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LECTURE 11- KINETIC THEORY 2Phys 124H- Honors Analytical Physics IBChapter 19Professor Noronha-Hostler
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TODAY’S OBJECTIVES
Kinetic theory
Distributions
Mean free path
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WHY THEN DOES SMELL TAKE TIME TO SPREAD?
Repeated collisions of particles prevents smell to spread as quick as the rms
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COLLISIONS WITH PARTICLES
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TRANSLATIONAL KINETIC ENERGY
Kavg = ( 12
mv2)avg
=12
m (v2)avg
=12
mv2rms, vrms =
3RTM
substitute in
=12
m3RTM
, NA =Mm
substitute in
=3RT2NA
, substitute in k =RNA
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TRANSLATIONAL KINETIC ENERGY
Kavg =32
kT
At a fixed temperature T, all types of gas have the same average translational kinetic energy
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Two identical, sealed containers have the same volume. Both containers are filled with the same number of moles of gas at the same temperature and pressure. One of the containers is filled with helium gas and the other is filled with neon gas. Which one of the following statements concerning this situation is true?
a.) The speed of each of the helium atoms is the same value, but this speed is different than that of the neon atoms.
b)
c) The pressure within the container of helium is less than the pressure in the container of neon.
d) The internal energy of the neon gas is greater than the internal energy of the helium gas.
e) The rms speed of the neon atoms is less than that of the helium atoms
KNeavg > KHe
avg
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Two identical, sealed containers have the same volume. Both containers are filled with the same number of moles of gas at the same temperature and pressure. One of the containers is filled with helium gas and the other is filled with neon gas. Which one of the following statements concerning this situation is true?
a.) The speed of each of the helium atoms is the same value, but this speed is different than that of the neon atoms.
b)
c) The pressure within the container of helium is less than the pressure in the container of neon.
d) The internal energy of the neon gas is greater than the internal energy of the helium gas.
e) The rms speed of the neon atoms is less than that of the helium atoms
KNeavg > KHe
avg
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ANSWER
vrms =3RTM
MNe > MHe
vHerms > vNe
rms
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A monatomic gas is stored in a container with a constant volume. When the temperature of the gas is T, the rms speed of the atoms is . What is the rms speed when the gas temperature is increased to 3T?
vrms
A.)
B.)
C.)
D.)
E.)
vrms
9vrms
33vrms
3vrms
9vrms
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A monatomic gas is stored in a container with a constant volume. When the temperature of the gas is T, the rms speed of the atoms is . What is the rms speed when the gas temperature is increased to 3T?
vrms
A.)
B.)
C.)
D.)
E.)
vrms
9vrms
33vrms
3vrms
9vrms
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NEXT WEEK
Distributions
Types of expansions
Degrees of freedom
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MEAN FREE PATHHow long does a
molecule travel before colliding?
λ
λ =1
2πd2N/V
m m
d
d d
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ASSUMPTIONS FOR λ
Assume molecule has and all other molecules at rest
Assume molecules are sphere with diameter d
Assume collisions collisions occur when molecules are with d of each other
v = const
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DENSITY AND MEAN FREE PATH
λ =1
2πd2N/VRecall ρ = N/V
λ =1
2πd2ρ
As ⇑ ρ, ⇓ λ
As ⇑ d, ⇓ λ
(more collisions)
(bigger molecules)
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DISTRIBUTIONS OF MOLECULAR SPEEDS
What fraction of molecules have
What fraction of molecules have
v > vrms?
v > 2vrms?
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MAXWELL’S SPEED DISTRIBUTION
P(v) = 4π ( M2πRT )
3/2
v2e− Mv22RTProbability distribution
∫∞
0P(v)dv = 1
This means the probability to find a molecule traveling at a certain speed
If you integrate over all velocities, the probability to find the particle has to
be one.
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DISTRIBUTION OF SPEEDS
vavg
vrms
vmode
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DISTRIBUTION OF HEIGHT (NBA)
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DISTRIBUTION OF INCOME
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SPEED DISTRIBUTION
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Closed containers A and B both contain helium gas at the same temperature. There are n atoms in container A and 2n atoms in container B. At time t = 0 s, all of the helium atoms have the same kinetic energy. The atoms have collisions with each other and with the walls of the container. After a long time has passed, which of the following statements will be true?
a) The atoms in both containers have the same kinetic energies they had at time t = 0 s.
b) Both containers have a wide range of speeds, but the distributions of speeds are the same for both A and B.
c) Average kinetic energy for atoms in container B is greater than A.
d) Average kinetic energy for atoms in container A is greater than B.
e) Both containers have a wide range of speeds, but the distributions of speeds has a greater range for container B than that for container A
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Closed containers A and B both contain helium gas at the same temperature. There are n atoms in container A and 2n atoms in container B. At time t = 0 s, all of the helium atoms have the same kinetic energy. The atoms have collisions with each other and with the walls of the container. After a long time has passed, which of the following statements will be true?
a) The atoms in both containers have the same kinetic energies they had at time t = 0 s.
b) Both containers have a wide range of speeds, but the distributions of speeds are the same for both A and B.
c) Average kinetic energy for atoms in container B is greater than A.
d) Average kinetic energy for atoms in container A is greater than B.
e) Both containers have a wide range of speeds, but the distributions of speeds has a greater range for container B than that for container A
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ANSWER
Kavg =32
kT
vrms =3RTM
M =Msamp
n
vrms =3nRTMsamp
Kinetic energy must be the same at the same T
The rms speeds are not the same. But the distribution is the same.
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HOW TO USE A DISTRIBUTION
P(v) = 4π ( M2πRT )
3/2
v2e− Mv22RT∫
∞
0P(v)dv = 1We know
What fraction of molecules have v > vrms?
∫∞
vrms
P(v)dv = 1
What fraction of molecules have v1 < v < v2?
∫v2
v1
P(v)dv = 1
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EXAMPLE
Given a simpler distribution P(v) = A ⋅ e−Av
Is this normalized? i. e. ∫∞
0P(v)dv = 1
∫∞
0A ⋅ e−Avdv = A∫
∞
0e−Avdv
= A−1A
e−Av |∞0 = − e−Av |∞
0 = − 0 + 1 = 1
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WHAT FRACTION ARE 2 TIMES THE RMS?
∫∞
2vrms
P(v)dv = ∫∞
2vrms
A ⋅ e−Avdv
We just found that the generic form of this integral is
∫ A ⋅ e−Avdv = − e−Av
putting in the new bounds we find
−e−Av |∞2vrms
= 0 + e−2Avrms = e−2Avrms
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AVERAGE VS. RMS
In order to get the average
vavg = ∫∞
0vP(v)dv
Add in velocity to get the average
vrms = ∫∞
0v2P(v)dv
In order to get the root mean squared
Add in velocity^2 to get the rms
These are known as moments of the
distribution. The average is the first moment and the root mean squared is the second moment.
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INTEGRATION BY PARTS
∫ udx = ux − ∫ xdu
∫ Av ⋅ e−Avdv = A ∫ v ⋅ e−Avdv
integrate this
∫ v ⋅ e−Avdv = v−1A
e−Av − ∫−1A
e−Avdv
Integration by parts
= −vA
e−Av +1A ∫ e−Avdv = −
vA
e−Av −1
A2e−Av
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FINDING THE AVERAGE
∫ Av ⋅ e−Avdv = A∫ v ⋅ e−Avdv = A (−vA
e−Av −1
A2e−Av)
vavg = − (v +1A ) e−Av
vrms = ∫∞
0v2P(v)dv Something to try at
home
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FOR THE MAXWELL’S DISTRIBUTION
vavg = ∫∞
0vP(v)dv
vrms = ∫∞
0v2P(v)dv
vavg =8RTπm
vrms =3RTM
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MODE OF THE DISTRIBUTION
vmode
mode=maximum of the distribution
dP(v)dv
= 0
After some math
vp =2RTM
Most probable speed from Maxwell’s speed
distribution
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MOLECULES IN THE TAIL
v ≫ vrms
Particles in the tail of the distribution
Rain. High speed tail of the distribution molecules evaporate -> clouds
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INTERNAL ENERGY OF A GAS
Eint = NKavgMonatomic ideal gas N = nNARecall
Kavg =32
kTEint = nNAKavg Recall
Eint = nNA32
kT R = kNARecall
Eint =32
nRTAlso only dependent
on T
The assumption here is that E only depends on the translational KE and not also rotational (needed for molecules)
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ISOCHORIC EXPANSION dV = 0
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MOLAR SPECIFIC HEAT AT CONSTANT VOLUME
Q = nCvΔTHeat vs. T
Molar specific heat at a constant volumeTo relate to the internal energy, 1st law of thermodynamics
ΔEint = Q − W
Eint =32
nRT
ΔEint = nCvΔT − W
W = 0Isochoric expansion
ΔEint = nCvΔT
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CONTINUED
ΔEint =32
nRΔTΔEint = nCvΔT Recall
32
nRΔT = nCvΔT
Cv =32
R
Cv = 12.5J
mol ⋅ K
monatomic gas in a constant-volume process
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OTHER SPECIFIC HEATS
Eint = nCVT
A change in the internal energy Eint of a confined ideal gas depends only on the change in the temperature, not on
what type of process produces the change.
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CHANGE IN INTERNAL ENERGY
ΔEint = ΔnCVT
Regardless of the process, all have the same change in the internal energy
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ISOBARIC EXPANSION dp = 0
Q = nCpΔT
Molar specific heat at a constant pressure
ΔEint = Q − W
What is work for an isobaric expansion?
W = pΔVFrom last week
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MOLAR SPECIFIC HEAT AT CONSTANT PRESSURE
Q = nCpΔTΔEint = Q − W
W = pΔV = nRΔT
substituting in
ΔEint = nCpΔT − W ΔEint = nCvΔTrecall
nCvΔT = nCpΔT − WIdeal gas law
substituting in
nCvΔT = nCpΔT − nRΔT
Cv = Cp − R
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SPECIFIC HEAT
Cv = Cp − R
Specific heat at constant volume vs. constant pressure
Cp > CV
Q = nCΔTRecalling
For a constant volume it’s easier to change the temperature than for a constant pressure!
In other words for p=const, you lose energy to work. For V=const you don’t lose energy to work.
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TYPES OF MOTION OF MOLECULES
translational motions (moving left-right and up-down)rotational motions (spinning about an axis like a top)oscillatory motions (atoms oscillating towards/aways from one another
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DEGREES OF FREEDOMdegrees of freedom (f), independent ways in which molecules can store energy.
Each such degree of freedom has associated with it an average energy
Kmolecule =12
kT Kmol =12
RT
Avg. energy per molecule Avg. energy per mole
Ktot =12
nRT
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SPECIFIC HEAT & DEGREES OF FREEDOM
For 3 spacial dimensions (translational degrees of freedom)
Eint =32
nRT
Generic f
Then, the specific heat is Cv =32
R →f2
R
Eint =32
nRT →f2
nRT
Cp = CV + RSince
Cp =f2
R + R
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PHASES OF MATTER AND DEGREES OF FREEDOM
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DEGREES OF FREEDOM
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ADIABATIC (Q=0)
Example sound waves through air
(very quick or well-insulated)
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DURING AN ADIABATIC PROCESS
pVγ = const where γ =Cp
CV
Before and after an adiabatic process
piVγi = pfV
γf
Substituting in the ideal gas law p =nRT
V
nR⏟const
TVγ−1 = const TVγ−1 = const
TiVγ−1i = TfV
γ−1f
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Summary of processes
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FREE EXPANSION
Free expansion is a different type of adiabatic process (previous equations don’t apply)
Ti = Tf
Then for an ideal class if there is no change in the temperature
piVi = pfVf
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NEXT WEEK
Last regular class of the semester (review follows)
Second law of thermodynamics
Refrigerators
Carnot cycle
Microscopic view