Root Mean Square Speed And Mean Free Path

Transcription

1 Disipli Cours-I Sstr-II Papr No: hral Physis : Physis-IIA Lsso: Root Ma Squar Spd Ad Ma Fr Path Lsso Dlopr: Sa Dabas Collg/ Dpartt: Shya Lal Collg, Uirsity of Dlhi

2 abl of Cotts Chaptr Root Ma Squar Spd Ad Ma Fr Path. Itrodutio. Mawll-oltza Law of Distributio of spds. Most probabl spd. Spd Distributio futio.5 Nubr of oluls orrspodig to aiu probabl spd ( N ( ) ) a.6 Ma or arag spd.7 Root a squar spd.8 Ma Fr path.9 Distributio of fr paths. How to dtri a fr path pritally Sold apls Qustios for prati Rfrs

3 Larig Objtis Aftr radig th lsso, you should b abl to ) udrstad th opt of Mawll-oltza Law of Distributio of spds ) dfi ost probabl spd ) dlop th opt of Spd Distributio futio ) Lar th opt of a or arag spd 5) Lar th opt of root a squar spd 6) udrstad th opt of a fr path. 7) dlop th thory of distributio of fr paths. 8) lar how to dtri a fr path pritally

4 Chaptr: itl Root Ma Squar Spd Ad Ma Fr Path. Itrodutio Aordig to iti thory (as w ha disussd i haptr ), a gas osists of a larg ubr of iut disrt partils alld oluls ad ths oluls ar otiuously ollidig with ah othr. Collisios btw ths oluls a hag idiidual olular spds, but th orall arag spd of th syst dos ot gt hagd. h root a squar spd C rais th sa at a fid alu of rs tpratur. At ay istat, all th oluls ar ot oig with th sa spd. So of th oluls ar haig spd gratr tha C ad othrs ar oig with a spd lowr tha this alu. ut th a iti rgy of all th oluls rais ostat at a gi tpratur. Moluls a frqut ollisios with ah othr ad hag th agitud ad dirtio of thir loitis. As th oluls rt o for o o aothr pt durig ollisio, thy tral i straight lis with uifor loity btw two sussi ollisios, this straight li path big alld th fr path. h arag dista tralld by a olul btw sussi ollisios is ow as a fr path.. Mawll-oltza Law of Distributio of spds Cosidr a gas haig ubr of oluls pr.. of th gas with all possibl loitis fro to. hus, th ubr d of oluls haig loity opots lyig btw u ad u du, u y ad u y du y, u z ad uz du z is gi by d f ( u ) f ( u ) f ( u ) du y z du y du z rs d u u u y z du du y du Lt b th spd of th olul whr u u y u z z y z (.) u, u, u ar its loity opots. h (.) his is also th quatio of a sphr of radius. I q (.), du d u d u rprsts olu lt i th loity spa. aig ito y z aout sphrial oordiats, w a prss olu lt as du d u d u y z dsi d d (.) otal olu of th shll btw ad +d is dsi d d d (.)

5 Now, th ubr of oluls with spds i th rag ad +d is (.5) ) ( d d (.6) ) ( d d or (.7) ) ( his is Mawll oltza s law whih plais distributio of spds.. Most probabl spd Most probabl spd is th spd orrspodig to whih th ubr of oluls is aiu i.. ) ( ) ( i.. At, a ) ( or (.8) ) ( d d d d (.9)

6 . Nubr of oluls orrspodig to aiu probabl spd At ( ) a ( ) a (.) As th tpratur of gas irass, th aiu probabl spd irass, but th ubr of oluls orrspodig to it drass. h potial fall of with ( ) bos slow. ut th ara losd by th ur (with -ais) rais th sa i.. th distributio ur sprads. Fig..5 Spd Distributio futio Wh, is or ffti tha, i.. o olul has th zro spd. For lowr alus of, th tr /, thrfor as irass, th ubr of oluls

7 orrspodig to it irass ad for, th ubr of oluls is aiu. If is furthr irasd, th potial tr / bos or prooud, ad thrfor th ubr of oluls ) ( th drass potially with. At, ubr of oluls aishs ad bos zro..6 Ma or arag spd (.) ) ( d whr ubr of oluls is gi by ) ( Substitutig its alu i q(.), w gt (.) d d Usig a stadard forula of a itgral, d w a alulat th alu of d. Hr, alu of is, so d

8 Substitutig this alu of itgral, arag spd bos (.) 8.7 Root a squar spd d d Usig a stadard itgral 8 d d 8 8 á ñ = p p æ è ç ö ø 8 æ è ç ö ø p = p p æ è ç ö ø 8 æ è ç ö ø = (.) rs d ()

9 .8 Eprssio for Ma Fr path Say a partiular olul show by th solid irl has b sigld out. Eah sgt of th path btw two sussi ollisios is alld a fr path. Hr w ha to fid out a fr path, rprstd by..swf Fig. Say, at a partiular ti, all of th oluls of a gas pt o, ar froz i positio. Rst of th oluls otiu to o aog th othrs with a spd qual to th arag spd (as w ha disussd i prious haptr). Lt us osidr oluls to b prftly lasti sphrs haig radius r. At th istat of a ollisio, th tr-to-tr dista of th ollidig oluls is r as show i Fig.. Fig.

10 h tr-to-tr dista would ot gt hagd i as th radius of th oig olul wr irasd to r, whil th statioary oluls wr shru to gotrial poits, as i Fig.. Fig. Collisio ross stio of th oig olul, a b prssd as r h oig olul trarss a dista (.5) u t i a ti itral t, alog a irrgular zigzag path, ad swps out a ylidrial olu of lgth l ad ross stio. Durig this ti itral t, this olul ollids with all th othr oluls whos trs li withi this ylidrial olu. If thr ar oluls pr uit olu, th ubrs with trs i th ylidr swpt out by th oig olul is u t. W ow that a fr path is th arag dista tralld by th olul btw ollisios, or i othr words w a also prss it as th total dista ord by th olul i ti t diidd by th ubr of ollisios durig this ti. H, u t u t (.6) o a it or lar, say,,,.. N ar th sussi fr paths trarsd by th olul i total ti t.... ut N whr u is th arag spd of a olul ad N th ubr of ollisios suffrd i.. fr paths trarsd i ti t, th

11 ... N N ollisios. ut N D N whr D is th total dista tralld i N Valu Additio: MEAN FREE PAH h followig li shows th aiatio of Ma Fr Path h followig aiatio dostrats th alulatio of a loitis of th oluls of gas _oluls_loity_tpratur_ssl_iti_thory_pag_.9 Distributio of fr paths W ha to dri th for of distributio futio of th fr paths. Say N A ar th larg ubr of oluls at a rtai istat. hy ar ollidig with othr oluls ad i ah ollisio, a olul gts rod fro th group. Cosidr N as th ubr that ha ot yt ad a ollisio aftr thos raiig i th group ha tralld a dista s, asurd alog th fr path of ah olul. I th t short dista ds, a rtai ubr of oluls will a ollisios ad gt rod fro th group. It is assud that this ubr is proportioal to th dista ds ad th ubr of oluls N raiig i th group. Si i ah ollisio, a olul is rod fro th group ad N drass, dn Z N ds (.7) whr dn N Z is a proportioality ostat alld th ollisio probability. h Z ds l N Z s ost Wh s=, N N A p N N A Z s (.8) h ubr raiig i th group thrfor falls off potially with s. Substitutig this alu for N, dn bos

12 dn Z N A p Z s ds (.9) Hr dn (osidrd with + sig) rprsts th ubr of oluls with fr paths of lgth btw s ad s+ds. sdn N A Z N sp( Z s) ds A N A Z (.) So, a fr path is irsly proportioal to th ollisio probability Also si, it as Z Z. i.. th ollisio probability is dirtly proportioal to th ollisio ross stio ad th ubr of oluls pr uit olu. Eq(.8) a b ow writt as (usig q(.)), N N A p s N s A dn p ds (.) (.) Fig.5

13 . How to dtri a fr path pritally Ma or dlopd a thiqu to dtri a fr path pritally. Q is a tub ad of quartz. Four brass diss D, D, D, D ar plad o abo th othr. hs diss ar a dista d of apart fro ah othr. Silr is hatd (at th poit X) whih aporats ad passs through th arrow slit ad th through a irular hol i ah of th diss. Eah of ths diss arris a glass quadrat o whih silr atos gt dpositd. his dpositio is du to th oolig fft of oolig itur C. h quadrats ar arragd i suh a way that ah quadrat gts o fourth of th iidt ba. Fig.6 Dsity of dposit o a quadrat is asurd with a irophototr wh all th gas fro th tub is pupd out ad th fr path is qual to th lgth of th habr. h th gas is agai pupd giig a fr path. h dsity o th sa quadrat is agai asurd. Fro th law of distributio of fr paths, r (.) whr r is th dista of th quadrat fro th slit.

14 I th sa way, th dsity ad o th t quadrat at a dista r fro th slit is asurd. r (.) Cobiig q(.) ad q(.), a fr path bos r r log Epritally it was foud that a fr path os out to b.7, wh prssur P 5.8. hus th produt of prssur ad a fr path is ostat as prditd by thory. his good ath btw thory ad prati rifis th law of distributio of fr paths. It hlps us i dtriig a fr path. Suary. Mawll oltza s spd distributio futio is ( ). Most probabl spd is gi by. Nubr of oluls orrspodig to aiu probabl spd ( ( ) ) a ( ) a. Ma or arag spd 8

15 5. Root a squar spd rs 6. Ma fr path is th arag dista whih a olul a tral through a gas without ollidig with aothr olul. 7. h arag dista btw ollisios, or th a fr path, quals th total dista ord i ti t, diidd by th ubr of ollisios i this ti. 8. whr ollisio ross stio is ad is ubr of oluls pr uit olu. Ma fr path is irsly proportioal to ollisio ross stio. 9. At a gi tpratur, th ubr of oluls pr uit olu is dirtly proportioal to th prssur, ad th a fr path is irsly proportioal to th prssur.. h ollisio probability is qual to riproal of a fr path. Sold Eapls Q Calulat root a squar spd at K. Gi th olar ass M of oyg as. Kg/ol. Sol. Root a squar spd is gi by R rs M Hr, =K, M=. Kg/ol, R=8. J/ol K, w gt 8. rs 8 / s. Q What is th ost probabl spd at K if th olar ass M of oyg is.5kg/ol. Sol. Most probabl spd is R M Substitutig alus of =K, M=.5 Kg/ol, R=8. J/ol K, w gt

16 / s.5 Q Fid th a traslatioal iti rgy pr olul of a gas at 7 C. Sol. Ma traslatioal iti rgy pr olul K. E. a whr N K R 8. J / ol K R 8. J / olul K N 6. So, K. E. a R N J. rgs Q Fid th total rado iti rgy of g of Hliu at K? Sol. Ergy for g of Hliu= Ergy for g of Hliu= R M R 5R J M M Q5 Calulat th alu of tpratur at whih will th root a squar spd r..s. of a gas b o-third of its alu at C wh prssur is pt ostat? Sol. Gi iitial alu of r..s. = C = C C C Hr C C

17 7 7 or. K Q6 What will b th root a squar spd of th hydrog oluls at 7 C? Sol. R.M.S. spd of oyg at N..P. = P dys / C P.89 g/ P C 878 / s.89 C C 6 C / s 7 Q7 Fid th a fr path of a gas olul, gi that th olular diatr is 9.5 ad th ubr of oluls pr is. Sol. d 5. (.5 8 ) 9.9 Q8 Calulat th alu of a fr path of th oluls of gas if followig paratrs ar gi. dys /. g / C / s

18 Sol. C... Q9 Fid th a fr path at NP if th diatr of th olul of a gas is.5a. Sol. Hr d.5 A.5.8 J / K Lt b th ubr of oluls pr ubi tr. PV R PV N N V P At NP Ma fr path d 8.(.5 ) Q Fid th frquy of ollisio if gi r..s ? C / s ad a fr path Sol. Frquy of ollisio = C MCQ Q h ratio of Uirsal gas ostat R ad Aagadro s ubr N is alld a) Stfa ostat b) Aagadro ostat

19 ) Rydbrg ostat d) oltza ostat Q At a tpratur, th root a squar loity, is proportioal to rs, of a gas olul haig ass a) b) ) d) Q At a gi tpratur, th arag iti rgy of gasous itrog oluls is a) qual to b) a littl gratr tha ) lss tha d) uh largr tha th arag iti rgy of gasous oyg oluls. Q I a gas th prssio for th arag spd of olul is gi by a) b) ) d) 8

20 Q5 I a gas th rlati agituds of th ost probabl spd ad root a squar spd rs of th olul ar p, th arag spd a) rs p b) rs p ) rs p d) rs p Q6 I a Mawllia gas, if loity p is rs is th root a squar loity, th th ost probabl a) r.. s. b) r.. s. 5 ) r.. s. d) r.. s. Q7 h root a squar spd, arag spd ad th ost probabl spd for a gas ar i th ratio a) : : b) : : ) : : d) :: Q8 If th root a squar spd of th oluls of a gas is doubld, th a) its tpratur will dras by a fator of.

21 b) its tpratur will iras by a fator of. ) its prssur will iras by a fator of. d) its tpratur will dras by a fator of. Q9 h total rado iti rgy of o g-olul of itrog at K is a) 8 J b) 6 J ) 7 J d) 7 J Q Udr th sa oditio of tpratur ad prssur, qual olu of all gass has qual ubr of oluls. his statt orrspods to a) Graha Law of diffusio b) Charls Law ) Aagadro Law d) oyls Law Q A idal gas is pt i a rigid otair. Wh its tpratur is K, th a fr path of th gas oluls is. What will b th a fr path of th oluls at K? a) b) ) d) Q h dista btw oluls at stadard oditios is of th ordr of th a fr path is of th ordr of tis th olular dista. 9, so a) 5 b) )

22 d) Q Collisio frquy i oyg at stadard oditios is ollisios/s. a) b) ) d) Q Ma fr path of th hydrog oluls at N..P. is, wh offiit of 5 5 isosity is 8.g.s. uit ad dsity of hydrog at N..P. is 9 g/. a) b) ) d) Q5 Ma fr path ad diatr d of th olul is rlatd by a) b) ) d d d d) d Aswrs. d. b.. 5. a 6. a 7. b 8. b 9. d.. a. b. a. d 5. a Fill i th blas. h path ord by a olul btw ay two osuti ollisios is alld th.

23 . It dpds upo ad.. Irasig th ubr of oluls or drasig th olu will aus th dsity to.. Irasig th radius of th olul will dras th spa btw th. So, a fr path will. 5. Ma fr path is proportioal to th prssur. 6. h ollisio probability is qual to riproal of. 7. Ma fr path is proportioal to dsity of th gas. 8. rasport phoa ar gord by of a olul. Aswrs. fr path. olular diatr, tpratur. iras. dras 5. irsly 6. a fr path 7. Irsly 8. Ma fr path Rfrs/ ibliography/ Furthr Radig Rfrs. Itrodutio to throdyais Sars. A Itrodutio to throdyais ad Statistial Mhais A.K.Saa. hrodyais, Statistial Physis ad Kitis Satyapraash. Egirig Physis Abhijit Naya 5. Hat ad throdyais rij Lal Subraaya 6. hral Physis S.O.Pillai