Surface Tension Notes

SURFACE TENSION Page # 1 SURFACE TENSION Pascal’s Law The French scientist Blaise Pascal observed that the pressure in a fluid at rest is the same at all points if they are at the same height. This fact may be demonstrated in a simple way. Fb sin = Fc, Fb cos = Fa (by equilibrium) Ab sin = Ac, Ab cos = Aa (by geometry) Thus, Fb F F  c  a ; Ab Ac Aa P b = P c = Pa Hence, pressure exerted is same in all directions in a fluid at rest. Hydraulic Machines whenever external pressure is applied on any part of a fluid contained in a vessel, it is transmitted undimin ished and equally in all directions. This is the Pascal’s law for transmission of fluid pressure and has many applications in daily life. A number of devices such as hydraulic lift and hydraulic brakes are based on the Pascal’s law. In these devices fluids are used for transmitting pressure. In a hydraulic lift as shown in Fig. 10.6 two pistons are separated by the space filled with a liquid. A piston of small cross section A1 is used F1 is transmitted throughout the liquid to the A1 larger cylinder attached with a larger piston of area A2, which results in an upward force of P × A2. Therefore, the piston is capable of supporting a large force (large weight of, say a car, or a truck, placed to exert a force F1 directly on the liquid. The pressure P = F1A 2 By changing the force at A1, the platform can be moved up or down. A1 A2 Thus, the applied force has been increased by a factor of A A and this factor is the mechanical 1 advantage of the device. The example below clarifies it. on the platform) F2 = PA2 = STREAMLINE FLOW So far we have studied fluids at rest. The study of the fluids in motion is known as fluid dynamics. The flow of the fluid is said to be steady if at any given point, the velocity of each passing fluid particle remains constant in time. The path taken by a fluid particle under a steady flow is a streamline. It is defined as a curve whose tangent at any point is in (a) The direction of the fluid velocity at that point. Consider the path of a particle as shown in the curve describes how a fluid particle moves with time. The curve PQ is like a permanent map of fluid flow, indicating how the fluid streams. No two streamlines can cross, for if they do, an oncoming fluid particle can go either one way or the other and the flow would not be steady. Hence, in steady flow, the map of flow is stationary in time. How do we draw closely spaced streamlines ? If we SURFACE TENSION Page # 2 intend to show streamline of every flowing particle, we would end up with a continuum of lines. Czonsider planes perpendicular to the direction of fluid flow e.g., at three points P, R and Q in (b).The plane pieces are so chosen that their boundaries be determined by the same set of streamlines. This means that number of fluid particles crossing the surfaces as indicated at P, R and Q is the same. If area of crosssections at these points are AP,AR and AQ and speeds offluid particles are P, R and Q, then mass of fluid mP crossing at AP in a small interval of time t is PAPP t. Similarly mass of fluid mR flowing or crossing at AR in a small interval of time At is RARR At and mass of fluid mQ is QAQQ At crossing at AQ. The mass of liquidflowing out equals the mass flowing in, holds in all cases. Therefore, PAPPt = RARRt = QAQQt. For flow of incompressible fluids P = R = Q reduces to PP = RR = QQ, which is called the equation of continuity and it is a statement of conservation of mass in flowof incompressible fluids. In general Av = constant. BERNOULLI'S PRINCIPLE Consider the flow at two regions 1 (i.e. BC) and 2 (i.e. DE). Consider the fluid initially lying between B and D. In an infinitesimal time interval t, this fluid would have moved. Suppose 1 is the speed at B and 2 at D, thenfluid initially at B has moved a distance 1 t to C (1t is small enough to assume constant crosssection along BC). In the same interval t the fluid initially at D moves to E, a distance equal to 2t. Pressures P1 and P2 act as shown on the plane faces of areas A1 and A2 binding the two regions. The work done on the fluid at left end (BC) is W1 = P1A1(1t) = P1V. Since the same volume V passes through both the regions (from the equation of continuity) the work done by the fluid at the other end (DE) is W2 = P2A2(2t) = P2V or, the work done on the fluid is –P2V. So the total work done on the fluid is W1 – W2 = (P1 P2) V Part of this work goes into changing the kinetic energy of the fluid, and part goes into changing the gravitational potential energy. If the density of the fluid is  and m = A11t = V is the mass passing through the pipe in time t, then change in gravitational potential energy is U = gV (h2 – h1) The change in its kinetic energy is   1 K     V 22  12 2 We can employ the work – energy theorem to this volume of the fluid and this yields 1 (P1 – P2) V =   V 22  12 + gV (h2 – h1) 2   We now divide each term by V to obtain 1 (P1 – P2) =    22  12 + g (h2 – h1) 2   We can rearrange the above terms to obtain 1 1 P1    12  gh1  P2   22  gh2 2 2 This is Bernoulli’s equation. Since 1 and 2 refer to any two locations along the pipeline, we may write 1 the expression in general as P +   2 + gh = constant 2 In words, the Bernoulli’s relation may be stated as follows: As we move along a streamline the sum of the  2    pressure (P), the kinetic energy per unit volume  2  and the potential energy per unit volume (gh)   remains a constant. SURFACE TENSION Page # 3 1. (a) COHESIVE FORCE The force of attraction between the molecules of the same substance is called cohesive force In case of solids, the force of cohesin is very large and due to this solids have definite shape and size. On the other hand, the force of cohesion in case of liquids is weaker than that of solids. Hence liquids do not have definite shape but have definite volume. The foce of cohesion is negligible in case of gases. Because of this fact, gases have neither fixed shape nor volume. Example. (i) Two drops of a liquid coalesce into one when brought in mutual contact because of the cohesive force. (ii) It is difficult to separate two sticky plates of glass wetted with water because a large force has to be applied against the cohesive force between the molecules of water. (iii) It is very difficult to break a drop of mercury into small droplets because of large cohesive force between mercury molecules. 1. (b) ADHESIVE FORCE The force at attraction between molecules of different substances is called adhesive force Examples. (i) Adhesive force enables us to write on the black board with a chalk. (ii) Adhesive force helps us to write on the paper with ink. (iii) Large force of adhesion between cement and bricks helps us in constrution work. (v) Fevicol and gum are used in gluing two surfaces together because of adhesive force. 2. SURFACE TENSION The property of a liquid at rest due to which its free surface tries to have minimum surface area and behaves as if it were under tension somewhat like a stretched elastic membrane is called surface tension. The molecules of the liquid exert attractive forces on each other. There is zero net force on a molecule inside the volume of the liquid. But a surface molecules is drawn into the volume. Thus, the liquid tends to minimize its surface area, just as a stretched membrane does. F F Surface tension of a liquid is measured by the force acting per unit length on either side of an imaginary line drawn on the free surface of liquid, the direction of this force being pependicular to the line and tangential to the free surface of liquid. So if F is the force acting on one side of imaginary line of length L. then T = (F/L) Regarding surface tension it is worth noting that : (1) It depends only on the nature of liquid and is independent of the area of surface or length of line considered. (2) It is a scalar as it has a unique direction which is not to be specified. (3) It has dimension [ML–2] and SI units N/m while CGS unit dyne/cm, so that one MKS unit of surface tension = 103 dyne/cm (4) Surface tension of a liquid decreases with rise in temperature 1 SURFACE TENSION Page # 4 (5) The surface tension of a liquid is very sensitive to impurities on the surface (called contamination) and decreases with contamination of surface. (6) In case of soluble impurities surface tension may increase or decrease depending on the nature of impurity. Usually highly soluble salt such as sodium chloride increases surface tension while sparingly soluble salt such as soap decreases surface tension. 2.1SURFACE ENERGY When the surface area of a liquid is increased, the molecules from the interior rise to the surface. This requires work against force of attraction of the molecules just below the surface. This work is stored in the form of potential energy. Thus, the molecules in the surface have some additional energy due to their positio. This additional energy per unit area of the surface is called ‘surface energy’. The surface energy is related to the surface tension as discussed below : dx 2Tl F Let a liquid flim be formed on a wire frame and a straight wire of length l can slide on this wire frame as shown in figure. The film has two surface and both the surface are in contact with the sliding wire and hence, exert forces of surface tension on it. If T be the surface tension of the solution, each surface will pull the wire parallel to itself with a force Tl. Thus, net force on the wire due to both the surface is 2Tl. One has to apply an external force F equal and opposite to it to keep the wire in equilibrium. Thus, F = 2Tl Now, suppose the wire is moved through a small distance dx, the work done by the force is, dW = F dx = (2Tl) dx But (2l) (dx) is the total increase in area of both the surface of the film. Let is be dA. Then, dW = T dA or T  dW dA Thus, the surface tension T can also be defined as the work done in increasing the surface area by unity. 2.2Excess Pressure Insider a liquid drop : Consider a liquid drop of radius 'R' and surface tension 'T' A liquid drop has only one surface film, hence the surface tension force is T(2R) Pin P0 lower half shows in figure Pin P0 Force due to inside pressure (Pin) is Pin × area i.e. Pin R2 similarly force due to outiside pressure (P0) is P0R2 since each half of the liquid drop is in equilibrium P0R2 + T (2R) = Pin (R2) Pin – P0 = 2T = Excess Pressure R 2.3Excess pressure inside a bubble Consider a bubble of radius 'R' and surface tension 'T'. A bubble consists of two spherical surface films with a thin layer of liquid between them. The total surface tension force for each surface inner and outer T (2R) for a total of (2T) (2R) Force due to inside pressure (Pin) is Pin R2 and due to outside pressure (P0) is P0R2 SURFACE TENSION Page # 5 Pin Pin P0 P0 Since each half of bubble is in equilibrium (lower half shown in figure) P0 R2 + 2T(2R) = Pin R2 Pin – P0 = 4T = Excess pressure R Note : (1) If we have an air bubble inside a liquid, a single surface is formed. There is air on the concave side and liquid on the convex side. The pressure in the concave side (that is in the air) is greater than the 2T pressure in the convex side (that is in the liquid) by an amount . R P1 P2 2T R The above expression has been written by assuming P1 to be constat from all sides of the bubble. For small size bubbles this can be assumed. (2) From the above discussion, we can make a general statement. The pressure on the concave side of a 2T spherical liquid surface is greater than the convex side by . R  P2  P1   1 1  (3) For any curved surface excess pressure on the concave side = T  where R1 & R2 are radius  R  R  2   1 of curvature of the surface in two perpendicular direction of instead of liquid surface, liquid film is given then above exression will be  1 1  P  2T    For spherical curved surface R1, R2 R R 2   1 3. CONTACT ANGLE AND SHAPE OF LIQUID SURFACE The surface of a liquid when meets a solid, such as the wall of a container, it usually curves up or down near the solid surface. The angle which the tangent to the is called the contact angle. The curved liquid surface at the pt. of surface of the liquid is called meniscus. The shape of the meniscus contact of liquid surface with (convex or concave) is determined by the relative strengths of solid cohesive and adhesive forces surface with the solid surface inside the liquid. Glass P R Q Water SURFACE TENSION Page # 6 When the adhesive force (P) between solid and liquid molecules is more than the cohesive force (Q) between liquid-liquid molecules (as with water and glass), shape of the meniscus is concave and the angle of contact  is less than 90º. In this case the liquid wets or adheres to the solid surface. The resultant (R) of P and Q passes through the solid. Glass P R Q Mercury On the other hand when P < Q (as with glass and mercury), shape of the meniscus is convex and the angle of contact  > 90º. The resultant (R) of P and Q in this case passes through the liquid. Let us now see why the liquid surface bends near the contact with a solid. A liquid in equilibrium can not sustain trangential stress. The resultant force on any small part of the surface layer must be perpendicular to the surface at that point. Basically three forces are acting on a small part of the liquid surface near its contact with solid. These forces are, (i) P, attraction due to the molecule of the solid surface near it i.e. adhesive force which acts outwards at right angle to the wall of tube. (ii) Q, attraction due to liquid molecules near this part and i.e. cohesive force which acts at an angle of 45º to the vertical. We have considered very small part, so weight of that part can be ignored for better understanding. As we have seen in the last figures, to make the resultant (R) of P and Q perpendicular to the liquid surface the surface becomes curved (convex or concave). Note : The angle of contact between water and clean glass is zero. 4. CAPILLARY RISE If a tube of very narrow bore (called capillary) is dipped in a liquid, it is found that the liquid in the capillary either ascends or descends relative to the surrounding liquid. This phenomenon is called capillarily. In order to calculate the height to which a liquid will rise in a capaillary, consider a glass capillary of radius R dipped in water as shown in Fig. shown. As the meniscus is concave and nearly spherical, the pressure below the meniscus will be [p0 – (2T/r)] with p0 as atmospheric pressure and r as radius of meniscus. Now as liquid flows from higher to lower pressure and at same level in a liquid pressure must be same (this is because a liquid cannot sustain tangential stress), so the liquid will ascends in the capillary till hydrostatic pressure of the liquid compensates for the decrease in pressure. i.e., r  R p0 p0 p0  2T   p0  p0 –  hg or r   h 2T rg p0 p0 h 2T t ...(1) But from figure shown it is clear that radius of meniscus r is related to the radius of capillary through the relation SURFACE TENSION Page # 7 (R/r) = cos , i.e., r = R/cos  where  is the anlge of contact. *So substituting the value of from Eqn. (2) in (1), we get h • 2T 2T cos   rg Rg ...(2) ...(3) Alternate Method As it can be seen from figure that T sin  cancels out : The force due to T cos  balances the weight of liquid (mg =  vg) vol. of the curve is negligible  vol. of liquid in r2h T cos  = 2r = r2hg  h = T T  h 2T cos  rg This is the desired result and from this it is clear that : 1. The capillarity depends on the nature of liquid and solid both, i.e., on T, ,  and R. If  > 90°, i.e., meniscus is convex, h will be negative, i.e., the liquid will descends in the capillary as actually happens in case of mercury in a Hg (A)  (B) 90;h   ve (C)   90;h  0   90;h  – ve glass tube. However, if  = 90°, i.e., meniscus is plane, h = 0 and so no capillarity. 2. For a given liquid and solid at a given place as , T,  and g are constant, (figure shown) hr = constant  lesser the radius of capillary greater will be the rise and vice-versa. (figure shown) 3. Here it is important to note that in equilibrium the height h is independent of the shape of capillary if the radius of meniscus remains the same. This is why the vertical height h of a liquid column in capillaries of different shapes and sizes will be same if the radius of meniscus remains the same and also the vertical height of the liquid in a capillary does not change, when it is inclined to the vertical. (figure shown)  (a) (b) (c) (d) h (e) 4. Capillarity has large number of applications in our daily life, e.g., (a) The oil in the wick of a lamp rise due to capillary action of threads in the wick. (b) Action of towel in soaking up moisture from the body is due to capillary action of cotton in the towel. (c) Water is retained in a piece of sponge on account of capillarity. (d) A blotting paper soaks ink by capillary action of the pores in the blotting paper. (e) The root-hairs of plants drawn water from the soil through capillary action. SURFACE TENSION Page # 8 5. In Case of glass and water  = 0 here force due to surface tension balances the weight of the liquid ( × v × g) T T r r 2 3 volume of the liquid = r h + r – r 3 2 3 h 2 where r – r3 is the volume of the curve which is 3 3 not negligible in this case  T.2r =  (r2h + r3 – 2T = rh g + 2 3  r )g 3 1 2 r g 3  T 6. If two parallel plates with the spacing 'd' are placed in water reservoir, then height or rise. 2T = hdg  h  h 2T dg d 7. If two concentric tubes of radius 'r1' and 'r2' (inner one is solid) are placed in water reservoir, then height of rise?  T[2r1  2r2 ]  [r22h – r12h] g h 2T (r2 – r1 ) g r1 r2 h r2 8. If weight of the liquid in the meniscus is to be consider : 1 T cos  × 2r = [r h + r2 × r]  g 3 2 r  2T cos   h  3   rg   T cos   T sin  T cos   r T sin  h 9. When capillary tube (radius, 'r') is in vertical position, the upper meniscus 2T is concave and pressure due to surface tension is directed vertically upward and is given by p1 = R 1 where R1 = radius of curvature of upper meniscus. The hydrostatic pressure p2 = h  g is always directed downwards. If p1 > p2 i.e. resulting pressure is directed upward. For equilibrium, the pressure due to lower meniscus SURFACE TENSION Page # 9 should be downward. This makes lower meniscus concave downward 2T (fig a). The radius of lower meniscus R2 can be given by R  (p1 – p 2 ) 2 If p1 < p2 i.e. resulting pressure is directed downward for equilibrium, the pressure due to lower meniscus should be upward. This makes lower (a) (b) (c) 2T meniscus convex upward (fig. b) The radius of lower meniscus can be given by R  p 2 – p1 2 If p1 = p2, then is no resulting pressure. then, p1 – p2 = 2T R2 = 0 or, R2 =  i.e. lower surface will be FLAT (fig c) 4.1CAPILLARY RISE IN A TUBE OF INSUFFICIENT HEIGHT O  C h  R'  ' h We know, the height through which a liquid rises in the capillary tube of radius r is given by  h 2T 2T or h R = = constant R g g When the capillary tube is cut an its length is less then h (i.e. h'), then the liquid rises upto the top of the tube and spreads in such a way that the radius (R') of the liquid meniscus increases and it becomes more flat so that hR = h' R' = Constant. Hence the liquid does not overflow. If h' < h then R' > R or r r  cos  ' cos   cos  < cos  ' >   5. VISCOSITY AND NEWTON'S LAW OF VISCOUS FORCE In case of steady flow of a fluid when a layer of fluid slips or tends to slip on adjacent layer in contact, the two layers exert tangential force on each other which tries to destroy the relative motion between them. The property of a fluid due to which it opposes the relative motion between its different layers is called viscosity (or fluid friction or internal friction) and the force between the layers opposing the relative motion viscous force. A briskly strirred fluid comes to rest after a short while because of viscosity. As a result of large number of experiments Newton found that viscous force F acting on any layer of a fluid is directly proportional to its area A and to the velocity gradient (dv/dy)* at the layer i.e., SURFACE TENSION Page # 10 Y v A F v dy X dv dv F  – A or ...(1) dy dy when  is a constant called coefficient of viscosity or simply viscosity of the fluid. The negative sign shows that viscous force on a liquid layer acts in a direction opposite to the relative velocity of flow of fluid. The Eq. (1) is known as Newton's law of viscous force. Here y is taken from the layer of which velocity is zero. Regarding viscosity of fluid it is worth noting that : (1) It depends only on the nature of fluid and is independent of area considered or velocity gradient. (2) Its dimensions are [ML–1 T–1] and SI unit poiseuille (PI) while CGS unit dyne-s/cm2 called poise (P) with 1 Pl = 10 poise (3) Viscosity of liquids is much greater (say about 100 times more) than that of gases i.e., L > G FA 6. STOKES LAW When a body moves through a fluid, the flui in contact with the body is dragged with it. This establishes relative motion in fluid layers near the body, due to which viscous force starts operating. The fluid exerts viscous force on the body to oppose its motion. The magnitude of the viscous force depends on the shape and size of the body, its speed and the viscosity of the fluid. Stokes established that if a sphere of radius r moves with velocity v through a fluid of viscosity , the viscous force opposing the motion of the sphere is F = 6   rv 7. TERMINAL VELCOITY (VT) Consider a small sphere falling from rest through a large column of viscous fluid. The forces acting on the shere are, (i) Weight W of the sphere acting vertically downwards (ii) Upthrust Ft acting vertically upwards (iii) Viscous force Fv acting vertically upwards, i.e., in a direction opposite to velocity of the sphere. Initially, Fv = 0 Ft + Fv and W > Ft v and the sphere accelerates downwards. As the velocity of the sphere increases, Fv increases, Eventually a stage is reached when W W = Ft + Fv After this net force on the sphere is zero and it moves downwards with a constant velocity called terminal velocity (vT). Substituting proper values in Eq. (i) we have, 4 3 4 r g  r 3 g  6rv T 3 3 v Here,  = density of sphere,  = density of fluid and  = coefficient of viscosity of fluid v T From Eq. (ii), we get vT  2 2 r (  ) g 9  Figure shows the variation of the velocity v of the sphere with time. O t