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9.6.5 - Capillary Rise

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Introduction to Capillary Rise

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Teacher
Teacher

Today, we are going to discuss capillary rise. Can anyone tell me what happens to water when you place a straw in it?

Student 1
Student 1

It goes up the straw!

Teacher
Teacher

Exactly! This is due to capillary rise, which occurs because of surface tension. Who can tell me why this might happen?

Student 2
Student 2

Is it because of the attraction between the water and the straw?

Teacher
Teacher

Correct! When a liquid wets a surface, such as water in a glass tube, it creates an adhesive force that pulls it upward against gravity.

Student 4
Student 4

So, does it mean that thinner straws will let more water rise?

Teacher
Teacher

Yes! The narrower the tube, the higher the water will rise. Remember this with the acronym 'WET' — Wider tubes go down, and Easier rise is Thinner!

Teacher
Teacher

Let's summarize: capillary rise is influenced by the balance of adhesive and cohesive forces.

Pressure Differences in Capillaries

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Teacher
Teacher

Now, let’s explore the pressure difference involved in capillary rise. The equation gives us a deeper understanding: (P_i - P_0) = (2S cos θ) / a. Can someone explain what this means in a simple way?

Student 3
Student 3

It means that the pressure inside the tube is lower than outside because of surface tension!

Teacher
Teacher

Exactly! The contact angle θ affects this pressure difference too. If the contact angle is small, the liquid rises higher.

Student 1
Student 1

So if the liquid doesn't wet the tube, like mercury, it won't rise?

Teacher
Teacher

Right! In this case, the liquid would even fall. To remember this, think about 'GLOW' — Greater angle, Lower rise, Opposite example for water!

Teacher
Teacher

That wraps up our discussion on pressure differences and how they affect capillary rise.

Mathematics of Capillary Rise

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Teacher
Teacher

Let's calculate the height of capillary rise using our equation: h = (2S cos θ) / (ρg a). What do you think happens if we change the radius?

Student 2
Student 2

If we make the radius smaller, the height will be larger!

Teacher
Teacher

That's right! Now, if I state that the surface tension S for water is approximately 0.072 N/m, and the water density ρ is about 1000 kg/m^3, let’s calculate it with a radius of 0.005 m.

Student 4
Student 4

So we plug the values into the equation for h, right?

Teacher
Teacher

Exactly! And you'll find that h is a few centimeters. This shows how practical applications can arise from these small changes!

Teacher
Teacher

In conclusion, understanding these calculations helps us see how nature works on a microscopic level.

Real-World Applications of Capillary Rise

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Teacher
Teacher

What do you think are some real-world applications of capillary rise?

Student 3
Student 3

Plant roots absorbing water!

Teacher
Teacher

Yes! Plants utilize capillarity to draw water from the ground through narrow roots.

Student 1
Student 1

What about ink in pens?

Teacher
Teacher

Great example! Ink rises in the nib of the pen through capillary action. So remember 'CATS': Capillary action is seen in Trees, And in pens, and in Straws.

Teacher
Teacher

This highlights how fundamental concepts in physics manifest in daily life and nature.

Introduction & Overview

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Quick Overview

Capillary rise occurs when water moves upward in a narrow tube due to surface tension, demonstrating the interplay between adhesive and cohesive forces.

Standard

Capillary rise is the phenomenon in which liquid rises or falls in a narrow tube due to surface tension effects. This section explores how liquid movement in narrow spaces is influenced significantly by surface tension and the resulting pressure differences, illustrating key principles of fluid mechanics.

Detailed

Capillary Rise

Capillary rise is a phenomenon where a liquid, such as water, rises in a narrow tube against the force of gravity due to the interactions between the liquid molecules and the tube's surface. This occurs primarily due to adhesive forces between the liquid and the walls of the tube, coupled with the cohesive forces within the liquid itself. When the liquid wets the surface of the tube (for example, water in a glass tube), it creates a concave meniscus where the surface tension results in a pressure difference.

In a circular capillary tube of radius a, the pressure difference across the liquid-air interface due to the curvature of the surface is given by the equation:

(P_i - P_0) = rac{2S}{a} ext{ cos } heta

where:
- P_i: pressure inside the tube,
- P_0: atmospheric pressure outside the tube,
- S: surface tension of the liquid,
- θ: contact angle between the liquid and tube.

The height of the liquid rise, also known as capillary height h, is related to the pressure difference by the equation:

(hρg = P_i - P_0)

Therefore, we can conclude that:

(h = rac{2S ext{ cos } heta}{a ρg})

The smaller the radius of the tube, the higher the liquid will rise due to greater curvature, which induces a larger pressure difference. In contrast, for a liquid that does not wet the surface, such as mercury in a glass tube, the meniscus is convex, and the liquid will descend rather than rise. This section emphasizes the significance of these phenomena in understanding fluid dynamics and interactions at interfaces.

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Audio Book

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Introduction to Capillary Rise

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One consequence of the pressure difference across a curved liquid-air interface is the well-known effect that water rises up in a narrow tube in spite of gravity. The word capilla means hair in Latin; if the tube were hair thin, the rise would be very large.

Detailed Explanation

Capillary rise occurs due to the pressure difference created at the curved surface of a liquid, like water, in contact with the sides of a narrow tube. This is primarily influenced by surface tension, a property of liquids that results in an excess force at the liquid's surface. The narrower the tube, the greater the rise of the liquid against gravity. This phenomenon is especially prominent in very narrow tubes (capillaries), where the height of the liquid column can be significantly higher than in wider tubes.

Examples & Analogies

An example of capillary rise is seen in how water travels through the tiny tubes of plants, allowing them to distribute water from their roots to leaves. Just like how a paper towel can absorb water by drawing it up through its fibers, plants use capillary action to move water upwards even against the force of gravity.

The Mechanics of Capillary Rise

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To see this, consider a vertical capillary tube of circular cross section (radius a) inserted into an open vessel of water. The contact angle between water and glass is acute. Thus the surface of water in the capillary is concave. This means that there is a pressure difference between the two sides of the top surface. This is given by (Pi – Po) =(2S/r) = 2S/(a sec θ) = (2S/a) cos θ.

Detailed Explanation

When a capillary tube is immersed in water, the water adheres to the glass, forming a meniscus that is concave. This concave shape results in a pressure difference between the liquid inside the tube and the atmospheric pressure outside. The formula (Pi - Po) = (2S/r) relates the pressure difference to the radius of the tube (r) and the surface tension (S) of the liquid. The smaller the radius of the tube, the greater the impact of surface tension, resulting in higher capillary rise.

Examples & Analogies

Imagine a thin straw in a drink. When you place the straw in the liquid, you may notice the liquid level inside the straw is higher than the liquid level outside it. This is the same principle; the thinner the straw (similar to a capillary), the higher the liquid rises inside due to surface tension pulling it upwards.

Equation for Capillary Rise

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Thus the pressure of the water inside the tube, just at the meniscus (air-water interface) is less than the atmospheric pressure. Consider the two points A and B. They must be at the same pressure, namely P0 + h ρ g = Pi = PA. Using Eq. (9.27) and (9.28), we have h ρ g = (Pi – P0) = (2S cos θ)/a.

Detailed Explanation

By using the relation of pressures at points A and B, we can establish that the height to which water rises in the tube (h) can be calculated using the equation hρg = (Pi - P0). When rearranged, this gives an expression for the height of rise as h = (2S cos θ)/(ρg a). This indicates that the height depends not only on the liquid's properties (density and surface tension) but also on the angle of contact θ and the radius of the tube.

Examples & Analogies

Consider a cotton wick in an oil lamp. The wick draws oil upward against gravity through capillary action, allowing the oil to reach the flame. The height to which the oil rises in the wick depends on the wick's thickness and the oil's surface tension. The narrower the wick, the higher the oil rises due to the principles of capillary action.

Practical Values and Implications

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Typically, the capillary rise is of the order of a few cm for fine capillaries. For example, if a = 0.05 cm, using the value of surface tension for water, we find that h = 2S/(ρ g a).

Detailed Explanation

Capillary action can lead to significant rises in liquid levels even within small tubes. By substituting actual values into the derived formula, one can calculate the height of liquid rise in practical scenarios. For instance, when the radius is about 0.05 cm, the height of the rise can be calculated to demonstrate how small adjustments in the radius of the tube can lead to measurable differences in the height of the liquid column.

Examples & Analogies

This principle is especially relevant in water conservation where small tubes are used in irrigation to allow water to flow slowly into the soil, effectively utilizing capillary rise to ensure that plants get enough water without wastage. This shows how understanding capillary action can lead to sustainable practices in agriculture.

Convex Meniscus and Mercury

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Notice that if the liquid meniscus is convex, as for mercury, i.e., if cos θ is negative then from Eq. (9.28) for example, it is clear that the liquid will be lower in the capillary.

Detailed Explanation

When dealing with mercury, which exhibits a convex meniscus, the behavior of liquids in capillaries differs from that of water. In this case, the liquid does not rise in the tube because the forces between the mercury molecules are stronger than the adhesive forces between mercury and the glass, leading to a negative value in the equation used for capillary rise. Thus, in such instances, the mercury level will be lower than the surrounding liquid.

Examples & Analogies

Think about using a mercury thermometer. Unlike water which climbs up a thin tube, mercury contracts back down. This occurs because the cohesive forces in mercury are stronger than the adhesive forces with the glass, causing the mercury to form a convex shape in the tube rather than rising like water. Understanding this helps us use mercury effectively in thermometers and other scientific instruments.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Capillary Rise: The phenomenon of liquid rising in narrow tubes due to surface tension and adhesive forces.

  • Surface Tension: The cohesive force among liquid molecules at the surface that creates an elastic-like force.

  • Pressure Difference: The difference in pressure between inside the capillary tube and outside, facilitating liquid rise.

  • Contact Angle: The angle that describes how a liquid meets a solid surface; it impacts capillary rise.

  • Meniscus: The curve formed at the surface of the liquid in a tube, which can be concave or convex based on cohesion and adhesion.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • The movement of water in a plant's roots is an example of capillary rise.

  • Ink rising in a pen due to capillarity illustrates real-world applications of pressure differences.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • In narrow tubes, water will climb, due to forces, every time.

📖 Fascinating Stories

  • Once a little water droplet decided to explore a tiny tube. With the help of surface tension, it bravely climbed up against gravity, eager to see the world.

🧠 Other Memory Gems

  • SHAKE - Surface Tension, Height, Adhesive forces, Kinetic energy, and Elevation for understanding capillary rise.

🎯 Super Acronyms

CATS - Capillary Action in Trees, And in tubes, and in Straws.

Flash Cards

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Glossary of Terms

Review the Definitions for terms.

  • Term: Capillary Rise

    Definition:

    The upward movement of liquid in a narrow tube due to surface tension and adhesive forces.

  • Term: Surface Tension

    Definition:

    The force per unit length acting at the liquid's surface, which causes it to behave like a stretched elastic membrane.

  • Term: Contact Angle

    Definition:

    The angle formed between the liquid surface and the solid surface at the contact line.

  • Term: Pressure Difference

    Definition:

    The difference in pressure between two points, particularly inside and outside the liquid column in a capillary.

  • Term: Cohesive Forces

    Definition:

    The intermolecular forces that attract like molecules to each other, contributing to liquid surface tension.

  • Term: Adhesive Forces

    Definition:

    The forces that attract different substances, such as liquid molecules to solid surfaces.

  • Term: Meniscus

    Definition:

    The curve of the liquid surface in a tube, which can be concave or convex depending on the liquid's interaction with the tube's material.