Peter Atkins • Julio de Paula
Atkins’ Physical Chemistry
Eighth Edition
Chapter 21 – Lecture 1
Molecules in Motion
Copyright © 2006 by Peter Atkins and Julio de Paula
Objectives:
• Describe the motion of all types of particles in all types
of fluids
• Concentrate of transportation properties:
• Diffusion ≡ migration of matter down a concentration
gradient
• Thermal conduction ≡ migration of energy down a
temperature gradient
• Electrical conduction ≡ migration of charge along a
potential gradient
• Viscosity ≡ migration of linear momentum down a velocity
gradient
Kinetic Molecular Theory of Gases
1. A gas is composed of widely-separated molecules. The
molecules can be considered to be points; that is, they
possess mass but have negligible volume.
2. Gas molecules are in constant random motion.
3. Collisions among molecules are perfectly elastic.
4. The average kinetic energy of the molecules is proportional
to the temperature of the gas in kelvins.
KE ∝ T
Fig 21.1 The pressure of a gas arises from the impact of
its molecules on the wall
Effect of Temperature on Molecular Speeds
urms ≡
root-mean-square
speed
urms =

3RT
(MM)
R = 8.314 J/(mol K)
Hot molecules are fast, cold molecules are slow
Fig 10.19 Effect of Molecular Mass on Molecular Speeds
The distribution of speeds
of three different gases
at the same temperature
urms =

3RT
(MM)
Heavy molecules are slow, light molecules are fast
Fig 21.3 Distribution of speeds with temperature and molar mass
Maxwell distribution for
fraction (f) of molecules with
speeds from v to v + dv
3
 M  2 2 Mv 2 / 2RT
f (v)  4 π
 v e
 2πRT 
Maxwell Distribution of Speeds
3
 M  2 2 Mv 2 / 2RT
f (v)  4 π
 v e
 2πRT 
• Decaying exponential – very few high speed molecules
• M/2RT forces exp to zero for high molar mass molecules
• M/2RT keeps exp high for high temperatures
• v2 exp goes to zero as v goes to zero: few slow molecules
• Remaining factors ensure that all speeds are normalized
Fig 21.4 To obtain probability, integrate f(v) between v1 and v2
3
 M  2 2 Mv 2 / 2RT
f (v)  4 π
 v e
 2πRT 
Fig 21.4 Summary of conclusions for Maxwell distribution
Most probable speed
 2RT 
c*  

 M 
1
2
Mean speed
 8RT 
c

 πM 
1
2
Relative mean speed
crel
 8kT 

 
 πμ 
1
2
Fig 21.8 Schematic of a velocity selector
• Fast rotation will
select fast molecules
• Slow rotation will
select slow molecules
The collision frequency:
where
Z  σc rel Ν
N
σ = πd2 ≡ collision cross-section
N = N/V ≡ number molecules / volume
In terms of pressure:
z
σcrel P
kT
Fig 21.9 Volume swept by a moving molecule
The mean free path:
λ
c
z
Substituting in terms of pressure:
The mean free path: λ 
z
σcrel P
kT
kT
2σP
• e.g., doubling the pressure decreases mean free path by half
• Typically λ ≈ 70 nm for nitrogen at 1 atm
• c ≈ 500 m s-1 at 298 K
Effusion - escape of gas molecules/atoms through a tiny hole
The rate of effusion
Graham’s law of effusion ≡ rate of effusion is inversely
proportional to the square root of the molar mass
Rate of effusion =
PA oN A
(2 πMRT )
1
2
Diffusion - the gradual mixing of molecules of one gas with
molecules of another by virtue of their kinetic properties
Brownian motion
NH4Cl
r1

r2
MM2
MM1
NH3
17 g/mol
HCl
36 g/mol
Fig 21.10 The flux of particles down a concentration gradient
Fick’s first law of diffusion:
If the concentration gradient
varies steeply with position,
then diffusion will be fast