Chapter 6. The surface phenomenon of liquid

Applications to life science

Summary to the last chapter

20

2 cmmcEk 42

0222 cmcpE

1. Momentum and mass

2. kinetic energy, rest energy and Total energy

vmp

0

1. Two postulates: (Equivalent principle: For all physical processes, the reference frame with uniform acceleration is equivalent to the local region of gravitational field and the inertial force is equivalent to the local region of gravitation.

General relativity principle: Physics law has the same form in all reference frames, no matter inertial or non-inertial.)

2. Space and time in general relativity

3. Gravitational collapse and Black hole

1. Sphere ( 球体 ), spherical ( 球形的 ) ， boundary, border

2. Curvature radius ( 曲率半径 ) ， anisotropic

3. metal frame ( 金属架 ) ， tensional pellicle

4. convex ( 凸起的 ), concave ( 凹入的 )

5. contact angle ( 接触角 ), lotus leaf ()

6. Capillary （毛细管） , capillarity ( 毛细管现象 )

7. embolism ( 栓塞 ), cohesion 聚合力

8. force of adherence 附着力 , transfuse ( 输血 )

9. vein ( 静脉 ), syringe ( 注射器 )

1. Macro-objects ( 宏观物体 )

2. microstructure ( 微观结构 )

3. gravitational and repulsive ( 引力的与斥力的 )

4. isotropic ( 各向同性的 ), anisotropic

5. Boundary ( 边界的）， border , edge

6. pellicle 薄膜 , film

7. Alveolus , alveoli /æl’viəlаi/ n 肺泡

6.1 The micro-structure of matter

Macro-objects consist of a great number of molecules or atoms which are all micro particles. These particles also have physical quantities such as sizes, mass, velocity and energy. The molecules or atoms in macro objects are permanently and irregularly in motion. Such a motion which contains a large number of molecules or atoms is called thermal motion.

It is well known that matter has three kinds of states: gas, liquid and solid. Liquid and solid are called condensed matter as they are hardly compressible.

There exists interaction force between molecules. The solid and liquid that can be formed explains that the gravitational force exists between molecules. On the other hand, solid and liquid cannot be further compressed and this means that the repulsive force exists among molecules. What is the relation between them and in what distance the gravitation dominates the system and in what distance the repulsive force controls the system?

r0

r0r

r

F

Ep

0

0

The gravitational and repulsive force among molecules is totally called Molecular Force. According to experiments and the analysis of the modern physics theory, the molecular force can be expressed as

nm r

C

r

CF 21 Gravitational & elastic potential

where C1, C2, m and n are constants. As m and n are quite big and m > n , the molecular force is short-distance force. r0 ~ 10-10 m. When r > r0, r is between 10-10 ~ 10-8 m. The above formula is semi-experienced one. The four parameters in this formula are determined by experiments. The positive term express repulsion with m between 9~15, and the negative term denotes gravitation with n between 4~7.

6.2 Thermal properties of matter

1. Equation of state for ideal gases

RTM

PV

Where R = 8.314 J·mol ·K-1, it is the universal gas constant or mole gas constant. is the mass of one mole molecules and M is the mass of gas in container with unit of kg, V is the volume of the container with units of m3, P is the pressure with units of N ·m-2.

2. The model of the ideal gas

(1). The size of molecules can be ignored comparing with the average distance between two molecules.

(2). The interaction forces between molecules and the wall of container could be ignored except the right-time of collision.

(3). All collisions are elastic

(4). Ignoring the gravity suffered by molecules

3. The pressure formula of ideal gases

nvmnP

3

2

2

1

3

2 2

Where 2

2

1vm

is the average translational kinetic energy ( 平均平动能 ), n is the number of molecules per unit volume and m is the mass of one molecule.

4. The energy formula of ideal gases

kTV

RT

N

N

N

V

RTM

nVV

RTM

n

A 2

3

2

3

1

2

31

2

3

,V

RTMPRT

MPV

As

AA N

RkN

MN

V

Nn

,,

molN A /10022.6 23

kTvm2

3

2

1 2

is Avogadro constant.

1231038.1 KJk is Boltzmann constant.

n is the number of molecules in unit volume.

5. The laws of ideal gases

(1). Avogadro law

nkTkTnnP

2

3

3

2

3

2

This law means that at the same temperature and the same pressure, the number of molecules contained in the same volume are the same. And also the pressure is relative to the temperature.

(2). Doltan law

Suppose that there are many kinds of molecules in one container and the number of molecules per unit volume are n1, n2, …respectively, The average translational kinetic energy are the same for all kinds of molecules and this means that

222

211 2

1

2

1vmvm

21

21

3

2

PP

nnP

Where P1, P2, P3 … are the pressures respectively for that kind of molecules to be present alone in the container. This is called Doltan’s law of partial pressures.

Example 6-1 (page 110 in Chinese-text book)

Finished here on Friday!

Summary to the last chapter

20

2 cmmcEk 42

0222 cmcpE

1. Momentum and mass

2. kinetic energy, rest energy and Total energy

vmp

0

3. Two postulates: (Equivalent principle: For all physical processes, the reference frame with uniform acceleration is equivalent to the local region of gravitational field and the inertial force is equivalent to the local region of gravitation.

General relativity principle: Physics law has the same form in all reference frames, no matter inertial or non-inertial.)

4. Space and time in general relativity

5. Gravitational collapse and Black hole

r0

r0r

r

F

Ep

0

0

6 The microstructure of matter

7 The thermal properties of matter

Equation of state for ideal gases

nRTPV

The model of the ideal gases (ignoring size, interaction moment, elastic collisions, ignoring gravity)

8. The pressure formula of ideal gases

knvmnP

3

2

2

1

3

2 2

9. The energy formula of ideal gases

kTvmk 2

3

2

1 2

10. The laws of ideal gases

(1). Avogadro law (a:və ｀ ga:dr əu/ )

nkTkTnnP k

2

3

3

2

3

2

(2). Doltan’s law of partial pressures

21213

2PPnnP

(3) Boltzmann’s energy distribution law

In the gravitational field, molecules not only have kinetic energies, but also potential energies. The number of molecules per unit volume is followed by Boltzmann’s energy distribution law:

kTE penn /0

Where n is the number of molecules per unit volume and n0 is the number for Ep =0.

New today:

As the pressure is proportional to the number of molecules per unit volume and Ep = mgh. So we have

kTmghen

n

P

P 00

Therefore: kTmghePP /

0

Where P0 is air pressure at the sea level, P is the pressure at a height of h.

6.3 Surface tension ( 张力 ) and surface energy

• The distance between molecules in a liquid is much shorter than in a gas, and the force between molecules increases obviously.

• A liquid molecule may move in every direction and this is isotropic ( 各向同性的 ). However, on a boundary ( 边界 ) of liquid, the properties of molecules are not isotropic and are quite different.

• In this section, we mainly discuss the process related to life science, surface tension and its effect.

• The surface of a liquid acts as a tensional ( 有张力的 ) film which always tends to contract ( 收缩 ) to a minimum area. It proves that surface of liquid has tension ( 张力 ). It is called surface tension ( 表面张力 ).

1. Surface tension

It is known that the equivalent ( 平衡的 ) distance between two molecules is denoted by r0 which is about 10-10 m in length. Generally speaking, when the distance between two molecules, denoted by d, is less than r0, the force between the two molecules is repulsion ( 斥力 ) and when the distance between two molecules

is larger than r0, the force between them should be gravitation ( 引力 ). However, when d > 10-9m, the gravitation between two molecules can be taken as zero.

Only if a molecule is in the sphere of radius of 10-9 m, it can be gravitated by the center of molecule. This sphere is called molecular action sphere, we draw a thin layer on the surface of a liquid. The thickness of the layer is equal to the radius of the molecular action sphere and the layer is called surface layer.

The molecules in the surface layer and those in the liquid experience different applied forces. In liquid, the molecules are gravitated by their neighbor molecules in all directions and the total force for each molecule is zero, while the molecules in the surface layer have different gravitations in different directions. The direction of total force acting on the molecules in the surface layer is perpendicular to the surface layer, pointing to the inside of the liquid.

So generally, the total forces point to the inner liquid. Due the total forces, the surface is in tension. This is so-called surface tension.

At the same time, the gravitational force ( 引力 ) is balanced by the repulsive force ( 斥力 ) of nearby molecules. So the molecules can stay on the surface.

If we want to move the lower molecules to the surface, we have to do some work opposing the gravitation from lower molecules. Therefore the potential energy of molecules increase. Obviously, the molecules on the surface layer have higher potential energy than the molecules in the inner liquid.

As any system always tends to ( 趋向于 ) the smallest potential energy, the molecules on the surface tend to move into the liquid. Then the surface area will reduce ( 减少 ) to minimum. Contrarily, if we want to increase the surface area, we have to do work to move molecules to the surface while the potential energy of the surface also increases.

The work done by increasing a unit area of liquid surface is called surface energy.

2. Surface energy ( 表面能 )

Surface tension F can be described by the surface tension coefficient ( 表面张力系数 ) . The relation between them is

LFtension where L is the length of a line on the liquid surface.

You can imagine ( 想象 ) that there are two forces (surface tensions) pulled ( 拉 ) from both sides of the line. (= Ftension/L) expresses the surface tension per unit length. It is found that different liquid has different , higher temperature corresponds to ( 对应于 ) lower value of . Let’s see the relationship between the magnitude of surface tension and surface energy.

Look at the diagram. ABCD is a metal frame ( 金属架 ) with a layer of liquid film (or pellicle 薄膜 ). The length of BC is L and it can move freely. However,

the pellicle tends to contract ， so there should be a force F to pull BC.

As there are two surfaces of the pellicle, twofold ( 双重的 ) surface tensions act on BC. Therefore, we have LFpull 2

L Fpull2L

A B

CD

B

CΔx

xLS 2

Let’s have a look at the relationship between and surface energy. Assume that BC moves a short distance because of Fpull. The surface area of film increases

The work done by applied force is

xFW pull This work should be the potential energy increased on the film. So the potential energy increased per unit area on the surface of the film should be

)/( 22

2mJL

F

xL

xF

S

W pullpull

So surface tension coefficient can be defined as the increased potential energy per unit increased area or the work done by increasing each unit area.

6.4 The additional pressure under a curved surface

Surface of liquid is similar to a layer of tensional pellicle ( 有张力的薄膜 ). If a surface is horizontal, the surface tension is also horizontal as it always along the tangential direction ( 切线方向 ) of the curved surface ( 曲面 ).

If a surface is convex ( 凸起的 ), the surface tension tends to pull the surface flat. Therefore the surface tension will put an positive additional pressure on the liquid under the surface.

Contrarily ( 相反 ), if the surface is concave ( 凹入的 ) a negative additional pressure acts on the liquid under the surface, that is, the pressure is lower under the curved surface.

This pressure change can be worked out by calculating the pressure difference between the inner and outer of the liquid surface.

It can be calculated that

T = α

p

TT

T = α

p

TT

R

Rsin

TTsin

Tension T is the tension per unit length. That is . So the total force downwards is

sinsin2 TR Total force upwards for the same volume is

PR 22 sin

RR

TP

22

This is the formula for spherical ( 球型的 ) liquid surface. Where R is the curvature radius ( 曲率半径 ) of liquid surface. If the surface is convex ( 凸起的 ), p is positive and inner pressure is greater than the outer pressure. However, if the surface is concave ( 凹入的 ), p is negative and the outer pressure is greater than the inner pressure. P is called the additional pressure under the curved surface of liquid.

Let’s have a look at the spherical ( 球型的 ) film which have the inner and outer surface layers.

R2

R1·c·b·a

As the pressure is inversely proportional ( 反比 ) to the curvature radius, the pressure at point a, b and c should have the following relations.

abc ppp

and21

22

Rpp

Rpp abbc

For a very thin film, the R1 should be equal to R2 and they are all equal to R. so difference of the pressure of inner and outer liquid surface can be obtained as

Rppp ac

4 Draw a graph, explain

experiment. See next.

A simple experiment can explain the additional pressure of spherical film. There are two bubbles at two ends of a tube, one is bigger and the other is smaller. There is a valve in the center of the tube.

If we open the valve, the big bubble will be getting bigger and bigger and the small one will be getting smaller and smaller because the additional pressure in the small bubble is greater than the big one. This is very important for us to know breath and the physical properties of alveolus ( 肺泡 ).

Example 6-1 Blow a bubble of diameter 10cm and its surface tension coefficient is 4010-3 N/m, Calculate: (1) Work done by blowing the bubble; (2) Pressure change of inner and outer of the bubble.

Solution: (1) According to the conservation principle, work done during the process of blowing the bubble should be equal to the surface change of the bubble. On the other hand, no matter how thin the bubble is, it has two surfaces, therefore

)(108

2

1.0410402

2422

4-

23

2

J

dSEW

(2) The pressure change of inner and outer of the bubble can be calculated using the additional pressure formula obtain previously as

)( 2.31.0

10408

84

23

mN

dRppp ac

The inner pressure of the bubble should be bigger than the outer pressure. The difference is related to the surface tension of the bubble. In such a case, the difference of the two surface areas is ignored.

6.4 Capillarity ( 毛细管现象 ) and air `embolism ( 栓塞 )

1. Contact angle

When liquid contacts with solid, liquid can wet solid sometimes but not always. As you know, water can wet many materials but not lotus leaf ( 荷叶 ). This phenomenon depends on whether the force between liquid molecules (force of cohesion 聚合力 ) is less or greater than the force between liquid and solid molecules (force of adherence 附着力 ).

• Liquid wets the solid.

If the force of cohesion ( 内聚力 )between liquid molecules is smaller than the force of adherence between solid and liquid molecules, the boundary of liquid and solid tends to enlarge. A drop of liquid outspreads a layer of pellicle.

• liquid can not wet solid.

Contrarily if the force of cohesion is greater than the force of adherence, the boundary tends to contract or reduce.

The boundary angle between solid and liquid is called contact angle. This angle is defined that the angle crosses the area which should contain liquid. The size of the contact angle depends on the force of cohesion and adherence. Usually we have

1800 We know that bigger adherence corresponds to smaller contact angle. Liquid can wet solid. In this case θ < 90. When θ =0, liquid can wet solid completely.

θ

Note that the contact angle of 90 is impossible as in such case, cohesion has to be equal to adherence and however, such a case can only happen when the solid and liquid is the same material.

For the 90 < θ <180, liquid cannot wet solid. When θ = 180, liquid cannot wet solid at all.

(finished here: from Boltzmann’s energy distribution law)

2. Capillarity

The tube with very short diameter is called a capillary tube. When a capillary tube is put in a liquid, the liquid surface in it will change.

If liquid can wet the wall of the capillary tube, the liquid surface in the tube will go up and if the liquid cannot wet the wall, the surface will go down.

This phenomenon is called capillarity. Let’s have a look at the case of liquid surface going up.

The liquid surface in the tube can be taken as part of a spherical surface. Because the surface in the tube is concave, the pressure in the tube under liquid surface is lower than the pressure out of liquid (atmospheric pressure).

··

From the figure, we know

cosRr Where r is the radius of the tube, R is the curvature radius of the liquid surface, is the contact angle. The pressure change of inner and outer liquid is

rRp

cos22

It is such a pressure that raises the liquid surface in the capillary tube and it is pointing upwards.

P0

Rr

h

·P0

B C

In a balance state, the pressure at point B should be equal to the pressure at point C which is atmospheric pressure P0. So

00

cos2pgh

rp

Where p0 is atmospheric pressure, h is the height of the liquid surface in the tube, is the density of liquid. The above equation can be solved as

grh

cos2

Therefore the height of liquid in a capillary tube is proportional to the surface tension coefficient, and is inversely proportional to the inside radius. This means that smaller inside diameter corresponds to higher liquid surface in a capillary tube.

For the opposite situation, the liquid in a capillary tube is lower and convex. The pressure in liquid is greater than the

atmospheric pressure. Now is greater than /2 and the formula we obtained can be still used. The height h is negative in such a case and this means that the surface in the capillary tube is lowering down.

Capillarity is a common phenomenon in our daily life, such as absorption and transportation of water in plant ( 植物 ), blood flowing in the capillary blood vessel （血管） and so on.

3. Air embolism ( 气体栓塞 ) When liquid flows in a capillary tube. liquid will be blocked if there are bubbles in the tube. This phenomenon is called air embolism. In figure (a), there is a bubble in a tube containing liquid. When the pressure at the left of the bubble is equal to the pressure from right, the two concave curvatures of the bubble are the same. So there are the same magnitude additional pressure on two curved surfaces but in opposite directions. The liquid will not flow.

pleftp ppright

Fig. (a)

The pressure in the bubble is the same but additional pressure caused by the surface tension might not be the same if they have different curvature radius. In such a case, the pressures pushing to the right and left are given respectively as

,2

leftRp

rightR

p2

pleftp ppright

Fig. (a)

For the case shown in figure (b), if the pressure on the left increases p, the radius of bubble on the left will increase and the radius on the right will decrease. if the bubble can move, the necessary condition should be

rightleft RRp

22

p+p pFig. (b)

In real case, there are some frictions between the liquid and the wall of the tube. The sufficient and necessary condition of the bubble moving is generally

rightleft RRp

22

That is, the pressure change on two sides of the tube is greater than a critical value .

The critical value depends on properties of liquid, wall of tube and radius of the capillary tube.

If there are n bubbles in the tube, the pressure difference between the two ends of the tube has to be greater than n times .

When we transfuse ( 输血 ) patients, we have to avoid air embolism in the transfusing tube. Especially, we have to avoid an air bubble in the syringe ( 注射器 ) when injecting veins ( 静脉 ), otherwise the air embolism might occur.

6.5 The surface active agent and the surface absorption

Surface tension of solution ( 溶液 ) changes with variation of solute ( 溶质 = a substance is dissolved in another substance.) Some solutes can decrease surface tension of solution and some can increase surface tension. The former is called surface-active agent ( 表 面 活 性 物质 ) and the latter is called non-surface-active agent. (solution = solute + solvent )

The surface-active agent is called surfactant ( 表面活性剂 ) or a wetting agent. For example, organic acid （有机酸） and soap are active agent of water; salt and gluside ( 糖精 ) are non-active agent of water. When the surface-active agent dissolves ( 溶解 ) in solvent ( 溶剂 , the liquid in which another substance (solute) is dissolved to form a solution), the attraction of solvent molecules is greater than attraction between solvent molecule and solute molecule.

In such a case, solvent molecules in the surface layer tend to move into the liquid because the greater attractions between solvent molecules. The solvent molecules move into the liquid as much as possible as in this way, the surface energy will be lower and the system will be more stationary. In the surface layer, the solute molecules will increases in order to stabilize the system.

Since the solute molecules concentrate in the surface layer of solution, a little bit of active substance can influence the properties of the surface and decrease the surface tension.

In some situations, the surface layer of the solution consists of solute molecules only and it can extend larger and larger. This phenomenon is called surface absorption. For example, the oil film spreads over water.

Contrarily, when a non-active agent is dissolved in a liquid, its molecules tend to leave the surface layer and move into solution. The surface tension in this case does not change very much and most of the solute molecules are mainly stayed in the solvent. Surface-active agent plays a very important role in respiration.

All these alveoli form a cystiform ( 囊状的 ) air chamber. As the alveoli do not have the same size and some of them are connected. Generally speaking, as the surface tension in the lungs are the same and the pressure in the small alveolus is bigger than that in big alveolus. The air in the small alveolus will go into the big ones. But such a case is not happened as the theory predicts because the surface-active agent exists in lungs. In lung, bronchus was divided into a great number of tiny bronchi. At each of the tiny bronchus, there forms an alveolus.

Lets consider a process of a respiration. There is a same amount of surface-active agent (SAA) in lungs. When inhaling, the alveoli becomes larger and less SAA on the alveoli and the surface tension will be getting larger. Therefore, this limits the enlargement of alveoli. On the other hand, breathing out (exhaling), the alveoli cannot become much smaller as more SAA will accumulate on the surface of the alveoli and reduce the surface tension of the alveoli. This will help the next time inhaling very much and will make the respiration process easier.

Summary to the last lecture

r0

r0r

r

F

Ep

0

0

6.1 The microstructure of matter

6.2 The thermal properties of matter

1. Equation of state for ideal gases

nRTPV

2. The model of the ideal gases (ignoring size, interaction moment, elastic collisions, ignoring gravity)

3. The pressure formula of ideal gases

knvmnP

3

2

2

1

3

2 2

4. The energy formula of ideal gases

kTvmk 2

3

2

1 2

5. The laws of ideal gases

(1). Avogadro law (a:və ｀ ga:dr əu/ )

nkTkTnnP k

2

3

3

2

3

2

(2). Doltan’s law of partial pressures

21213

2PPnnP

2. Surface energy ( 表面能 )

)( 1 mNL

Ftension

Surface tension F can be described by the surface tension coefficient ( 表面张力系数 ) is defined as that

Where F is the surface tension, L is the considered length on the surface of liquid.

6.3 surface tension and surface energy

1. Surface tension （ surface layer ）