microscopic model of gas
DESCRIPTION
Physics I. Microscopic Model of Gas. Prof. WAN, Xin [email protected] http://zimp.zju.edu.cn/~xinwan/. The Naïve Approach, Again. N particles r i (t), v i (t); interaction V(r i -r j ). Elementary Probability Theory. - PowerPoint PPT PresentationTRANSCRIPT
Physics IPhysics I
Microscopic Model of GasMicroscopic Model of Gas
Prof. WAN, Xin
[email protected]://zimp.zju.edu.cn/~xinwan/
The Naïve Approach, AgainThe Naïve Approach, Again
N particles ri(t), vi(t); interaction V(ri-rj)
Elementary Probability TheoryElementary Probability Theory
Assume the speeds of 10 particles are 0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0 m/s
m/s5.410
0.90.10.0
N
vv i
m/s5.710
0.90.10.0 2222
N
vv i
rms
When we have many particles, we may denote pa the probability of finding their velocities in the interval [va, va+1].
Elementary Probability TheoryElementary Probability Theory
Now, the averages become
1where, a
aaa ppvv
arms pvva
2
In the continuous version, we may denote p(v)dv the probability of finding particles’ velocities in the interval [v, v+dv].
1)(where,)(
dvvpdvvvpv
dvvpvvrms )(2
Assumptions of the Ideal Gas ModelAssumptions of the Ideal Gas Model
Large number of molecules and large average separation (molecular volume is negligible).
The molecules obey Newton’s laws, but as a whole they move randomly with a time-independent distribution of speeds.
The molecules undergo elastic collisions with each other and with the walls of the container.
The forces between molecules are short-range, hence negligible except during a collision.
That is, all of the gas molecules are identical.
The Microscopic ModelThe Microscopic Model
xF
LWDALV
AWD
xL
A
tvm
A
F
A
Fp
moleculesonxpistononx
,,
Pressure, the Microscopic ViewPressure, the Microscopic View
Pressure that a gas exerts on the walls of its container is a consequence of the collisions of the gas molecules with the walls.
xvmvmtAp 2
2
1 tvA x
22
3mvmvp x
= N / Vhalf of molecules
moving right
Applying the Ideal Gas LawApplying the Ideal Gas Law
22
33mv
nNmv
NVp A
TknN
Vpmv B
nRTpV
A 2
3
2
3
2
1 2
KJmole
KmoleJ
N
Rk
AB /1038.1
)(/1002.6
)/(31.8 2323
Boltzmann’s
constant
TemperatureTemperature
Temperature is a measure of internal energy (kB is the conversion factor). It measures the average energy per degree of freedom per molecule/atom.
Equipartition theorem: can be generalized to rotational and vibrational degrees of freedom.
Tkmvmvmv Bzyx
2
1
2
1
2
1
2
1 222
Heat Capacity at Constant VHeat Capacity at Constant V
We can detect the microscopic degrees of freedom by measuring heat capacity at constant volume.
Internal Energy U = NfkBT/2
Heat capacity
Molar specific heat cV = (f/2)R
degrees of freedomBV Nk
f
T
UC
2Vfixed
Specific Heat at Constant VSpecific Heat at Constant V
• Monoatomic gases has a ratio 3/2. Remember?
• Why do diatomic gases have the ratio 5/2?
• What about polyatomic gases?
Specific Heat at Constant VSpecific Heat at Constant V
A Simple Harmonic Oscillator
2
2)(
dt
xdmkx
dx
xdUF
x
2
2
1)( kxxU
xdt
xd 22
2
FF
m
kω 2
)cos0 t(xx
Two Harmonic Oscillators
)(' 21121
2
xxkxkdt
xdm
1x
)(' 12222
2
xxkxkdt
xdm
)(')(
21221
2
xxm
k
dt
xxd
)(2')(
21221
2
xxm
kk
dt
xxd
2x
Two Harmonic Oscillators
)(' 21121
2
xxkxkdt
xdm
1x
)(' 12222
2
xxkxkdt
xdm 201010
2 )'( kxxkkxm
2010202 )'( xkkkxxm
)cos0 t(xx ii
2x
Assume
Two Harmonic Oscillators
0'
'
20
10
2
2
x
x
mkkk
kmkk
2010102 )'( kxxkkxm 201020
2 )'( xkkkxxm
)cos0 t(xx iiAssume
1x 2x
Vibrational Mode
20100' xxk
2/
2' 0'
m
k
m
kk k
Solution 1:
Vibration with the reduced mass.
1x 2x
Translational Mode
2010 xx
0' 0' k
m
kSolution 1:
Translation!
1x 2x
Two Harmonic Oscillators
20
102
20
10
'
'
x
xm
x
x
kkk
kkk
1x 2x
In mathematics language, we solved an eigenvalue problem.
The two eigenvectors are orthogonal to each other. Independent!
Mode Counting – 1DMode Counting – 1D
1D: N-atom linear molecule – Translation: 1
– Vibration: N – 1
A straightforward generalization of the two-atom problem.
From 1D to 2D: A Trivial ExampleFrom 1D to 2D: A Trivial Example
rotation
vibration
translation
1x
2x
2y
1y
Mode Counting – 2DMode Counting – 2D
2D: N-atom (planer, nonlinear) molecule– Translation: 2
– Rotation: 1
– Vibration: 2N – 3
Mode Counting – 3DMode Counting – 3D
3D: N-atom (nonlinear) molecule– Translation: 3
– Rotation: 3
– Vibration: 3N – 6
Vibrational Modes of COVibrational Modes of CO22
N = 3, linear– Translation: 3
– Rotation: 2
– Vibration: 3N – 3 – 2 = 4
Vibrational Modes of HVibrational Modes of H22OO
N = 3, planer– Translation: 3
– Rotation: 3
– Vibration: 3N – 3 – 3 = 3
Contribution to Specific HeatContribution to Specific Heat
i
iii i
i qkm
pE 2
2
2
1
2
Equipartition theorem: The mean value of each independent quadratic term in the energy is equal to kBT/2.
Specific Heat of HSpecific Heat of H22
Quantum mechanics is needed to explain this.
Specific Heat of SolidsSpecific Heat of Solids
nRTTNkU B 33
DuLong – Petit law
RdT
dU
nc
VV 3
1
spatial dimension
vibration energy
Molar specific heat
Again, quantum mechanics is needed.
Root Mean Square SpeedRoot Mean Square Speed
m
Tkv B
rms
3root mean square speed
Estimate the root mean square speed of water molecules at room temperature.
m/s6003
m
Tkv B
rms
Distribution of SpeedDistribution of Speed
slow
fast
rotating drum
to pump
oven
Speed SelectionSpeed Selection
Can you design an equipment to select gas molecules with a chosen speed?
to pump
?
Maxwell DistributionMaxwell Distribution
kTmvevkT
mNvN 2/2
2/32
24)(
)(vN
vv dvv
dvvN )(
dvvNv
v 2
1
)(
Maxwell DistributionMaxwell Distribution
)(vN
v1v 2v
kTmvevkT
mNvN 2/2
2/32
24)(
number of molecules v [v1, v2]
Maxwell DistributionMaxwell Distribution
kTmvevkT
mNvN 2/2
2/32
24)(
)(vN
v
Total number of molecules
0
)( dvvNN
Characteristic SpeedCharacteristic Speed
0)(
dv
vdN
02 2/3
2/ 22
kTmvkTmv ekT
mvve
m
kTvp
2
Most probable speed
kTmvevkT
mNvN 2/2
2/32
24)(
Characteristic SpeedCharacteristic Speed
dvevkT
m
N
dvvNv
v kTmv
0
2/42/3
0
2
2 2
24
)(
m
kTvvrms
32
Root mean sqaure speed
kTmvevkT
mNvN 2/2
2/32
24)(
Characteristic SpeedCharacteristic Speed
dvevkT
m
N
dvvvN
v kTmv
0
2/32/3
02
24
)(
m
kTv
8
Average speed
kTmvevkT
mNvN 2/2
2/32
24)(
Varying TemperatureVarying Temperature
kTmvevkT
mNvN 2/2
2/32
24)(
)(vN
v
321 TTT T1
T2
T3
Boltzmann DistributionBoltzmann Distribution
Continuing from fluid statics
The probability of finding the molecules in a particular energy state varies exponentially as the negative of the energy divided by kBT.
TkmgyV
Tknppgh BBV ennepp /0
/0
00
potential energy
TkhvEV
Benhvn /),,(0),,(
Boltzmann distribution law
How to cool atoms?How to cool atoms?
Laser CoolingLaser Cooling
Figure: A CCD image of a cold cloud of rubidium atoms which have been laser cooled by the red laser beams to temperatures of a millionth of a Kelvin. The white fluorescent cloud forms at the intersection of the beams.
Bose-Einstein CondensationBose-Einstein Condensation
Velocity-distribution data for a gas of rubidium atoms, confirming the discovery of a new phase of matter, the Bose–Einstein condensate.
Left: just before the appearance of a Bose–Einstein condensate. Center: just after the appearance of the condensate. Right: after further evaporation, leaving a sample of nearly pure condensate.
Earlier BEC ResearchEarlier BEC Research
BEC in ultracold atomic gases was first realized in 1995 with 87Rb, 23Na, and 7Li. This pioneering work was honored with the Nobel prize 2001 in physics, awarded to Eric Cornell, Carl Wieman, and Wolfgang Ketterle.
For an updated list, check http://ucan.physics.utoronto.ca/
BEC of Dysprosium BEC of Dysprosium
Strongly dipolar BEC of dysprosium, Mingwu Lu et al., PRL 107, 190401 (2011)
Brownian MotionBrownian Motion
Mean Free PathMean Free Path
dd
v
d
v
d
Average distance between two collisions
During time interval t, a molecule sweeps a cylinder of diameter 2d and length vt.
Mean Free PathMean Free Path
pd
Tk
dnvtdn
vtl B
VV222
1
Tknp BV
Tkpn BV /
vtdnz V2
Volume of the cylinder
vtdV 2Average number of collisions
Mean free path
During time interval t, a molecule sweeps a cylinder of diameter 2d and length vt.
Mean Free PathMean Free Path
pd
Tk
dntvdn
vtl B
VV222 22
1
)2(
vtdnz V2
Average number of collisions
Mean free path Relative motion vv 2
Q&A: Collision FrequencyQ&A: Collision Frequency
Consider air at room temperature. – How far does a typical molecule (with a
diameter of 2 10-10 m) move before it collides with another molecule?
Q&A: Collision FrequencyQ&A: Collision Frequency
Consider air at room temperature. – How far does a typical molecule (with a
diameter of 2 10-10 m) move before it collides with another molecule?
Q&A: Collision FrequencyQ&A: Collision Frequency
Consider air at room temperature. – Average molecular separation:
Q&A: Collision FrequencyQ&A: Collision Frequency
Consider air at room temperature. – On average, how frequently does one
molecule collide with another?
m
kT
m
kTv ~
8
l
vf
Expect ~ 500 m/s
Expect ~ 2109 /s
Try yourself!
Fluid flows layer by layer with varying v.
F = A dv/dy : coefficient of viscosity
Transport: Viscous FlowTransport: Viscous Flow
A
A
F, vy
Cylindrical Pipe, NonviscousCylindrical Pipe, Nonviscous
v 2Rr
constvrv 0)(
02vRQ (volumetric flow rate)
Cylindrical Pipe, ViscousCylindrical Pipe, Viscous
V(r) 2Rr
22
4)( rR
L
Prv
L
RPrdrrvQ
82)(
4 (Poiseuille Law)
“current”“voltage”
HomeworkHomework
CHAP. 22 Exercises 7, 8, 10, 21, 24 (P513)