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Chapter 12 Lecture Pearson Physics Gases, Liquids, and Solids Prepared by Chris Chiaverina © 2014 Pearson Education, Inc. Chapter Contents • • • • Gases Fluids at Rest Fluids in Motion Solids © 2014 Pearson Education, Inc. Gases • A substance that can flow from one location to another and has no set shape of its own is referred to as a fluid. • Gases and liquids are both fluids. They differ in that gases expand to fill their container, whereas liquids have a definite volume and may only partially fill their container. © 2014 Pearson Education, Inc. Gases • As the figure below shows, the pressure in a gas (or any fluid) acts equally in all directions. © 2014 Pearson Education, Inc. Gases • In addition, the pressure is always at right angles to any surface it acts on. © 2014 Pearson Education, Inc. Gases • If a car runs over a nail and a tire goes flat, it may be reasonable to think that the pressure inside the tire is zero. This is not so. • Since the hole in the tire allows air to pass freely from the inside of the tire to the atmosphere, the pressure in the tire is equal to atmospheric pressure. • When the tire is patched and inflated to a typical value of 241 kPa (about 35 lb/in2), the pressure inside the tire is greater than atmospheric pressure by 241 kPa. © 2014 Pearson Education, Inc. Gases • In fact, the pressure in the inflated tire is P = 241 kPa + Patmospheric = 241 kPa + 101 kPa = 342 kPa • However, the pressure you read on the gauge is—appropriately enough—described as gauge pressure. Gauge pressure is defined as follows: Pgauge= P − Patmospheric © 2014 Pearson Education, Inc. Gases • In this equation, the pressure P is the pressure inside the tire. The pressure that you read on a tire gauge is the gauge pressure, which is less than the pressure in the tire. © 2014 Pearson Education, Inc. Gases • In physics, idealized cases are often well approximated by many real-world situations. This is the case with the air surrounding us, in which the particles have little effect on one another. Behavior such as this can be considered ideal. • Thus an ideal gas is one in which the particles have no effect on one another. Like the air around us, most real gases are good approximations to an ideal gas. © 2014 Pearson Education, Inc. Gases • Pressure is the key to understanding the behavior of an ideal gas. The following are three common ways of changing the pressure exerted by a gas. – Increasing the number of gas particles in an enclosed space increases the pressure. – Decreasing the volume of an enclosed gas increases the pressure. – Heating an enclosed gas increases the average kinetic energy of its particles, causing an increase in pressure. © 2014 Pearson Education, Inc. Gases • These three observations can be combined into one simple equation for the pressure of an ideal gas: © 2014 Pearson Education, Inc. Gases • In this equation, N is the number of gas particles, T is the Kelvin temperature, and V is the volume of gas. The k is known as the Boltzmann constant. It is named for the Austrian physicist Ludwig Boltzmann (1844–1906). Its numerical value is 1.38 x 10-23 J/K © 2014 Pearson Education, Inc. Gases • The ideal gas equation is also sometimes written in an equivalent form: PV = NkT • The ideal gas equation shows that the gas pressure increases if the number of gas molecules increases, the temperature of the gas increases, or the volume of the gas decreases. © 2014 Pearson Education, Inc. Gases • In many cases it's more convenient to talk about the amount of gas in terms of weight rather than in terms of the number of particles. • This way of measuring the amount of gas uses moles. • A mole (mol) is the amount of a substance that contains as many particles as there are atoms in 12 grams of carbon-12. © 2014 Pearson Education, Inc. Gases • Experiments show that the number of particles in 12 grams of carbon-12 is 6.022 x 1023. This number is known as Avogadro's number, NA, named for the Italian physicist Amedeo Avogadro (1776−1856). • The photograph below offers examples of a mole of various substances. © 2014 Pearson Education, Inc. Gases • If n is used to represent the number of moles in an amount of gas, then the number of particles, N, is the number of moles times Avogadro's number: N = nNA • Using this expression for N in the ideal gas equation (PV = NkT) yields PV = nNAkT • This equation may be simplified by replacing NAk with the universal gas constant, R. © 2014 Pearson Education, Inc. Gases • The value of R is defined below. © 2014 Pearson Education, Inc. Gases • Thus, the ideal gas equation in terms of moles is © 2014 Pearson Education, Inc. Gases • The following example illustrates a calculation using the ideal gas equation. © 2014 Pearson Education, Inc. Gases • A mole of anything has precisely the same number of particles. What differs from substance to substance is the mass of 1 mole. • In general, the molar mass, M, of a substance is the mass in grams of 1 mole of that substance. For example, 1 mole of helium atoms has a mass of 4.00260 g, and 1 mole of copper atoms has a mass of 63.546 g. © 2014 Pearson Education, Inc. Gases • Molar mass provides a convenient bridge between the macroscopic world, where we measure the mass of a substance in grams, and the microscopic world, where the number of particles in a sample of a substance is typically 1023 or more. • If you measure out a mass of copper equal to 63.546 g, you have, in effect, counted out NA = 6.022 x 1023 atoms of copper. © 2014 Pearson Education, Inc. Gases • The picture of a gas consisting of innumerable particles flying about randomly at high speeds is known as the kinetic theory of gases. • Experiments show that the average kinetic energy of the particles in a gas is directly proportional to the Kelvin temperature. Thus, when a gas is heated, its particles move faster. This in turn increases the pressure exerted by the gas. © 2014 Pearson Education, Inc. Gases • Similarly, phase transitions can be understood in terms of the kinetic behavior of molecules. This explanation is sometimes referred to as the kinetic molecular theory. • For example, as the temperature of a solid is increased, the molecules oscillate about their fixed positions with more and more energy. When the temperature is high enough, the molecules have enough energy to break free of one another and move about more or less freely in the liquid state. • Increasing the temperature even more gives the molecules enough energy to form the gas phase. © 2014 Pearson Education, Inc. Fluids at Rest • Fluids flow from place to place and can change their shape and hence may seem somewhat difficult to describe and analyze. • One of the best ways to describe a fluid is in terms of amount of mass it has per volume. In general, the density of a substance (fluid or not) is the mass m of the substance divided by its volume, V. • The denser the substance, the more mass it has in any given volume. © 2014 Pearson Education, Inc. Fluids at Rest • Using the Greek letter rho, ρ (pronounced "row"), to stand for density, the definition of density is as follows: © 2014 Pearson Education, Inc. Fluids at Rest • The cylindrical flask in the figure below contains three colored liquids with different densities. The helium-filled blimp floats because helium is less dense than air. © 2014 Pearson Education, Inc. Fluids at Rest • A container 1 meter on a side encloses 1 cubic meter (1 m3). It takes 1000 kilograms of water to fill the container. Therefore, the density of water is ρ = 1000 kg/1 m3 = 1000 kg/m3 • The table below gives the densities of a variety of solids, liquids, and gases. © 2014 Pearson Education, Inc. Fluids at Rest • The following example shows how the density equation can be used to solve for the mass of a given volume or for the volume of a given mass. © 2014 Pearson Education, Inc. Fluids at Rest • The pressure in a fluid increases with depth. The increase in pressure is due to the added weight of the fluid pressing down as the depth increases. • The figure below shows the how the force varies with depth. © 2014 Pearson Education, Inc. Fluids at Rest © 2014 Pearson Education, Inc. Fluids at Rest • The top of the fluid-filled container in the figure is open to the atmosphere, which has a pressure Patmospheric. • If the cross-sectional area of the container is A, the downward force exerted on the top surface by the atmosphere is Ftop = PatmosphericA • At the bottom of the container, the downward force is Ftop plus the weight of the fluid, where the weight of the fluid is W = mg = ρVg = ρ(hA)g © 2014 Pearson Education, Inc. Fluids at Rest • It follows that the total force at the bottom of the container is Fbottom = PatmosphericA + ρ(hA)g • Dividing the force by the area equals the pressure at the bottom: Pbottom = Patmospheric + ρgh • This equation holds not only for the bottom of the container, but also for any depth below the surface. © 2014 Pearson Education, Inc. Fluids at Rest • The following example illustrates how pressure can be calculated for a specific depth. © 2014 Pearson Education, Inc. Fluids at Rest • The equation Pbottom = Patmospheric + ρgh can be applied to any two points in a fluid. • Suppose the pressure at one point is P1 and the pressure is P2 at a depth h below point 1. As shown in the figure below, the pressure at point 2 is greater than that at point 1 by the amount ρgh. © 2014 Pearson Education, Inc. Fluids at Rest © 2014 Pearson Education, Inc. Fluids at Rest • The barometer is an interesting application of the change of pressure with depth. • The basic idea of the barometer is that the height difference is directly related to the atmospheric pressure that pushes down on the fluid in the bowl. • The figure below shows that the pressure in the vacuum at the top of the tube is zero. © 2014 Pearson Education, Inc. Fluids at Rest © 2014 Pearson Education, Inc. Fluids at Rest • Thus the pressure in the tube at a depth h below the vacuum is ρgh. • At the level of the fluid in the bowl, the pressure is 1 atmosphere. Therefore, Patmospheric = ρgh. • Thus, a measurement of the height difference (h) gives the atmospheric pressure. • Mercury is a fluid that is often used in a barometer of the type shown above. Using h = Patmospheric/ρg and appropriate values for ρ and g, the height of a column of mercury at normal atmospheric pressure is found to be 760 mm. © 2014 Pearson Education, Inc. Fluids at Rest • The unit in the above result, millimeters of mercury, is used to define normal atmospheric pressure: 1 atmospheric = Patmospheric = 760 mmHg • The table below summarizes the various units in which atmospheric pressure can be expressed. © 2014 Pearson Education, Inc. Fluids at Rest © 2014 Pearson Education, Inc. Fluids at Rest • Squeezing a balloon causes the pressure to increase everywhere in the balloon. This is an example of Pascal's principle. • Pascal's principle states that an external pressure applied to an enclosed fluid is transmitted unchanged to every point within the fluid. © 2014 Pearson Education, Inc. Fluids at Rest • A classic example of Pascal's principle is the hydraulic lift, such as the one shown in the figure below. © 2014 Pearson Education, Inc. Fluids at Rest • Pushing down on the small piston increases the pressure in that cylinder by the amount ΔP = F1/A1. • By Pascal's principle, the pressure in cylinder 2 increases by the same amount: ΔP = F2/A2. • Combining the two equations, we find F2 = F1(A2/A1) • Since the figure shows A2 to be greater than A1, the force, F1, exerted on the small piston causes a larger force, F2, on the large piston. • The lift magnifies force F2, but the distance d2 is smaller than distance d1. Energy is conserved. © 2014 Pearson Education, Inc. Fluids at Rest • As the figure below shows, a fluid surrounding an object exerts a buoyant force in the upward direction. © 2014 Pearson Education, Inc. Fluids at Rest • The direction of the buoyant force is due to the fact that the pressure increases with depth, and hence the upward force on the object, F2, is greater than the downward force, F1. Forces acting to the left and to the right cancel out. © 2014 Pearson Education, Inc. Fluids at Rest • It can be shown that the buoyant force acting on the object is equal to the weight of the fluid that the object displaces. This is referred to as Archimedes' principle. • The figure below illustrates an example of Archimedes' principle. Note that the decrease in scale reading is equal to weight of the water that the block displaces. © 2014 Pearson Education, Inc. Fluids at Rest © 2014 Pearson Education, Inc. Fluids at Rest • When an object floats in water, the upward buoyant force is equal in magnitude to the downward force of gravity. • High-density objects can also float. For example, a steel block that normally sinks will float if is it is molded into the shape of a bowl (see figure below). This is the principle that allows ships to be made of steel and iron. © 2014 Pearson Education, Inc. Fluids at Rest © 2014 Pearson Education, Inc. Fluids in Motion • When the opening of a hose is made smaller with a nozzle or a thumb, the velocity of flow increases (see figure below). © 2014 Pearson Education, Inc. Fluids in Motion • This increase in speed illustrates what is referred to as the equation of continuity. • The equation of continuity states that if the cross-sectional area through which a fluid flows is reduced, the speed of the fluid increases. © 2014 Pearson Education, Inc. Fluids in Motion • That is, cross-sectional area 1 x velocity 1 = cross-sectional area 2 x velocity 2 or A1v1 = A2v2 • The figure below illustrates the increase in velocity due to the narrowing of a pipe. © 2014 Pearson Education, Inc. Fluids in Motion © 2014 Pearson Education, Inc. Fluids in Motion • When a fluid flows at high speed, its pressure is less than when it flows at low speed. • The fact that the pressure of water—or any fluid—is reduced when it flows at a higher speed is referred to as Bernoulli's principle. • As the figure below shows, blowing across the top of a piece of paper reduces the pressure there, resulting in a net upward force that lifts the paper. © 2014 Pearson Education, Inc. Fluids in Motion © 2014 Pearson Education, Inc. Fluids in Motion • Practical applications of Bernoulli's principle include: – the lift provided by the flow over the top of an airplane wing – the modification of roof design to prevent high-speed winds from lifting roofs off houses – the natural air conditioning provided by the design of a prairie dog burrow. A burrow typically has a high mound and a low mound. Greater air speed over the higher mound reduces the pressure over the higher mound. The difference in pressure causes air to be drawn through the burrow. © 2014 Pearson Education, Inc. Fluids in Motion • A fluid flowing past a stationary surface experiences a force opposing the flow. This tendency of a fluid to resist flow is referred to as the viscosity of the fluid. • Fluids like air have low viscosities, thick fluids like water are more viscous, and fluids like honey and molasses are characterized by high viscosity. • Real fluids always have some viscosity, and hence work must be done to force a fluid to flow through the tube. • This is the reason the blood in our arteries has a relatively high pressure. If a blood vessel is partially blocked, the resistance to blood flow increases, resulting in a higher than normal blood pressure. © 2014 Pearson Education, Inc. Fluids in Motion • Real fluids also exhibit surface tension, the force that tries to minimize the surface area of a fluid. • It is surface tension that makes the surface of water behave like an elastic sheet. • Surface tension enables some insects to walk on the surface of a pond (see figure below). © 2014 Pearson Education, Inc. Fluids in Motion • Even a needle or razor blade can be supported by surface tension if it is put into place gently. © 2014 Pearson Education, Inc. Solids • Unlike fluids, solids maintain their shape. • A solid may be thought of as a collection of small balls representing molecules. In this model, the molecules are connected to one another by springs (see figure below). © 2014 Pearson Education, Inc. Solids • When you pull on a solid rod with a force, you stretch each of the intermolecular springs in the direction of the force. The result is that the entire solid increases in length. © 2014 Pearson Education, Inc. Solids • In the case of a spring, the amount of stretch varies with the force applied. The greater the force, the greater the stretch. • The figure below shows how a spring scale responds to increasing weight. © 2014 Pearson Education, Inc. Solids • As the figure indicates, the change in the spring's length is directly proportional to the applied force. © 2014 Pearson Education, Inc. Solids • The proportionality between stretch and applied force may be summarized with the following equation: F = kΔL • This equation is known as Hooke's law. • The change in length ΔL can be either a stretch or a compression. • The constant k is called the spring constant, and its units are newtons per meter (N/m). © 2014 Pearson Education, Inc. Solids • The change in length is generally represented with the symbol x, as is shown in the figure below. © 2014 Pearson Education, Inc. Solids • When x = 0, the spring is relaxed, and there is no change in length. When stretched or compressed, x represents the distance from equilibrium. Replacing ΔL with x yields Hooke's law for the specific case of a spring: F = kx. © 2014 Pearson Education, Inc. Solids • Objects that return to their original shape and size after being deformed are said to be elastic. • Stretching a spring too far, however, causes permanent deformation. In this case, which is referred to as exceeding the elastic limit of the spring, the spring never returns to its original state. © 2014 Pearson Education, Inc. Solids • The stretch-force graph in the figure below shows how a spring responds to forces of various magnitudes. © 2014 Pearson Education, Inc. Solids • As the figure below illustrates, the amount that a wire, or any object, will stretch is proportional to the original length of the wire. A convenient way to increase the initial length of a wire is to wrap it into a coil of many turns. This is how a spring is made. © 2014 Pearson Education, Inc. Solids © 2014 Pearson Education, Inc.