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Background:

Ideal Gas Law:

Pressure x Volume = Moles x Ideal Gas Constant x Temperature

Substituting in variables, te !ormula is:

PV=n"T

Explanation and Discussion: 

Te Ideal Gas Law ma# be te largest and most $omplex o! te gas laws% Tis is in part be$auseo! te number o! variables in te e&uation, and in part to te abstra$tion o! an 'ideal' gas tat telaw is built on% Te Ideal Gas Law is also designed as a sort o! umbrella !or (o#le)s, Carles),and *vogadro)s laws%

+irst, we)ll go over te parts o! te e&uation, PV=n"T% P is pressure% Pressure $an be in eiteratmosperes atm- or .ilopas$als .Pa-% V is volume in liters L-% n is te number o! moles o! tegas% (e$ause moles o! a substan$e are determined b# mass divided b# mole$ular mass, it $an$reate an interesting variant we will dis$uss later% " is te Ideal Gas Constant% /epending onweter atmospers or .ilospas$als were used, te value is eiter 0%0123 L4atm5mol46 or 1%73 L4.Pa5mol46, respe$tivel#% Temperature is in absolute degrees 6elvin%

*n interesting aspe$t o! te Ideal Gas Law is its !lexibilit#% It $ontains elements tat allow #ou tosolve !or oter &uantities, su$ as densit# or mole$ular mass% To solve !or mole$ular mass:

PV=n"T 4 start wit te e&uationPV=mass5mol% mass x "T 4 $ange moles to massm- in grams divided b# mole$ular mass ingrams

mol% mass x PV = m"T 4 multipl# b# mole$ular massmole$ular mass = m"T5PV 4 divide b# pressure and volume%

8e $an also see densit# in tat last e&uation, m5V grams5liter-% Te same e&uation, but witdensit#d- in pla$e o! mass per volume m5V-, is:

mole$ular mass = d"T5P

To solve 9ust !or densit#, te e&uation would be$ome:densit# = mole$ular mass x pressure-5$onstant x temperature-

So !ar, we ave been s.irting te $on$ept o! an ideal gas% 8at exa$tl# is an ideal gas *n idealgas is one tat exa$tl# $on!orms to te .ineti$ teor#% Te .ineti$ teor#, as stated b# "udol!Clausius in 31;<, as !ive .e# points% Tese are:

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3% Gases are made o! mole$ules in $onstant, random movement% Gases li.e *rgon ave 34atom mole$ules%

2% Te large portion o! te volume o! a gas is empt# spa$e% Te volume o! all gas mole$ules,in $omparison, is negligible%

7% Te mole$ules sow no !or$es o! attra$tion or repulsion%

% >o energ# is lost in $ollision o! mole$ules? te impa$ts are $ompletel# elasti$%

;% Te temperature o! a gas is te average .ineti$ energ# o! all o! te mole$ules%

Non-Ideal Behavior

Te 6ineti$ Teor# ma.es several assumptions about an ideal gas% Tese $ause problems be$ause real gases are not ideal% Te main $auses o! error are related to pressure and temperature%

Pressure*t ig pressures, te beavior o! real gases $anges dramati$all# !rom tat predi$ted b# teIdeal Gas Law% @nder 30 atmosperes o! pressure or less, Ideal Gas Law predi$tions are ver#$lose to real amounts and do not generate serious error%

Temperature8en te temperature o! a gas is $lose to its li&ue!a$tion point, te beavior is ver# di!!erent!rom Ideal Gas Law predi$tions% 8it in$reasing temperatures, te Ideal Gas Law predi$tions be$ome $lose to real values%

8#Te answer is simple: ideal gases ave mole$ular volume and sow no attra$tion betweenmole$ules at an# distan$e? real gas mole$ules ave volume and sow attra$tion at sortdistan$es% Let us !irst $onsider wat pressure does% Pressure at ig degrees will bring temole$ules ver# $lose togeter% Tis $auses more $ollisions and also allows te wea. attra$tive!or$es to $ome into pla#% 8it low temperatures, te mole$ules do not ave enoug energ# to$ontinue on teir pat to avoid tat attra$tion%

The van der Waals Equation

In order to over$ome te errors in te Ideal Gas Law, Aoannes van der 8aals developed an

e&uation to predi$t te beavior o! real gases% Aoannes van der 8aals) e&uation in$luded$orre$tions !or te !inite volume o! te mole$ules o! te gas and te attra$tive !or$es between temole$ul

es% Two new $onstants, a and b, were added% Te $orre$ted e&uation is:

P = n"T-5V4nb- 4 n2a-5V2-

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Te $orre$tion nb subtra$ts te volume o! te mole$ules% b is measured in liters5mole% Te$orre$tion wit a re!le$ts te strengt o! attra$tion and is measured in liters24atmosperes permoles2%

Te e&uation is generall# put in te !orm:

Values o! a and b are di!!erent !or ea$ gas% Te values o! a and b generall# in$rease witin$reased mass o! te mole$ule and $omplexit# o! te mole$ule%

Gas Las•  prev

• dis$ussion

• summar#

•  pra$ti$e

•  problems

• resour$es

• next

Discussion

introduction

Te gas laws are a set o! intuitivel# obvious statements to most ever#one in te 8estern world

toda#% It)s ard to believe tat tere was ever a time wen te# weren)t understood% *nd #etsomeone ad to noti$e tese relationsips and write tem down% +or tis reason, man# studentsare taugt te tree most important gas laws b# te names o! teir dis$overers% Bowever, sin$ete laws are .nown b# di!!erent names in di!!erent $ountries and more importantl#- sin$e I $annever remember wo gets $redit !or wi$ law witout re!erring to notes, I will not !ollow tis$onvention%

pressure-volu!e "constant te!perature#

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8at appens to te volume o! a gas as te pressure on it $anges% Let)s tr# te !ollowingexperiment using e&uipment tat migt be !ound in #our .it$en%

Marsmallows in a .it$en va$uum pump% Do tr# tis experiment at ome% Do it 

Te volume o! a marsmallow in$reases as te pressure on itde$reases D

and vi$e versa%

Marsmallows are a mixture o! sugar, air, and gelatin% Sugar ma.es tem sweet, air ma.es tem!lu!!#, and gelatin ma.es tem elasti$% Marsmallows are a !roEen !oam and are mostl# air b#volume% 8en pla$ed in a va$uum pump, te# expand as te pressure de$reases% (rea. te seal

on teir $ontainer and te# srin. during te return to normal atmosperi$ pressure% Sin$e teva$uum pump pulls on te marsmallows ard enoug to burst some o! te air bubbles, te# area$tuall# a bit smaller and more sriveled at te end o! tis experiment% Tis illustrates a!undamental, #et important, propert# o! gases% Te pressure o! a gas is inversel# proportional toits volume wen temperature is $onstant% S#mboli$all# D

 P  ∝ 3

 T  $onstant-V 

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or 

 P 3V 3 = P 2V 2 = constant 

Tis $orrelation was dis$overed independentl# b# "obert (o#le 3F2<43F3- o! Ireland in 3FF2

and Hdme Mariotte 3F2043F1- o! +ran$e in 3F<F% In Great (ritain, *meri$a, *ustralia, te8est Indies and oter remnants o! te (ritis Hmpire it is $alled (o#le)s law, wile inContinental Hurope and oter pla$es it is $alled Mariotte)s law%

Mariotte added te important provision tat temperature remain $onstant% (o#le negle$ted tomention it, but te data e used to derive is law were most li.el# $olle$ted during a period inwi$ te temperature did not experien$e an# signi!i$ant $ange% Sin$e te gas needs to be intermal e&uilibrium wit its environment or some oter eat reservoir- to maintain an eventemperature, te pressure4volume relationsip normall# applies onl# to 'slow' pro$esses% Temarsmallow4va$uum experiment sown above is an example o! a 'slow' pro$ess% Te pressureis redu$ed at a rate slow enoug tat eat !rom te environment is able to .eep te 9ar and its

$ontents at nearl# room temperature% Su$ a trans!ormation tat ta.es pla$e witout a $ange intemperature is said to be isotermal%

Pumping a bi$#$le tire wit a and pump is an example o! a '!ast' pro$ess% Te wor. done pusing te piston trans!orms into an in$rease in te internal energ# and tus an in$rease in tetemperature- o! te air mole$ules witin te pump% People !amiliar wit and bi$#$le pumps willattest to te !a$t tat te# get ot a!ter use% Li.ewise, wen a gas is allowed to expanded into aregion o! redu$ed pressure it does wor. on its surroundings% Te energ# to do tis wor. $omes!rom te internal energ# o! te gas and so te temperature o! te gas drops% ou $an experien$etis #oursel! witout te aid o! an# apparatus oter tan #our mout% Purse #our lips so tat #ourmout as onl# a tin# opening to te outside and blow ard% Te air rusing !rom #our mout

will be &uite $ool despite $oming !rom te $ore o! #our bod#, wi$ is normall# &uite otaround 7< ℃-% /uring a '!ast' pro$ess li.e te ones 9ust des$ribed, pressure and volume are$anging so rapidl# tat eat doesn)t ave enoug time to get into or out o! te gas to .eep tetemperature $onstant% Su$ a trans!ormation tat ta.es pla$e witout an# !low o! eat is said to be adiabati$%

volu!e-te!perature "constant pressure#

8at appens to te volume o! a gas wen its temperature $anges Let)s tr# anoter .it$enexperiment%

(read doug be!ore and a!ter ba.ing% Do tr# tis experiment at ome% Do it JdougK JbreadKIn$reasing te temperature o! bread doug in$reases its volume%

(read is made !rom weat !lour, water, #east, and a bit o! sugar% east are tin# mi$roorganisms%Te# are &uite possibl# te ver# !irst domesti$ated animals and, mu$ li.e dogs and orses, #eastave been bred !or di!!erent purposes% Aust as we ave guard dogs, lap dogs, and unting dog?dra!t orses, ra$e orses, and war orses? we also ave brewer)s #east, $ampagne #east, and

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 bread #east% (read #east ave been sele$tivel# bred to eat sugar and burp $arbon dioxide C2-%8en weat !lour and water are mixed togeter and .neaded, te protein mole$ules are masedand stret$ed until te# line up neatl# to !orm a substan$e $alled gluten tat, li.e $ewing gum,is bot elasti$ and plasti$% Let tis spe$ial matrix sit and te te C2 vented !rom te #east gettrapped in tousands o! tin# resilient, stret$# po$.ets% *s tis pro$ess $ontinues tese tin#

 po$.ets expand, wi$ $auses te volume o! te doug to expand or rise in a pro$ess $alled proo!ing% 8e now ave a !lu!!# gumm# blob read# !or te oven%

8ile tere te doug expands again, but is time it)s not due to te a$tion o! mi$roorganismste# all die around te boiling point o! water-% Tis time it)s te eat, or rater te temperature%Te temperature inside a bread oven is rougl# ;0 greater in absolute terms- tan tetemperature outside% *nd similarl#, te ba.ed bread tat $omes out o! a bread oven is alsorougl# ;0 greater tan te room temperature doug tat goes in% Tis domesti$ exampleillustrates &uite ni$el# a !undamental propert# o! gases% Te volume o! a gas is dire$tl# proportional to its temperature wen pressure is $onstant% S#mboli$all# D

V  ∝ T   P  $onstant-8ile no doubt .nown and understood in!ormall# b# billions o! ba.ers sin$e te dawn o!$iviliEation, te pre$ise matemati$al relationsip was !irst dis$overed b# te +ren$ p#si$istGuillaume *montons 3FF743<0;- in 3F% Te experiment was repeated mu$ later b# Aa$&ues*lexander CNsare Carles 3<F43127- in 3<1< and mu$, mu$ later b# Aosep Louis Ga#4Lussa$ 3<<1431;0- in 3102% Carles did not publis is !indings, but Ga#4Lussa$ did% It is most!re&uentl# $alled Carles) law in te (ritis spere o! in!luen$e and Ga#4Lussa$)s law in te+ren$, but never *monton)s law%

*n isobari$ pro$ess is one tat ta.es pla$e witout an# $ange in pressure%

Let)s re$all wat it means wen two &uantities are dire$tl# proportional li.e volume andtemperature% Beat up a gas and it)s volume will expand% Cool it down and it)s volume will$ontra$t% Te two &uantities $ange in te same dire$tion% More spe$i!i$all#, an in$rease in oneresults in a proportional in$rease in te oter and a de$rease in one results in a proportionalde$rease in te oter% +or example D

• /oubling te absolute temperature o! te air in an engine $#linder will double its volume%

• Balving te absolute temperature o! te air in a bag o! potato $ips will $ause it to srin.to one al! its original volume%

• Te absolute temperature o! a bread oven is one and a al! times tat o! roomtemperature% Tere!ore, te loa! o! ba.ed bread tat $omes out o! an oven as ;0 morevolume tan te ball o! doug tat went into it%

Tere)s a s#mmetr# at wor. ere somewere% * s#mmetr# is a $ange in one &uantit# tat leavesanoter, more !undamental &uantit# un$anged% It)s someting li.e multipl#ing bot tenumerator and denominator o! a !ra$tion b# te same ting%

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a 

⎛⎝

 x ⎞⎠  =

a  x =

a

b x b  x b

 >o wait, it)s exa$tl# li.e tat% Te onl# wa# two &uantities $an $ange in dire$t proportion is i!teir ratio remains $onstant% Tus D

V 3  =V 2  = constant 

T 3 T 2

pressure-te!perature "constant volu!e#

+ix tis, too%

Te pressure o! a gas is dire$tl# proportional to its temperature wen volume is $onstant%S#mboli$all# D

 P  ∝ T 

*n iso$ori$ pro$ess is one tat ta.es pla$e witout an# $ange in volume%

Tis relationsip doesn)t reall# ave a name, but I ave eard it $alled te 'pressure law' ormista.enl#- 'Ga#4Lussa$)s law'%

Temperatures drop F ℃ !or ever# 3000 m o! altitude%

 P 3  = P 2  = constant 

T 3 T 2

a$solute te!perature

In 3<07, *montons stated D

Jmagni!#K%

/ouble room temperature, 27 6 = 20 ℃, and #ou get ;1F 6 = 737 ℃ not 0 ℃%

a co!plete ideal gas la

Proportionalit# statements aren)t as popular toda# in te Twent# !irst Centur# as te# were in te >ineteent Centur# and earlier% 8e live in an era were it)s all about te e&uation% Tere)s goodand bad in tis !o$us% H&uations $onve# a lot o! in!ormation in a ver# !ew s#mbols, wi$ is w#te#)re so popular, but te#)re also a $rut$? a devi$e used to support a wea. understanding andma.e it seem strong% H&uations $an be used b# a student wit no understanding to !a.e$ompeten$#%

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'I put te numbers into te e&uation and I got te rigt answer% Sin$e I ave te rigt answer, Iam smart%'

S.illed Certainl#% Smart >ot ne$essaril#%

Still, it would be ni$e to ave an e&uation around !or tose times wen all #ou want to do is 9ustget te 9ob done wit a minimum o! assle%

Combine te tree togeter%

 P 3V 3  = P 2V 2  = constant 

T 3 T 2

Tere are two wa#s to write te $omplete statement o! te ideal gas law as an e&uation D

 functional  termod#nami$s  statistical  termod#nami$s

 PV  = nRT PV   = NkT were D

 P  = absolute pressureT  = absolute temperatureV  = volume

and D or Dn = number o! moles  N  = number o! parti$les

 R = gas $onstant = 1%73; A5mol 6  k  = (oltEmann)s $onstant = 3%712 O 3027A56  Mass is one o! te most !undamental &uantities in p#si$s and it)s di!!i$ult to get troug a

 p#si$s $ourse witout getting a ang o! basi$ mass !ormulas% *lmost ever# !undamental p#si$se&uation as mass as a variable in it be$ause it is an inseparable &uantit#%

8at is Mass

Mass is an inerent propert# o! matter in all its !orms% Mass sould not be $on!used wit weigt%8eigt is te !or$e o! gravit# a$ting on a mass, wereas mass is a measure o! matter $ontent tatresponds to gravit# and as te propert# o! inertia% Tere are two de!ined t#pes o! mass% ne isinertial mass and te oter is gravitational mass% Inertial mass is te propert# o! matter tat o!!ersresistan$e to $ange b# a !or$e% Gravitational mass is te propert# o! matter wi$ exerts tegravitational !or$e% Te inertial mass and gravitational mass were sown to be e&ual b# te

Hotvos experiment% In te SI s#stem o! units, mass as te unit o! )6ilogram 6g-)%

+ormulas

Tere are various p#si$s e&uations tat $an #ield te value o! mass% Ha$ one o! tem is use!ulin deriving a value o! mass given $ertain .nown &uantities%

Cal$ulating Mass @sing /ensit# and Volume

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8en te densit# and volume o! an# substan$e are .nown, it is possible to $al$ulate its mass% Itis made possible due to te relation between densit# mass and volume% Te densit# !ormula,itsel! provides te relation%

Mass in 6ilograms- = /ensit# in 6g per m7- x Volume in m7-

+ormula +or Center o! MassCenter o! mass is te point in a bod# were te wole mass $an be #poteti$all# assumed to be$on$entrated% Center o! mass is a !un$tion o! te individual masses o! parti$les and teir relative positions wit respe$t to ea$ oter% Te $enter o! mass $an onl# be de!ined !or rigid bodies woave !ixed positions o! mass parti$les%

" Center o! Mass- = m3r3 Q m2r2 Q m7r7% % Q miri- 5 m3 Q m2 Q m7% % Q mi-

= R miri- 5 R mi-

Tus te $enter o! mass is $al$ulated as a ratio o! individual masses weiged b# position ve$torsand total mass o! te wole bod# o! parti$les%

+ormula +or Mole$ular MassTe atomi$ mass !ormula derives te masses o! atoms in $omparison to 3532t te mass o! a$arbon C32 atom% *ll tese masses are derived experimentall#% * mole$ule is a $olle$tion o!atoms bonded togeter% So te mole$ular mass is te sum o! individual masses o! atoms bondedin it%

Mole$ular Mass = R Mass o! an *tom x >umber o! *toms-

Molar Mass +ormulaMolar mass o! a mole$ule or substan$e is simple te mole$ular weigt expressed in grams% Itsunit is gm5mole%

Molar Mass = Mole$ular 8eigt- x 3 g5mol

+ormula +or Mass in >ewtonian Me$ani$s

Mass in 6g- = +or$e in >ewtons- 5 *$$eleration Meter5s2-

So, .nowing te a$$eleration o! a bod# and te !or$e a$ting on it, #ou $an easil# $al$ulate itsmass%

Spe$ial "elativisti$ +ormulaSpe$ial relativit# $anged te wa# we see te world% It demolised te notion o! absolute timewi$ was prevalent in >ewtonian me$ani$s and time was now .nown to be relative% Inme$ani$s, based on te spe$ial relativit#, ever#ting is dependent on te !rame o! re!eren$eex$ept te speed o! ligt in va$uum, wi$ remains $onstant% Bere is te relativisti$ !ormula:

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m = m0-53 4 v25$2--

were m0 is rest mass o! te parti$le, v is te velo$it# o! te parti$le and $ is te velo$it# o! ligt%

"e$entl#, p#si$ists ave provided a #potesis wi$ tries to explain ow parti$les o! matter

a$&uire mass% Te teor# sa#s tat tere is a Biggs !ield pervading all spa$e wi$ intera$ts witall matter% * parti$le a$&uires mass troug its intera$tion wit Biggs boson% Currentexperiments at CH">)s )Large Badron Collider LBC-) are aimed at !inding te Biggs boson%ne da# we will .now ow matter a$&uires mass%"ead more at (uEEle: ttp:55www%buEEle%$om5arti$les5mass4!ormula%tml

3 Counting atoms: *vogadro)s number 

wing to teir tin# siEe, atoms and mole$ules $annot be

$ounted b# dire$t observation% (ut mu$ as we do wen'$ounting' beans in a 9ar, we $an estimate te number o!  parti$les in a sample o! an element or $ompound i! we avesome idea o! te volume o$$upied b# ea$ parti$le and tevolume o! te $ontainer%

n$e tis as been done, we .now te number o! !ormula units to use te most general term !oran# $ombination o! atoms we wis to de!ine- in an# arbitrar# weigt o! te substan$e% Tenumber will o! $ourse depend bot on te !ormula o! te substan$e and on te weigt o! tesample% (ut i! we $onsider a weigt o! substan$e that is the same as its formula (molecular)weight expressed in grams, we have onl one number to know: !vogadro"s number , F%02233;2<

O 3027, usuall# designated b# N *%

Jimage  K

%!adeo %vogadro "&'((-&)*(# never kne his on nu!$er+

*vogadro onl# originated te concept  o! tis number, wose a$tual value was !irst estimated b#Aose! Los$midt, an *ustrian $emistr# tea$er, in 31;%

ou sould .now it to tree signi!i$ant !igures: N * = F%02 O 3027

F%02 O 3027 o! what  8ell, o! an#ting #ou li.e: apples, stars in te s.#, burritos% (ut te onl# practical  use !or N * is to ave a more $onvenient wa# o! expressing te uge numbers o! te tin# parti$les su$ as atoms or mole$ules tat we deal wit in $emistr#% *vogadro)s number is acollective number , 9ust li.e a doEen%

Tin. o! F%02 O 3027 as te '$emist)s doEen'%

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 videos on %vogadro,s Nu!$er and the !ole

Te basi$ idea Curtis8ang, min- UUUUTe Mole Hxplained utaustin$em, min- UUUUSimple mole $al$ulations boEemans$ien$e, ;m- UUUU

Mole and *vogadro)s number  6an, m- UUTe mole $on$ept Isaa$sTea$, ; min- UUUUMr Cause# < m- UUUMole problems review LindaBanson, 27 min- UUUU

(e!ore we get into te use o! *vogadro)s number in problems, ta.e a moment to $onvin$e#oursel! o! te reasoning embodied in te !ollowing examples%

 

ro$le! Exa!ple &: !ass ratio .ro! ato!ic eights 

Te atomi$ weigts o! ox#gen and o! $arbon are 3F%0 and 32%0, respe$tivel#% Bow mu$ eavieris te ox#gen atom in relation to $arbon

 Solution: *tomi$ weigts represent te relative masses o! di!!erent .inds o! atoms% Tis meanstat te atom o! ox#gen as a mass tat is 3F532 = /01 W 3%77 as great as te mass o! a $arbonatom%

ro$le! Exa!ple 2: 3ass o. a single ato! 

Te absolute mass o! a $arbon atom is 32%0 uni!ied atomi$ mass units 8at are tese-% Bowman# grams will a single ox#gen atom weig

 Solution: Te absolute mass o! te $arbon atom is 32%0 u,or 32 O 3%FF0; O 30 X2< g = 3% O 30 X2< .g% Te mass o! te ox#gen atom will be 57 greater, or24(( 5 &6 72( kg%

*lternativel#: 32 g5mol- Y F%022 O 3027 mol X3- O 57- = 24(( 5 &6 721 g%

ro$le! Exa!ple 1: 8elative !asses .ro! ato!ic eights 

Suppose tat we ave N  $arbon atoms, were N  is a number large enoug to give us a pile o!$arbon atoms wose mass is 32%0 grams% Bow mu$ would te same number, N , o! ox#genatoms weig

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 Solution: Te $olle$tion o! N  ox#gen atoms would ave a mass o!57 O 32 g = &(46 g%

 

I! te !oregoing problems don)t ma.e sense, #ou sould review te previous lesson%-Things to understand a$out %vogadro,s nu!$er N  A

Z It is a number , 9ust as is 'doEen', and tus is dimensionless%

Z It is a huge number, !ar greater in magnitude tan we $an visualiEe? see ere !or someinteresting $omparisons wit oter uge numbers%

Z Its pra$ti$al use is limited to $ounting tin# tings li.e atoms, mole$ules, '!ormula units',ele$trons, or potons%

Z Te value o! N  ! $an be .nown onl# to te pre$ision tat te number o! atoms in a measurableweigt o! a substan$e $an be estimated% (e$ause large numbers o! atoms $annot be $ounteddire$tl#, a variet# o! ingenious indire$t measurements ave been made involving su$ tings as brownian motion and [4ra# s$attering%

Z Te $urrent value was determined b# measuring te distan$es between te atoms o! sili$on inan ultrapure $r#stal o! tis element tat was saped into a per!e$t spere% Te measurement wasmade b# [4ra# s$attering%- 8en $ombined wit te measured mass o! tis spere, it #ields*vogadro)s number% (ut tere are two problems wit tis: 3- Te sili$on spere is an arti!a$t,rater tan being someting tat o$$urs in nature, and tus ma# not be per!e$tl# reprodu$ible% 2-

Te standard o! mass, te .ilogram, is not pre$isel# .nown, and its value appears to be $anging%+or tese reasons, tere are proposals to revise te de!initions o! bot N  ! and te .ilogram% Seeere !or more, and sta# tuned

Bistor# o! te determination o! *vogadro)s number  

8i.ipedia as a good dis$ussion o! *vogadro)s number 2 Moles and teir uses

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The Kinetic Molecular Theory Postulates

Te experimental observations about te beavior o! gases dis$ussed so !ar $an be explained

wit a simple teoreti$al model .nown as te kinetic !olecular theor9% Tis teor# is based onte !ollowing postulates, or assumptions%

3% Gases are $omposed o! a large number o! parti$les tat beave li.e ard, speri$al ob9e$tsin a state o! $onstant, random motion%

2% Tese parti$les move in a straigt line until te# $ollide wit anoter parti$le or te wallso! te $ontainer%

7% Tese parti$les are mu$ smaller tan te distan$e between parti$les% Most o! te volumeo! a gas is tere!ore empt# spa$e%

% Tere is no !or$e o! attra$tion between gas parti$les or between te parti$les and tewalls o! te $ontainer%

;% Collisions between gas parti$les or $ollisions wit te walls o! te $ontainer are per!e$tl#elasti$% >one o! te energ# o! a gas parti$le is lost wen it $ollides wit anoter parti$leor wit te walls o! te $ontainer%

F% Te average .ineti$ energ# o! a $olle$tion o! gas parti$les depends on te temperature o!te gas and noting else%

Te assumptions beind te .ineti$ mole$ular teor# $an be illustrated wit te apparatus sownin te !igure below, wi$ $onsists o! a glass plate surrounded b# walls mounted on top o! treevibrating motors% * and!ul o! steel ball bearings are pla$ed on top o! te glass plate to representte gas parti$les%

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8en te motors are turned on, te glass plate vibrates, wi$ ma.es te ball bearings move in a$onstant, random !asion postulate 3-% Ha$ ball moves in a straigt line until it $ollides witanoter ball or wit te walls o! te $ontainer postulate 2-% *ltoug $ollisions are !re&uent, teaverage distan$e between te ball bearings is mu$ larger tan te diameter o! te ballspostulate 7-% Tere is no !or$e o! attra$tion between te individual ball bearings or between te

 ball bearings and te walls o! te $ontainer postulate -%Te $ollisions tat o$$ur in tis apparatus are ver# di!!erent !rom tose tat o$$ur wen a rubber ball is dropped on te !loor% Collisions between te rubber ball and te !loor are inelastic, assown in te !igure below% * portion o! te energ# o! te ball is lost ea$ time it its te !loor,until it eventuall# rolls to a stop% In tis apparatus, te $ollisions are per!e$tl# elastic% Te ballsave 9ust as mu$ energ# a!ter a $ollision as be!ore postulate ;-%

*n# ob9e$t in motion as a kinetic energ9 tat is de!ined as one4al! o! te produ$t o! its masstimes its velo$it# s&uared%

 #$  = 352 mv2 

*t an# time, some o! te ball bearings on tis apparatus are moving !aster tan oters, but tes#stem $an be des$ribed b# an average kinetic energ% 8en we in$rease te 'temperature' o!te s#stem b# in$reasing te voltage to te motors, we !ind tat te average .ineti$ energ# o! te ball bearings in$reases postulate F-%

 How the Kinetic Molecular Theory Explains the Gas Laws

Te .ineti$ mole$ular teor# $an be used to explain ea$ o! te experimentall# determined gaslaws%

The Link Beteen P  and n 

Te pressure o! a gas results !rom $ollisions between te gas parti$les and te walls o! te$ontainer% Ha$ time a gas parti$le its te wall, it exerts a !or$e on te wall% *n in$rease in tenumber o! gas parti$les in te $ontainer in$reases te !re&uen$# o! $ollisions wit te walls andtere!ore te pressure o! te gas%

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%!ontons, La " P T #

Te last postulate o! te .ineti$ mole$ular teor# states tat te average .ineti$ energ# o! a gas parti$le depends onl# on te temperature o! te gas% Tus, te average .ineti$ energ# o! te gas parti$les in$reases as te gas be$omes warmer% (e$ause te mass o! tese parti$les is $onstant,

teir .ineti$ energ# $an onl# in$rease i! te average velo$it# o! te parti$les in$reases% Te !astertese parti$les are moving wen te# it te wall, te greater te !or$e te# exert on te wall%Sin$e te !or$e per $ollision be$omes larger as te temperature in$reases, te pressure o! te gasmust in$rease as well%

Bo9le,s La " P   &0v#

Gases $an be $ompressed be$ause most o! te volume o! a gas is empt# spa$e% I! we $ompress agas witout $anging its temperature, te average .ineti$ energ# o! te gas parti$les sta#s tesame% Tere is no $ange in te speed wit wi$ te parti$les move, but te $ontainer is smaller%Tus, te parti$les travel !rom one end o! te $ontainer to te oter in a sorter period o! time%

Tis means tat te# it te walls more o!ten% *n# in$rease in te !re&uen$# o! $ollisions witte walls must lead to an in$rease in te pressure o! te gas% Tus, te pressure o! a gas be$omeslarger as te volume o! te gas be$omes smaller%

;harles, La "V   T #

Te average .ineti$ energ# o! te parti$les in a gas is proportional to te temperature o! te gas%(e$ause te mass o! tese parti$les is $onstant, te parti$les must move !aster as te gas be$omeswarmer% I! te# move !aster, te parti$les will exert a greater !or$e on te $ontainer ea$ timete# it te walls, wi$ leads to an in$rease in te pressure o! te gas% I! te walls o! te$ontainer are !lexible, it will expand until te pressure o! te gas on$e more balan$es te pressure

o! te atmospere% Te volume o! te gas tere!ore be$omes larger as te temperature o! te gasin$reases%

%vogadro,s <9pothesis "V N #

*s te number o! gas parti$les in$reases, te !re&uen$# o! $ollisions wit te walls o! te$ontainer must in$rease% Tis, in turn, leads to an in$rease in te pressure o! te gas% +lexible$ontainers, su$ as a balloon, will expand until te pressure o! te gas inside te balloon on$eagain balan$es te pressure o! te gas outside% Tus, te volume o! te gas is proportional to tenumber o! gas parti$les%

Dalton,s La o. artial ressures " P t  P & = P 2 = P 1 = 444# Imagine wat would appen i! six ball bearings o! a di!!erent siEe were added to te mole$ulard#nami$s simulator % Te total pressure would in$rease be$ause tere would be more $ollisionswit te walls o! te $ontainer% (ut te pressure due to te $ollisions between te original ball bearings and te walls o! te $ontainer would remain te same% Tere is so mu$ empt# spa$e inte $ontainer tat ea$ t#pe o! ball bearing its te walls o! te $ontainer as o!ten in te mixtureas it did wen tere was onl# one .ind o! ball bearing on te glass plate% Te total number o!

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$ollisions wit te wall in tis mixture is tere!ore e&ual to te sum o! te $ollisions tat wouldo$$ur wen ea$ siEe o! ball bearing is present b# itsel!% In oter words, te total pressure o! amixture o! gases is e&ual to te sum o! te partial pressures o! te individual gases%

Graha!s Laws o" #i""usion an$ E""usion

* !ew o! te p#si$al properties o! gases depend on te identit# o! te gas% ne o! tese p#si$al properties $an be seen wen te movement o! gases is studied%

In 312 Tomas Graam used an apparatus similar to te one sown in te !igure below to stud#te di..usion o! gases te rate at wi$ two gases mix% Tis apparatus $onsists o! a glass tube

sealed at one end wit plaster tat as oles large enoug to allow a gas to enter or leave tetube% 8en te tube is !illed wit B2 gas, te level o! water in te tube slowl# rises be$ause teB2 mole$ules inside te tube es$ape troug te oles in te plaster more rapidl# tan temole$ules in air $an enter te tube% (# stud#ing te rate at wi$ te water level in tis apparatus$anged, Graam was able to obtain data on te rate at wi$ di!!erent gases mixed wit air%

Graam !ound tat te rates at wi$ gases di!!use is inversel# proportional to te s&uare root o!teir densities%

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Tis relationsip eventuall# be$ame .nown as Graha!,s la o. di..usion%

To understand te importan$e o! tis dis$over# we ave to remember tat e&ual volumes o!di!!erent gases $ontain te same number o! parti$les% *s a result, te number o! moles o! gas perliter at a given temperature and pressure is $onstant, wi$ means tat te densit# o! a gas isdire$tl# proportional to its mole$ular weigt% Graam)s law o! di!!usion $an tere!ore also bewritten as !ollows%

Similar results were obtained wen Graam studied te rate o! e..usion o! a gas, wi$ is terate at wi$ te gas es$apes troug a pinole into a va$uum% Te rate o! e!!usion o! a gas isalso inversel# proportional to te s&uare root o! eiter te densit# or te mole$ular weigt o! tegas%

Graha!,s la o. e..usion $an be demonstrated wit te apparatus in te !igure below% * ti$.4walled !ilter !las. is eva$uated wit a va$uum pump% * s#ringe is !illed wit 2; mL o! gas and

te time re&uired !or te gas to es$ape troug te s#ringe needle into te eva$uated !ilter !las. ismeasured wit a stop wat$%

*s we $an see wen data obtained in tis experiment are graped in te !igure below, te time re&uired !or 2;4mL samples o! di!!erent gases to es$ape into a va$uum is proportional to te

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s&uare root o! te mole$ular weigt o! te gas% Te rate at wi$ te gases e!!use is tere!oreinversel# proportional to te s&uare root o! te mole$ular weigt% Graam)s observations aboutte rate at wi$ gases di!!use mix- or e!!use es$ape troug a pinole- suggest tat relativel#ligt gas parti$les su$ as B2 mole$ules or Be atoms move !aster tan relativel# eav# gas parti$les su$ as C2 or S2 mole$ules%

The Kinetic Molecular Theory an$ Graha!s Laws 

Te .ineti$ mole$ular teor# $an be used to explain te results Graam obtained wen e studiedte di!!usion and e!!usion o! gases% Te .e# to tis explanation is te last postulate o! te .ineti$teor#, wi$ assumes tat te temperature o! a s#stem is proportional to te average .ineti$

energ# o! its parti$les and noting else% In oter words, te temperature o! a s#stem in$reases i!and onl# i! tere is an in$rease in te average .ineti$ energ# o! its parti$les%

Two gases, su$ as B2 and 2, at te same temperature, tere!ore must ave te same average.ineti$ energ#% Tis $an be represented b# te !ollowing e&uation%

Tis e&uation $an be simpli!ied b# multipl#ing bot sides b# two%

It $an ten be rearranged to give te !ollowing%

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Ta.ing te s&uare root o! bot sides o! tis e&uation gives a relationsip between te ratio o! te

velo$ities at wi$ te two gases move and te s&uare root o! te ratio o! teir mole$ularweigts%

Tis e&uation is a modi!ied !orm o! Graam)s law% It suggests tat te velo$it# or rate- at wi$gas mole$ules move is inversel# proportional to te s&uare root o! teir mole$ular weigts%

3.1 The Physics of Diusion

/i!!usion is te .ineti$ pro$ess tat leads to te omogeniEation, or uni!orm mixing, o! te$emi$al $omponents in a pase% *ltoug mixing in a !luid li&uid or gas- ma# o$$ur on man#lengt s$ales, as indu$ed b# ma$ros$opi$ !low, di!!usive mixing in solids, b# $ontrast, o$$ursonl# on te atomi$ or mole$ular level% *s time in$reases, te extent o! omogeniEation b#

di!!usion also in$reases, and te lengt s$ale over wi$ $emi$al omogeneit# persists witin a pase graduall# extends to ma$ros$opi$ distan$es%

In te sili$on pro$ess te$nolog#, depending on te $omplexit# o! te model, dopantredistribution does not onl# in$lude dopant atoms itsel!, but also point de!e$ts% Te point de!e$ts$an be divided into simple and extended point de!e$ts%

*dditionall#, te presen$e o! te large number o! te $arged dopant atoms $auses an ele$tri$al!ield in te inside o! te wa!er, wi$ reversel# in!luen$es te di!!usion o! dopants%

Te $ontinuum4based di!!usion models are su$$ess!ull# applied in man# pro$ess simulation programs JFK% Tese models use te so4$alled metodolog# o! di!!usion4rea$tion e&uations,wi$ was sown to be ver# e!!i$ient !or te numeri$al simulation o! te di!!usion penomena%Te metod itsel! $an be generaliEed to design a set o! $oupled partial di!!erential e&uations !romsome assumption about possible rea$tions between intrinsi$ point de!e$ts and dopants%

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Prin$ipal ingredients o! te di!!usion4rea$tion models are: +i$.)s di!!usion laws, ele$tri$al !ieldenan$ed di!!usion, and spe$ies generation and re$ombination teor#% In te !ollowing we will brie!l# dis$uss and illustrate ea$ o! tese basi$ $on$epts%

>ick,s di..usion las 

Te laws o! di!!usion are matemati$al relationsips wi$ relate te rate o! di!!usion to te$on$entration gradient !or net mass trans!er% Su$ laws are $onsidered to be penomenologi$al%

8e de!ine ve$tor to be a mass !lux o! te di!!using $omponent% In Cartesian

$oordinates, wit te unit $oordinates , and , te mass !lux ma# be written as a ve$torialsum:

(3.1)

 The individual u !a"ni#udes a$$ea%in" in (3.1) can &e e$%essed &y co!$onen#e'ua#ions #h%ou"h Ficks's frst law

(3.)

were denotes te di!!usion $oe!!i$ient and is te $on$entration o! te di!!using$omponent%(# imposing a mass $onservation on te di!!using s#stem we obtain %icks"s second law in te!orm:

(3.3

)

Te interpretation o! - is $lear: I! tere is a $onverging !low o! spe$ies at a point, so tat

, te $on$entration rises wit time? i! tere is a divergent !low, so tat , te

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$on$entration !alls%

In.luence o. the Electrical .ield 

8e $onsider a single $arged parti$le moving in te ele$tri$al !ield wit potential %Te gradient o! tis potential des$ribes te !or$e on te parti$le J31K,

(3.*)

were is te valen$e o! te $arged spe$ies% * potential gradient tends to produ$e a !lux o! tespe$ies, and tis !lux must be added to tat produ$ed b# te $on$entration gradient to arrive atte e&uation !or te total !lux% It is !ound empiri$all# tat a potential gradient gives rise to amean di!!usion velo$it# !or te a!!e$ted atoms% Tis !a$t is matemati$all# expressed b# tee&uation,

(3.+)

in ,hich is called !o&ili#y. The fo%ce "ives %ise #o a s#eady-s#a#e veloci#y ins#eadof con#inuin" accele%a#ion.

In appl#ing a potential gradient instead o! $on$entration gradient we are simpl# repla$ing onesmall !or$e wit anoter% Tus is plausible tat te mobilit# is proportional to te di!!usion$oe!!i$ient % Tis !a$t is expressed b# $instein"s relationsip,

(3.)

were is te (oltEmann $onstant% Te !lux tat results in a omogenous s#stem is tus,

(3./)

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0o!&inin" #he las# e'ua#ion ,i#h ic2s diusion la, ,e o&#ain a $a%#ial die%en#iale'ua#ion desc%i&in" #he &ehavio% of #he cha%"ed $a%#icle in #he concen#%a#ion and

elec#%ical "%adien# 4eld5

(3.6)

7lec#%ic 4elds a%ise f%o! #he la%"e die%ences in diusion cons#an#s of #he ioni8eddo$an#s and #he cha%"e ca%%ie%s associa#ed ,i#h #he! 9*:. The cha%"e ca%%ie%s ,ouldo%dina%y #end #o diuse a,ay fas#e% #hen #he ions5 #hus c%ea#in" a s$ace cha%"e.

 The elec#%ic 4eld o%i"ina#in" f%o! #his s$ace cha%"e ac#s on #he ions and cha%"eca%%ie%s #o coun#e%ac# #he se$a%a#ion.

Generall#, in te $ase o! non4uni!orm $on$entrations o! dopants, te ele$tri$ !ield $al$ulation is based on tird Maxwell)s e&uation,

div(3.;

)

,he%e is #he elec#%ic dis$lace!en# vec#o% and #he s$ace cha%"e concen#%a#ion. The elec#%ic dis$lace!en# vec#o% is %ela#ed #o #he elec#%ic 4eld vec#o% and #hus #o#he elec#%os#a#ic $o#en#ial5

(3.1<)

inally5 unde% #he assu!$#ion of co!$le#e ioni8a#ion5 (3.;) #a2es #he fo%! 9*:5

div sinh

(3.11)

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deno#es #he elec#%on cha%"e associa#ed ,i#h #he defec#s and is #hei%concen#%a#ion.

*ssuming lo$al $arge neutralit#,

(3.1)

i# can &e sho,n #ha# elec#%on and hole concen#%a#ion is e$%essed &y 9 *:5

(3.1

3)

,he%e is in#%insic ca%%ie% concen#%a#ion (see =ec#ion 3.3.1).

Species generation and recombination theory Du%in" #he diusion $%ocess s$ecies !ay a%%an"e in $ai%s ,i#h s$ecies of ano#he%#y$e and #he%e&y #he ne, s$ecies a%e c%ea#ed. >e# ,e #a2e #h%ee s$ecies 5 and

5 ,he%eas #he #hi%d one is cons#%uc#ed &y $ai%in" of 4%s# and second. Thische!ical %eac#ion can &e sy!&olically dis$layed as 91;:5

(3.1*)

 The in#ensi#y of #hese %eac#ion is e$%essed &y a %a#e coe?cien# @lso #he %eve%se

$%ocess is #hin2a&le5 dissolu#ion of #he co!$ound s$ecies in#o #he o%i"inals$ecies and 5

(3.1+)

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 The %a#e coe?cien# of #he %eve%se %eac#ion is . o% #hese #,o %eac#ions ,e canalso de4ne #he rate o reaction &y5

(3.1

)

Ae say #ha# #he %eac#ions a%e in "lo&al #he%!al e'uili&%iu! if eve%y,he%e is full4led5

(3.1/)

@ #he%!al de$endence of #he %eac#ion %a#es is al,ays assu!ed5 and ,i#h #a2in"

5 (3.1/) can &e ,%i##en in #he fo%! of #he law o mass action5

(3.16)

Bo,eve%5 if #he %eac#ions a%e no# in "lo&al #he%!al e'uili&%iu!5 #he%e is o&viously a

chan"e in concen#%a#ion of each s$ecies and #he follo,in" e'ua#ions hold5

(3.1;)

 

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