atkins’ physical chemistry eighth edition chapter 21 – lecture 1 molecules in motion copyright...
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Atkins’ Physical ChemistryEighth Edition
Chapter 21 – Lecture 1
Molecules in Motion
Copyright © 2006 by Peter Atkins and Julio de Paula
Peter Atkins • Julio de Paula
Objectives:
• Describe the motion of all types of particles in all typesof fluids
• Concentrate of transportation properties:
• Diffusion ≡ migration of matter down a concentrationgradient
• Thermal conduction ≡ migration of energy down atemperature gradient
• Electrical conduction ≡ migration of charge along apotential gradient
• Viscosity ≡ migration of linear momentum down a velocitygradient
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∝
Effect of Temperature on Molecular SpeedsEffect of Temperature on Molecular Speeds
urms ≡root-mean-square
speed
Hot molecules are fast, cold molecules are slow
urms = 3RT (MM)
R = 8.314 J/(mol K)
The distribution of speedsof three different gases
at the same temperature
urms = 3RT (MM)
Fig 10.19 Effect of Molecular Mass on Molecular Speeds
Heavy molecules are slow, light molecules are fast
Fig 21.3 Distribution of speeds with temperature and molar mass
Maxwell distribution forfraction (f) of molecules withspeeds from v to v + dv
RT2/Mv2 223
evRT2
M4)v(f
π
π
Maxwell Distribution of Speeds
RT2/Mv2 223
evRT2
M4)v(f
π
π
• 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 Summary of conclusions for Maxwell distribution
21
M
RT8c
π
Most probable speed
Mean speed
Relative mean speed
21
M
RT2*c
21
kT8crel
πμ
Fig 21.8 Schematic of a velocity selector
• Fast rotation willselect fast molecules
• Slow rotation will select slow molecules
The collision frequency:
where σ = πd2 ≡ collision cross-sectionN = N/V ≡ number molecules / volume
Ν cσZ rel N
kTPcσ relz
In terms of pressure:
The mean free path:
P2
kT
σλ
kTPcσ relz Substituting in terms of pressure:
The mean free path:
z
cλ
• 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
The rate of effusion
21
)MRT2(
NPA Ao
π
Graham’s law of effusion ≡ rate of effusion is inverselyproportional to the square root of the molar mass
Rate of effusion =
Diffusion - the gradual mixing of molecules of one gas with molecules of another by virtue of their kinetic properties
NH3
17 g/molHCl36 g/mol
NH4Cl
Brownian motion
1
2
2
1
MMMM
rr