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Kinetic Theory of Gases
Chapter 33
Kinetic Theory of Gases
Kinetic theory of gases envisions gases as a collection of atoms or molecules in motion. Atoms or molecules are considered as particles.
This is based on the concept of the particulate nature of matter, regardless of the state of matter.
The Kinetic Theory relates the 'micro world' to the 'macro world’.
A particle of a gas could be an atom or a group of atoms (molecule). Observations Gas density is very low
Pressure is uniform in all directions IGL is independent of particle type
Dalton’s Law of Partial Pressures
KT Postulate Particles are far apart
Particle motion is random Gas particles do not interact
Gas particles do not interact
KT (IGL): Applicable when particle density is such that the inter-particle distance >> particle size (point masses). Low pressures and high temperatures e.g 1atm and room temp.
For gases following the relationship, PV = nRT (IGL);
V
P nRT
Postulates
Gas particles in constant random motion.
Pressure in a gas is due to particle collisions (elastic) with the walls of the container from translational motion
- the microscopic explanation of pressure.
Gas particles do not exert forces on each other due to their large intermolecular distances.
Gas particles are very far apart.
Collisions with the wall are elastic, therefore, translational energy of the particle is conserved with these collisions.
Each collision imparts a linear momentum to the wall, which results the gaseous pressure. In Newtonian mechanics force defined as the change of momentum, here, due to the collision; pressure is force per unit area.
In KT, the pressure arising from the collision of a single molecule at the wall is derived and then scaled up to the collection of molecules in the container, to obtain the ideal gas law (IGL);
PV = nRT
Gas kinetic theory derives the relationship between root-mean-squared speed and temperature.
The particle motions are random , therefore velocities along all directions are equivalent. Therefore the average velocity (vector) along any dimension/direction will be zero.
Now, the root-mean-squared velocity = root-mean-squared speed ; it is nonzero.
A distribution of translational energies; therefore, many velocities would exist for a collection of gaseous particles.
What is the distribution of the particle velocities?
Velocity is a vector quantity (v). Speed is a scalar ().
v 2 = v•v = v (^) x^2 + v (^) y^2 + v (^) z^2
v 2 = ^2 = x^2 + y^2 + z^2
<v 2 > = <^2 >
Most properties of gases depend on molecular speeds.
; symbol = average of x.
The translational movements of particles are amenable to treatment with classical Newtonian mechanics (Justification, later).
Root mean square velocity, translational energy:
m
Assume: (Chapter 31) v 2 x B
k T
m
2 2 2 2
2 2 2
2 2 2 2 2 2 2
2
2
2 2
2
2
2
2
v v v v
v v v
v = v + v + v = v + v + v
3
v v
v 3 v
v 3 v 3
v 3
3 v
x (^) B
x y z
x y z
x y z x x x
x
x B
x
y
B
z
m m k T
m
m (^) k T k T
k
m
remember
lastslide ︶
by definition
random motion Therefore,
Using
1 2 v 2 3 B^3 B
k T (^) kT T m m
^
/ ;
=
For a gas sample of n moles occupying a volume V (cube), with an area of each side A. Consider a single particle of mass m, velocity v.
Particle collides with the wall. (elastic collisions)
Change of momentum p =
v v 2 v
v v 2 v
x (^ x ) x
m m m
m m m
t
v x t
Number density of particles = N ^ nNA V
Half of the molecules moving on x axis with a (velocity component in the x direction) within the volume v (^) x t collides with one surface in the x direction.
N (^) coll = number of collisions on the wall of area A in time t.
where N (^) A = Avagadro Number
Similarly
In general
Because derivatives of three independent variables are equal, the derivatives must be constant, say = - ;(>0).
Upon rearrangement and integration,
Note the distribution (probability) function!
where A = integration constant
Evaluating A:
Math supplement
even function
Mean/average
(Assumption slide 9)
Distribution function
Distribution function: probability of a gas particle having a velocity within a given range, e.g. v (^) x and v (^) x +dvx.
use tables
averaging
Math supplement
Substituting for in f (v j);
kT
where m Now,
velocity distribution function F ( ) v
Deriving the distribution function for v
Changing the ‘volume’ element (in Cartesian ) to variable v, spherical coordinates.
2
2 2 2 2
replace by 4
by
x y z x y z
dv dv dv d
and v v v v
rise exponential decay
Notice the shape, blue. (^) ave’s of Xe, H 2 , He? Earth and Jupiter (300)
N (^) c = number of particles colliding.
Number of collisions per unit time on the wall of area A:
<v (^) x >
average x component of velocity
<v (^) x > =
A
molecules in ‘light blue volume’
of the cube colliding per unit time =
Rate of collisions on surface =
Collisional Flux Z (^) c :
Number of collisions on the wall per unit time per unit area.
Substituting in Z (^) c ;
Particle collision rates: (Hard sphere model)
Particles interact when spheres attempt to occupy the same region of the phase. (consider one moving particle – orange, label 1; all other particles stationary are red – label 2
V cyl vave dt
= collisional cross-section.
Because the collisional ‘partners are moving too’ in reality, an “effective speed”, < v 12 >, of orange particle will be considered in the model to emulate the collisions the orange particle encounters;
Because the collisional ‘partners are moving’, an effective speed < v 12 > used to model the system;
1 2 1 2
; reduced mass
m m
m m
V cyl vave dt
Collisional partner (red) density = N^2 V
Particle collisional frequency of it = z 12
For a sample of one type of gas ;
Volume covered by orange in dt = Vcyl
Collisions by it in time dt = V
N
Vcyl^2
Total collisional frequency, two types of gases Z 12 :
Total number of collisions in the gaseous sample.
For a sample of one type of gas we have;
Accounts for double counting
Mean Free Path:
Average distance a particle would travel between two successive collisions two types of molecules , say 1 and 2.
For one type of molecules,
Effusion:
Effusion is the process in which a gas escapes through a small aperture. This occurs if the diameter of the aperture is considerably smaller than the mean free path of the molecules (effusion rate = number of molecules that pass through the opening (aperture) per second). Once the particle passes through it generally wont come back because of the low partial pressure on the other side
Pressure of the gas and size of the aperture is such the molecules do not undergo collisions near or when passing through the opening.
-v (^) x0 v (^) x
0 0 0 0
xo xo x x x x x x x x
f v v v f v dv f v dv
2 2 2
mv x kT
(^0 ) 0 0 0
f ( vx vx v (^) x ) e d
2 (^2 ) (^0 )
mv x kT
2 0 (^0 )
mv x kT
(^0 ) 0 0 0
f ( vx vx vx ) e d
2
0
z erf z e xdx
^ f^ (^ ^ z^ ^ vx^ z )
erf ( ) z (^) erf ( ) z
0 2 2 0
x x x
v v kT m
mv kT
mvx kT
2 0
z erf z e xdx
(^12)
0
erf e xdx
covers vx 2 kT / m
probability vx 2 kT / m?
