Download Thermodynamics: Lecture 8, Kinetic Theory

Thermodynamics: Lecture 8, Kinetic Theory
Chris Glosser
April 15, 2001
1
OUTLINE
I. Assumptions of Kinetic Theory
(A) Molecular Flux
(B) Pressure and the Ideal Gas Law
II. The Maxwell-Boltzmann Distributuion
(A) Equipartion of Energy
(B) Specific Heat Capacity
(C) Speed Distribution
III. Mean Free Path and Effusion
2
Assumptions of Kinetic Theory
The fundamental assumptions of Kinetic Theory are as follows;
1. A reasonably sized system contains a vary large number of molecules;
2. The seperation of the molecules is large in comparison with the characteristic distance of the intermplecular forces;
3. The intermolecular forces do not significantly determine the dynamics
of the molecules;
4. No energy is lost during collisions; That is the collision s are elastic;
5. The molecules are uniformly distributed throughout the volume;
1
6. The directions of motion of the molecules are evenly distributed: That
is, the center of mass of the gas remains motionless.
To substantiate item 3, bit, we consider the Lennard-Jones Potential,
" 12
d
r
V (r) = 4 (∆E)
−
6 #
d
r
.
(1)
One may derive a potential of this form in quantum mechanics. Typically,
the parameter ∆E is of the order of 10− 2 electron Volts, and the parameter
d is of the order of angstroms. Therefore, if the average kinetic energy of the
molecules is much larger than the characteristic energy, and the avergage
spacing between molecules (that is the cube root of V /n) is much larger
than the characteristic distaance, d, then we may apply kinetic theory.
We assume the existance of a distribution of speeds, f (v), which tells
us the fraction of molecules with speeds between v and v + dv. The fraction of molecules over a large ranges of speeds is the integral of this speed
distribution,
Z
v2
dv f (v).
(2)
v1
f (v) therefore has dimensions of inverse velocity, and satisfies the following
normalization condition;
Z
∞
dv f (v) = 1.
(3)
0
The mean speed v and the mean square speed v 2 are thus given as;
v =
Z
∞
Z
∞
0
v2 =
dv v f (v),
(4)
dv v 2 f (v),
(5)
0
and, the RMS speed is defined through the usual relation, vrms =
2.1
p
v2.
Molecular Flux
Let’s define the molecular flux as the number of molecules striking a surfance
per unit time. Consider an infintessimal area dA. The number of molecules
with a speed between v and v + dv that strike the infinessimal area is is just
the number of molecules in a cylender of length v dt. That is;
dN =
ρ
f (v)dv × (cos(θ)dAvdt) ×
m
2
sin(θ) dθ dφ
4π
(6)
Therefore, the differential flux is;
dΦ =
dN
ρ
cos(θ) sin(θ) dθ dφ
= f (v)vdv
.
dA dt
m
4π
(7)
In the preceding equations, m is the molecular mass, and ρ is the density of
the gas. We can integrate this to obtain the average flux (remember, you
are only integrating over half the unit sphere);
1ρ
v
4m
Φ=
2.2
(8)
Pressure and the Ideal Gas Law
We may extend the molecular flux idea to derive an equation for the differential momentum;
dp =
Z
1
2mv cos(θ) dN = ρv 2 dA dt,
3
(9)
and therefore, the pressure is;
P =
dp
1
= ρv 2 .
dA dt
3
(10)
Since ρ is related to the total number of particles and the volume, we may
rewrite this expression as;
P =
2
3V
1
N mv 2 .
2
(11)
The term in parenthesis is just the total energy, U; therefore,
2
P V = U.
3
(12)
This also serves as a definition of temperatre:
3
1
kT = mv 2 .
2
2
(13)
At room temperature, you can easily check that the average kinetic energies
of the molecules are about an order of magnitude larger than the average
potential energies.
3
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The Maxwell-Boltzmann Distributuion
Up until now, we have avoided any specifying any details about f (v). As
you can imagine, it might be advantageous to have an explicit expression
for the distribution. To do so, we will make a number of assumptions, which
will be supported later when we get into statistical mechanics.
3.1
Equipartion of Energy
The first thing that we will assume is that the energy is evenly distributed
over each of teh degrees of freedom. This turns out to be a decent assumption
for rotational and translational degrees of freedom, but not vibrational ones.
Hence, we have:
v 2 = vx2 + vy2 + vz2 ,
(14)
and therefore;
v 2 = 3vx2 .
(15)
We can therefore assign a set amount of energy per degree of freedom;
1
= kT.
2
(16)
Again this works reasonably well for rotational and translational degrees of
freedom, but not vibrational ones. We therefore take the following expression as our model of an ideal gas with f translational and rotational degrees
of freedom.
f
U = N kT
(17)
2
The specific heat capacity at constant volume is easily derived;
cv =
∂u
∂T
=
v
f
R
2
(18)
By Mayer’s equation, we have cp ;
cp =
f
f +2
R+R=
R
2
2
(19)
f +2
.
f
(20)
The ratio γ = cp /cv is;
γ=
4
3.2
Speed Distribution
Now, let’s try to derive an expression for the speed distribution f (v). We
will assume that three particle collisions are extremely improbable. Two
particle collisions, of course satisfy momentum conservation;
v1 + v2 = v3 + v4
(21)
Now the rate at which collisions occur is directly proportional to the number
of particles in a given volume. Thus,
F (v1 )F (v2 ) = F (v3 )F (v4 ).
(22)
By inspection, the following solves both of these conditions;
F (v) = A exp(−αv · v).
(23)
This gives the velocity distribution. To get the speed distribution, integrate
thsi over the solid angle, as we did for the flux;
f (v)dv = (4πv 2 ) F (v 2 ) dv = (4πv 2 ) A exp(−αv 2 ) dv.
(24)
The constants A, α can now be determined through the normalization condition and the the defintion of temperature to be;
α = m/(2kT ),
A = N
m
2πkT
(25)
3
2
.
(26)
We thus have the Maxwell-Boltzmann speed distribution,
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f (x)dx = √ N x2 exp −x2 dx
π
(27)
√
where x = αv.
We may now write down explicit expressions for the mean velocity (and
the remaining modes) in terms of Gamma functions and the parameter α:
vn = √
2
3+n
Γ
.
π αn
2
5
(28)
4
Mean Free Path and Effusion
Now, Let’s discuss some applications of Kinetic Theory. First, we write
down an order of magnitude expression for the mean free path, that is , the
average distance that a molecule travels before it collides with another. If
we assume that the molecules can be modeled with a characteristic crosssectional area of exclusion σ, then the molecule excudes a cylinder of volume
σvt. If a molecule has a center in this volume, then it collides whith moving
one. Therefore, the mean free path is just
l=
mvt
m
=
.
ρvtσ
ρσ
(29)
The collision frequency is the mean velocity divided by this expression;
f =v
ρσ
.
m
(30)
Effusion is the process by which gas under pressure escapes from a small
hole in a container. Let’s try and calculate the average velocity of such
molecules. Since we are interested in the average velocity of molecules incident on an area, we use the expression for the differential flux to calculate
the average velocity of the escaping gas:
dΦ =
1
xf (x)dx
4α
Therefore, the average velocity of the gas incident on the hole is;
R
√
1 0∞ xdΦ
3 π
ve =
=
.
α
4α
Φ
Note that this is slightly larger than the average velocity of the gas.
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(31)
(32)