Capillary Action

Capillary action is the phenomenon in which a liquid rises or falls in a narrow tube or porous material due to the combined effect of adhesive force, cohesive force, and surface tension. This movement can occur even against the force of gravity. The phenomenon mainly depends on the interaction between the liquid molecules and the surface of the tube.

It occurs in narrow tubes or porous materials.
Adhesive and cohesive forces are responsible for this phenomenon.
Liquids rise when adhesive force is greater than cohesive force.
The height of rise depends on tube diameter, surface tension, and gravity.

Formula

If a liquid rises to a height "h" in the capillary tube of radius 'r' and surface tension on the surface of the liquid is 'T' then its rise in height is given by the formula,

h = \frac{2T cosθ}{ρgh}
where,
h = Height the liquid rises in the capillary tube
T = Surface tension of the liquid
θ = Angle of contact (or contact angle) between the liquid and the tube's surface
r = Radius of the capillary tube
ρ = Density of the liquid
g= Acceleration due to gravity

Derivation of Capillary Action Formula

The pressure difference caused by the height h of the liquid column is balanced by the pressure due to the surface tension. The pressure difference at the liquid-air interface is given by the equation for capillary rise or fall:

ΔP=\frac{2T}{ R}
where,
T is the surface tension of the liquid,
R is the radius of curvature of the meniscus.

So, the balance of forces gives the equation:

ρgh=\frac{2T}{R}
where,
ρ is the density of the liquid,
g is the acceleration due to gravity,
h is the height of the liquid column.

The radius of curvature of the meniscus is related to the radius r of the capillary tube and the angle of contact θ between the liquid and the tube. The relation is:

R=\frac{r}{cosθ}
where,
r is the radius of the capillary tube,
θ is the angle of contact (or wetting angle).
Substituting R into the pressure balance equation:
hρg=\frac{\frac{2T}{r}}{cosθ}
Rearranging this equation to solve for the height h =\frac{2Tcosθ}{ρgr }
This final equation is the correct expression for the capillary rise or fall in a capillary tube. So, your expression:
h=\frac{2Tcosθ}{ρgr}

Forces in Capillary Action

There are three main factors that influence capillary action are:

Cohesive Force
Adhesive Force
Surface Tension

1. Cohesive Force: Cohesion refers to the attractive force between molecules of the same substance. In the case of liquids, these forces hold the molecules together, creating surface tension. Example: the formation of raindrops is due to the cohesive forces between water molecules. Surface tension allows objects denser than the liquid, like a needle or insect, to float on the surface without sinking, as the liquid resists the force of gravity.

2. Adhesive Force: Adhesion refers to the attractive force between different substances, such as a liquid and a solid surface. In capillary action, adhesion causes a liquid to adhere to the walls of a container, pulling the liquid upwards. Water, Example: adheres to glass surfaces, causing the liquid to "climb" the sides of a container.

3. Surface Tension: Surface tension is the result of both cohesive and adhesive forces. On the surface of a liquid, molecules experience stronger cohesive forces from the molecules inside the liquid, as they are surrounded by similar molecules. This creates a "skin" on the liquid's surface, allowing it to resist external forces and making the surface behave like a stretched elastic membrane.

When adhesion is stronger than cohesion, the liquid will spread over the surface, as seen with water on glass. In contrast, when cohesion is stronger than adhesion, such as with mercury, the liquid forms an inward curve, causing the liquid to pull away from the surface and creating a convex meniscus.

Factors Affecting Capillary Action

Capillary action is influenced by various factors, including the properties of the liquid and solid, the angle of contact, the diameter of the capillary tube, and the surface tension of the liquid. These factors collectively determine how high or low the liquid will rise in a capillary tube.

1. Nature of Liquids and Solids: The type of liquid and solid surface (e.g., water in glass vs. mercury in glass) determines whether the liquid will rise or fall in the capillary.

2. Angle of Contact:

If θ < 90°: The meniscus is concave, and the liquid rises (e.g., water in a glass tube).
If θ = 90°: The meniscus is flat, and there is no capillary action.
If θ > 90°: The meniscus is convex, and the liquid falls (e.g., mercury in a glass tube).

Capillary-Action-based on angle of contact

3. Diameter of the Capillary Tube: The liquid rises higher in narrower tubes. The rise is inversely proportional to the tube's radius.

4. Surface Tension: Liquids with higher surface tension (e.g., water) show more capillary action compared to those with lower surface tension (e.g., alcohol).

Liquid Meniscus in Capillarity

A liquid in a capillary tube can exhibit three different types of meniscus shapes, depending on the interactions between the liquid and the tube surface. These menisci include concave, convex, and plane meniscus.

1. Concave Meniscus: A concave meniscus occurs when the pressure below the meniscus is less than the pressure above it. This is observed when liquids like water are in contact with materials like glass.

The pressure difference between the liquid above and below the meniscus is given by:

P_A - P_B = \frac{2T}{r}
Where P_A is the pressure above the meniscus, P_Bis the pressure below the meniscus, T is the surface tension, and r is the radius of the meniscus.

2. Convex Meniscus: A convex meniscus occurs when the pressure below the meniscus is greater than the pressure above it. This is seen with liquids like mercury in contact with glass.

The excess pressure is given by:

P_B - P_A = \frac{2T}{r}

3. Plane Meniscus: A plane meniscus occurs when the pressure difference between the liquid above and below the meniscus is zero, meaning there is no capillary action. In this case, the pressure above and below the meniscus is equal, and the excess pressure is zero.

These three types of menisci (concave, convex, and plane) play an important role in determining the behavior of liquids in capillary tubes, influencing the liquid's rise or fall depending on the nature of the liquid and the solid surface.

Concave vs Convex vs Plane Meniscus

Concave Meniscus	Convex Meniscus	Plane Meniscus
In concave meniscus the pressure below the meniscus (P₀ - 2T/r) is less than pressure above the meniscus (P_o)	In convex meniscus the pressure below the meniscus (P₀ + 2T/r) is greater than pressure above the meniscus (P_o)	In plane meniscus the pressure below the meniscus is equal to pressure above the meniscus.
Excess Pressure is given as, P = P_above - P_below P = 2T/r	Excess Pressure is given as, P = P_below - P_above P = 2T/r	Excess Pressure is given as, P = 0
Liquid will wet the solid surface.	Liquid will not wet the solid surface.	No capillary action observed.
Angle of contact is acute angle, i.e. (θ < 90°)	Angle of contact is obtuse angle, i.e. (θ > 90°)	Angle of contact is right angle, i.e. (θ = 90°)
Liquid level in the capillary ascends	Liquid level in the capillary descends	No capillarity

Relation between Excess Pressure and Surface Tension

Let's take a liquid drop of radius (r) having internal pressure P_Iand external pressure P_o, then the relation between surface tension and excess pressure for a liquid drop is given as,

Excess Pressure(P) = P_I - P_O
To change the radius of the drop of liquid from r to r + dr, external work(W) is required.
W = F.dr
F = P.A
W = P.A.dr...(i)
A = 4πr²
W = P.(4πr²).dr...(ii)
If the radius of droprs changes from r to r + dr then its area is changed as,
dA = 4π(r + dr)² - 4πr²
dA = 8πrdr...(iii)
Now, we know that workdone (W) is also given as,
W = T.dA...(iv)
where T is surface tension
T =\frac{W}{dA}..(v)
From eq (ii), (iii), and (iv)
T = \frac{P.4πr^2.dr}{8πrdr}
P = P_i - P_o = \frac{2T}{r}
This is the relation between Excess Pressure(P) and Surface Tension(T)

Applications of Capillarity

Water moves from roots to leaves in plants through xylem due to capillary action.
Oil rises in a lamp wick and reaches the flame due to capillary action.
Towels and cotton clothes absorb water because of capillary action in fine pores.
Ink flows in fountain pens through narrow channels due to capillary action.
Water spreads through soil particles and helps in supplying moisture to plants.

Solved Problems

Question 1: A liquid of surface tension T= 0.06 N/m, density ρ=900 kg/m³, rises in a capillary tube of radius r=0.4 mm. Angle of contact is 0^o. Find height.

Solution: Formula:
h = \frac{2T\cos\theta}{\rho g r}
h = \frac{2 \times 0.06 \times \cos 0^\circ}{900 \times 9.8 \times 0.4 \times 10^{-3}}
h = \frac{0.12}{900 \times 9.8 \times 0.4 \times 10^{-3}}
h = \frac{0.12}{3.528}
h = 0.0340\,m = 3.4\,cm

Question 2: A capillary tube of radius 0.3 mm shows a rise of 5 cm. Find surface tension of water. Given: ρ = 1000 kg/m³, g = 9.8 m/s², θ = 0^o

Solution: Formula
T = \frac{h\rho g r}{2}
T = \frac{0.05 \times 1000 \times 9.8 \times 0.3 \times 10^{-3}}{2}
T = \frac{0.147}{2}
T = 0.0735\,N/m

Question 3: A liquid rises to a height of 3 cm in a capillary tube of radius 0.3 mm. The surface tension of the liquid is 0.06 N/m. Find the density of the liquid. Assume θ = 0^o, g = 9.8 m/s².

Solution: Formula:
h = \frac{2T\cos\theta}{\rho g r}
Since θ = 0^o ⇒ cos⁡θ = 1
Rearranging:
\rho = \frac{2T}{h g r}
Substituting values:
\rho = \frac{2 \times 0.06}{0.03 \times 9.8 \times 0.3 \times 10^{-3}}
\rho = \frac{0.12}{0.0000882}
\rho \approx 1360.5\,kg/m^3

Question 4: A liquid rises to height h₁ in a capillary tube when the angle of contact is 30^o. If the angle of contact changes to 60^o, find the ratio h₂/ h₁. Assume all other quantities remain constant.

Solution: Formula:
h \propto \cos\theta
\frac{h_2}{h_1} = \frac{\cos 60^\circ}{\cos 30^\circ}
\cos 60^\circ = \frac{1}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}
\frac{h_2}{h_1} = \frac{1/2}{\sqrt{3}/2}
\frac{h_2}{h_1} = \frac{1}{\sqrt{3}}

Unsolved Problem

Question 1: A liquid of surface tension 0.07 Nm and density 900 kg/m³ rises in a capillary tube of radius 0.5 mm. Find the height of rise if the angle of contact is 0^o.

Question 2: A liquid rises to a height of 4 cm in a capillary tube of radius 0.2 mm. If the radius is doubled, find the new height of rise.

Question 3: The surface tension of a liquid is 0.06 Nm. It rises to 3cm in a capillary tube of radius 0.3 mm. Find the density of the liquid.

Question 4: A liquid has angle of contact 60^o. If the capillary rise for 0^o is h, find the rise for 60^o in terms of h.

Question 5: In a capillary tube, a liquid rises to height 5 cm. If surface tension is doubled and all other factors remain constant, find the new height of rise.

Tension Force
Gravitational Force
Buoyant Force
Adhesion and Cohesion