# Thermal Properties of Matter

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1 Chapter 18 Thermal Properties of Matter PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson

2 Goals for Chapter 18 To relate the pressure, volume, and temperature of a gas To see how molecular interactions determine the properties of a substance To relate temperature and pressure to the kinetic energy of gas molecules To learn what the heat capacities of a gas tell us about the motion of its molecules To discover what determines if a substance is a gas, liquid, or solid

3 Introduction How does the speed of the molecules in the air above the frying pan compare with that of the ones in the rest of the kitchen? How do the atoms of a gas determine its temperature and pressure? We ll see how the microscopic properties of matter determine its macroscopic properties.

4 Equations of state and the ideal-gas law Quantities such as pressure, volume, temperature, and the amount of a substance are state variables because they describe the state of the substance. The equation of state relates the state variables. The ideal-gas law, pv = nrt, is an equation of state for an ideal gas. The molar mass M (molecular weight) is the mass per mole. The total mass of n moles is m total = nm. Read Problem-Solving Strategy Follow Example 18.1.

5 There are relationships among the large-scale or macroscopic properties of a substance, such as pressure, volume, temperature, and mass. But we can also describe a substance using a microscopic perspective. This means investigating small-scale quantities such as the masses, speeds, kinetic energies,and momenta of the individual molecules that make up a substance

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7 In this chapter we ll begin our study of the thermal properties of matter by looking at some macroscopic aspects of matter in general. We ll pay special attention to the ideal gas, one of the simplest types of matter to understand. Using our knowledge of momentum and kinetic energy, we ll relate the macroscopic properties of an ideal gas to the microscopic behavior of its individual molecules.

8 The variables like pressure, volume, temperature, and amount of substance describe the state of the material and are called state variables. These variables are not independent. In a few cases the relationship among p, V, T, and m (or n) is simple enough that we can express it as an equation called the equation of state. When it s too complicated for that, we can use graphs or numerical tables. Even then, the relationship among the variables still exists; we call it an equation of state even when we don t know the actual equation.

9 Here s a simple (though approximate) equation of state for a solid material Measurements of the behavior of various gases lead to three conclusions: 1. The volume V is proportional to the number of moles n. If we double the number of moles, keeping pressure and temperature constant, the volume doubles.

10 2. The volume varies inversely with the absolute pressure p. If we double the pressure while holding the temperature T and number of moles n constant, the gas compresses to one-half of its initial volume. In other words, pv = constant when n and T are constant. 3. The pressure is proportional to the absolute temperature. If we double the absolute temperature, keeping the volume and number of moles constant, the pressure doubles. In other words, p = (constant) T when n and V are constant. These three relationships can be combined neatly into a single equation, called the ideal-gas equation: pv = nrt (ideal-gas equation) (18.3)

11 This is an idealized model; it works best at very low pressures and high temperatures, when the gas molecules are far apart and in rapid motion. Why? We might expect that the constant R in the ideal-gas equation would have different values for different gases, but it turns out to have the same value for all gases, at least at sufficiently high temperature and low pressure. It is called the gas constant (or ideal-gas constant). The numerical value of R depends on the units of p, V, and T. In SI units, in which the unit of p is Pa ( N/m 2 ) and the unit of V is m 3,the current best numerical value of R is R = (15) J/mol. K

12 pv =(N/m 2 ) (m3) =N.m = Joule

13 For a constant mass (or constant number of moles) of an ideal gas the product nr is constant, so the quantity pv/t is also constant. If the subscripts 1 and 2 refer to any two states of the same mass of a gas, then p 1 V 1 /T 1 = p 2 V 2 /T 2 (ideal gas, constant mass) (18.6) Notice that you don t need the value of R to use this equation.

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19 At the summit of Mount Everest, where y = 8863m, Mgy/RT = (28.8x10-3 kg/mol)(9.80 m/s 2 )/(8863m)/(8.314J/mol.K)(273K) = 1.10 p = (1.013 x 10 5 Pa) e = 0.337x 10 5 Pa = 0.33 atm The assumption of constant temperature isn t realistic, and g decreases a little with increasing elevation. Even so, this example shows why mountaineers need to carry oxygen on Mount Everest. It also shows why jet airliners, which typically fly at altitudes of 8000 to 12,000 m, must have pressurized cabins for passenger comfort and health.

20 The van der Waals equation The model used for the ideal-gas equation ignores the volumes of molecules and the attractive forces between them. (See Figure 18.5(a) below.) The van der Waals equation is a more realistic model. (See Figure 18.5(b) below.)

21 The ideal-gas equation can be obtained from a simple molecular model that ignores the volumes of the molecules themselves and the attractive forces between them (Fig. 18.5a). Meanwhile, we mention another equation of state, the van derwaals equation, that makes approximate corrections for these two omissions (Fig. 18.5b). This equation was developed by the 19th-century Dutch physicist J. D. van der Waals; the interaction between atoms that we discussed in Section 14.4 was named the van der Waals interaction after him. The van der Waals equation is

22 The constants a and b are empirical constants, different for different gases. Roughly speaking, b represents the volume of a mole of molecules; the total volume of the molecules is then nb, and the volume remaining in which the molecules can move is V-nb

23 The constant a depends on the attractive intermolecular forces, which reduce the pressure of the gas for given values of n, V, and T by pulling the molecules together as they push on the walls of the container. The decrease in pressure is proportional to the number of molecules per unit volume in a layer near the wall (which are exerting the pressure on the wall) and is also proportional to the number per unit volume in the next layer beyond the wall (which are doing the attracting). Hence the decrease in pressure due to intermolecular forces is proportional to n 2 /V 2

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25 We could in principle represent the p-v-t relationship graphically as a surface in a threedimensional space with coordinates p, V, and T. This representation sometimes helps us grasp the overall behavior of the substance, but ordinary two-dimensional graphs are usually more convenient. One of the most useful of these is a set of graphs of pressure as a function of volume, each for a particular constant temperature. Such a diagram is called a pvdiagram. Each curve, representing behavior at a specific temperature, is called an isotherm, or a pvisotherm.

26 pv-diagrams A pv-diagram is a graph of pressure as a function of volume. Figure 18.6 at the right is a pv-diagram for an ideal gas at various temperatures.

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28 At temperatures below the isotherms develop flat regions in which we can compress the material (that is, reduce the volume V) without increasing the pressure p. Observation shows that the gas is condensing from the vapor (gas) to the liquid phase. The flat parts of the isotherms in the shaded area of Fig represent conditions of liquid-vapor phase equilibrium. As the volume decreases, more and more material goes from vapor to liquid, but the pressure does not change. (To keep the temperature constant during condensation, we have to remove the heat of vaporization, discussed in Section 17.6.)

29 When we compress such a gas at a constant temperature T 2 in Fig. 18.7, it is vapor until point a is reached. Then it begins to liquefy; as the volume decreases further, more material liquefies, and both the pressure and the temperature remain constant. At point b, all the material is in the liquid state. After this, any further compression requires a very rapid rise of pressure, because liquids are in general much less compressible than gases. At a lower constant temperature T 1 similar behavior occurs, but the condensation begins at lower pressure and greater volume than at the constant temperature T 2

30 At temperatures greater than T c no phase transition occurs as the material is compressed; at the highest temperatures, such as T 4 the curves resemble the ideal-gas curves of Fig We call T c the critical temperature for this material. In Section 18.6 we ll discuss what happens to the phase of the gas above the critical temperature.

31 We will use pv-diagrams often in the next two chapters. We will show that the area under a pv-curve (whether or not it is an isotherm) represents the work done by the system during a volume change. This work, in turn, is directly related to heat transfer and changes in the internal energy of the system

32 Molecules and Intermolecular Forces Any specific chemical compound is made up of identical molecules. The smallest molecules contain one atom each and are of the order of in size; the largest contain many atoms and are at least 10,000 times larger. In gases the molecules move nearly independently; in liquids and solids they are held together by intermolecular forces. These forces arise from interactions among the electrically charged particles that make up the molecules. Gravitational forces between molecules are negligible in comparison with electrical forces.

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35 Molecules are always in motion; their kinetic energies usually increase with temperature. At very low temperatures the average kinetic energy of a molecule may be much less than the depth of the potential well. The molecules then condense into the liquid or solid phase with average intermolecular spacings of about r 0 But at higher temperatures the average kinetic energy becomes larger than the depth IU 0 I of the potential well. Molecules can then escape the intermolecular force and become free to move independently, as in the gaseous phase of matte

36 In solids, molecules vibrate about more or less fixed points. In a crystalline solid these points are arranged in a crystal lattice. Figure 18.9 shows the cubic crystal structure of sodium chloride, and Fig shows a scanning tunneling microscope image of individual silicon atoms on the surface of a crystal.

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39 One mole is the amount of substance that contains as many elementary entities as there are atoms in kilogram of carbon-12.

40 Moles and Avogadro s number One mole of a substance contains as many elementary entities (atoms or molecules) as there are atoms in kg of carbon-12. One mole of a substance contains Avogadro s number N A of molecules. N A = x molecules/mol The molar mass M is the mass of one mole. M = N A m, where m is the mass of a single molecule. Follow Example 18.5, which illustrates atomic and molecular mass.

41 Find the mass of a single hydrogen atom and of a single oxygen molecule.

42 Kinetic-molecular model of an ideal gas The assumptions of the kinetic-molecular model are: 1. A container contains a very large number of identical molecules. 2. The molecules behave like point particles that are small compared to the size of the container and the average distance between molecules. 3. The molecules are in constant motion and undergo perfectly elastic collisions. 4. The container walls are perfectly rigid and do not move.

43 Collisions and gas pressure Pressure is caused by the forces that gas molecules exert on the walls of the container during collisions.

44 Our program is first to determine the number of collisions that occur per unit time for a certain area A of wall. Then we find the total momentum change associated with these collisions and the force needed to cause this momentum change. From this we can determine the pressure, which is force per unit area, and compare the result to the ideal-gas equation.

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56 Heat Capacities of Gases The basis of our analysis is that heat is energy in transit. When we add heat to a substance, we are increasing its molecular energy. In this discussion the volume of the gas will remain constant so that we don t have to worry about energy transfer through mechanical work

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62 A transition from one phase to another ordinarily requires phase equilibrium between the two phases, and for a given pressure this occurs at only one specific temperature. We can represent these conditions on a graph with axes p and T, called a phase diagram; Fig shows an example. Each point on the diagram represents a pair of values of p and T.

63 Phases of matter Figure below shows a typical pt phase diagram, and Table 18.3 gives triple-point data for some substances.

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65 A liquid-vapor phase transition occurs only when the temperature and pressure are less than those at the point lying at the top of the green shaded area labeled Liquid-vapor phase equilibrium region. This point corresponds to the endpoint at the top of the vaporization curve in Fig It is called the critical point, and the corresponding values of p and T are called the critical pressure and temperature, A gas at a pressure above the critical pressure does not separate into two phases when it is cooled at constant pressure (along a horizontal line above the critical point in Fig.18.24). Instead, its properties change gradually and continuously from those we ordinarily associate with a gas (low density, large compressibility) to those of a liquid (high density, small compressibility) without a phase transition.

66 pvt-surface for a substance that expands on melting A pvt-surface graphically represents the equation of state. Projections onto the pt- and pv-planes are shown.

67 pvt-surface for an ideal gas The pvt-surface for an ideal gas is much simpler than the pvt-surface for a real substance, as seen in the previous slide.

68 Collisions between molecules We model molecules as rigid spheres of radius r as shown at the right. The mean free path of a molecule is the average distance it travels between collisions. The average time between collisions is the mean free time. Follow the derivation of the mean free path. Follow Example 18.8.

69 Heat capacities of gases The degrees of freedom are the number of velocity components needed to describe a molecule completely. A monatomic gas has three degrees of freedom and a diatomic gas has five. Figure at the right illustrates the motions of a diatomic molecule. The equipartition of energy principle states that each degree of freedom has 1/2 kt of kinetic energy associated with it. Follow the derivation of the molar heat capacity at constant volume, C V, for monatomic and diatomic gases and for solids. The results are C V = 3/2 R for an ideal monatomic gas, C V = 5/2 R for a diatomic gas, and C V = 3R for an ideal monatomic solid.

70 Compare theory with experiment Table 18.1 shows that the calculated values for C V for monatomic gases and diatomic gases agree quite well with the measured values.

71 Molecular speeds The Maxwell-Boltzmann distribution f(v) gives the distribution of molecular speeds. Figure at the right helps to interpret f(v). Part (a) shows how the shape of the curve depends on temperature. Part (b) shows the fraction of molecules within certain speed ranges. The most probable speed for a given temperature is at the peak of the curve.

72 Phases of matter Figure below shows a typical pt phase diagram, and Table 18.3 gives triple-point data for some substances.

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