OSECH - MSE calphad@postch.ac.kr
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 0 0 0 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
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
Einstin and Dby Modl or Hat Capacity Background
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 < >
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
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 (
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
Einstin and Dby Modl or Hat Capacity Comparison
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
Einstin and Dby Modl or Hat Capacity Mor about Cp c γ ' or << F
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
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 /
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 ]
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 /
Eusion: angmuir Equation A * * / τ ( atm 05 4 ( cm sc 0 ( πm Aτ M 0 / π.8 0 6.0 0 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.7 0-9 g/s Knudsn usion mthod * 0.7 0 Aτ τ Aτ vapor quil ( πm vapor quil (πm Aτ 7.0 0 atm / / 5
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 (.8 0 0 ( at atm, 0-7 m at 0-9 atm, 00 m / a > Knudsn low / a < 0.0 iscous low ( atm 00.0 0 5.0 0 ( atm 7 ( mtr
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
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 4.85 0 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 0 4.6 0 atom / m atoms / m s
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
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
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
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
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
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
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 /
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
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
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