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∝

Fig 21.1 The pressure of a gas arises from the impact ofits molecules on the wall

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 To obtain probability, integrate f(v) between v1 and v2

RT2/Mv2 223

evRT2

M4)v(f

π

π

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:

Fig 21.9 Volume swept by a moving molecule

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

Effusion - escape of gas molecules/atoms through a tiny hole

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

Fig 21.10 The flux of particles down a concentration gradient

Fick’s first law of diffusion:

If the concentration gradientvaries steeply with position,then diffusion will be fast