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Chemical Thermodynamics : Georg Duesberg
Chapter 2
Kinetic gas theory
http://www.tcd.ie/Chemistry/staff/people/duesberg/ASIN/2
0web/2027-10-09/teaching.html
Or also via my chemistry staff page - link to ASIN page
Chemical Thermodynamics : Georg Duesberg
Kinetic Molecular Theory of Gases
Maxwell
(1831-1879)
Boltzmann
(1844-1906)
macroscopic
(gas cylinder)
microscopic
(atoms/molecules)
Chemical Thermodynamics : Georg Duesberg
Physical properties of gases can be described by motion of
individual gas atoms/molecules
Assumptions:
1) each macroscopic and microscopic
particle in motion holds an kinetic
energy according to Newton’s law
2) They undergo elastic collisions
3) They are large in number and are
randomly distributed
4) They can be treated as points of
mass (diameter<< mean free path)
Kinetic Molecular Theory of Gases
-v v
Δt
2vm
Δtime
ΔvelocitymassForce
1) According to Newton's law of action–reaction, the force on
the wall is equal in magnitude to this value, but oppositely
directed.
2.) Elastic collision with wall: vafter = -vbefore
Kinetic Molecular Theory of Gases: Assumptions
3. Avogardo Number – Brownian motion
4. Gases are composed of atoms/molecules which are
separated from each other by a distance l much more than their
own diameter d
d = 10-10 m
L = 10-3 m….. few m
molecules are mass points with
negligible volume
Kinetic Molecular Theory of Gases: Assumptions
Chemical Thermodynamics : Georg Duesberg
L
reactionF
Collisions of the gas molecules with a wall
As a result of a collision with the wall
the momentum of a molecule changes by
Small volume, v=LA, adjacent to
wall where L is less than the mean
free path
Chemical Thermodynamics : Georg Duesberg
Pressure = Forcetotal/Area P=F/A
• Ftotal = F1 collision x number of collisions in a particular time interval
Assume that in a time Δt every molecule (atom) in the original
volume, v=LA, within the range of velocities
will collide with the wall.
Kinetic Molecular Theory of Gases
All molecules within a distance xt
with x 0 can reach the wall on the
right in an interval t.
tvL x
This means that Δt is given by:
Chemical Thermodynamics : Georg Duesberg
Collisions of the gas molecules with a wall
The “reaction force” of a molecule on the wall is the
negative of the average rate of change in the
momentum of gas molecules in the volume v that
collide with the wall in the time Δt.
The total force on the wall is the sum of the average rate of
momentum change for all molecules in the volume v=LA that
collide with the wall
Here we have divided by 2 since only ½ of the molecules in our
volume have a positive velocity toward the wall
dt
mvd
dt
dpF
)2(
with
L
vm
L
vmF
xx
x
22
2
2
Chemical Thermodynamics : Georg Duesberg
L
We do the sum by noting that the total number of
molecules in the volume V is (N/ V)xLA
Remembering Pascal’s law
dividing by A yields the pressure
everywhere.
V=LA
N/ V = density
AFP
Collisions of the gas molecules with a wall
Chemical Thermodynamics : Georg Duesberg
L
Kinetic theory: go from 1 to 3 dimensions
Velocity squared of a molecule: 2222
zyx vvvv
The average of a sum is equal to
the sum of averages…
All the directions of motion
(x, y, z) are equally
probable.
Remember homogeneous
and isotropic! Equipartition principle
Chemical Thermodynamics : Georg Duesberg
Kinetic theory
Combing these results yields
From the ideal gas law
And with c = <v>
m
kTcv
32
Relation between the absolute temperature and average kinetic
energy of a molecule.
m
kT
Nm
NkTv
332
22
3
1vm
V
Nvm
V
NP x
2
3
1vNmNkTPV
Chemical Thermodynamics : Georg Duesberg
vrms of a molecule is “thermal speed”:
The absolute temperature is a measure of the average kinetic
energy of a molecule.
Example:
What is the thermal speed of hydrogen molecules at 800K?
Kinetic theory
Chemical Thermodynamics : Georg Duesberg
Chapter 1 : Slide 13
This is roughly correlated to the speed of sound in those media –
as a results your voice has a higher timbre in He…
Gas Temperature (°C) Speed in m/s
Air 0 331.5
Air 20 344
Hydrogen 0 1270
Carbon dioxide 0 258
Helium 20 927
Water vapor 35 402
m
kTvsound
m
kTvrms
3
γ = the adiabatic constant,
characteristic of the specific
gas
Chemical Thermodynamics : Georg Duesberg
Kinetic theory
The average kinetic energy per molecule is
kTvm2
32
1 2
m
kTv
m
kTv
33 22
kTkE transkin 23
Chemical Thermodynamics : Georg Duesberg
Measurement of molecular speeds
# of molecules striking various locations along drum is
directly related to speed distribution inside gas
What is the distribution of molecular speeds?
Chemical Thermodynamics : Georg Duesberg
Histogram of measured speed distributions
N is total number of atoms (molecules)
∆N is number in a particular speed range v + dv
Chemical Thermodynamics : Georg Duesberg
Maxwell-Boltzmann Distribution
Number of molecules having
speeds in an interval of width
v around v. It is proportional
to v, the total number of
molecules, N, and to the height
of the distribution curve.
P(v)
P(v)
Chemical Thermodynamics : Georg Duesberg
IG-09
Distribution of velocities in a gas
, ,x y z x y z
transKkTP v v v dv dv dve
2
2 2 2 2
1
2trans
x y z
K mv
v v v v
2 2 2
2 2 2
1
2
1 1 1
2 2 2
x y z
x y z
m v v v
kTx y z
mv mv mv
kT kT kTx y z
P dv dv dv
dv dv dv
e
e e e
P is the probability, at temperature T , of finding a molecule with
velocity in the range (vx+dvx, vy+dvy, vz+dvz)
Chemical Thermodynamics : Georg Duesberg
IG-09 19
Velocity probability distribution in polar coordinates
2 2 2
2
1
2
1
22 4
x y zm v v v
kTx y z
mv
kT
P dv dv dv
v dv
e
e
Volume element (dvx)(dvy )(dvz) in rectangular coordinates is
(4v2 dv) in polar coordinates a spherical shell of radius v and
thickness dv
Because kinetic energy depends on v,
the volume of this velocity space is
proportional to the number of ways of
obtaining a particular kinetic energy
(within a small range), i.e., all points
in this thin spherical shell of fixed
thickness Δv correspond to the same
kinetic energy. The greater the radius
of the shell, the more points it
encloses.
Maxwell-Boltzmann Distribution
Velocity components of molecule (vx, vy, vz). N molecules
represented by N points in velocity space. Volume of space between
v and v + Δv is ~4v2 Δv.
However, all else is not equal. The
factor
accounts for the decreased likelihood
that a molecule will have a given
speed.
Molecular Speeds
kTmv 2exp 2
We have 4v2 in the distribution function. This says that, all else
being equal, we expect more molecules to have speeds with a range
between v and v + Δv, where Δv is fixed, the larger the value of v.
Chemical Thermodynamics : Georg Duesberg
1P v dv
12
2
m ma
kT kT
2
2
mkT
kT
m
2
13
1
2
( ) xmv
kTxP v const e
1
3
22
1x x
kTP v dv const
m
32
2
mconst
kT
Normisation factor for the 3D case
See also: Atkins (8th Ed.) Justification 21.2
Normalisation factor
and
aeax
2With Standard
Integral:
This integral is the fraction of molecules with speeds lying
between the limits v1 and v2
Maxwell-Boltzmann Distribution
3 2
2 24 exp 22
mP v v mv kT
kT
molecular mass (kg) Boltzmann’s constant
1.3810-23 J/K
temperature in K
(not ºC!!!)
Chemical Thermodynamics : Georg Duesberg
Distribution of speeds in helium gas
Fraction of helium atoms, at 293 K, with speeds between 500
m.s-1 and 600 m.s-1
3 2
2 24 exp 22
mP v v mv kT
kT
Chemical Thermodynamics : Georg Duesberg
Average, most probable and rms speed
•Root mean square speed: (rms) 2
rmsv v
2 21 1 3
2 2 2rmsmv mv kT
3rms
kTv
m
( )0mp
P vv
dv
0
8 8 3( ) 0.92
3rms
kT kTv P v dv v
m m
20.82mp rms
kTv v
m
•Average speed:
•Most probable speed: vmp The most probable speed, is that for which P(v)
has a maximum. By differentiating our distribution function and setting it = 0
Chemical Thermodynamics : Georg Duesberg
P(v) = 4v2 (M/2RT)3/2e-Ek/RT
Maxwell-Boltzmann Distribution
<v> = (8RT/πM)1/2
vrms = (3RT/M)½
vmp = (2RT/M)1/2 3 2
2 24 exp 22
mP v v mv kT
kT
Chemical Thermodynamics : Georg Duesberg
Maxwell-Boltzmann Speed Distribution vs Temperature for Helium
20.82mp rms
kTv v
m
Chemical Thermodynamics : Georg Duesberg
Maxwell-Boltzmann Speed Distribution vs mass
20.82mp rms
kTv v
m
Chemical Thermodynamics : Georg Duesberg
Chapter 1 : Slide 29
A container filled with N molecules of oxygen gas
is maintained at 300K . What fraction of the
molecules has speeds in the range 599-601 m/s?
The molar mass M of oxygen is 32g/mol.
The fraction in that interval is ,
where , .
From
smv /2
N
vvN
N
dvvNf
)()(0
601
599
smv /6000
3106.2 f
Solution:
Sample calculation
3 2
2 24 exp 22
mP v v mv kT
kT
Chemical Thermodynamics : Georg Duesberg
Summary of Kinetic Theory
Physical meaning of the absolute
temperature is a measure of the
average kinetic energy of a
molecule.
From this we can express
the pressure of an ideal
gas as
Chemical Thermodynamics : Georg Duesberg Chemical Thermodynamics : Georg Duesberg
Intermolecular Collisions in Hard-sphere Gases
Quantitative picture of the events that take place in a
collection of gaseous molecules.
Frequency of collisions?
Distance between successive collisions?
Rate of collisions per unit volume?
Definition:
A pair of molecules will collide whenever the centres of the two
molecules come within a distance d (the collision diameter)
of one another.
No collision. Collision occurs.
d
Chemical Thermodynamics : Georg Duesberg Chemical Thermodynamics : Georg Duesberg
The Collision Cylinder
Stationary particles inside the collision tube.
2d d
Imagine one particle flying through stationary (frozen) particles.
Within the area , the collision cross section, it will
have collisions. The volume of the cylinder is give by:
2d
2dtvtvLV
L
Chemical Thermodynamics : Georg Duesberg Chemical Thermodynamics : Georg Duesberg
Collision Frequency
To determine the collision frequency Z we have to consider the
relative speed of the colliding particles.
vvvvrel
22
12
2
2
1
2
1
12
8
RTv
rel
The reduced mass µ of two identical particles is m/2 and therefore
2112
11
1
MMwith
21
8RT
Mv
Chemical Thermodynamics : Georg Duesberg Chemical Thermodynamics : Georg Duesberg
V
N
V
NNd
1
Collision Frequency Z
V
Nd
mV
Nv
t
tNvz
drel 22
1
1
8kT22
21
8kT2 2
mvv
rel
To determine the collision frequency Z we determine the total number
of molecules that have a collision in the time interval ∆t.
The number of centres Nd are the volume of the collision tube (with
their relative velocities) multiplied with their density minus 1.
2dtvtvLV relrel
Chemical Thermodynamics : Georg Duesberg Chemical Thermodynamics : Georg Duesberg
Collision Frequency
kT
pvz
V
Nvz
2
2
1
1
kT
p
V
Nwith
The Collision Density
22
1
111
4
21
V
N
m
kT
V
NzZ
We define the collision density as the total rate of collisions per unit volume,
and therefore multiply with the density Nd .The factor ½ stems from the fact
that only AB and not BA collisions or counted
Typical Numbers: Z=5x1034 s-1m-3 for N2 at RT, 1Bar
Chemical Thermodynamics : Georg Duesberg Chemical Thermodynamics : Georg Duesberg
The Mean Free Path
p
kT
p
kT
v
v
vzwithz
v
kT
p
kT
p
707.02
2
21
1
1
The Mean Free Path is inverse proportional with pressure.
The mean free path - the average distance traveled between successive collisions:
velocity divided by collision frequency.
Chemical Thermodynamics : Georg Duesberg
Diameter of Molecules, D 2 Å = 2 x10-10 m
Collision Cross-section: 1 x 10-19 m
Mean Free Path at Atmospheric Pressure:
m0.3or m1031041.110
3001038.1
21 7
195
23
p
kT
At 1 Torr, 200 mm;
at 1 mTorr, 20 cm
The Mean Free Path : Examples
2d
Chemical Thermodynamics : Georg Duesberg
Collisions flux with Walls and Surfaces Zw
• Rate at which molecules collide with a wall
of area A
dxvfvtAV
NZ xw )(
0
0
2
12
axewith ax
A = area
t = interwall
vm
kT
m
kT
kT
m
dvevkT
mdxvfv x
kT
mv
xx
x
4
1
2
2
2
2)( 2
00
2
“Bolzmann for 1D”
Chemical Thermodynamics : Georg Duesberg
Collisions Flux with Walls and Surfaces
• Rate at which molecules collide with a wall
of area A 2
1
24
1
kTm
p
V
NvZw
It is also called the impingement rate (molecules cm-2 sec-1)
shows up in a large number of calculations and is
important in MBE, ALD, CVD…
kT
p
V
N
vm
kT
4
1
2
Chemical Thermodynamics : Georg Duesberg
Chemical Thermodynamics : Georg Duesberg Chemical Thermodynamics : Georg Duesberg
Effusion
2
1
000
24
1
kTm
pA
V
NvAAZw
• Rate at which molecules
pass through a small hole
of area Ao, r
A gas under pressure goes (escapes) from one compartment of a
container to another by passing through a small opening.
Chemical Thermodynamics : Georg Duesberg Chemical Thermodynamics : Georg Duesberg
The Effusion Equation
Graham’s Law - estimate the ratio of the effusion rates for two different gases.
Effusion rate of gas 1 r1.
2
1
1
01,1
2
r
kTm
pAAZ ow
Effusion rate of gas 2 r2.
2
1
2
02,2
2
r
kTm
pAAZ ow
2
1
2
1
2
1
2
0
2
1
1
0
1
2
22
r
m
m
kTm
pA
kTm
pA
r
Effusion Ratio
Chemical Thermodynamics : Georg Duesberg
General Chemistry: Chapter 6
Graham’s Law
• Only for gases at low pressure (natural escape, not a jet).
• Tiny orifice (no collisions)
• Does not apply to diffusion.
A
B
B
A
Brms
Arms
M
M
3RT/M
3RT/M
)(v
)(v
Bofeffusionofrate
Aofeffusionofrate
• Ratio used can be:
– Rate of effusion (as above)
– Molecular speeds
– Effusion times
– Distances traveled by molecules
– Amounts of gas effused.
Chemical Thermodynamics : Georg Duesberg
Kinetic Theory of Energy Transport
Specific heat Velocity Mean free path
vCk3
1Thermal conductivity: