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this document is my lecture notes on the kinetic theory of gases
Typology: Lecture notes
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๏ก Kinetic Molecular Model of Gases
o Assumption
Gas molecules in ceaseless random motion (no phase change).
Molecular size is negligible (no repulsion).
Molecules do not interact (no attraction).
All gases at same P and T have the sam properties.
o Definition of Pressure
โgas pressure is caused by collective collision of gas molecules with containerโs wallโ
Gas molecule has mass m and velocity v
๐น
1 โ ๐๐๐๐๐๐ข๐๐
= ๐น
ฬ
=
๐๐
ฬ
๐๐ก
=
๐
ฬ
(๐ก = 1 ) โ ๐
ฬ
(๐ก = 0 )
1 โ 0
= ๐๐ฃฬ
๐ฅ
โ
( โ๐๐ฃฬ
๐ฅ
)
= 2 ๐๐ฃฬ
๐ฅ
Middle of container
Let V = volume of container and N = number of molecules at equilibrium
๐ 2
โ travel left with +
| ๐ฃโ
| speed
๐ 2
โ travel right with โ
| ๐ฃโ
| speed
Number of molecules that hit wall per second = number of molecules at distance
| ๐ฃ
๐ฅ
โโโโโ
| for walls.
In volume V there are N molecules
In volume |๐ฃ
๐ฅ
โโโโโ |๐ด there are
๐|๐ฃ
๐ฅ
โโโโโ |๐ด
๐
molecules
๐ด โก surface area of a wall
Force exerted on right wall per second
๐น =
1
2
๐|๐ฃ
๐ฅ
โโโโโ |๐ด
๐
( 2 ๐
| ๐ฃ
๐ฅ
โโโโโ
|)
Only
1
2
๐|๐ฃ
๐ฅ
|
โโโโโโโโ
๐ด
๐
( 2 ๐|๐ฃ
๐ฅ
โโโโโ |) hit right wall ( N/2 out of N travel right)
๐น =
๐๐|๐ฃ
๐ฅ
โโโโโ |
2
๐ด
๐
๐ =
๐น
๐ด
=
๐๐|๐ฃ
๐ฅ
โโโโโ |
2
๐
Vector can be written as:
๐ฃโ = ๐ฃ
๐ฅ
๐ฬ + ๐ฃ
๐ฆ
๐ฬ + ๐ฃ
๐ง
๐
ฬ
๐
2
=
| ๐ฃโ
|
2
= ๐ฃ
๐ฅ
2
๐ฆ
2
๐ง
2
๐ = gas speed
Assume velocity in all 3 axes are equal
๐ฃ
๐ฅ
= ๐ฃ
๐ฆ
= ๐ฃ
๐ง
or ๐
2
= 3 ๐ฃ
๐ฅ
2
โด ๐ฃ
๐ฅ
2
=
1
3
๐ฃ
๐ฅ
2
๐ =
1
3
๐๐๐
2
๐
Or ๐๐ =
1
3
2
2
n = number of moles
M = molar mass
Every gas does not travel at the same speed
Maxwell distribution of gas velocity
Distribution of gas velocity can be presented by a mathematical function cal โMaxwell
Distributionโ.
3 / 2
2
โ๐๐
2
/ 2 ๐ ๐
โ
0
3 / 2
3
โ๐๐
2
/ 2 ๐ ๐
โ
0
3
โ๐๐ฅ
2
โ
0
2
2
from
2
๐๐ = ๐๐ ๐ ideal gas law
2
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๐๐๐