Question

$\huge{\underline{\underline{\red{\mathfrak{Question :}}}}}$

1) Explain Rise of Liquid in A Capillary tube. (Derivation)

2) Expain Surface Energy and Surface Tension

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Answer 1

Answer 1)

Capillarity: Rise or fall of liquid in a capillary tube is known as capillarity.

Consider a capillary tube of radius (R) is emerged in a liquid of density ($\sf{\rho})$ such that the shape of meniscus becomes concave.

[Refer the attachment for figure]

Let R be the radius of meniscus and $\sf{P_o}$ be the atmospheric pressure.

Then,

P(A) = $\sf{P_o}$

P(B) = $\sf{P_o\:-\:\frac{2s}{R}}$ (excess pressure = 2s/R)

Also,

P(C) = $\sf{P_o}$

P(D) = $\sf{P_o}$

Since, B and D are at same horizontal level. P at (B) must be equal to P at (D).

Therefore,

In order to attain equilibrium. The liquid in the capillary rises to height (h) as shown in figure (2)

P(E) = P(B) + $\sf{h \rho g}$

$\implies\:\sf{P_o\:=\:(P_o\:-\:\frac{2s}{R})\:+\:h \rho g}$

$\implies\:\sf{P_o\:-\:P_o\:+\:\frac{2s}{R}\:=\:h \rho g}$

$\implies\:\sf{\frac{2s}{R}\:=\:h \rho g}$

$\implies\:\sf{h\:=\:\frac{2s}{R \rho g}}$

Answer 2)

Surface Energy: It is the amount it work done against the force of surface tension in forming a liquid surface at given temperature.

(Refer the attachment for figure)

Consider a rectangular surface film of surface tension (s) where, S= F/2L

Therefore, F = S× 2L

Small work done to stretch this film by length ∆x will be

⇒ ∆W = F.∆x

⇒ ∆W = S× (2L ∆x)

⇒ ∆W = S× A

This work done is stored as energy. So,

⇒ E = S× ∆A

Surface Tension: It is property of liquid due to which the free surface of the liquid tends to acquire minimum surface area as if covered by a thin membrane.

⇒ S= F/L

Answer 2

Answer:

• h = [ 2 T cos ∅ / r ρ g] (or) h = [ 2 T / R ρ g]

Explanation:

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1) Explain Rise of Liquid in A Capillary tube. (Derivation)

Assumption:

Let there be a Capillary tube of large length and Radius ' r ' is dipped in a liquid having Angle of Contact ' ∅ '  & Surface Tension ' T '. And, Let the radius of Meniscus be ' R '.

# Refer the Attachment for Figure.

Now, Surface Tension (T) acts Tangentially as shown in Figure making an angle ∅ with the Meniscus.

Now, the Upward Force (F) will be,

$\longmapsto \large{ \text{F} = \text{Circumfrence of Capillary T.} \times \text{Surface Tension}}$

Substituting,

$\longmapsto\large{\tt F = 2 \pi r \times T \cos \theta}$

As,

• Surface Tension which causes Upward Movement of Water is T cos ∅.
• And, Circumference is 2 Π r.

Therefore,

$\longmapsto \large{\tt F = 2 \pi r \times T \cos \theta \; \; \quad\dfrac{\quad}{} \tt{[1]}}$

Liquid rises in the capillary tube till the upward force due to Surface Tension is balanced by the Weight of Liquid acting downwards.

Now,

$\longmapsto\large{\tt \text{F} = \text{Mass} \times \text{Accelration}}$

Substituting,

$\longmapsto\large{\tt F = M \times g}$

As,

• Mass is M
• And, Acceleration = Acceleration due to gravity (g)

$\longmapsto\large{\tt F = M \times g \; \; \quad\dfrac{\quad}{} \tt{[2]}}$

From (1) and (2)

$\longmapsto\large{\tt 2 \pi r \times T \cos \theta = M \times g}$

∵ [ Mass (M) = Volume (v) x Density (ρ) ]

Substituting,

$\longmapsto\large{\tt 2 \pi r \times T \cos \theta = V \times \rho\times g}$

As Capillary is Cylindrical Volume (V) = Π r² h

Now,

$\longmapsto\large{\tt 2 \pi r \times T \cos \theta = \pi \; r^2 \; h \times \rho\times g}$

$\longmapsto\large{\tt 2 \cancel{\pi r} \times T \cos \theta = \cancel{\pi \; r^2} \; h \times \rho\times g}$

$\longmapsto\large{\tt 2 \times T \cos \theta = r \times h \times \rho\times g}$

$\longmapsto\large{\tt 2T \cos \theta = h \times r \; \rho\; g}$

$\longmapsto\large{\underline{\boxed{\red{\tt h = \dfrac{2 \; T \cos \theta}{r \; \rho \; g} }}}}$

(OR)

We Know, r = R cos ∅

Substituting,

$\longmapsto\large{\underline{\boxed{\red{\tt h = \dfrac{2 \; T}{R \; \rho \; g} }}}}$

• R = radius of Meniscus.
• r = radius of Capillary tube.

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2) Explain Surface Energy and Surface Tension.

Surface Energy:

Surface Energy is Defined as the Mechanical Work required to increase area of 1 m² i.e. Unit area of a liquid Provided that the Temp. is Constant.

Formula of Surface Energy :-

• E = W / ΔA

S.I unit :-

• J m⁻².

Surface Tension:

Surface Tension is defined as Force per unit length acting on the Interference between Plane of liquid & Other Surfaces.

Formula of Surface Tension :-

• T = F / L

S.I unit :-

• N m⁻¹

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