# States of matter. Density high > high >> low (pressure dependent)

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1 Fluids States of matter Solids Fluids crystalline amorphous liquids gasses Inter-atomic forces strong > strong >> very weak Density high > high >> low (pressure dependent)

2 Density is an important material parameter. For a uniform material the density is defined as its total mass divided by its total volume. Density: m V kg m ρ= 3 (a scalar, Greek symbol: ρ, rho) For a non-uniform material we can still define the local density as that of the mass m per very small volume element V m V m ρ (x) = V x r(x)

3

4 How much volume is there around atoms (or molecules) in solids, liquids and gasses (typically)? Consider a mole (Avogadro s number: class: atomic or Density molec. weight iron (solid) mercury (liquid) air (N 2 ) at STP g 55.8 mole g 7.86 cm ) from each volume mole g mole g cm g g g mole cm 3 g mole cm 3 cm = 7.1 mole 3 cm = 14.8 mole 3 g g 28.0 liter cm = 22,400 mole cm 3 mole mole 23 work backwards

5 Since each of these volumes contain the same number (Avogadro s #) of objects (atoms or molecules) the ratio of these volumes gives the relative volumes around the objects in each. iron (solid) mercury (liquid) volume mole 3 cm 7.10 mole 3 cm 14.8 mole So an atom of liquid mercury has 14.8/7.1 ~ 2 times the volume around it that an atom of Fe has in solid iron air (N 2 ) at STP 3 cm 22,400 mole But a molecule of N 2 in air (at STP) has 22,400/7.1 ~ 3150 times the volume around it that an atom of Fe has in solid iron This makes gasses far more compressible than solids or liquids.

6 If we try to compress a gas in a container with a piston the volume will change with the applied force. A gas is said to be compressible. For a liquid, because its molecules are already effectively in contact, it s volume will change little, so to reasonable approximation (and certainly compared to gasses) liquids are often treated as being incompressible. F piston A defining characteristic of a fluid (liquid or gas) is a material that cannot support a shear stress. Consider an unusual split container with a smooth interface between the top and bottom halves. F A force F applied to its top will cause its acceleration (in accord with Newton s 2 nd law).

7 If the space of in the container is filled with a solid like e.g. iron the force will act to try and shear the material at the split but unless the force is very large (and the container material very strong) the iron will resist being sheared. F If however the volume is instead filled by a fluid the resistance to the shearing will be largely the inertial resistance to the additional mass. F The fluid does not resist being sheared.

8 Because of this fluids will adjust their shape to conform to the forces acting on them. Under gravity they will fill the bottom of any container shape in which they are placed. The inability of fluids to support a shear stress makes it difficult to talk about the force on a fluid. If your hand pushes on a fluid it yields, flowing around you. This ability to flow and change shape makes it difficult to talk of fluids as discrete objects. If two bodies of fluid meet they merge and mix making them no longer distinguishable. Parameters that are more useful in (quantitatively) describing fluids are the density and pressure.

9 To explore forces in a fluid we resort to a type of gauge described in the text consisting of piston cylinder arrangement evacuated behind the piston, but containing a calibrated spring. F = k x, (k known) By reading off x, we can determine force acting on the piston. For experimentation we also make these with different piston areas.

10 With these at our disposal we can perform a series of experiments in a tall tank of water. We would find that: 1) The force (as read off by the spring compression), is greater the deeper we go into the tank. Depending linearly on the depth. 2) The force at a given level is independent of the orientation of the pressure gauge. 3) The lateral dimensions of the tank don t matter (double the tank width and all force readings are un-changed).

11 4) For each depth, if we use a device that has double the piston area, the force measured by the spring is doubled. The last of these tells us that if we define the pressure as the force per area of the piston p = F A Then two different sized gauges read the same values of pressure at the same depth. The bigger piston is pushed in twice as far because the fluid exerts its force on a per area basis.

12 With this realization we can understand the lack of dependence on the tanks lateral dimensions and the linear dependence on depth. Consider the column of water having depth d and area at the bottom A. The pressure at the bottom of this column is the force of the weight of the water above it pressing down on the bottom face, divided by its area. F mg p = = A A Area A m But since ρ= m = ρv = ρad, V Then, F mg ( ρad)g Which depends linearly on the depth, p = = = = ρgd A A A d, but not the area, A, of the column. Independent of tank width. d

13 The density does depend slightly on the depth however we ignore this for liquids, assuming them to be incompressible (careful not a good assumption for gasses). The orientation independence of the gauge reading tells us that pressure is a scalar quantity. Finally, if we watched carefully we would have observed when our pressure gauge was evacuated (not under water) that the piston sank in a little bit. Considering this we realize that we are living at the bottom of an enormous tank of fluid: the atmosphere. What we observed was the pressure due to the atmosphere (initially equal on both sides of the piston) pushing the piston in as the atmosphere inside was removed.

14 This pressure of the atmosphere is also due to the weight of the column of the atmosphere sitting above our column of water. If the pressure of the atmosphere at the water surface is p o then the absolute pressure at the bottom of our water column, at depth d is d p = po +ρgd Area A The pressure ρgd, referenced to atmospheric pressure taken as zero is called the gauge pressure. N The SI unit of pressure is the Pascal: 1 Pa = 1 m 2

15 There are also a number of other pressure units in common use that have the following equivalence, atm = 1.01x10 Pa = 760 Torr = 14.7 lb / in And there are a number of different types of pressure gauges in use.

16 HITT Superman (of course) has super suction, capable of pulling an infinite vacuum. Given an appropriate straw how high could he suck the fresh water from a lake? A) as high as he wants B) 32.0 m C) 10.3 m h p o = 1.01x10 5 Pa ρ = 1000 kg/m 3

17 A device for measuring atmospheric pressure is the barometer. This is a close bottomed tube, filled to overflowing with a fluid and then turned over in an open bath of the same fluid. Done on the moon which has essentially no atmosphere, the fluid would just run out until h = 0. vacuum On earth where the pressure on the exterior fluid surface is around 1 atm the fluid at point x inside the tube has pressure equal to p o, from the atmospheric pressure p o outside the tube. p o x h That means a force up given by solving p o F = with A the cross-section of the tube. A

18 Since the fluid is stationary, there must be an equal but opposite force down at point x having the same magnitude, but where we recognize the force as being due to the weight of the fluid column, p o F = = A mg A But since the density (rho) this can be rearranged to give, m = ρv = ρah, ρ= m V h vacuum which substituted above gives p o F mg ρgah = = = A A A p o = ρgh p o x

19 p o = ρgh vacuum Taking h = 0 as zero pressure (absolute) the height of the fluid column is proportional to the pressure. Mercury is often used. The height of a column of mercury at sea level is on average 760 mm, so you will often hear atmospheric pressures quoted in mm of mercury (also called Torr) around this value. p o x h The density of mercury depends on the temperature, and since the pressure reading depends on density, for accurate readings of atmospheric pressure this must be corrected for the local temperature.

20 HITT Superman (of course) has super suction, capable of pulling an infinite vacuum. Given an appropriate straw how high could he suck the fresh water from a lake? A) as high as he wants B) 32.0 m C) 10.3 m h p o = 1.01x10 5 Pa ρ = 1000 kg/m 3

21 HITT The barometer also has a vacuum in the upper part of the tube. Making that vacuum more perfect changes the height of the water negligibly. There we found p o = ρgh so, h p ρg kg m m s 5 o Pa = = = 10.3 m h p o = 1.01x10 5 Pa ρ = 1000 kg/m 3

22 The open tube manometer works similarly to measure the gauge pressure of a gas inside an otherwise closed volume. Before the gas volume is attached the liquid level on the two sides of the open U tube are equal. After the gas volume is attached, if the gas pressure is > 1 atm, the liquid level on that side will be forced down by this added pressure, causing the liquid level on the other side to climb. When they are in equilibrium the pressure at B and B must be equal.

23 But at B the pressure due to the column of height d above B (and B) is, p B F mg ρgad = = = A A A p =ρ gd = p B g Which is the gauge pressure (i.e. relative to the atmospheric pressure) because before connecting the pressurized gas, when the liquid was at the same level on the two sides, the atmosphere pressed equally on the two sides.

24 Pascal s Principle Consider the circumstance to the right in which we have a cylinder of cross-section A filled with fluid. The pressure at a depth d below the surface depends on the depth. d p 1 Let s now add a piston and apply an additional force to it. The pressure at depth d increases by F F p = A but that s true at every point of the fluid. d p 2 The pressure change is transmitted throughout the entire fluid.

25 Such a change in pressure would be transmitted independent of the source of the pressure change. For example, if the temperature rises and the fluid expands, the resulting change in the pressure would occur throughout the volume of fluid. This is Pascal s principle: the change in pressure occurring in an incompressible fluid, in a closed container, is transmitted undiminished to every portion of the fluid and to the walls of the container.

26 This allows for a hydraulic lift, which consists of different area pistons/cylinders connected together as shown here. A force F 1 is applied at the small area piston and the force F occurs 2 at the large area piston. By Pascal s principle, the change in pressure is the same everywhere so But then, F p = = A F = A F A A1 1 2 F So force F 2 is force F 1 multiplied by the ratio of the piston areas.

27 For round cylinder/pistons: 2 2 A2 πr 2 R 2 2 = 1 = 2 1 = 1 A1 R 1 R 1 F F F F π So if the output cylinder has a 10 cm radius and the input cylinder a 1 cm radius the force multiplier is (10/1) 2 = 100. This is how hydraulic lifts and the brakes in your car work. Does this scheme defy conservation of energy? (A wise thing to ask whenever we seem to be getting something seemingly extraordinary)

28 If piston 1 moves down a distance x 1 the volume of fluid it displaces is V = x 1 A 1. Since the fluid is incompressible this must be the same volume displaced by the opposing piston so, Or, V= xa 1 1 = xa 2 2 A x = x A2 This is the inverse ratio of the force multiplier so the distance moved by the output piston is proportionately smaller than the distance moved by the input piston. Since work is force times displacement the same work is done on both sides so that energy is conserved.

29 U-tubes (determining the density of an immiscible fluid) Tube cross-sectional area A. h p H-h p H The pressure on the two sides at the lowest dashed line must be equal (or the fluid would move). p = p p o left mg A m = right u + = po + u m k mg k A Initially fluid of known density ρ k Add column h of fluid of unknown density ρ u ( ρ >ρ u k case) ρ Ah =ρ u ρ = u k H h AH ρ k

30 If ρ <ρ u k h-h The pressure on the two sides at the lowest dashed line must be equal (or the fluid would move). p = p left right h p p H p o mg A m = u + = po + ρ Ah =ρ u u m k k AH mg k A Initially fluid of known density ρ k Add column h of fluid of unknown density ρ u So again, ρ = u H h ρ But now H < h. k

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