Download Ch18 The Micro/Macro Connection

講者： 許永昌 老師
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Contents
 Molecular Speeds and Collisions
 Pressure
 Temperature
 Thermal energy and Specific heat
 Thermal interaction and Heat
 Irreversible Processes & the 2nd Law of
thermodynamics.
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The aim of this chapter
 Understand
 Collision
 Average translational
 Pressure
 Temperature
 Energy transfer




kinetic energy
 Microscopic energies
 Molecular basis
 Probability
Thermal energy
Ideal-gas law
Specific Heat
Heat and Thermal
equilibrium
 Entropy
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Molecule Speeds (請預讀P542)
 Changing the (1)
or
changing to a (2)
changes the
does
distribution.
P  v  dv 
Probability
where  
e  
Boltzmann distribution
, but it
of the
* v 2 dv
states
1
1
and   v   mv 2 .
k BT
2
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Mean Free Path (請預讀P543)
 The average speed of N2 at 20oC is about 500 m/s
~0.1 s (of course not)
~10m
 There must be some collisions happened.
 Reference: http://en.wikipedia.org/wiki/Diffusion
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Mean Free Path (continue)
 N coll   N / V Vcyl   N / V    2r 2 L
area
assume that all other molecules are fixed 
L
1

N coll 4  N / V  r 2
 More careful calculation (all
molecules move)
1

4 2  N / V  r 2
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Stop to Think
 What would happen in the room if the molecules of
the gas were not moving?
 What would happen in an isolated room if the
molecular collisions were not perfectly elastic?
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Pressure in a Gas (請預讀P544~P545)
 Objects:
 A wall whose normal is x direction.
 Molecules in the left hand side of this wall.
 Condition: perfectly elastic. J wall on molecule  2mvx xˆ
=
Favg 
 N  t  J
i
i
i

  N / V  Av
ix
t * p  vi    vix  *  2mvix xˆ 
+
+…
velocity: vi
# of particles collide with the wall: Ni  t 
i
t
Favg N
N
P
  2mvix2 p  vi    vix  
A
V i
V
t
N
 mv p  v   V m  v 
2
ix
i
i
where p  v  is the probability of a particle whose velocity is v ,
2
x avg
,if p  vi   p  vi  .
 p v   1
i
i
1,
and   x   
0,
x0
x0
, respectively.
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The Root-Mean-Square Speed
(請預讀P545~P546)
 vrms 
 v
2

v 
2
avg
avg
  vi2 p  vi 
i
  vix2 p  vi    viy2 p  vi    viz2 p  vi 
i
  vx2 
avg
 3  vx2 
  v y2 
i
avg
  vz2 
i
avg
avg
assume p  Rvi   p  vi  , R  Rotation Group
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Stop to Think & Exercise
 What are the definitions of
1. rms speed
2. Average speed
3. Average velocity
 What are the benefits of the definition of rms speed?
:
 2 particles: v=3êx+êy,
3 particles: v=2êx+2êy,
4 particles: v=2êy.
 Find: (1) rms speed (2) average speed (3) average velocity.
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Homework
 Student workbook:
 18.5, 18.6
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Temperature (請預讀P546~P548)
 Microscopic avg
Macroscopic T.
 Average translational kinetic energy:

avg(½mv2)avg.
 Kinetic Theory:


PV=Nm(vx2)avg.
(v2)avg=3(vx2)avg.
 Ideal-gas Law: PV=NkBT.
3
2
 We get  avg  k BT
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Temperature (continue)
  avg
3
 k BT
2
 For a gas, this thing we call
measures
the
.
 This concept of temperature also gives meaning to
as the temperature at which avg=0
and all molecular motion ceases.
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Homework
 Student workbook:
 18.10
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Thermal Energy for monatomic
Gases (請預讀P549~P550)
 Eth=Kmicro+Umicro.
 For monatomic gases:
 Eth=Kmicro.
 Eth=Navg=3/2NkBT=3/2nRT.
 Owing to the 1st Law of thermodynamics,
Eth  QV  nCV T
 We get CV=3/2R=12.5 J/mol K.
 Q: How about other systems?
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The Equipartition Theorem (請預讀
P550~P551)
 The thermal energy of a system of particles is equally divided among
. For a system of N particles at temperature
T, the energy stored in each mode (each degree of freedom) is ½NkBT.
 It is not proved here.
 To prove it, you need the concepts of
 Probability
 States
 Phase space
 Boltzmann distribution
 Example: Solid (for high enough temperature)
 Dulong-Petit law

Detail: Solid State Physics.
 It has 6 degrees of freedom
 Eth=6*N*½kBT  C=3R=25.0 J/mol K~6.00 cal/mol K.
*Solid State Physics, Ashcroft/Mermin, P463
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Specific Heat of diatomic
molecules (請預讀P552~P553)
  micro
2
2
L
L2x
L
1
1
1
1
1
y
2
2
2
z
 MvCM , x  MvCM , y  MvCM , z 


  v122  k r122
2
2
2
2 I zz 2 I yy 2 I xx 2
2
 avg
Why?
rotational kinetic energy
vibrational energy
A: In quantum mechanics,
1.
<Li>=nћ
2.
Discrete energy levels
for bounded states.
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Additional Remark (補充)
 For two particles system
1
1
m1v12  m2v22
2
2
1
1
2
 MvCM
  v122 , where M  m  m , v  v  v ,   m1m2 .
1
2 12
2
1
2
2
M
2
K tot 
 <rot,z>~
 mH=1.66*10-27 kg,
 r~3.7*10-11 m,
 ћ=1.05*10-34 Js,
 kB=1.38*10-23 J/K
 Teff~180K (n=1)
1 Lz
1 n2 2
1

 k BTeff
2
2 Iz
2 2mH r
2
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Thermal Interactions and Heat
(請預讀P554~P555)
System 1
Th
System 2
Heat
Tc
 Thermal interaction
 Thermal Equilibrium:
 Collisions
 Thermal Equilibrium:
 (1)avg= (2)avg.
 T1=T2.
 Energy can transfer
 Energy can transfer
from 2 to 1:
from 2 to 1:
Probability
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Exercise
 Conditions:
 System 1: 4.00 mol N2 at
T1=27oC.
 System 2: 1.00 mol H2 at
T2=327oC.
 3 s.f.
 Find:
 Thermal energies
 Tf =?
 Heat transfer=?
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Homework
 Student Workbook:
 18.13, 18.15, 18.16
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Irreversible Processes and the 2nd
Law of Thermodynamics (請預讀P556~P558)
(irreversible)
(reversible)
 One particle
Probability
 Many particles
 Go to reach Equilibrium.
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Order, Disorder and Entropy (請預
讀P558~P560)
 Scientists and engineers use a state
variable called entropy to
that a macroscopic state
.
 The second Law of thermodynamics:
 The entropy of an
. The entropy either
increases, until the system reaches
equilibrium, or, if the system began in
equilibrium, stays the same.
 Th  Tc spontaneously.
(Heat)
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Homework
 Student Workbook:
 18.18
 Student Textbook:
 15, 65
 製作Terms and Notation的卡片，以方便自我練習。
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