1. Physics 52 - Heat and Optics Dr. Joseph F. Becker Physics Department San Jose State University © 2005 J. F. Becker

2. Chapter 18 Thermal 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 matter 4. Kinetic-molecular model of an ideal gas 5. Heat capacities (C v ) – theory 6. Molecular speeds 7. Phases of matter

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 be varied and measured.

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

6. Ideal gas model assumptions: No molecular volumeNo attractive forces No potential energy 100% elastic collisions Ideal gas equation: p V = n R T P = absolute pressure = p gauge + 1 atm V = volume in m 3 n = number of moles; mole = 6.022 (10) 23 T = 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 an ideal gas. p = nRT / V HOT COLD

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

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

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. 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  P = m 2v x Elastic collision of a

14. A molecule moving toward the wall with speed v x collides with the area A during the time interval dt if it is within a distance v x dt of the wall at the beginning of the interval. All such molecules are contained within a volume = A v x dt.

15. The number of collisions with the area A of the wall is ½ (N/V) (A v x dt). For all the molecules in the cyl.  (momentum):  P x = (# of collisions) x (  P x per collision)  P x = ½ (N/V)A v x dt x (m 2v x ) =NAm v x 2 dt /V Force on wall =  P x / dt = N A m v x 2 /V, so the pressure p = F/A = N m v x 2 /V and pV = N m v x 2 We need to take the average (or mean) speed of all the molecules, so v 2 = v x 2 + v y 2 + v z 2 and (v 2 ) avg = (v x 2 ) avg + (v y 2 ) avg + (v z 2 ) avg = 3 (v x 2 ) avg (v x 2 ) avg = (v 2 ) avg /3 and pV = N m (v 2 ) avg /3. Now multiply through by 2/2 to get pV = (2/3) N [½ m (v 2 ) avg ] = (2/3) K tr © 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics

16. pV = n R T = (2/3) K tr T  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) K tr K tr = (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 (v 2 ) avg ] = (3/2) k T v rms = [(v 2 ) avg ] ½ = [3 kT/m] ½ = [3 RT/M] ½ where the molar mass M = N A 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. COLLISIONS BETWEEN MOLECULES

18. 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 is dN = (N/V) (  (2r) 2 v dt) (See Fig. 18.12) Collisions / time = dN/dt = (N/V)  (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  (2r) 2 v and = v t mean = V / N 1.41  (2r) 2 From pV = NkT  = kT / p 1.41  (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 dK tr = dQ, and the temperature increases by dT = dQ / n C v dQ = n C v dT dQ C v = molar heat capacity at constant volume GASES

20. MOLECULAR PROPERTIES OF MATTER Heat 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) K tr we get K tr = (3/2) n R T or dK tr = (3/2) n R dT. Comparison with dQ = n C v dT recall dK tr = dQ so C v = (3/2) R ideal gas of point particles C v = (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 C v for hydrogen gas (H 2 ). 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. © 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics MOLECULAR PROPERTIES OF MATTER Heat Capacities of Solids The 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 = k H x 2 /2 (and average KE = average PE). So the total energy is KE + PE = 3kT/2 + 3kT/2 = 3kT/molecule or 3RT/mole. C v = 3 R for a diatomic solid (3/2 + 3/2)

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

26. MOLECULAR PROPERTIES OF MATTER Molecular Speeds v rms = [3 kT/m] ½ The molecules don’t all have the same speed! Maxwell-Boltzmann distribution function: f(v) = 4  (m/2  kT) 3/2 v 2 exp {-mv 2 /kT} The number of molecules dN having speeds in the range between v and v+dv is given by dN = N f(v) dv v avg = V v f(v) dv (v 2 ) avg = V v 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 T 3 the fraction of molecules having speeds in the range v 1 to v 2 is shown by the shaded area under the T 3 curve. The fraction with speeds greater than v A is shown by the area from v A to infinity.

28. MOLECULAR PROPERTIES OF MATTER Phases of Matter Now 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 (p 3 T 3 ) at which gas, liquid and solid can coexist. (H 2 O: 0.01 atm, 273.16 K) Critical point - the point (p C T C ) above which liquid and vapor do not undergo a phase transition, only continuous gradual changes from one phase to the other. (H 2 O: 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. © 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics A THERMODYNAMIC SYSTEM is 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.

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 . If v =  x / , the molecule passes through the second slit and reaches the detector.

34. Review