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20 B Week II Chapters 9 -10)

• Macroscopic Pressure •Microscopic pressure( the kinetic theory of gases: no potential energy) • Real Gases: van der Waals Equation of State London Dispersion Forces: Lennard-Jones V(R ) and physical bonds Chapter 10 • 3 Phases of Matter: Solid, Liquid and Gas of a

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20 B Week II Chapters 9 -10)

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  1. • Macroscopic Pressure •Microscopic pressure( the kinetic theory of gases: no potential energy) • Real Gases: van der Waals Equation of State London Dispersion Forces: Lennard-Jones V(R ) and physical bonds Chapter 10 • 3 Phases of Matter: Solid, Liquid and Gas of a single component system( just one type of molecule, no solutions) Phase Transitions: A(s) A(g) Sublimation/Deposition A(s) A(l) Melting/Freezing A(l) A(g) Evaporation/Condensation 20 B Week II Chapters 9 -10)

  2. For a fixed mass( of gas, e.g., Air) how does the Volume occupied by the gas change when the gas is cooled or heated? Lets do an experiment! • Define a Thermodynamic Temperature scale to make a thermometer( to measure T) Assumes we know nothing about the Boltzmann distribution so T is a parameter • Charles’ Law V/T = const

  3. Quantifying the temperature scale requires some material Property that changes if the material is heated or cooled. The thermal expansion coefficient = (1/V)(V/T) assume fractional rate of change of V with respect to T is ideal. The temperature scale should be material independent (Universal) appears to be Universal for low pressure gases Over the T range where water freezes and boils, Charles observed that = (1/V)(V/T) ~const for low pressure gases. = (1/V)(V/T)=(1/V0){V-V0}/{T-T0} V0 and T0, are the initial volume and Temperature, thermometer material and V and T, , are the final volume and Temperature, respectively!

  4. The Experiment: we need 3 T’s to be accurate The material for the experiment is air 20±? Room Temp! Assigned values

  5. = (1/V0){V-V0}/{T-T0} Thermal expansion Let V0 be the gas Volume at T0 = 0 The Freezing point of water! T = (1/{(V/V0) – 1} Re-arrange to V=V0 T} If T has units of °C (Celsius), the boiling point of water must be set to 100 °C! All measurements are for air at low pressure

  6. Fitting the V vs T the equation to the data, we can extrapolate to absolute zero! V0= 1.5 L V=V0 T} solve for T at V=0 gives T=C V0= 1.0 L

  7. However, the mass of gas is fixed! So V can approach zero but cannot be zero! V=V0 T} so as V goes to zero T goes to The absolute T=0 K T=C V0= 1.5 L V0= 1.0 L

  8. In terms of an absolute temperature T=-273.15 C= 0 K (Kelvin) V~T since as V approaches 0, T  0 V = TConst. This is most useful form of Charles’ Law V~ N (the number of gas atoms/Molecules) (Since the more molecules the greater the volume) V~1/P from Boyle’s Law Therefore V~ NT/P or V=kNT/P: PV=kNT The is Ideal or Perfect Gas Law where k = Boltzmann Constant : PV=kNT

  9. PV=kNT: The Idea Gas Law! Appliedtwo different gases, e.g., N2 and O2 (Or even better! The Law describing the behavior of a Theoretically Perfect/Ideal Gas)! Can be used for all gases at Low Pressure and high enough temperature ~ ultrahigh vacuum ultralow pressures! ~ very small hole or area A. If P=PO2 + PN2(partial pressures) P=NO2kT/V + NN2kT/V P = (NO2 + NN2 )kT/V=NkT/V N=NO2 + NN2 which is true Effusion Cell

  10. Partial Pressures add. P= P1 + P2 + P3 + etc + = Fig. 9-9, p. 377

  11. When dealing with the behavior of large numbers N~NA NA=6.022 x 1023 mole-1 then we must average over the behavior of the individual atom/molecule in the system For example. If i is the total energy of the ith atom/molecule, Kinetic + Potential, then the average energy per atom/molecule is <>= (1 + 2 + 3 ………. + N )/N=∑i/N This is the behavior we observe and is the domain of Statistical Mechanics: the science that describes the behavior of a system with large numbers of atoms/molecules based on the behavior of individual atoms/molecules, e.g., Gases!

  12. The Kinetic Theory of Gases uses this idea to describe the Behavior of an ideal gas! Think about this for a min! The average behavior can be determined by watching one particle for a very long time(infinitely long in principle), or a large number of particles (infinitely large in principle) for a short time, i.e., snapshot or an instant <(vx)2

  13. The Kinetic Theory Nanoscopic theory of gas pressure watch the average behavior of one particle and describe the behavior of a system with a large number of particles L

  14. The Kinetic Theory Nanoscopic theory of gas pressure watch the average behavior of one particle and describe the behavior of a system with a large number of particles - FA =F = ma=d(mv)/dt mv = linear momentum (mv)=(mvx)f - (mvx)i (mv)=-mvx - mvx=-2mvx t=2L/vx L

  15. The Kinetic Theory Microscopic pressure one particle at a time V=AL -FA = F = ma=d(mv)/dt force atom/molecule mv = linear momentum (mv)=(mvx)f - (mvx)i (mv)=-mvx - mvx=-2mvx t=2L/vx L • FA =F = ma=d(mv)/dt ~ -2mvx/(2L/vx)= - m(vx)2/L • FA=2m(vx)2/L; <FA>=m<(vx)2>/L average force on the wall • Px=P= N<FA>/A=Nm<(vx)2>/AL=(N/V) m<(vx)2>

  16. Motion in all directions x,y, and z are equally likely: <vx>=<vy>=<vz> v2 = (vx)2 + (vy)2 +(vz)2 =u2 then < v2 >=<u2>=<(vx)2 + (vy)2 +(vz)2>=3<(vx)2> or <(vx)2> = <u 2 >/3 The average of speed in all directions must be the same <(vx)2>=<(vy)2>=<(vz)2> for the random motion of an atom/molecule P= (N/V) m<(vx)2>= (N/3V) m< u2 > but <KE>= (m/2)< u2 > So P=(N/V)<KE>(2/3) recall that PV=NkTso <KE>=(3/2)kT Since <>=<PE> + <KE> = <KE> for a prefect gas, that is a gas described by PV=NkT Fig. 9-11, p. 379

  17. Partial Pressures add. P= P1 + P2 + P3 + etc + = Fig. 9-9, p. 377

  18. • Macroscopic Pressure •Microscopic pressure( the kinetic theory of gases: no potential energy) • Real Gases: van der Waals Equation of State London Dispersion Forces: Lennard-Jones V(R ) and physical bonds Chapter 10 • 3 Phases of Matter: Solid, Liquid and Gas of a single component system( just one type of molecule, no solutions) Phase Transitions: A(s) A(g) Sublimation/Deposition A(s) A(l) Melting/Freezing A(l) A(g) Evaporation/Condensation 20 B Week II Chapters 9 -10)

  19. <V(R )> = 0 For R Very Large Density N/V is low Therefore P=(N/V)kT is low A A R Fig. 9-18, p. 392

  20. Real Gases and Intermolecular Forces Lennard-Jones Potential V(R ) = 4{(R/)12 -(R/)6} kT >>  Ar+ Ar  well depth ~ Ze or Mass but it’s the # of e

  21. Real Gases and Intermolecular Forces Lennard-Jones Potential V(R ) = 4{(R/)12 -(R/)6} ~ hard sphere diameter • well depth Bond Dissociation D0= h

  22. The London Dispersion or Induced Dipole Induced Dipole forces Weakest of the Physical Bonds but is always present!

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