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Physics 52 - Heat and Optics Dr. Joseph F. Becker Physics Department San Jose State University

Physics 52 - Heat and Optics Dr. Joseph F. Becker Physics Department San Jose State University. © 2005 J. F. Becker. Chapter 18 Thermal Properties of Matter.

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Physics 52 - Heat and Optics Dr. Joseph F. Becker Physics Department San Jose State University

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  1. Physics 52 - Heat and OpticsDr. Joseph F. BeckerPhysics DepartmentSan Jose State University © 2005 J. F. Becker

  2. Chapter 18Thermal Properties of Matter © 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics

  3. Thermal Properties of Matter 2. Equation of state -Ideal gas equation -Van der Waals equation 3. Molecular properties of matter4. Kinetic-molecular model of an ideal gas5. Heat capacities (Cv) – theory6. Molecular speeds7. Phases of matter © 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics

  4. A hypothetical apparatus for studying the behavior of gases. The pressure p, volume V, temperature T, and number of moles n, of a gas can bevaried and measured.

  5. Cutaway of an automobile engine showing the intake and exhaust valves.

  6. Ideal gas model assumptions: No molecular volume No attractive forces No potential energy 100% elastic collisionsIdeal gas equation: p V = n R T P = absolute pressure = pgauge + 1 atmV = volume in m3n = number of moles; mole = 6.022 (10)23T = absolute temperature (Kelvin)R = gas constant = 8.3145 J/mole K = 0.0821 L atm / mole K © 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics

  7. Van der Waals gas equation: ( p + a n 2/ V 2 )(V – n b ) = n R T where a and b are empirical constants and are different for different gasses.b = volume on one mole of molecules, so nb = the total volume of the molecules.a = attractive intermolecular forces (called “Van der Waals forces”) which reduce the pressure of the gas for a given n, V, and T. Ideal gas p p + a (n 2/ V 2) © 200 J. F. Becker San Jose State University Physics 52 Heat and Optics

  8. Isotherms, or constant-temperature curves, for a constant amount of anideal gas. p = nRT / V HOT COLD

  9. pV diagram for a non-ideal gasisotherms for temperatures above and below the critical temperature Tc.

  10. The force between two molecules (blue curve) is zero at a separation r = ro, where the potential energy (red curve)is a minimum. U = Uo[(Ro/r)12 - 2(Ro/r)6] The force is attractive when the separation is greater than ro and repulsive when the separation is less than ro F= -dU/dr

  11. Schematic representation of the cubic crystal structure of sodium chloride.

  12. Kinetic-molecular model of an ideal gas • Container of volume V contains a large number N of identical molecules mass m. • The molecules are point particles – average distance between molecules and walls is large. • Molecules are in motion, obey Newton’s laws, undergo 100% elastic collisions with walls - no heat, friction, etc.O Container walls are perfectly rigid and don’t move. © 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics

  13. Elastic collision of a molecule with an idealized container wall. The velocity component parallel to the wall does not change; the component perpendicular to the wall reverses direction.The change in momentum is DP = m 2vx

  14. A molecule moving toward the wall with speed vxcollides with the area A during the time interval dtif it is within a distance vx dtof the wall at the beginning of the interval. All such molecules are contained within a volume = Avx dt.

  15. The number of collisions with the area A of the wall is ½ (N/V) (Avx dt). For all the molecules in the cyl. D(momentum): DPx = (# of collisions) x (DPx per collision)DPx = ½ (N/V)Avx dt x (m 2vx ) =NAm vx2dt/VForce on wall = DPx /dt = N A m vx2 /V, so the pressure p = F/A = N m vx2 /V and pV = N m vx2 We need to take the average (or mean) speed of all the molecules, so v2 = vx2 + vy2 + vz2 and (v2)avg = (vx2)avg + (vy2)avg + (vz2)avg = 3 (vx2)avg (vx2)avg = (v2)avg/3 and pV = N m (v2)avg/3.Now multiply through by 2/2 to getpV = (2/3) N[½ m (v2)avg] = (2/3) Ktr © 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics

  16. pV = n R T = (2/3) KtrT a total translational KE of all N molecules.We can also show that n R = N k where k is Boltzmann’s constant k = 1.38 (10)-23 J/K so pV = n R T = N k T = (2/3) Ktr Ktr = (3/2) n R T = (3/2) N k T so Average transl. KE / mole = (3/2) R T and Average transl. KE / molecule = (3/2) k T or [½ m (v2)avg] = (3/2) k T v rms = [(v2)avg] ½ = [3 kT/m] ½ = [3 RT/M] ½ where the molar mass M = NA n © 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics

  17. In a time dt a molecule with a radius r will collide with any other molecule within a cylindrical volume of radius 2r and length v dt. COLLISIONSBETWEEN MOLECULES

  18. COLLISIONS BETWEEN MOLECULES Now we can get an estimate of the mean free time & the mean free path between collisions:If the molecules are not points, but rather rigid spheres of radius r, the number of molecules inside a cylinder of radius 2r isdN = (N/V) (p (2r)2 v dt) (See Fig. 18.12)Collisions / time = dN/dt = (N/V) p (2r)2 v Now, if ALL the molecules are moving (not just one) there are 1.41 times more collisions and:time/collision = dt/dN = V / N 1.41 p (2r)2 vand l = v tmean = V / N 1.41 p (2r)2From pV = NkT  l = kT / p 1.41 p (2r)2 © 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics

  19. Heat dQ is added to (a) a constant volume of monatomic ideal gas molecules. (b) The total translational kinetic energy increases by dKtr = dQ, and the temperature increases by dT = dQ / n CvdQ = n Cv dT GASES Cv=molar heat capacity at constant volume dQ

  20. MOLECULAR PROPERTIES OF MATTERHeat Capacities of Gases For now we assume constant volume so we can avoid taking into account work done by the gas on the atmosphere.From pV = n R T = N k T = (2/3) Ktrwe get Ktr = (3/2) n R T or dKtr = (3/2) n R dT. Comparison withdQ = n CvdTrecall dKtr = dQ soCv= (3/2) R ideal gas of point particles Cv= (5/2) R diatomic gas (3/2 + 2/2) © 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics

  21. A diatomic molecule - almost all the mass of each atom is in its tiny nucleus. (a) The center of mass has 3 independent velocity components. (b) The molecule has 2 independent axes of rotation through its c.m. (c) The atoms and “spring” have additional kinetic and potential energies of vibration.

  22. Experimental values of Cv for hydrogen gas (H2). Appreciable rotational motion begins to occur above 50 K, and above 600 K the molecule begins to increase its vibrational motion.

  23. The forces between neighboring particles in a crystal may be visualized by imagining every particle as being connected to its neighbors by springs. SOLIDS

  24. MOLECULAR PROPERTIES OF MATTERHeat Capacities of SolidsThe atoms in a crystal can vibrate in 3 directions with energy per degree of freedom kT/2 per molecule or RT/2 per mole  3kT/2.In addition to KE, each atom vibrating in a solid has PE = kHx2/2 (and average KE = average PE). So the total energy is KE + PE = 3kT/2 + 3kT/2 = 3kT/molecule or 3RT/mole.Cv = 3 R for a diatomic solid (3/2 + 3/2) © 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics

  25. At high temperatures Cv for each solid approaches approx. 3R, in agreement with the rule of Dulong and Petite.

  26. MOLECULAR PROPERTIES OF MATTERMolecular Speeds v rms = [3 kT/m] ½The molecules don’t all have the same speed! Maxwell-Boltzmann distribution function:f(v) = 4p (m/2pkT) 3/2 v 2 exp {-mv2/kT}The number of molecules dN having speeds in the range between v and v+dv is given bydN = N f(v) dv v avg = Vv f(v) dv (v 2) avg = Vv 2 f(v) dv © 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics

  27. Curves of the Maxwell-Boltzmann distribution function f(v) for various temperatures. As the temperature increases, the maximum shifts to higher speeds. (b) At temperature T3 the fraction of molecules having speeds in the range v1 to v2 is shown by the shaded area under the T3 curve. The fraction with speeds greater than vA is shown by the area from vA to infinity.

  28. MOLECULAR PROPERTIES OF MATTERPhases of MatterNow we consider phases (gas, liquid, solid) of matter at various pTV. An ideal gas has no phase transitions because there is no interaction between the molecules, but real matter does have these transitions. Triple point – the point (p3T3) at which gas, liquid and solid can coexist. (H2O: 0.01 atm, 273.16 K) Critical point - the point (pCTC) above which liquid and vapor do not undergo a phase transition, only continuous gradual changes from one phase to the other. (H2O: 200 atm, 650K) © 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics

  29. A typical pT phase diagram, showing regions of temperature and pressure at which the various phases exist and where phase changes occur. GAS

  30. pVT-surface for substance that expands on melting.

  31. pVT-surface for an ideal gas.

  32. A THERMODYNAMIC SYSTEMis described by its STATE VARIABLES, like pVT. Each state is described as a point (pVT ) on the surface of a phase diagram. A PROCESS takes a system through changes in its state variables. © 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics

  33. A molecule with a speed v passes through the first slit. When it reaches the second slit, the slits have rotated rotated through offset angle q. If v = w x / q, the molecule passes through the second slit and reaches the detector.

  34. Review

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