NYU CHEM GA 2600: Saisical Mechanics Miderm Glen Hocky Ocober 18, 2018 nsrucions: This miderm is open book/open noe No elecronic devices besides a calculaor Please wrie your name on each page and ry o wrie your soluions legibly Try o use scrach paper for figuring hings ou 1 van der Waal s Equaion of Sae (0 ps oal) n class we showed ha you can approximae he pressure of a nonideal gas by P nrt V nb a n2 V 2 (1) (a) Describe in words he meanings of he erms a and b (10 ps) (b) Example sysem Real daa says ha 1 mole of carbon dioxide a 373 K occupies 36 ml when a 00 am Wha is he calculaed value of he pressure for CO 2 a 373 K in a box of sized 36 ml using: i The ideal gas equaion (remember, n is he number of moles of gas such ha Nk B nr, and R008206 L am / (K mol)) ( ps) ii The van der Waal s equaion, wih a 361 L 2 am/mol 2 and b 00428L/mol ( ps) n boh cases, compue he percen deviaion from he rue pressure ( ps each) (c) The Canonical pariion funcion corresponding o our van der Waal s equaion of sae is Q(N, V, T) 1 N! 2pmk B T/h 2 3N/2 (V Nb) N e an2 /(Vk B T) Compue he average energy and he hea capaciy from his pariion funcion (20 ps) (2)
Miderm, NYU Saisical Mechanics 2018 2 MaxwellBolzmann disribuion (90 poins oal) The MaxwellBolzmann disribuion gives he probabiliy of a paricle having velociy v in an ideal gas wih consan (N, V, T) For 1dimension, i is given by: r m P(v) 2pk B T e (a) Some properies of a paricle in 1 dimension: mv2 2k B T (3) i For a paricle in a 1D box of lengh L, wrie he Canonical pariion funcion, and do he all he inegrals o find a consan facor relaed o wha is seen in Eqn 3 (1 ps) ii Using Eqn 3, wha is he average velociy of a paricle, ie hvi R vp(v)dv (10 ps) iii Using Eqn 3, wha is he meansquared velociy of a paricle, ie hv 2 i (Hin: remember how we solved similar problems in he firs homework) (10 ps) iv Wha is he variance of he velociy of a paricle? (10 ps) v Skech he 1D MaxwellBolzmann probabiliy disribuion (Eqn 3) ( ps) (b) n hree dimensions, he probabiliy of finding a velociy vecor is given by he produc of he probabiliies in each direcion, ie m 3/2 P(~v) e 2pk B T m(v 2 x +v2 y +v2 z ) 2k B T m 3/2 e 2pk B T m~v 2 2k B T (4) Changing q o spherical coordinaes, he probabiliy disribuion for he speed s v 2 x + v 2 y + v 2 z is given by: m 3/2 P(s) 4ps 2 e 2pk B T ms 2 2k B T () Reminder: since we re in spherical coordinaes, he average of a variable A is only an inegral over posiive speeds: hai Z 0 dsa(s)p(s) (6) i Skech he 3D MaxwellBolzmann probabiliy disribuion for speed (Eqn ) (10 poins) ii Wha is he roomeansquared speed ( p hs 2 i)? (1 ps) iii Wha is he mos probable speed? How does his compare o he roomeansquared speed? Hin: solve dp(s) ds 0 (1 ps) 2/3
Miderm, NYU Saisical Mechanics 2018 3 One dimensional laice model (60 poins oal) n he homework, we sudied N independen 2 level sysems n his problem, we will line up all of our N sysems and call each one a sie on a laice An example sae of a sysem of size N 9 migh look like he following: Sae (n i ): 1 0 0 1 1 0 0 0 1 Sie (i): 1 2 3 4 6 7 8 9 Figure 1: An example configuraion of a laice wih N 9 saes Sie i can have a value eiher n i 0 or n i 1 (in he example above, n 1 1, n 4 1, n 1 and n 9 1) n one dimension, define he energy of a given sae as H N Â i1 µn i N 1 Â i1 Jn i n i+1 (7) Every ime a sie has value n i 1, he energy goes up by µ, and when 2 sies nex o eachoher have value n i 1 and n i+1 1, hen he energy goes down by J The energy of he configuraion in Fig 1 is H 4µ J (a) Lis all configuraions for a laice of lengh N 3 and heir corresponding energies (16 poins) (b) How many configuraions does a laice of lengh N have? (4 poins) (c) Wrie he Canonical pariion funcion of a sysem of size N a emperaure T (20 poins) (d) Suppose we wan o simulae a sysem wih N 9 using Meropolis Mone Carlo A move is flipping a sie s value from 1 o 0 or 0 o 1 We will sar in he configuraion given in Fig 1 Wrie he Meropolis accepance rule for he following moves (20 poins oal): i Changing he value of sie 7 o a 1 ii Changing he value of sie o a 0 iii Changing he value of sie 6 o a 1 3/3
Miderm GA 2600 Oc 18, 2018 la ) a quanifies how araced molecules Leang are o eachoher i b quanifies how much space hey ake up S b ) i ) p e nr C me Ko 08206 7, )( 373k ) C c ) 149 m %% error 171,0001 1001 1 10010 i ) limo ll am 08206 p ) ( 373k ) mw, L9 4%1 soak 36 36L 62 06 126 am 49 36 % error 0 am % 13
cu v n exp c ) Q LE > Hoi, ( 713? B h b) N Caf p ) off f 3 cons iogpafp ] no p dependence ZNp 2 312 N k BT a N% 1 Cu Off en nky 2) a i ) Q h p g exp c 4h ffdp e T [ Noe : o ge full prob dis, coun Pcu ) f fdpidg e 13714 S ( u Es l Pkm ) ) P weigh S ) fda : e gous/gefephmi1a ( peu five P ) Q, le P mu Jiya 3 of all Saes w/ velociy v e THE
427 iii us a F n o f T e! he five ifengve w o q a 49 10 iii ) Luz > fpulpcudujef?v2eeedv remember Karnik Efi f e C l 3/2 f % KE krslm iu) Var ( u ) Luz ) a krslm 0 krshn o v # peu ) A s b) i ) Pls ) o a i s w XD
, Ere ( m ii ) C 7 foods PCs ) % kno 4 ( as ) f s s e 9 as even so 2 f?s fjass J Yds s4 e m 42kg T remember f!dx Me ( r a %) % Fez a Sk S O a 2 Fas rn ( 2k ) rn a sie 22412 ( Furs ) 2 ) 317 RMS speed FF if 1 K 3 K XD
iii ) mos probable speed, dpao o s ( 4h s ( Fas ) e ms%ks ) s ( S2 e h SYZKBT ) ( divide by cons on boh sides ) s? ( s ) Em las + c hskksqgs, none 23 2g, e m 42k T + e riskies o oher side & divide 2 s e by zg riskies 2 fp sm 12 k mos probable Smos prob FbRF ) ( mos prob s smaller ) 3 XD
3) a ) for N 3, here are 8 can figs energy 000 O o 0 860, } 16 0 2K 110 O } 2µ J 3M FS N XZ Tl 2 2X y i / N 2J b) Every box has 2 con figs & c) Zx 2x all independen so # config, Zq o expffl?iuniiefninid) 20 d) Meropolis accepance prob mind i ) ee µ, expc g pee ) ] ii ) ee iii ) be r µ J 6