MODERN PHYSICS - 1. Work function Metal (ev)

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1 MODERN PHYSICS - PHOTOELECTRIC EFFECT : We eletomageti adiatios of suitable wavelegt ae iidet o a metalli sufae te eletos ae emitted, tis peomeo is alled poto eleti effet.. Potoeleto : Te eleto emitted i potoeleti effet is alled potoeleto.. Potoeleti uet : If uet passes toug te iuit i potoeleti effet te te uet is alled potoeleti uet..3 Wok futio : Te miimum eegy equied to make a eleto fee fom te metal is alled wok futio. It is ostat fo a metal ad deoted by o W. It is te miimum fo Cesium. It is elatively less fo alkali metals. Wok futios of some potosesitive metals Metal Wok futio Wok futio Metal (ev) (ev) Cesium.9 Calium 3. Potassium. Coppe.5 Sodium.3 Silve.7 Litium.5 Platium 5.6 To podue poto eleti effet oly metal ad ligt is eessay but fo obsevig it, te iuit is ompleted. Figue sows a aagemet used to study te potoeleti effet. A C V Reostat itesity fequey A ell, few volts Hee te plate () is alled emitte o atode ad ote plate () is alled olleto o aode.. Satuatio uet : We all te poto eletos emitted by atode ea te aode te uet flowig i te iuit at tat istat is kow as satuated uet, tis is te maximum value of potoeleti uet..5 Stoppig potetial : Miimum magitude of egative potetial of aode wit espet to atode fo wi uet is zeo is alled stoppig potetial. Tis is also kow as utoff voltage. Tis voltage is idepedet of itesity..6 Retadig potetial : Negative potetial of aode wit espet to atode wi is less ta stoppig potetial is alled etadig potetial.

2 . OBSERVATIONS : (MADE BY EINSTEIN). A gap betwee itesity of ligt ad potoeleti uet is foud to be a staigt lie as sow i figue. Potoeleti uet is dietly popotioal to te itesity of iidet adiatio. I tis expeimet te fequey ad etadig potetial ae kept ostat. Potouet PHYSICS O Itesity of ligt. A gap betwee potoeleti uet ad potetial diffee betwee atode ad aode is foud as sow i figue. P S > V S S satuatio uet V V I ase of satuatio uet, ate of emissio of potoeletos ate of flow of potoeletos, ee, v s stoppig potetial ad it is a positive quatity Eletos emitted fom sufae of metal ave diffeet eegies. Maximum kieti eegy of potoeleto o te atode ev s KE max ev s Weeve potoeleti effet takes plae, eletos ae ejeted out wit kieti eegies agig fom to K.E max i.e. KE C ev s Te eegy distibutio of potoeleto is sow i figue. A C No. of Potoeletos Kieti eegy ev S.3 If itesity is ieased (keepig te fequey ostat) te satuatio uet is ieased by same fato by wi itesity ieases. Stoppig potetial is same, so maximum value of kieti eegy is ot effeted.. If ligt of diffeet fequeies is used te obtaied plots ae sow i figue. O It is lea fom gap, as ieases, stoppig potetial ieases, it meas maximum value of kieti eegy ieases.

3 .5 Gaps betwee maximum kieti eegy of eletos ejeted fom diffeet metals ad fequey of ligt used ae foud to be staigt lies of same slope as sow i fiuge k max fo metal's m m m 3 Gap betwee K max ad m, m, m 3 : Tee diffeet metals. t t t 3 It is lea fom gap tat tee is a miimum fequey of eletomageti adiatio wi a podue potoeleti effet, wi is alled tesold fequey. t Tesold fequey Fo potoeleti effet t fo o potoeleti effet < t Miimum fequey fo potoeleti effet t mi t Tesold wavelegt ( t ) Te maximum wavelegt of adiatio wi a podue potoeleti effet. t fo poto eleti effet Maximum wavelegt fo potoeleti effet max t. Now witig equatio of staigt lie fom gap. We ave K max A + B We t, K max ad B A t Hee [K max A( t )] ad A ta J-s (fom expeimetal data) late o A was foud to be..6 It is also obseved tat potoeleti effet is a istataeous poess. We ligt falls o sufae eletos stat ejetig witout takig ay time. 3. THREE MAJOR FEATURES OF THE PHOTOELECTRIC EFFECT CANNOT BE EXPLAINED IN TERMS OF THE CLASSICAL WAVE THEORY OF LIGHT. Itesity : Te eegy ossig pe uit aea pe uit time pepediula to te dietio of popagatio is alled te itesity of a wave. Coside a ylidial volume wit aea of osssetio A ad legt t alog te X-axis. Te eegy otaied i tis ylide osses te aea A i time t as te wave popagates at speed. Te eegy otaied. U u av (. t)a U Te itesity is A t u. av I te tems of maximum eleti field, t E. If we oside ligt as a wave te te itesity depeds upo eleti field. If we take wok futio W. A. t, te t A W so fo potoeleti effet tee sould be time lag beause te metal as wok futio. But it is obseved tat potoeleti effet is a istataeous poess. Hee, ligt is ot of wave atue. A x 3

4 3. Te itesity poblem : Wave teoy equies tat te osillatig eleti field veto E of te ligt wave ieases i amplitude as te itesity of te ligt beam is ieased. Sie te foe applied to te eleto is ee, tis suggests tat te kieti eegy of te potoeletos sould also ieased as te ligt beam is made moe itese. Howeve obsevatio sows tat maximum kieti eegy is idepedet of te ligt itesity. 3. Te fequey poblem : Aodig to te wave teoy, te potoeleti effet sould ou fo ay fequey of te ligt, povided oly tat te ligt is itese eoug to supply te eegy eeded to ejet te potoeletos. Howeve obsevatios sows tat tee exists fo ea sufae a aatesti utoff fequey t, fo fequeies less ta t, te potoeleti effet does ot ou, o matte ow itese is ligt beam. 3.3 Te time delay poblem : If te eegy aquied by a potoeleto is absobed dietly fom te wave iidet o te metal plate, te effetive taget aea fo a eleto i te metal is limited ad pobably ot mu moe ta tat of a ile of diamete ougly equal to tat of a atom. I te lassial teoy, te ligt eegy is uifomly distibuted ove te wavefot. Tus, if te ligt is feeble eoug, tee sould be a measuable time lag, betwee te impigig of te ligt o te sufae ad te ejetio of te potoeleto. Duig tis iteval te eleto sould be absobig eegy fom te beam util it ad aumulated eoug to esape. Howeve, o detetable time lag as eve bee measued. Now, quatum teoy solves tese poblems i povidig te oet itepetatio of te potoeleti effet. PLANCK S QUANTUM THEORY : Te ligt eegy fom ay soue is always a itegal multiple of a smalle eegy value alled quatum of ligt.ee eegy Q NE, wee E ad N (umbe of potos),,3,... Hee eegy is quatized. is te quatum of eegy, it is a paket of eegy alled as poto. E ad ev Å 5. EINSTEIN S PHOTON THEORY I 95 Eistei made a emakable assumptio about te atue of ligt; amely, tat, ude some iumstaes, it beaves as if its eegy is oetated ito loalized budles, late alled potos. Te eegy E of a sigle poto is give by E, If we apply Eistei s poto oept to te potoeleti effet, we a wite W + K max, (eegy osevatio) Equatio says tat a sigle poto aies a eegy ito te sufae wee it is absobed by a sigle eleto. Pat of tis eegy W (alled te wok futio of te emittig sufae) is used i ausig te eleto to esape fom te metal sufae. Te exess eegy ( W) beomes te eleto s kieti eegy; if te eleto does ot lose eegy by iteal ollisios as it esapes fom te metal, it will still ave tis mu kieti eegy afte it emeges. Tus K max epesets te maximum kieti eegy tat te potoeleto a ave outside te sufae. Tee is omplete ageemet of te poto teoy wit expeimet. A Now A N N o. of potos iidet pe uit time o a aea A we ligt of itesity is iidet omally. If we double te ligt itesity, we double te umbe of potos ad tus double te potoeleti uet; we do ot age te eegy of te idividual potos o te atue of te idividual potoeleti poesses. Te seod objetio (te fequey poblem) is met if K max equals zeo, we ave t W, Wi assets tat te poto as just eoug eegy to ejet te potoeletos ad oe exta to appea as kieti eegy. If is edued below t, will be smalle ta W ad te idividual potos, o matte ow may of tem tee ae (tat is, o matte ow itese te illumiatio), will ot ave eoug eegy to ejet potoeletos. Te tid objetio (te time delay poblem) follows fom te poto teoy beause te equied eegy is supplied i a oetated budle. It is ot spead uifomly ove te beam oss setio as i te wave teoy.

5 Hee Eistei s equatio fo potoeleti effet is give by t + K max K max Ex. I a expeimet o poto eleti emissio, followig obsevatios wee made; (i) Wavelegt of te iidet ligt.98 7 m; (ii) Stoppig potetial.5 volt. Fid : (a) Kieti eegy of potoeletos wit maximum speed. (b) Wok futio ad () Tesold fequey; Sol. (a) Sie v s.5 V, K max ev s so, K max.5 ev (b) Eegy of iidet poto E 98 ev 6.6 ev W E K max 3.76 ev t Ex () t W J t 9. Hz A beam of ligt osists of fou wavelegt Å, 8 Å, 6 Å ad 7 Å, ea of itesity.5 3 Wm. Te beam falls omally o a aea m of a lea metalli sufae of wok futio.9 ev. Assumig o loss of ligt eegy (i.e. ea apable poto emits oe eleto) alulate te umbe of potoeletos libeated pe seod. Sol. E 3. ev, E.58 ev E3.6 ev 8 6 ad E.77 ev 7 Teefoe, ligt of wavelegts Å, 8 Å ad 6 Å a oly emit potoeletos. Numbe of potoeletos emitted pe seod No. of potos iidet pe seod) A E + (.5 A E 3 )( As. 3A E ) A E E E Ex. 3 A small potassium foil is plaed (pepediula to te dieito of iidee of ligt) a distae (.5 m) fom a poit ligt soue wose output powe P is.w. Assumig wave atue of ligt ow log would it take fo te foil to soak up eoug eegy (.8 ev) fom te beam to ejet a eleto? Assume tat te ejeted potoeleto olleted its eegy fom a iula aea of te foil wose adius equals te adius of a potassium atom (.3 m). Sol. If te soue adiates uifomly i all dietios, te itesity of te ligt at a distae is give by P. W (.5m).3 W/m. Te taget aea A is (.3 m) o 5.3 m, so tat te ate at wi eegy falls o te taget is give by P A (.3 W/m ) (5.3 m ).7 J/s. If all tis iomig eegy is absobed, te time equied to aumulate eoug eegy fo te eleto to esape is.8ev 9 t.7 J/ s.6 J ev 7 s. Ou seletio of a adius fo te effetive taget aea was some-wat abitay, but o matte wat easoable aea we oose, we sould still alulate a soak-up time witi te age of easy measuemet. Howeve, o time delay as eve bee obseved ude ay iumstaes, te ealy expeimets settig a uppe limit of about 9 s fo su delays. 5

6 Ex. A metalli sufae is iadiated wit mooomati ligt of vaiable wavelegt. Above a wavelegt of 5 Å, o potoeletos ae emitted fom te sufae. Wit a ukow wavelegt, stoppig potetial is 3 V. Fid te ukow wavelegt. Sol. Usig equatio of potoeleti effet K max E W (K max ev s ) Ex. 5 Sol. Ex. 6 3 ev 5 o 6 Å.8 ev Illumiatig te sufae of a etai metal alteately wit ligt of wavelegts.35 m ad.5 m, it was foud tat te oespodig maximum veloities of poto eletos ave a atio. Fid te wok futio of tat metal. Usig equatio fo two wavelegts mv mv Dividig Eq. (i) wit Eq. (ii), wit v v, we ave W W W 3W ev.88 ev As. W W 5...(i)...(ii) 5.6 ev 35 A potoell is opeatig i satuatio mode wit a potouet.8 A we a mooomati adiatio of wavelegt 3 Å ad powe mw is iidet. We aote mooomati adiatio of wavelegt 65 Å ad powe 5 mw is iidet, it is obseved tat maximum veloity of potoeleto ieases to two times. Assumig effiiey of potoeleto geeatio pe iidet to be same fo bot te ases, alulate, (a) tesold wavelegt fo te ell (b) effiiey of potoeleto geeatio. [(No. of potoeletos emitted pe iidet poto) ] () satuatio uet i seod ase Sol. (a) K W.3 W...(i) 3 K W 7.5 W...(ii) 65 Sie v v so, K K...(iiii) Solvig above equatios, we get W 3 ev Tesold wavelegt 33 Å As. 3 (b) Eegy of a poto i fist ase.3 ev 3 o E J Rate of iidet potos (umbe of potos pe seod) P 3 E pe seod 6.8 Numbe of eletos ejeted 9.6 pe seod 6

7 () 3. 3 pe seod Effiiey of potoeleto geeatio Eegy of poto i seod ase 3 3. () 5.5 % As. Teefoe, umbe of potos iidet pe seod E 7.5 ev 9 J 65 P 3 5. E pe seod Ex. 7 Sol. Numbe of eletos emitted pe seod pe seod Satuatio uet i seod ase i (9. 3 ) (.6 9 ) amp 5 A As. Ligt desibed at a plae by te equatio E ( V/m) [si (5 5 s ) t + si (8 5 s )t] falls o a metal sufae avig wok futio. ev. Calulate te maximum kieti eegy of te potoeletos. Te ligt otais two diffeet fequeies. Te oe wit lage fequey will ause potoeletos wit lagest kieti eegy. Tis lage fequey is 8 s Te maximum kieti eegy of te potoeletos is K max W 5 5 (. 5 ev-s) 8 s. ev 5.7 ev. ev 3.7 ev. 6 FORCE DUE TO RADIATION (PHOTON) Ea poto as a defiite eegy ad a defiite liea mometum. All potos of ligt of a patiula wavelegt ave te same eegy E / ad te same magitude of mometum p /. We ligt of itesity falls o a sufae, it exets foe o tat sufae. Assume absoptio ad efletio oeffiiet of sufae be a ad ad assumig o tasmissio. Assume ligt beam falls o sufae of sufae aea A pepediulaly as sow i figue. Fo alulatig te foe exeted by beam o sufae, we oside followig ases. Case : (I) a, iitial mometum of te poto fial mometum of poto age i mometum of poto (upwad) P eegy iidet pe uit time A o. of potos iidet pe uit time A A total age i mometum pe uit time P 7

8 A A (upwad) foe o potos total age i mometum pe uit time A (upwad) A foe o plate due to potos(f) (dowwad) F A pessue A A Case : (II) we, a itial mometum of te poto fial mometum of poto (dowwad) (upwad) age i mometum + eegy iidet pe uit time A A o. of potos iidet pe uit time total age i mometum pe uit time foe total age i mometum pe uit time pessue F A F A P A A Case : (III) We o < < a +. P A age i mometum of poto we it is efleted age i mometum of poto we it is absobed. A C (upwad o potos ad dowwad o te plate) (upwad) (upwad) o. of potos iidet pe uit time No. of potos efleted pe uit time A A. No. of poto absobed pe uit time A ( ) foe due to absobed poto (F a ) A ( ). A ( ) (dowwad) Foe due to efleted poto (F ) total foe F a + F (dowwad) A ( ) + A A. A (dowwad) 8

9 A A Now pessue P ( + ) ( + ) A ( + ) Ex. 8 Sol. Ex. 9 A plate of mass gm is i equilibium i ai due to te foe exeted by ligt beam o plate. Calulate powe of beam. Assume plate is pefetly absobig. Sie plate is i ai, so gavitatioal foe will at o tis F gavitatioal mg (dowwad) 3 N fo equilibium foe exeted by ligt beam sould be equal to F gavitatioal F poto F gavitatioal Let powe of ligt beam be P F poto P P P 3. 8 P 3 7 W Calulate foe exeted by ligt beam if ligt is iidet o sufae at a agle as sow i figue. Coside all ases. Sol. Case - I a, iitial mometum of poto (i dowwad dietio at a agle wit vetial) fial mometum of poto age i mometum (i upwad dietio at a agle wit vetial) eegy iidet pe uit time A os Itesity powe pe uit omal aea P A os P A os A os No. of potos iidet pe uit time. total age i mometum pe uit time (i upwad dietio at a agle wit vetial) A os. A os. Foe (F) total age i mometum pe uit time A os F (dietio o poto ad Pessue omal foe pe uit Aea Fos Pessue A P Case II We, a age i mometum of oe poto No. of potos iidet pe uit time os (upwad) A os A o te plate) os si [ ] [ os [ ] ] os si 9

10 eegy iidet pe uit time A os total age i mometum pe uit time A os A os os (upwad) foe o te plate A os (dowwad) Pessue A os A P os Case III < <, a + age i mometum of poto we it is efleted os (dowwad) age i mometum of poto we it is absobed (i te opposite dietio of iidet beam) eegy iidet pe uit time A os o. of potos iidet pe uit time o. of efleted poto ( ) A os. A os A os. o. of absobed poto ( a ) ( ) foe o plate due to absobed potos F a a. P a A os. ( ) A os ( ) (at a agle wit vetial foe o plate due to efleted potos F P A os os (vetially dowwad) A os ) ow esultat foe is give by F R F a F F F os a ad, pessue P A os Fa os F A ( ) () os ( )os A os( )os A + A os A os ( ) + os os ( + )

11 Ex. A pefetly efletig solid spee of adius is kept i te pat of a paallel beam of ligt of lage apetue. If te beam aies a itesity, fid te foe exeted by te beam o te spee. Sol. Let O be te ete of te spee ad OZ be te lie opposite to te iidet beam (figue). Coside a adius OP of te spee R p makig a agle wit OZ. Rotate tis adius about OZ to get a Q ile o te spee. Cage to + d ad otate te adius about OZ to get aote ile o te spee. Te pat of te spee betwee tese iles is a ig of aea si d. Coside a small pat Aof tis ig at P. Eegy of te ligt fallig o tis pat i time t is U t(a os ) o Z Te mometum of tis ligt fallig o Ais U/ alog QP. Te ligt is efleted by te spee alog PR. Te age i mometum is U p os t (A os ) (dietio alog OP ) Te foe o A due to te ligt falig o it, is p A os. (dietio alog t PO ) Te esultat foe o te ig as well as o te spee is alog ZO by symmety. Te ompoet of te foe o A alog ZO p os A os3. (alog t ZO ) Te foe atig o te ig is df ( si d)os 3. Te foe o te etie spee is F / / os 3 si d / os 3 d(os ) os Note tat itegatio is doe oly fo te emispee tat faes te iidet beam. 7. de-broglie WAVELENGTH OF MATTER WAVE A poto of fequey ad wavelegt as eegy. / E By Eistei s eegy mass elatio, E m te equivalet mass m of te poto is give by, E m...(i) o o m p...(ii) Hee p is te mometum of poto. By aalogy de-boglie suggested tat a patile of mass m movig wit speed v beaves i some ways like waves of wavelegt (alled de-boglie wavelegt ad te wave is alled matte wave) give by,... (iii) mv p wee p is te mometum of te patile. Mometum is elated to te kieti eegy by te equatio, p Km ad a age q we aeleated by a potetial diffeee V gais a kieti eegy K qv. Combiig all tese elatios Eq. (iii), a be witte as, mv p Km qvm (de-boglie wavelegt)...(iv)

12 7. de-boglie wavelegt fo a eleto If a eleto (age e) is aeleated by a potetial of V volts, it aquies a kieti eegy, K ev Substitutig te values of, m ad q i Eq. (iv), we get a simple fomula fo alulatig de-boglie wavelegt of a eleto. 5 ( i Å) V(i volts)...(v) 7. de-boglie wavelegt of a gas moleule : Let us oside a gas moleule at absolute tempeatue T. Kieti eegy of gas moleule is give by K.E. 3 kt ; k Boltzma ostat gas moleule 3mkT Ex. A eleto is aeleated by a potetial diffeee of 5 volt. Fid te de-boglie wavelegt assoiated wit it. Sol. Fo a eleto, de-boglie wavelegt is give by, 5 5 V Å As. Ex. Fid te atio of De-Boglie wavelegt of moleules of ydoge ad elium wi ae at tempeatues 7ºC ad 7ºC espetively. Sol. de-boglie wavelegt is give by H He m m He T T He H H (7 73). (7 73) 8. THOMSON S ATOMIC MODEL : J.J. Tomso suggested tat atoms ae just positively age lumps of matte wit eletos embedded i tem like aisis i a fuit ake. Tomso s model alled te plum puddig model is illustated i figue. 8 3 Eleto Positively aged matte Tomso played a impotat ole i disoveig te eleto, toug gas disage tube by disoveig atode ays. His idea was take seiously. But te eal atom tued out to be quite diffeet. 9. RUTHERFORD S NUCLEAR ATOM : Rutefod suggested tat; All te positive age ad ealy all te mass wee oetated i a vey small volume of uleus at te ete of te atom. Te eletos wee supposed to move i iula obits oud te uleus (like plaets oud te su). Te eletostati attatio betwee te two opposite ages beig te equied etipetal foe fo su motio. mv kze Hee ad total eegy potetial eegy + kieti eegy kze Rutefod s model of te atom, altoug stogly suppoted by evidee fo te uleus, is iosistet wit lassial pysis. Tis model suffe s fom two defets

13 9. Regadig stability of atom : A eleto movig i a iula obit oud a uleus is aeleatig ad aodig to eletomageti teoy it sould teefoe, emit adiatio otiuously ad teeby lose eegy. If total eegy deeases te adius ieases as give by above fomula. If tis appeed te adius of te obit would deease ad te eleto would spial ito te uleus i a fatio of seod. But atoms do ot ollapse. I 93 a effot was made by Neil Bo to oveome tis paadox. 9. Regadig explaatio of lie spetum : I Rutefod s model, due to otiuously agig adii of te iula obits of eletos, te fequey of evolutio of te eletos must be agig. As a esult, eletos will adiate eletomageti waves of all fequeies, i.e., te spetum of tese waves will be otiuous i atue. But expeimetally te atomi speta ae ot otiuous. Istead tey ae lie speta.. THE BOHR S ATOMIC MODEL I 93, Pof. Niel Bo emoved te diffiulties of Rutefod s atomi model by te appliatio of Plak s quatum teoy. Fo tis e poposed te followig postulates () A eleto moves oly i etai iula obits, alled statioay obits. I statioay obits eleto does ot emit adiatio, otay to te peditios of lassial eletomageti teoy. () Aodig to Bo, tee is a defiite eegy assoiated wit ea stable obit ad a atom adiaties eegy oly we it makes a tasitio fom oe of tese obits to aote. If te eegy of eleto i te ige obit be E ad tat i te lowe obit be E, te te fequey of te adiated waves is give by E E E E o (3) Bo foud tat te magitude of te eleto s agula mometum is quatized, ad tis magitude fo te eleto must be itegal multiple of. Te magitude of te agula mometum is L mv fo a patile wit mass m movig wit speed v i a ile of adius. So, aodig to Bo s postulate, mv (,, 3...) Ea value of oespods to a pemitted value of te obit adius, wi we will deote by. Te value of fo ea obit is alled piipal quatum umbe fo te obit. Tus, mv mv...(ii) Aodig to Newto s seod law a adially iwad etipetal foe of magitude F eeded by te eleto wi is beig povided by te eletial attatio betwee te positive poto ad te egative eleto. mv e Tus, Solvig Eqs. (ii) ad (iii), we get...(iii) me...(iv) ad e v...(v) Te smallest obit adius oespods to. We ll deote tis miimum adius, alled te Bo adius as a. Tus, a me Substitutig values of,, p, m ad e, we get a.59 m.59 Å...(vi) mv is Eq. (iv), i tems of a a be witte as, a o...(vii) 3

14 Similaly, substitutig values of e, ad wit i Eq. (v), we get v.9 6 m/s...(viii) PHYSICS Tis is te geatest possible speed of te eleto i te ydoge atom. Wi is appoximately equal to /37 wee is te speed of ligt i vauum. Eq. (v), i tems of v a be witte as, v v o v Eegy levels : Kieti ad potetial eegies K ad U i t obit ae give by me K mv 8 ad U e (assumig ifiity as a zeo potetial eegy level) me Te total eegy E is te sum of te kieti ad potetial eegies. me so, E K + U 8 Substitutig values of m, e, ad wit, we get te least eegy of te atom i fist obit, wi is 3.6 ev. Hee, E 3.6 ev...(x) E 3.6 ad E ev...(xi) Substitutig, 3,,..., et., we get eegies of atom i diffeet obits. E 3. ev, E 3.5 ev,... E. Hydoge Like Atoms Te Bo model of ydoge a be exteded to ydoge like atoms, i.e., oe eleto atoms, te ulea age is +ze, wee z is te atomi umbe, equal to te umbe of potos i te uleus. Te effet i te pevious aalysis is to eplae e evey wee by ze. Tus, te equatios fo,, v ad E ae alteed as ude: a mze z o z wee a.59 Å (adius of fist obit of H) ze z z v v o v wee v.9 6 m/s (speed of eleto i fist obit of H)...(i)...(ii) mz e z z E 8 E o E...(iii) wee E 3.6 ev (eegy of atom i fist obit of H). Defiatios valid fo sigle eleto system () Goud state : Lowest eegy state of ay atom o io is alled goud state of te atom. Goud state eegy of Hatom 3.6 ev Goud state eegy of He + Io 5. ev Goud state eegy of Li ++ Io. ev () Exited State : State of atom ote ta te goud state ae alled its exited states. fist exited state 3 seod exited state tid exited state + t exited state

15 (3) Ioisatio eegy (E.) : Miimum eegy equied to move a eleto fom goud state to is alled ioisatio eegy of te atom o io Ioisatio eegy of H atom 3.6 ev Ioisatio eegy of He + Io 5. ev Ioisatio eegy of Li ++ Io. ev () Ioisatio potetial (I.P.) : Potetial diffeee toug wi a fee eleto must be aeleated fom est su tat its kieti eegy beomes equal to ioisatio eegy of te atom is alled ioisatio potetial of te atom. I.P of H atom 3.6 V I.P. of He + Io 5. V (5) Exitatio eegy : Eegy equied to move a eleto fom goud state of te atom to ay ote exited state of te atom is alled exitatio eegy of tat state. Eegy i goud state of H atom 3.6 ev Eegy i fist exited state of H-atom 3. ev st exitatio eegy. ev. (6) Exitatio Potetial : Potetial diffeee toug wi a eleto must be aeleated fom est so tat its kieti eegy beomes equal to exitatio eegy of ay state is alled exitatio potetial of tat state. st exitatio eegy. ev. st exitatio potetial. V. (7) Bidig eegy o Sepeatio eegy : Eegy equied to move a eleto fom ay state to is alled bidig eegy of tat state. o eegy eleased duig fomatio of a H-like atom/io fom to some patiula is alled bidig eegy of tat state. Bidig eegy of goud state of H-atom 3.6 ev Ex. 3 Fist exitatio potetial of a ypotetial ydoge like atom is 5 volt. Fid tid exitatio potetial of te atom. Sol. Let eegy of goud state E E E 3.6 Z ev ad E give E, E E E 5 3E 5 E fo, E 6 E tid exiitatio eegy 6 E E ev 75 tid exitatio potetial is V.3 Emissio spetum of ydoge atom : H-gas Pism See Ude omal oditios te sigle eleto i ydoge atom stays i goud state ( ). It is exited to some ige eegy state we it aquies some eegy fom exteal soue. But it adaly stays tee fo moe ta 8 seod. 5

16 A poto oespodig to a patiula spetum lie is emitted we a atom makes a tasitio fom a state i a exited level to a state i a lowe exited level o te goud level. Let i be te iitial ad f te fial eegy state, te depedig o te fial eegy state followig seies ae obseved i te emissio spetum of ydoge atom. O See : A potogap of spetal lies of te Lyme, Balme, Pase seies of atomi ydoge. Pase seies Lyma seies Balme seies 3 3 Wavelegt (ieasig ode),, 3... epesets te I, II & III lie of Lyme, Balme, Pase seies. 6

17 Te ydoge spetum (some seleted lies) Name of seies Lyme Balme Pase Numbe of Lie Quatum Numbe i (Low e State f (Uppe State) W avelegt (m) Eegy I.6. ev II ev III ev seies limit (seies limit) ev I ev II ev III ev seies limit (seies limit) ev I ev II ev III ev seies limit 3 (seies limit) 8.5 ev Seies limit : Lie of ay goup avig maximum eegy of poto ad miimum wavelegt of tat goup is alled seies limit. Lyme Pfud seies Bakett seies seies 7.8eV 6.38eV 5.5eV.85eV 3 Balme seies Pase seies Fo te Lyme seies f, fo Balme seies f ad so o..5ev 3.eV 3.6eV. Wavelegt of Poto Emitted i De-exitatio Aodig to Bo we a atom makes a tasitio fom ige eegy level to a lowe level it emits a poto wit eegy equal to te eegy diffeee betwee te iitial ad fial levels. If E i is te iitial eegy of te atom befoe su a tasitio, E f is its fial eegy afte te tasitio, ad te poto s eegy is, te osevatio of eegy gives, Ei E f (eegy of emitted poto)...(i) By 93, te spetum of ydoge ad bee studied itesively. Te visible lie wit logest wavelegt, o lowest fequey is i te ed ad is alled H, te ext lie, i te blue-gee is alled H ad so o. I 885, Joa Balme, a Swiss teae foud a fomula tat gives te wave legts of tese lies. Tis is ow alled te Balme seies. Te Balme s fomula is, R...(ii) Hee, 3,, 5..., et. R Rydbeg ostat.97 7 m ad is te wavelegt of ligt/poto emitted duig tasitio, Fo 3, we obtai te wavelegt of H lie. Similaly, fo, we obtai te wavelegt of H lie. Fo, te smallest wavelegt ( 366 7

18 Å) of tis seies is obtaied. Usig te elatio, E we a fid te poto eegies oespodig to te wavelegt of te Balma seies. R R E R Tis fomula suggests tat, R E,,, (iii) Te wavelegts oespodig to ote spetal seies (Lyme, Pase, (et.) a be epeseted by fomula simila to Balme s fomula. Lyme Seies : Pase Seies : Bakett Seies : Pfud Seies : R,, 3,... R 3 R R 5,, 5, 6..., 5, 6, 7..., 6, 7, 8 Te Lyme seies is i te ultaviolet, ad te Pase. Bakett ad Pfud seies ae i te ifaed egio. Ex. Calulate (a) te wavelegt ad (b) te fequey of te H lie of te Balme seies fo ydoge. Sol. (a) H lie of Balme seies oespods to te tasitio fom to level. Te oespodig wavelegt fo H lie is, 7 (.97 ) m As. 3. (b) Hz As. 8 Ex. 5 Fid te lagest ad sotest wavelegts i te Lyme seies fo ydoge. I wat egio of te eletomageti spetum does ea seies lie? Sol. Te tasitio equatio fo Lyme seies is give by, R () fo lagest wavelegt,, 3, max 7 max.5 7 m 5 Å As. Te sotest wavelegt oespods to.97 7 max o mi.9 7 m 9 Å As. Bot of tese wavelegts lie i ultaviolet (UV) egio of eletomageti spetum. Ex. 6 How may diffeet wavelegts may be obseved i te spetum fom a ydoge sample if te atoms ae exited to states wit piipal quatum umbe? Sol. Fom te t state, te atom may go to ( )t state,..., d state o st state. So tee ae ( ) possible tasitios statig fom te t state. Te atoms eaig ( )t state may make ( ) diffeet tasitios. Similaly fo ote lowe states. Te total umbe of possible tasitios is ( ) + ( ) + ( 3) ( ) 8

19 Ex. 7 (a) Fid te wavelegt of te adiatio equied to exite te eleto i Li ++ fom te fist to te tid Bo obit. (b) How may spetal liea ae obseved i te emissio spetum of te above exited system? Sol. (a) Te eegy i te fist obit E Z E wee E 3.6 ev is te eegy of a ydoge atom i goud state tus fo Li ++, E 9E 9 ( 3.6 ev). ev Te eegy i te tid obit is E E E ev 9 Tus, E 3 E ev 8.8 ev. Eegy equied to exite Li ++ fom te fist obit to te tid obit is give by E 3 E ev 8.8 ev. Te wavelegt of adiatio equied to exite Li ++ fom te fist obit to te tid obit is give by o, E 3 E E 3 E ev m.m 8.8 ev (b) Te spetal lies emitted ae due to te tasitios 3, 3 ad. Tus, tee will be tee spetal lies i te spetum. Ex. 8 Fid te kieti eegy potetial eegy ad total eegy i fist ad seod obit of ydoge atom if potetial eegy i fist obit is take to be zeo. Sol. E 3.6 ev K E 3.6 ev U E 7. ev E E () 3. ev K 3. ev ad U 6.8 ev Now U, i.e., potetial eegy as bee ieased by 7. ev wile kieti eegy will emai uaged. So values of kieti eegy, potetial eegy ad total eegy i fist obit ae 3.6 ev,, 3.6 espetively ad fo seod obit tese values ae 3. ev,. ev ad 3.8 ev. Ex. 9 A litium atom as tee eletos, Assume te followig simple pitue of te atom. Two eletos move lose to te uleus makig up a speial loud aoud it ad te tid moves outside tis loud i a iula obit. Bo s model a be used fo te motio of tis tid eleto but states ae ot available to it. Calulate te ioizatio eegy of litium i goud state usig te above pitue. Sol. I tis pitue, te tid eleto moves i te field of a total age + 3e e + e. Tus, te eegies ae te same as tat of ydoge atoms. Te lowest eegy is : E 3.6 ev E 3. ev Tus, te ioizatio eegy of te atom i tis pitue is 3. ev. Ex. Te eegy levels of a ypotetial oe eleto atom ae sow i te figue. (a) Fid te ioizatio potetial of tis atom. (b) Fid te sot wavelegt limit of te seies temiatig at () Fid te exitatio potetial fo te state 3. (d) Fid wave umbe of te poto emitted fo te tasitio 3 to. (e) Wat is te miimum eegy tat a eleto will ave afte iteatig wit tis atom i te goud state if te iitial kieti eegy of te eleto is (i) 6 ev (ii) ev ev ev.8 ev.5 ev 3.8 ev 5.3 ev 5.6 ev 9

20 Sol. (a) Ioizatio potetial 5.6 V (b) mi 3 Å 5.3 () E ( 5.6).5 ev Teefoe, exitatio potetial fo state 3 is.5 volt. (d) 3 E 3.5 Å Å. 7 m (e) (i) E E.3 ev > 6 ev. Hee eleto aot exite te atoms. So, K mi 6 ev. (ii) E E.3 ev < ev. Hee eleto a exite te atoms. So, K mi (.3).7 ev. Ex. A small patile of mass m moves i su a way tat te potetial eegy U a wee a is a ostat ad is te distae of te patile fom te oigi. Assumig Bo s model of quatizatio of agula mometum ad iula obits, fid te adius of t allowed obit. Sol. Te foe at a distae is, du F a d Suppose be te adius of t obit. Te eessay etipetal foe is povided by te above foe. Tus, mv a Fute, te quatizatio of agula mometum gives, mv Solvig Eqs. (i) ad (ii) fo, we get 8am / Ex. A imagiay patile as a age equal to tat of a eleto ad mass times te mass of te eleto. It moves i a iula obit aoud a uleus of age + e. Take te mass of te uleus to be ifiite. Assumig tat te Bo s model is appliable to te system. (a) Deive ad expessio fo te adius of t Bo obit. (b) Fid te wavelegt of te adiatio emitted we te patile jumps fom fout obit to te seod. mpv ze Sol. (a) We ave Te quatizatio of agula mometum gives, m p v Solvig Eqs. (i) ad (ii), we get zm e Substitutig m p m wee m mass of eleto ad z p As....(i)...(ii) (b) we get, me As. As we kow, Eegy of ydoge atom i goud state 3.6 ev z ad E m

21 (3.6)() Fo te give patile, E () 36 ev (3.6)() ad E () DE E E 8 ev 5 ev (i Å) 3. Å As. 8 Ex. 3 A patile kow as -measo, as a age equal to tat of a eleto ad mass 8 times te mass of te eleto. It moves i a iula obit aoud a uleus of age +3e. Take te mass of te uleus to be ifiite. Assumig tat te Bo s model is appliable to tis system, (a) deive a expessio fo te adius of te t Bo obit, (b) fid te value of fo wi te adius of te obit is appoximately te same as tat of te fist Bo obit fo a ydoge atom ad () fid te wavelegt of te adiatio emitted we te meso jumps fom te tid obit to te fist obit. Sol. (a) We ave, o, mv Ze Ze v m Te quatizatio ule is v Te adius is (v) v m m Ze Zme Fo te give system, Z 3 ad m 8 m e. Tus 6m e e...(i)...(ii) (b) Fom (ii), te adius of te fist Bo obit fo te ydoge atom is m ee Fo, 6m e e mee o, 6 o, 5 () Fom (i), te kieti eegy of te atom is mv Ze 8 ad te potetial eegy is Ze Te total eegy is E Ze 8 Z me Usig (ii), E 8 9 8m 8 e 87 mee 8

22 But mee 8 is te goud state eegy of ydoge atom ad ee is equal to 3.6 ev eV Fom (iii), E 3.6 ev Tus, E 559. ev ad E 3 9 E 88.8 ev. Te eegy diffeee is E3 E 63.eV. Te wavelegt emitted is ev m E 63.eV 55 pm. Ex. A gas of ydoge like atoms a absob adiatios of 68 ev. Cosequetly, te atoms emit adiatios of oly tee diffeet wavelegt. All te wavelegts ae equal o smalle ta tat of te absobed poto. (a) Detemie te iitial state of te gas atoms. (b) Idetify te gas atoms. () Fid te miimum wavelegt of te emitted adiatios. (d) Fid te ioizatio eegy ad te espetive wavelegt fo te gas atoms. ( ) Sol. (a) 3 3 i.e., afte exitatio atom jumps to seod exited state. Hee f 3. So i a be o If i te eegy emitted is eite equal to, geate ta o less ta te eegy absobed. Hee te emitted wavelegt is eite equal to, less ta o geate ta te absobed wavelegt. Hee i. If i, te E e E a. Hee e (b) E 3 E 68 ev () mi (3.6) (Z ) 9 68 Z 6 E 3 E (3.6)(6) As. 35. (d) Ioizatio eegy (3.6) (6) 89.6 ev As Å As. Ex. 5 A eleto is obitig i a iula obit of adius ude te ifluee of a ostat mageti field of stegt B. Assumig tat Bo s postulate egadig te quatisatio of agula mometum olds good fo tis eleto, fid (a) te allowed values of te adius of te obit. (b) te kieti eegy of te eleto i obit () Te potetial eegy of iteatio betwee te mageti momet of te obital uet due to te eleto movig i its obit ad te mageti field B. (d) Te total eegy of te allowed eegy levels. Sol. (a) adius of iula pat mv Be...(i) mv Solvig tese two equatios, we get...(ii) Be Be ad v m

23 (b) K mv Be m As. e ev () M ia ( ) T Now potetial eegy e Be U M. B Be m e m eb m (d) E U + K eb m. EFFECT OF NUCLEUS MOTION ON ENERGY OF ATOM Let bot te uleus of mass M, age Ze ad eleto of mass m, ad age e evolve about tei ete of mass (CM) wit same agula veloity () but diffeet liea speeds. Let ad be te distae of CM fom uleus ad eleto. Tei agula veloity sould be same te oly tei sepaatio will emai uaged i a eegy level. Let be te distae betwee te uleus ad te eleto. Te M m + m M ad M m M m Cetipetal foe to te eleto is povided by te eletostati foe. So, m Ze M o m M m o Mm M m 3 Ze o 3 e Mm wee M m Momet of ietia of atom about CM, Ze. M + m Mm M m M CM m Aodig to Bo s teoy, o Solvig above equatios fo, we get e Z ad (.59 Å) m Z μ Fute eletial potetial eegy of te system, 3

24 U Ze Z e U ad kieti eegy, K ad K v v-speed of eleto wit espet to uleus. (v ) Ze ee 3 K Ze 8 Total eegy of te system E K + U Z 8 e tis expessio a also be witte as e E 8 E (3.6 ev) Z μ m me Te expessio fo E witout osideig te motio of poto is E, i.e., m is eplaed by 8 wile osideig te motio of uleus. Ex. 6 A positoium atom is a system tat osists of a posito ad a eleto tat obit ea ote. Compae te wavelegt of te spetal lies of positoium wit tose of odiay ydoge. Sol. Hee te two patile ave te same mass m, so te edued mass is mm m m m M m wee m is te eleto mass. We kow tat Hee E' E m E m eegy of ea level is alved. Tei diffeee will also be alved.. ATOMIC COLLISION I su ollisios assume tat te loss i te kieti eegy of system is possible oly if it a exite o ioise. Ex. 7 euto K, v H atom at est i goud state ad fee to move Sol. ead o ollisio Wat will be te type of ollisio, if K ev,. ev, ev,.8 ev (elasti/ielasti/peetly ielasti) Loss i eegy (E) duig te ollisio will be used to exite te atom o eleto fom oe level to aote. Aodig to quatum Meais, fo ydoge atom. E {,. ev,.9 ev,..., 3.6 ev) Aodig to Newtoio meais miimum loss. (elasti ollsio) fo maximum loss ollisio will be pefetly ielasti m m v f if euto ollides pefetly ielastially te, Applyig mometum osevatio m m f

25 v f v fial K.E. m m K maximum loss K Aodig to lassial meais (E) [, (a) If K ev, Aodig to quatum meais (E) {,.ev,.9 ev} Aodig to lassial meais E [, 7 ev] loss, ee it is elasti ollisio speed of patile ages. (b) If K. ev Aodig to lassial meais loss [,. ev] Aodig to quatum meais loss {,.ev,.9ev,...} loss elasti ollisio. loss.ev pefetly ielasti ollisio () If K ev Classial meais E [, ] Quatum meais E {,.ev,.9ev,...} loss elasti ollisio loss. ev ielasti ollsio (d) If K.8 ev Aodig to lassial meais E [,.9eV] Aodig to quatum meais E {,.ev,.9ev, eV} loss elasti ollisio loss. ev ielasti ollisio loss.9 ev pefetly ielasti ollisio K ] Ex. 8 A He + io is at est ad is i goud state. A euto wit iitial kieti eegy K ollides ead o wit te He + io. Fid miimum value of K so tat tee a be a ielasti ollisio betwee tese two patile. Sol. Hee te loss duig te ollisio a oly be used to exite te atoms o eletos. So aodig to quatum meais loss {,.8eV, 8.3eV,..., 5.eV}...() Z E 3.6 ev m m K He + Now aodig to ewtoio meais Miimum loss maximum loss will be fo pefetly ielasti ollisio. let v be te iitial speed of euto ad v f be te fial ommo speed. v so by mometum osevatio mv mv f + mv f v f 5 wee m mass of Neuto mass of He + io m so fial kieti eegy of system K.E. m v f + m v f.(5m). v 5.( mv K ) 5 5 5

26 maximum loss K 5 K so loss will be K 5 K,...() 5 Fo ielasti ollisio tee sould be at least oe ommo value ote ta zeo i set () ad () K >.8 ev 5 K > 5 ev miimum value of K 5 ev. Ex. 9 A movig ydoge atom makes a ead o ollisio wit a statioay ydoge atom. Befoe ollisio bot atoms ae i goud state ad afte ollisio tey move togete. Wat is te miimum value of te kieti eegy of te movig ydoge atom, su tat oe of te atoms eaes oe of te exitatio state. Sol. Let K be te kieti eegy of te movig ydoge atom ad K, te kieti eegy of ombied mass afte ollisio. Fom osevatio of liea mometum, p p o Km K'(m) o K K...(i) Fom osevatio of eegy, K K + E...(ii) Solvig Eqs. (i) ad (ii), we get E K Now miimum value of E fo ydoge atom is. ev. o E. ev K. K. ev Teefoe, te miimum kieti eegy of movig ydoge is. ev E. ev Ex. 3 A euto movig wit speed v makes a ead-o ollisio wit a ydoge atom i goud state kept at est. Fid te miimum kieti eegy of te euto fo wi ielasti (ompletely o patially) ollisio may take plae. Te mass of euto mass of ydoge.67 7 kg. Sol. Suppose te euto ad te ydoge atom move at speed v ad v afte te ollisio. Te ollisio will be ielasti if a pat of te kieti eegy is used to exite te atom. Suppose a eegy E is used i tis way. Usig osevatio of liea mometum ad eegy. mv mv + mv...(i) ad mv mv + mv + E...(ii) Fom (i), v v + v + v v, E Fom (ii), v v + v + m E Tus, v v m Hee, (v v ) v v v As v v must be eal, v E m E m o, mv > E. Te miimum eegy tat a be absobed by te ydoge atom i goud state to go i a exited state is. ev. Tus, te miimum kieti eegy of te euto eeded fo a ielasti ollisio is mvmi. ev. ev As. 6

27 Ex. 3 How may ead-o, elasti ollisios must a euto ave wit deuteium uleus to edue its eegy fom MeV to.5 ev. Sol. Let mass of euto m ad mass of deuteium m iitial kieti eegy of euto K Let afte fist ollisio kieti eegy of euto ad deuteium be K ad K. Usig C.O.L.M. alog dietio of motio mk mk + mk veloity of sepeatio veloity of appoa mk mk m m Solvig equaito (i) ad (ii) we get mk m K K 9 Loss i kieti eeyg afte fist ollisio K K K Afte seod ollisio K 9 8 K... () K 8 8 K K Total eegy loss K K + K K 8 8 As, K K + K K 8 K K ( K 8 9 K ) Hee, K 6 ev, K ( 6.5) ev K K 9 K Takig log bot sides ad solvig, we get o 9 7 Ex. 3 A euto wit a eegy of.6 MeV ollides wit potos ad is etaded. Assumig tat upo ea ollisio euto is defleted by 5º fid te umbe of ollisios wi will edue its eegy to.3 ev. Sol. Mass of euto mass of poto m m Neuto K m Poto Neuto Poto 5º º K y x K Fom osevatio of mometum i y-dietio mk si 5º mk si...(i) I x-dietio mk mk os 5º mk os...(ii) Squaig ad addig equatio (i) ad (ii), we ave K K + K K K...(iii) 7

28 Fom osevatio of eegy K K K Solvig equatios (iii) ad (iv), we get...(iv) K K i.e., afte ea ollisio eegy emais alf. Teefoe, afte ollisios, K K.3 (.6 6 ) Takig log ad solvig, we get As Calulatio of eoil speed of atom o emissio of a poto (a) (b) mometum of poto m fixed H-atom H-atom i fist exited state fee to move m - mass of atom Aodig to mometum osevatio mv... (i) ' Aodig to eegy osevatio. ev m. ev ' Sie mass of atom is vey lage ta poto ee m a be egleted '. ev '. ev m. ev. m. eoil speed of atom m 3. X-RAYS It was disoveed by ROENTGEN. Te wavelegt of x-ays is foud betwee. Å to Å. Tese ays ae ivisible to eye. Tey ae eletomageti waves ad ave speed 3 8 m/s i vauum. Its potos ave eegy aoud times moe ta te visible ligt. ieases Rw mw IR v uv x We fast movig eletos avig eegy of ode of seveal KeV stike te metalli taget te x-ays ae podued. 8

29 3. Podutio of x-ays by oolidge tube : Taget (of Mo o w) oppe blok oppe od To vaum pump ollimate filamet V filamet voltage x-rays Aeleatig voltage ~ kv Te meltig poit, speifi eat apaity ad atomi umbe of taget sould be ig. We voltage is applied aoss te filamet te filamet o beig eated emits eletos fom it. Now fo givig te beam sape of eletos, ollimato is used. Now we eleto stikes te taget te x-ays ae podued. We eletos stike wit te taget, some pat of eegy is lost ad oveted ito eat. Sie, taget sould ot melt o it a absob eat so tat te meltig poit, speifi eat of taget sould be ig. Hee oppe od is attaed so tat eat podued a go beid ad it a absob eat adtaget does ot get eated veyig. Fo moe eegeti eleto, aeleatig voltage is ieased. Fo moe o. of potos voltage aoss filamet is ieased. Te x-ay wee aalysed by mostly takig tei spetum 3. Vaiatio of Itesity of x-ays wit is plotted as sow i figue : otiuous mi. Te miimum wavelegt oespods to te maximum eegy of te x-ays wi i tu is equal to te maximum kieti eegy ev of te stikig eletos tus ev max mi mi Å. ev V(ivolts) We see tat utoff wavelegt mi depeds oly o aeleatig voltage applied betwee taget ad filamet. It does ot deped upo mateial of taget, it is same fo two diffeet metals (Z ad Z ) Ex. 33 A X-ay tube opeates at kv. A patiula eleto loses 5% of its kieti eegy to emit a X-ay poto at te fist ollisio. Fid te wavelegt oespodig to tis poto. Sol. Kieti eegy aquied by te eleto is K ev 3 ev. Te eegy of te poto.5 3 ev 3 ev. Tus, 3 ev (. 5 ev s) (3 m / s) 3 ev 8 ev m.m 3 ev 9

30 . Caatisti X-ays Te sap peaks obtaied i gap ae kow as aateisti x-ays beause tey ae aateisti of taget mateial.,, 3,,... aeteisti wavelegt of mateial avig atomi umbe Z ae alled aateisti x-ays ad te spetum obtaied is alled aateisti spetum. If taget of atomi umbe Z is used te peaks ae sifted. PHYSICS V, Z 3 mi Caateisti x-ay emissio ous we a eegeti eleto ollides wit taget ad emove a ie sell eleto fom atom, te vaay eated i te sell is filled we a eleto fom ige level dops ito it. Suppose vaay eated i iemost K-sell is filled by a eleto dopig fom ext ige level L-sell te K V, Z aateisti x-ay is obtaied. If vaay i K-sell is filled by a eleto fom M-sell, K lie is podued ad so o similaly L, L,...M, M lies ae podued. mi 3 Ex. 3 Fid wi is K ad K 5 3 l l K K N K M M N M L L L L x-ays Sol. E, E sie eegy diffeee of K is less ta K E k < E k k < k is K ad is K O K 3 V, Z' < Z Ex. 35 Fid wi is K ad L Sol E K > E L is K ad is L. MOSELEY S LAW : Moseley measued te fequeies of aateisti x-ays fo a lage umbe of elemets ad plotted te sqaue oot of fequey agaist positio umbe i peiodi table. He disoveed tat plot is vey losed to a staigt lie ot passig toug oigi. 3

31 , ', '', ''', ', '', ''' Z l l Z l ' l ' Z 3 l " l '' Z l "' l ''' Z Wavelegt of aatisti wavelegts. Ex. 36 Moseley s obsevatios a be matematially expessed as a(z b) a ad b ae positive ostats fo oe type of x-ays & fo all elemets (idepedet of Z). Moseley s Law a be deived o te basis of Bo s teoy of atom, fequey of x-ays is give by CR. (Z b) by usig te fomula R z wit modifiatio fo multi eleto system. b kow as seeig ostat o sieldig effet, ad (Z b) is effetive ulea age. fo K lie, Hee a K 3RC a(z b) K (Z b) 3RC, [b fo K lies] Z Z Fid i Z ad Z wi oe is geate. Sol. R. (Z b) If Z is geate te will be geate, will be less < Z > Z. Ex. 37 A obalt taget is bombaded wit eletos ad te wavelegt of its aateisti spetum ae measued. A seod, faite, aateisti spetum is also foud beause of a impuity i te taget. Te wavelegt of te K lies ae 78.9 pm (obalt) ad 3.5 pm (impuity). Wat is te impuity? Sol. Usig Moseley s law ad puttig / fo (ad assumig b ), we obtai az a ad Dividig yields x az x a 3

32 Substitutig gives us x Z Z x 78.9pm 3.5pm Z x. 7 Solvig fo te ukow, we fid Z x 3.; te impuity is zi. Ex. 38 Fid te ostats a ad b i Moseley s equatio v a(z b) fom te followig data. Elemet Z Wavelegt of K X-ay Mo 7 pm Co pm Sol. Moseley s equatio is v a(z b) Tus, a(z b) ad a(z b) Fom (i) ad (ii) a(z Z ) o, a (Z Z ) 8 (3 m/ s) 7 / Dividig (i) by (ii), / (7 m) ( (Hz) / / Z Z b b m)...(i)...(ii) o, 78.5 b 7 7 b o, b.37 Poblem. Solutio : Fid te mometum of a. MeV poto. p E MeV/. Poblem. Solutio : Mooomati ligt of wavelegt 3 Å is iidet oally o a sufae of aea m. If te itesity of te ligt is 5 W/m, detemie te ate at wi potos stike te sufae. Rate at wi potos stike te sufae 5 A 6 J/ s / 3 poto/s J/ poto 3

33 Poblem 3. Te kieti eegies of potoeletos age fom zeo to. 9 J we ligt of wavelegt 3 Å falls o a sufae. Wat is te stoppig potetial fo tis ligt? ev Solutio : K max. 9 J 9.5 ev..6 J Te, fom ev s K max, V s.5 V. Poblem. Wat is te tesold wavelegt fo te mateial i above poblem? Solutio :.5 ev 3. ev.å. 3 Å Solvig, t 759 Å. t 3 ev.å Poblem 5. Fid te de Boglie wavelegt of a. kg pellet avig a veloity of m/s. Solutio : /p J.s.kg m/ s Å. Poblem 6. Detemie te aeleatig potetial eessay to give a eleto a de Boglie wavelegt of Å, wi is te size of te iteatomi spaig of atoms i a ystal. Solutio : V m e Poblem 7. Solutio. Poblem 8. 5 V. Detemie te wavelegt of te seod lie of te Pase seies fo ydoge. (.97 3 Å ) 3 5 o,8 Å. How may diffeet potos a be emitted by ydoge atoms tat udego tasitios to te goud state fom te 5 state? Solutio : No of possible tasitio fom 5 ae 5 C As. potos. Poblem 9. Solutio : A eleto otates i a ile aoud a uleus wit positive age Ze. How is te eletos veloity eleated to te adius of its obit? Te foe o te eleto due to te ulea povides te equied etipetal foe Ze.e mv v Ze.m As. v Ze..m Poblem. Solutio : (i) Calulate te fist tee eegy levels fo positoium. (ii) Fid te wavelegt of te H a lie (3 tasitio) of positoium. I positoium eleto ad posito evolve aoud tei ete of mass p e mv () / mv k/ () Fom () & () V.. e p e e TE mv m. 6 33

34 (i) 6.8 ev E 6.8 ev E 6.8 ev.7 ev E ev.76 ev (ii) E (3 ) E 3 E.76 (.7) ev.9 ev Te oespodig wave legt. Å 33 Å.9 As. (i) 6.8 ev,.7 ev,.76 ev ; (ii) 33 Å. Poblem. Solutio : Poblem. Solutio : A H-atom i goud state is movig wit itial kieti eegy K. It ollides ead o wit a He + io i goud state kept at est but fee to move. Fid miimum value of K so tat bot te patiles a exite to tei fist exited state. K Eegy available fo exitatio 5 Total eegy equied fo exitatio. ev +.8 ev 5. ev k 5 5 k ev A TV tube opeates wit a kv aeleatig potetial. Wat ae te maximum eegy X ays fom te TV set? Te eletos i te TV tube ave a eegy of kev, ad if tese eletos ae bougt to est by a ollisio i wi oe X ay poto is emitted, te poto eegy is kev. Poblem 3. I te Moseley elatio, v a(z b) wi will ave te geate value fo te ostat a fo K o K tasitio? Solutio : A is lage fo te K tasitios ta fo te K tasitios. 3