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1 OSECH - MSE calphad@postch.ac.kr
2 Equipartition horm h avrag nrgy o a particl pr indpndnt componnt o motion is ε ε ' ε '' ε ''' U ln Z Z ε < ε > U ln Z β ( ε ' ε '' ε ''' / Z' Z translational kintic nrgy : mv rotational kintic nrgy : Iωω vibrational nrgy : mv k kintic nrgy or ach indpndnt componnt o motion has a orm o ε b p b p b p y i β / p i βb p βb p dp dp βbi pi / bi yi / dpi β dyi Ki 0 β 0 βb '' Z b i p i p ''' dp Z β β / / / K K / β β K KK K
3 Equipartition horm h avrag nrgy o a particl pr indpndnt componnt o motion is Z β / K K K < ε > ln β β or a monoatomic idal gas : u R or diatomic gass : 5 u R or polyatomic molculs which ar sot and vibrat asily with many rquncis, say, q: u ( q R or liquids and solids, th quipartition principl dos not work
4 Einstin and Dby Modl or Hat Capacity Background
5 Einstin and Dby Modl or Hat Capacity Concpt indpndnt (wakly intracting but distinguishabl simpl harmonic oscillators. ε i i ( ln ln( or simpl harmonic vibrators U ln avrag nrgy pr vibrator ε < >
6 Einstin and Dby Modl or Hat Capacity numbr dnsity t d v b th numbr o oscillators whos rquncy lis btwn v and v dv d g( d whr g(v, th numbr o vibrators pr unit rquncy band, satisy th condition d g ( d h nrgy o particls o th crystal U < ε > d g( h d C U k( ( g( d
7 Einstin and Dby Modl or Hat Capacity Einstin All th quivalnt harmonic oscillators hav th sam rquncy v E C U k( ( g( d C k( ( E h E E Dining Einstin charactristic tmpratur θ E k h E θe / c θ E θe / R (
8 Einstin and Dby Modl or Hat Capacity Dby A crystal is a continuous mdium supporting standing longitudinal and transvrs wavs 9 g ( m C U k( ( g( d C k ( / m ( d ( m 0 st m h m Θ c R ( Θ/ ( Θ 4 / 0 d
9 Einstin and Dby Modl or Hat Capacity Comparison
10 Einstin and Dby Modl or Hat Capacity Mor about Dby Bhavior o c R ( Θ / 4 ( Θ / 0 d at and 0 4 ( at c R at 0 c R 4 4π 5 Θ : Dby s law
11 Einstin and Dby Modl or Hat Capacity Mor about Cp c γ ' or << F
12 Eusion: angmuir Equation Qustion: h rat at which particls strik a unit surac o a containr pr unit tim, givn th prssur and tmpratur o th gas Application:. Estimat o th tim ndd or a totally clan surac to b covrd with a monolayr o atoms or molculs, assuming that all th molculs that hit th surac stick to it. Calculat how many atoms will scap rom a small hol in a vssl pr unit tim, givn th ara o hol (masur o vapor prssur. How may particls may vaporat rom a surac pr unit tim
13 Mawll Distribution o Spd in Dilut Gass n( ε Z ( ε h ε 8m ε g g( ε dε 4πr dr 8 π (m h / ε / dε ( n n n y z h 8m r / n( ε dε π ε / p π ε dε Z πm h / n( v dv 4π m π / v p mv dv ε mv < vn( v dv 0 8 v> π m / < v > m v * m /
14 Eusion: angmuir Equation A - ais v τ umbr o atoms * that collid with sid walls within a tim τ * i v, i τ A [ probability h p * 8m v, iτ A i Z / o i v, i ]
15 Eusion: angmuir Equation h p * 8m v, iτ A i Z v, i γ i m / i / / γ πm Z h h 8m Z / / πm h / / * γ Aτ Z m 0 i p( γ i di i p( γ i di 0 γ * Aτ πm / * Aτ ( πm /
16 Eusion: angmuir Equation A * * / τ ( atm 05 4 ( cm sc 0 ( πm Aτ M 0 / π Assum 0 5 atoms pr cm in th surac monolayr. For O (M g/mol at 00K and at 0-0 atm, about 7sc is ncssary or monolayr dposition. o kp th surac clan or hour, th prssur should b 0 - atm. At quilibrium, rat o vaporation is th sam as th rat o dposition. or liquid Al in vacuum at 50 K, mass loss du to a hol o 0 - cm g/s Knudsn usion mthod * Aτ τ Aτ vapor quil ( πm vapor quil (πm Aτ atm / / 5
17 Eusion: angmuir Equation Knudsn usion mthod: (valid or Knudsn low quil Aτ vapor ( πm / / Man r path ( vt / vtπσ πσ ( / πσ ( / > a or idal gas πσ O gas (diamtr 0-0 m at 00 K π ( 0 ( ( at atm, 0-7 m at 0-9 atm, 00 m / a > Knudsn low / a < 0.0 iscous low ( atm ( atm 7 ( mtr
18 Diusion in Gass J m C D C C * C * o d J 6 * < v> C C < v> D < v> < v > 8 πm / πσ / / k / π or << m σ D
19 Flu during D Evaporation (angmuir or Diusion Controlld? Considr W vaporation at 47K onto cold substrat along cm path. in vacuum 5 quil. 0 0 atm 0 cm πσ * 6 J W atoms / m s / Aτ ( πm in 0. atm gas 0 4 cm / k / D.66 0 m / s / π m σ C C J W D atom / m atoms / m s
20 Flu during D Gnral angmuir Equation J W * Aτ W ( πmw / Diusion Controlld J W J W C D W / W D C In Gnral J W (πm W / W / D
21 Siz Distribution o Molculs in olymr For a polymr with A mrs (sgmnt : A (Avogadro numbr numbr o sgmnts in molculs numbr o molculs with siz total numbr o molculs n raction o molculs with siz / b numbr o bonding btwn sgmnts 고분자도 ( 중합도, dgr o polymrization b / A ( b A
22 Siz Distribution o Molculs in olymr Siz Distribution o Molculs in olymr n A n n n ln Maimiz With Constraints n A ak Siz? ( A A ( A ( ( ( /( ( n A A ( W A
23 Entropy o Miing in olymr Solutions - Flory Huggins hory S M R i ln i n Considr th numbr o ways or distribution o sgmnts o th (ith n-mr w n ni ni n i ( ni Z ( Z Z( Z Ω! i w i S k ln! i c w i ni S n c ln ln [ln Z ( n ln( Z ( n ln n] k S M k R ( lnφ lnφ ln n ln R ln n ln n
24 Elasticity o Rubbr Elasticity o Rubbr Scop Scop R. R. Castllan, Gilbrt W., hysical Chmistry rd Ed., Bnjamin/Cummings, Castllan, Gilbrt W., hysical Chmistry rd Ed., Bnjamin/Cummings, w York, 98 (Chap. 9 w York, 98 (Chap. 9 d ds du U S U Idal rubbr: 0 U S olymr molculs thmslvs ar not strtchd, but th dgr o alignmnt is changd. osition o on nd o an -mr with rspct to th position o th othr nd
25 Elasticity o Rubbr Elasticity o Rubbr Statistics o Random Walk Statistics o Random Walk R R R R!!!, ( l R l l l! /! /!, ( l l ln ln, ( ln l p p, ( l l l K π l l l l ln ln, ( ln ln l l l, ( ln l
26 Elasticity o Rubbr Statistics o Random Walk (, y, z, ddydz, (, y, z, ddydz π l y, / z, ddydz ( y z p l ddydz y z R ddydz 4πR dr ( R, dr π l π / R 4 R p l dr / 4 R R 4π π l R p dr l 0 l l R rms /
27 Elasticity o Rubbr Dormation & Entropy o / o ε / o ε Whn a solid with original dimnsions 0, y 0, z 0 is strtchd in th z dirction at constant volum z 0 z 0 0 y y / 0 0 / 0
28 Elasticity o Rubbr Dormation & Entropy Whn a solid with original dimnsions 0, y 0, z 0 is strtchd in th z dirction at constant volum z 0 z 0 0 / 0 y 0 / y 0 ddydz u π l / ( y p l z ddydz ddydz s / y z ( / / πl p l ddydz Flory, aul J., rincipls o olymr Chmistry, Cornll Univrsity rss, Ithaca, Y, 960. S S S k (lnω lnω s u s u S i k y l ( z
29 Elasticity o Rubbr Elasticity o Rubbr Elasticity Elasticity k S k S i o o S S o σ A o σ d E d d d d o o o o σ ρ M c R E c A M ρ
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