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Molecular Collisions: R.M.S. Speed Z Collision Frequency

This document discusses molecular collisions and the behavior of real gases. It introduces concepts like mean free path, collision cross-section, and the Lennard-Jones potential to model interactions. The critical point where gas and liquid densities become equal is described. Deviations from ideal gas behavior are quantified using compression factor and the van der Waals and virial equations of state. Real gases can liquefy through Joule-Thomson expansion due to attractive intermolecular forces.

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Maria Zvolinskaya
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84 views8 pages

Molecular Collisions: R.M.S. Speed Z Collision Frequency

This document discusses molecular collisions and the behavior of real gases. It introduces concepts like mean free path, collision cross-section, and the Lennard-Jones potential to model interactions. The critical point where gas and liquid densities become equal is described. Deviations from ideal gas behavior are quantified using compression factor and the van der Waals and virial equations of state. Real gases can liquefy through Joule-Thomson expansion due to attractive intermolecular forces.

Uploaded by

Maria Zvolinskaya
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Molecular collisions

Mean free path


λ between two collisions
r.m.s. speed c= = λz z = collision frequency
1
z Time of flight
between two collisions
σ = collision cross-
section (target area)
= πd2

RT 2 N Aσpc
Þ λ= ;z =
2 N Aσp RT Nils Walter: Chem 260
In reality: Gases have attractive and
repulsive forces
Lennard-Jones
6-12 potential At high T: perfect
gas isotherms

Þ
e.g.,
CO2 At low T:
liquefaction

Nils Walter: Chem 260


The critical point: Gas and liquid
density become equal
Heating a liquid in a container

At critical point
(for water 373oC
@ 218 atm!) the
boundary is lost

Þ Application:
Extraction of caffeine
from coffee with
supercritical CO2

Nils Walter: Chem 260


Describing the deviation from the perfect gas
Introducing the molar volume of real gas
compression factor Z: Z= molar volume of perfect gas

Vm Vm pVm
Z= = =
Vm
perfect
RT RT
p
Z = 1 Þ perfect gas
Z < 1 Þ molecules cluster, attractive
forces are dominant
Z > 1 Þ molecules repel each other,
repulsive forces are dominant

Nils Walter: Chem 260


The virial equation of state
virial coefficients
B C
Empirically: Z = 1 + + 2 + ...
Vm Vm
B > 0 Þ Z > 1, e.g., H2
B < 0 Þ Z < 1, e.g., CH4, NH3

C > 0 Þ Z > 1 at high pressure (Vm small)

pVm
and Z = very accurate
RT
nRT æ nB n 2C ö
Þ p= çç1 + + 2 + ...÷÷
V è V V ø
Nils Walter: Chem 260
Physically more palpable:
The van der Waals equation
[Johannes van der Waals 1873]
Lennard-Jones
molecules have a molecules have 6-12 potential
non-zero volume attractive forces

Þ reduction in exerted
pressure: a(n/V)2
[molecules strike less
frequently and with
reduced force]

æ an 2 ö
Þ additional volume Þ çç p + 2 ÷÷(V − nb ) = nRT
needed: nb è V ø
Nils Walter: Chem 260
Plotting the van der Waals equation:
In reasonable agreement with reality

only the van


der Waals
loops are
unrealistic

in 3D
p

T
Nils Walter: Chem 260
V
Liquefaction of real gases:
The Joule-Thomson effect
Linde refrigerator
Real gases have attractive
forces

Þ if they are allowed to


expand through a throttle
without outside heat
entering (“adiabatic”
process) they will use their
kinetic (heat) energy to
escape each other’s
attraction
Þ they will cool down
Nils Walter: Chem 260

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