THERMAL CONDUCTIVITY OF GASES, MOLECULAR COLLISIONS AND MEAN FREE PATH FROM TRANSPORT PROPERTIES.

1. Physical Chemistry
1. Thermal Conductivity in gases
2. Molecular collisions and Mean free path.
23PGC01

2. Transport processes (flow) in which some physical quantity
such as mass or energy or momentum or electrical charge is
transported from one region of a system to another
In all cases the flow(the amount of physical quantity
transported in unit time through a unit area perpendicular to
the direction of flow is proportional to the negative gradient of
some other physical property such as T/P/I/V
Jz = L( -∂Y/∂z)
Jz -- flow, the amount of the quantity
transported/m2/s,
L -- proportionality constant,
(- ∂Y/ ∂ z) -- negative gradient of Y in the direction of flow.
Y can be T/P/I/V
Heat flow  Jz = - KT ∂T/∂z (Fourier`s law)…………1
KT - Coefficient of thermal conductivity.

3. General equation for flow:
j = n.<c>.dt.q
j  flow
n  No. of molecules/m3
q  Amount of physical quantity carried.
<c>  Average velocity
If only a fraction of molecules move in the direction then, the
fraction() should be multiplied along
j = α.N.<c>dt.q………2

4. Derivation of the coefficient of thermal
conductivity
Consider three large planes parallel
to the xy-plane at a distance z apart from each other. The
metal planes A and C are at temperature T1 and T2. C is
placed above A. B is placed in between the other two
plates. A steady state is applied and a downward low of
heat at constant rate is observed. The molecules moving
downwards carry more energy than those moving upward.

5. We imagine a large number of
horizontal layers in the gas,
each successive layer being at
slightly higher temperature
than the one below it. ∂T/ ∂z is
the temperature gradient.
∂T = dT = T2 – T1
∂z dZ z - 0
if the lower plate lies at the position z = 0, the upper one
at z = Z. The gradient, ∂T/ ∂z , is
T = T1 + (∂T/ ∂z)z …….3
C
B
A

6. If the gas is monatomic with an average thermal energy <ε> =
3/2kT, then the average energy of the molecules at the height z is
<ε> = 3kT  3k [T1 + (∂T/ ∂z) z] (using 3)
2 2
on the average, the molecules have traveled a distance λ since
their last collision. If the surface of interest lies at a height z, the
molecules going down made their last collision at a height z+λ,
while those going up made their last collision at a height z–λ.
ε↓ = 1/6(n<c>)z + λ. 3k [T1 + (∂T/ ∂z).(z+λ)]
2
ε↑ = 1/6(n<c>)z - λ. 3k [T1 + (∂T/ ∂z).(z–λ)]
2
The net flow = ε↓ -- ε↑
if the gas is not to have net motion through the surface we require
that the number of molecules going up in unit time must equal the
number going down, so that

7. 1/6.(n.<c>)z + λ= 1/6.{n.<c>} z - λ
Jε = 1/6.(n<c>).3k.(∂T/ ∂z).(z–λ– z-λ)]  -1.n.<c>.k.λ.(∂T/ ∂z)
2 2
As the molecules are moving in all possible
directions due to thermal agitation, it may be supposed that 1/3 of
the molecules are moving in each of the three directions parallel to
the axes, so that on average 1/6 of the molecule move parallel to
any one axis in one particular direction.
Comparing with Fourier law and using Cv=3/2.k.NA
KT = 1 . n .Cv.<c>.λ
…………4
3 NA
we get the <c> from Maxwell distribution of velocity,
<c> = √8kT
πm

8. The mean free path of the molecule
by definition is the average distance traveled
between collisions. In one second, a molecule
travels ( c) x 1s meters and makes Z collisions.
2. Molecular collisions and mean
free path.
λ=<c>…………………5
Z1
To calculate the Z1(no. of collisions), consider a cylinder.

9. Consider a cylinder of radius (σ and
height <c>. The number of molecules
in the cylinder is πσ 2<c>n; this is the
number of collisions made by one
molecule in one second The formula
Zn = πσ 2<c>n must be multiplied by
the factor √2 to account for the fact
that it is the average
velocity along the line of
centers of two molecules
that matters and not the
average velocity of a
molecule.

10. Consider two molecules that have their velocity vectors
in the orientations. For molecules moving in the same
direction with the same velocity, the relative velocity of
approach is zero. In the second case, where they
approach head-on, the relative velocity of approach is
2(c). If they approach at 90°, the relative velocity of
approach is the sum of the velocity components along
the line joining the centers; this is ½.√2<c>+½.√2<c> =
√2<c>
Zn = √2.σ2.π<c>.n……………6

11. From 5 and 6,
λ = 1 ................................7
√2.σ2.π.n
The mean free path depends on 1/n and is proportional to
l/p by the gas law 1/n = RT/Na.P. The lower the pressure,
the fewer collisions in unit time and the longer will be the
mean free path. Since there are n molecules/m3 and each
makes Z1 collisions per second, the total number of
collisions per cubic meter in one second is
Z11 = ½.Z1.n = ½.√2.σ2.π<c>.n2
The factor ½ is introduced because a simple multiplication
of Z1 by n would count every collision twice.

12. the number of collisions in one cubic meter per second
between unlike molecules in a mixture is
Z12 = πσ2
12.√8 KT T.n1.n2
√πμ
where n1 and n2 are the numbers of molecules per cubic
meter of kind 1 and kind 2, σ12is the average of the
diameters of the two kinds of molecules, and μ
is the reduced mass, 1/μ = 1/m1 + 1/m2 . These values
for collision numbers will be useful later in the
calculation of the rates of chemical reactions. A chemical
reaction between two molecules can occur only when
the molecules collide.

13. Final expression for thermal
conductivity
Using 7 in 4,
KT = <c>.Cv
3√2NAπσ2
1. The thermal conductivity is independent of the
pressure.
i. because, KT α n; KT α λ
ii. however, λ α 1/n
so that the product n.λ is a constant and
independent of pressure

14. If CV is independent of temperature, then everything
on the RHS is constant except <C>, since <c> α T½
Thus, KT α T½
 We have assumed that the pressure is
high so that A is much smaller than the distance
separating the two plates.
 At very low pressures where λ is much
larger than the distance between the plates, the
molecule bounces back and forth between the plates
and only barely collides with another gas molecule.
In this case the mean free path does not enter the
calculation, and the value of KT depends on the

15. the separation of the plates. At these low pressures the
thermal conductivity is proportional to the pressure, since it
must be proportional to n and λ does not appear in the
formula to compensate for the pressure dependence of n.

16. Applications:
Insulation materials:
The thermal conductivity of gases is an
important factor in the design of insulation materials. Gases with
low thermal conductivity, such as argon or krypton, are often
used to fill the gaps between panes in double or triple-glazed
windows.
Nuclear Reactors: In nuclear power plants, thermal conductivity
plays a role in reactor coolant systems and in the design of fuel
rods. It helps manage heat transfer and ensure reactor safety.
Heat Transfer in HVAC Systems: The thermal conductivity of
gases is critical in designing heating, ventilation, and air
conditioning (HVAC) systems. It helps engineers determine the
appropriate materials and insulation needed to efficiently transfer
or contain heat in these systems.

17. Thermal modeling and simulation:
In various engineering fields, thermal
conductivity data for gases is used in computer simulation
programs to model and predict heat transfer phenomena.
These simulations aid in the design of systems that require
accurate temperature control, such as electronics cooling,
aerospace applications, and energy systems.
Metrology and standards:
It is essential for establishing
metrological standards, ensuring the accuracy and traceability
of testing and measurement equipment. These standards are
crucial for industries that rely on precise thermal
measurements, such as scientific research, quality control,
and process optimization.

18. Reference
• GILBERT W. CASTELLAN 1983, Physical Chemistry 3rd edition.
• PETER ATKINS . JULIO DE PAULA, 2006, Atkins`s Physical
Chemistry 8th edition.
• PURI . SHARMA . PATHANIA, 2019, Principles of Physical
Chemistry 48th edition.
• ‘High Temperature Thermal Conductivity of Gases’ – American
Chemical Society.
ALBERT J. ROTHMAN1 AND LEROY A. BROMLEY Radiation
Laboratory and Division of Chemical Engineering, University of
California, Berkeley, Calif.
• ‘Analysis of gas thermal conductivity at low pressures using a
mathematical-physical model’ - Journal of Physics:
Conference Series
P ŠABAKA 11, Department of Electrical and Electronic
Technology, Brno University of Technology, Technická 10, 616 00
Brno, Czech Republic