The Molecular Model
For simple kinetic theory, we ignore the internal structure of
molecules and treat them as elastic hard spheres. Thus, there are
no intermolecular forces except the repulsive force of a collision.
Later, we will include some of the molecular structure. The
structure will affect quantities such as internal energy and specific
heat. For now we introduce the simple kinetic theory with the
hard-sphere “billiard-ball” molecule.
In order for the hard-sphere model to be appropriate, we assume a
simple dilute gas.
- All molecules have identical structure
-Most of the time they are in free flight
-The random motion is characterized by an average speed:
2


c
∑
i
1/2
=
 root mean square
i
=
c2
 N  molecular speed




c= c=
molecular velocity, N = number of molecules
i
( )
(
)
-Macroscopic motion can be subtracted to examine the random
(thermal) motion of the molecules
2
Intermolecular Force Diagram
∞
F
F = intermolecular force
actual
attractive
repulsive
d = diameter
hard elastic sphere
(billiard ball)
r = separation distance
r
d
Sutherland model
(purely attractive)
Thermodynamics
A = cross sectional area
gas
F
V = volume of gas
N = number of molecules
m = molecular mass
A = cross-sectional area
F = force
p = pressure
Mechanical pressure: p = F / A
Thermodynamics allows us to relate this mechanical definition to
the condition or state of the gas
cx = average speed in x-direction
4
A = cross sectional area
gas
F
− cx
cy
c
m
y
c
m
 momentum=
 mc − (− mc =
2mcx
 transfer 
x
x)


x
cy
cx
cx∆t
Molecules within this distance impact wall
in ∆t without intermolecular collisions
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 momentum=
 mc − (− mc =
2mcx
 transfer 
x
x)


collisions  N
 1 N
= (cx ∆t ) A = cx A
time
V
 ∆t V
 momentum   collisions 
N

×
=
F= 
(2
mc
)
c
A
x 
x 
 

 collision   time 
V

F
mN 2
⇒ p=
= 2
cx
A
V
mN
Now define mass density ρ =
V
Since cx can be positive or negative, p = ρ cx2 , and with random motion
2
2
2
c=
c
=
c
x
y
z
c = c + c + c = c + c + c ⇒ c = 3c
2
2
x
2
y
2
z
2
x
2
y
2
z
2
2
x
1 2 1 mN 2
⇒ p = ρc =
c
3
3 V6
We can relate this kinetic result to the ideal gas result from
classical thermodynamics, p = ρ RT .
^ indicates per mole quantity
 kg 
ˆ
M = molecular weight 
 kmol 
molec
Avogadro's number = 6.02 ×10
mol
J
Rˆ = universal gas constant = 8134.3
kmol ⋅ K
m=
molecular mass = 1.660 ×10−24 Mˆ g =
1.660 ×10−27 Mˆ kg
 J 
Rˆ
R=
= specific gas constant 

ˆ
kg
K
⋅
M


Nˆ
23
7
T = temperature [K]
 kg 
ρ = density  3 
m 
N
=
p pressure
 m 2  [Pa]
1 atm = 1.01325 ×105 N/m 2
Mˆ = mNˆ
R
8.3143 J
8314.3 m 2
=
2
Mˆ mol ⋅ K
Mˆ s ⋅ J
8
Now using the classical equation of state and the other definitions
just introduced, we can develop an expression relating the
translational temperature to the average kinetic energy of an ideal gas


 N = number of molecules 


mN Rˆ
kNT
,
p =
T
k Boltzmann constant 
=

V mNˆ
V


ˆ
R
J 
 =
=
1.380 ×10−23

N
K 

1 mN 2 kNT
p =
c
=
V
3 V
⇒
1 2 3
mc = kT ⇒ average kinetic energy  T
2
2
9
This explicitly shows that the average kinetic energy of the simple,
dilute gas is proportional to temperature, so for a gas of fixed mass,
the “hotter” it is, the greater the average molecular (random)
kinetic energy
Using the nomenclature of Vincenti and Kruger,
(
) indicates a per molecule quantity
etr = average kinetic energy per molecule
Nˆ
3 Nˆ Rˆ
=
etr =
etr
T
2 Mˆ Nˆ
Mˆ
= translational specific internal energy
10
The specific heats are a measure of “thermal capacity,”
change of internal energy
cv =
unit change of abs. temperature const. vol. process
 ∂e   ∂e   J 
=
cv =
 
 

∂
∂
T
T

v 
 ρ  kg ⋅ K 
detr 3
= = =
R specific heat at
constant volume
dT 2
(
)
c p cv + R 5
=
=
γ =
3
cv
cv
11
change of internal energy
cp =
unit change of abs. temperature const. press. process
5
c p − cv = R → c p = R
2
c p cv + R 5
=
=
γ =
cv
cv
3
{
ideal, no internal
structure
12
Mean Free Path
stationary molecules
σT = π d 2
sphere of influence
z
d
V π d 2 c ∆t
=
d
c ∆t
A collision occurs when the center of a field molecule lies inside
the cylinder. For this simple model we assume all the molecules
have the same average speed and we can freeze the motion and
focus on the motion of molecule z.
13
cz= c= mean molecular speed
ν = collision frequency = coll/time
number of molecules inside V
=
time
π d 2 c ∆t N / V
=
= π d 2 cn
∆t
The average distance between collisions is the mean free path
λ=
c
ν
A more accurate result that accounts for the relative motion is
=
λ
1
=
2π d 2 n
m
2π d 2 ρ
14
Transport Phenomena
x2
Kinetic theory provides a
mechanism for continuum
transport relations
region 1
region 2
x1
dna
du1
dT
, q =−κ
, Γ A =− DAB
τ =µ
dx2
dx2
dx2
15
τ = shear stress from velocity gradient du1 / dx2
µ = coefficient of shear viscosity
q = heat flux in x2 direction from temperature gradient dT / dx2
K = coefficient of thermal conductivity
Γ A =Flux of A-molecules in x 2 -direction from
concentration gradient dnA / dx2
DAB = Diffusion coefficient of A-B mixture
x2
x20
u1 = u ( x2 )
Example: Shear Stress
δ x2 ≅ λ / 2
x1
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Assume the gradient is resolved such that δ x2 = O (λ ) , which will
vary depending on the approximation.
Average number of molecules crossing the x20-plane per unit time
is nc , evaluated at x20.
Average momentum in the x1-direction per molecule is mu(x2).
Shear stress τ on the x20-plane is proportional to the rate of
change in momentum across the plane,
τ  nc [ mu1 ( x20 + λ / 2) − mu1 ( x20 − λ / 2) ]
Using the Taylor series: u1 ( x20 ± λ / 2) = u1 ( x20 ) ±
⇒ τ  nmc λ
λ du1
2 dx2
+
du1
dx2
du1
m
or τ µ
where µ ∼ ρ c λ ∼ 2 c .
=
dx2
d
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