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Chapter 13

CHEM 131B Chapter Notes - Chapter 13: Diatomic Molecule, Partial Derivative, Chemical Potential


Department
Chemistry
Course Code
CHEM 131B
Professor
Shaul Mukamel
Chapter
13

Page:
of 47
The$Distribution$of$Molecular$States:$
$
The$closest$we$can$come$to$a$description$of$the$distribution$$
of$energy$is$the$report$the$population$of$a$state,$what$is$a$population?$
$
The$avg.$#$of$molecules$that$occupy$it,$and$to$say$on$avg.$$
there$are$Ni$molecules$in$a$state$of$energy$εi.$
$
They$remain$almost$constant.$$
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The$Distribution$of$Molecular$States:$
$
What$is$the$only$restriction$when$addressing$the$calculation$$
of$the$populations$of$states$for$any$type$of$molecule$in$$
any$mode$of$motion$at$any$temperature?$
$
The$molecules$should$be$independent,$in$the$sense$that$$
the$total$energy$of$the$system$is$a$sum$of$their$individual$energies.$$
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The$Distribution$of$Molecular$States:$
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What$is$the$Principle$of$equal$a"priori$probabilities?$
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The$assumption$that$all$possibilities$for$the$$
distribution$of$energy$are$equally$probable.$$
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‘A$priori’$=$as$far$as$one$knows$
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The$Distribution$of$Molecular$States:$
13.1$Configurations$and$Weights$
(a)$Instantaneous$configurations:$
$
What$Symbol$represents$the$total$number$of$$
molecules$in$the$system?$
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N$
where$
N$=$N0$+N1+$N2+…+Nz$
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The$Distribution$of$Molecular$States:$
13.1$Configurations$and$Weights$
(a)$Instantaneous$configurations:$
$
What$does$it$mean$to$say$the$$
Instantaneous$configuration$of$the$system?$
$
The$specification$of$the$set$of$populations$$
N0,$N1,$…$in$the$form${N0,$N1,$…}$$
$
Fluctuates$with$time$b/c$the$population$changes$
$
EX:$$
{U2,2,0,0,…}$means$2$molecules$are$in$the$$
first$excited$state$! can$be$achieved$in$
$!
!
𝑁𝑁1$different$ways.$
$
The$Distribution$of$Molecular$States:$
13.1$Configurations$and$Weights$
(a)$Instantaneous$configurations:$
$
How$can$a$general$configuration$of${N0,$N1,$N2,…}$$
can$be$achieved$in$W$different$ways.$What$is$W?$
$
𝑊=!
𝑁!
𝑁!!𝑁
!!𝑁!!$
$
with$𝑥!=𝑥𝑥1.1!!𝑏𝑦!𝑑𝑒𝑓.0!=1$
$
The$Distribution$of$Molecular$States:$
13.1$Configurations$and$Weights$
(a)$Instantaneous$configurations:$
$
What$is$lnW?$$
Why$is$it$easier$to$use$the$lnW$rather$$
than$with$the$weight$itself?$
$
𝑙𝑛𝑊 =𝑙𝑛𝑁!!𝑙𝑛𝑁!!
!
$
$
It$is$easier$to$make$approximations.$$
In$particular$we$can$simplify$the$factorials$by$using$$
Stirling’s$approximation$in$the$form$𝑙𝑛𝑥!!𝑥𝑙𝑛𝑥 𝑥$
$
$
The$Distribution$of$Molecular$States:$
13.1$Configurations$and$Weights$
(a)$Instantaneous$configurations:$
$
What$is$the$approximate$expression$for$the$$
weight$using$Stirling’s$Approximation?$
$
𝑙𝑛𝑊 =𝑁𝑙𝑛𝑁 𝑁𝑁!𝑙𝑛𝑁!𝑁!
!
$
$
=𝑁𝑙𝑛𝑁 𝑁!𝑙𝑛𝑁!!!!!!!!!!!!!!!!
!
$
$
The$Distribution$of$Molecular$States:$
13.1$Configurations$and$Weights$
(b)$The$most$probable$distribution:$
$
How$can$the$dominating$configuration$be$found?$
$
By$looking$for$the$values$of$Ni$that$lead$to$a$max.$value$of$W.$$
Can$do$this$by$looking$for$the$values$that$correspond$to$dW$=$0.$$
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The$Distribution$of$Molecular$States:$
13.1$Configurations$and$Weights$
(b)$The$most$probable$distribution:$
$
When$looking$for$the$max.$value$of$W,$$
what$conditions$must$be$followed?$