (e.g., Pure water in silver).

4. Jurin's Law (Formula for Capillary Rise/Fall):

$$h = \frac{2T\cos\theta}{r\rho g}$$

Where: * $h$: Height of liquid column (m) * $T$: Surface tension of liquid (N/m) * $\theta$: Angle of contact * $r$: Radius of capillary tube (m) * $\rho$: Density of liquid (kg/m$^3$) * $g$: Acceleration due to gravity (m/s$^2$)

5. Proportionalities from Jurin's Law:

* $h \propto T$ (Directly proportional to surface tension) * $h \propto \cos\theta$ (Directly proportional to $\cos\theta$) * $h \propto 1/r$ (Inversely proportional to tube radius) * $h \propto 1/\rho$ (Inversely proportional to liquid density) * $h \propto 1/g$ (Inversely proportional to gravity)

6. Special Cases & Important Points:

* Inclined Capillary Tube: If a tube is inclined at an angle $\alpha$ with the vertical, the vertical height of rise ($h$) remains the same. The length of the liquid column along the tube ($L$) is $L = h/\cos\alpha$.

* Insufficient Length: If the actual length of the tube ($L_{tube}$) is less than the calculated $h$, the liquid will rise to the top of the tube but will not overflow. The meniscus will flatten (its radius of curvature increases) to adjust the upward surface tension force to balance the weight of the liquid column.

* **Weightless Condition ($g=0$):** In space, $h \to \infty$. Liquid rises to fill the entire tube, as there's no gravity to oppose surface tension. * Effect of Temperature: Generally, $T$ decreases with increasing temperature, so $h$ decreases.

* Effect of Impurities: Impurities can increase or decrease $T$, thus affecting $h$.

7. Units: Ensure consistency, preferably SI units for all calculations.

Vyyuha Quick Recall

How Tall Can Radius Drop Gravity? (H = 2TCos$\theta$ / RDG)

Height ($h$) is proportional to:
Tension ($T$)
Cosine of angle of contact ($\cos\theta$)
Inversely proportional to:
Radius ($r$)
Density ($\rho$)
Gravity ($g$)