Data Provided: A formula sheet and table of physical constants are attached to this paper.

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1 PHY51A Dt Provided: A formul sheet nd tble of physicl constnts re ttched to this pper. DEPARTMENT OF PHYSICS AND ASTRONOMY Spring Semester ( ) From Thermodynmics to Atomic nd Nucler Physics Pper A: Therml Physics nd Solids 3 HOURS There re three sections to this pper. Ech section contins compulsory question nd two optionl questions. Section A is worth 40 mrks. Sections B nd C re worth 30 mrks ech. You must ttempt the compulsory question nd one optionl question from ech section. Answers to different sections must be written in seprte books, the books tied together nd hnded in s one. Plese clerly indicte the question numbers on which you would like to be exmined on the front cover of your nswer book. Cross through ny work tht you do not wish to be exmined. PHY51A 1 TURN OVER

2 PHY51A COMPULSORY QUESTION SECTION A THERMAL PHYSICS 1 1. () Wht is the second lw of thermodynmics in terms of entropy? Explin how this cn be used to find the equilibrium of n isolted system. [4] In Crnot cycle the hot source is held t 500 C nd the cold sink t 30 C. i) Wht is the efficiency of the Crnot cycle? ii) If 300 J of het is dded t the hot source how much het is removed t the cold sink? iii) Wht is the totl internl energy chnge in the cycle? [4] (c) Explin wht is ment by microstte nd by mcrostte. Wht is the fundmentl ssumption of sttisticl physics? [4] (d) A given system in contct with het bth cn tke n energy ɛ = 0 or ɛ = ɛ m. Write down n expression for the probbility tht the energy of the system is zero, nd for the probbility tht the energy of the system is ɛ m s function of temperture T. [4] (e) A litre of n idel montomic gs initilly t pressure of P is heted from 0 C to 300 C in seled vessel. How much energy is required for this process? [4] PHY51A CONTINUED

3 PHY51A ANSWER EITHER QUESTION OR 3. A spce shuttle (volume 50 m 3, ir temperture 5 C, p = P) docks onto the loding by (volume 50 m 3 ) of its mother-ship nd opens its irlock. Unfortuntely the crew of the mother-ship hve not ensured tht the loding by ws properly prepred nd it is under vcuum. The ir in the spce shuttle rushes out to fill the vilble spce. () (i) Wht is the internl energy chnge of the ir in this process? Explin your nswer. (ii) Assuming the ir cn be treted s n idel gs, find the temperture of the gs. (iii) Qulittively, how would your nswer to (ii) differ if the gs ws not idel? Explin why. [6] Wht is the chnge of entropy for this process? Show tht the sme nswer is obtined both by clculting the entropy chnge from f S ds nd from considering the number of vilble sttes nd S k B ln. [6] i (c) It turns out tht the mother-ship hs been tken over by crew of crbon monoxide brething liens. They open door connecting the loding by to the rest of the mother-ship. The volume of the crgo by plus spce shuttle is equl to the volume of the rest of the mother-ship. Just before they opened the door the rest of the mother-ship contined only crbon monoxide, the crgo by plus spce shuttle contined only ir, nd the concentrtion (i.e. number of moles per unit volume) of crbon monoxide in the rest of the mother-ship ws equl to the concentrtion of ir (i.e. number of moles per unit volume) in the crgo by plus shuttle. (i) Explin wht hppens nd why. [3] (ii) Clculte the entropy chnge ssocited with this process. [] (iii) Wht would the entropy chnge be if the mother-ship hd contined ir rther thn crbon monoxide? Explin your nswer. [3] PHY51A 3 TURN OVER

4 PHY51A 3. An Einstein solid consists of collection of microscopic systems, ech ble to store ny number of energy units where energy units re quntised into. () For n Einstein solid of N oscilltors with q energy units derive n expression for the number of vilble microsttes ( Nq, ). Wht is the multiplicity for such system when N = 10 nd q = 30? [5] For rel solid both N nd q re very lrge. If such n Einstein solid is t high temperture (i) Wht dditionl ssumption cn we mke? [] (ii) Show tht ln ( q N)ln( q N) qln q N ln N, nd tht, becuse the solid is t high T, q N ln Nln N. N q [11] (iii) Hence find n expression for the entropy, S, of n Einstein solid t high T in terms of N nd q. [] PHY51A 4 CONTINUED

5 PHY51A COMPULSORY QUESTION SECTION B SOLIDS 4. () (c) (d) (e) (f) Sketch the phonon density of sttes g(ω) for the Einstein model of 3-dimensionl solids. [1] Tungsten hs Debye temperture of 400 K. If the molr het cpcity t 30 K is 1.35 J K -1 mol -1, wht is the molr het cpcity of tungsten t (i) 600 K nd (ii) 15 K? Stte ny ssumptions you mke. [4] Are Umklpp processes more likely to occur t low or high temperture? Justify your nswer. [] Sketch N (E), the number of occupied sttes s function of E, for metl t (i) bsolute zero temperture; (ii) temperture T > 0 K. Mrk the loction of the Fermi energy, ε F, on your sketches. [3] Explin why only very smll frction of the totl number of free electrons in solid cn contribute to the het cpcity. [] For -dimensionl squre lttice, show tht the kinetic energy of free electron t the midpoint of side fce of the Brillouin zone is hlf tht of n electron t corner of the first Brillouin zone. [3] PHY51A 5 TURN OVER

6 PHY51A ANSWER EITHER QUESTION 5 OR 6 5. () Give one similrity nd one difference between the Einstein nd Debye models for the specific het of solids. [] In the Debye model of -dimensionl solid, show tht the phonon density of sttes is given by L g, v where L is the side length of the solid, ω is the vibrtionl ngulr frequency nd v is the speed of sound. [5] (c) The temperture-dependent verge therml energy of solid cn be written s ωmx g d U, 0 exp k T 1 B where ω is the vibrtionl ngulr frequency, kb is the Boltzmnn constnt nd T is the bsolute temperture. Using g(ω) from prt nd the chnge of vrible x kbt, show tht the het cpcity C T t low tempertures for -dimensionl solid. [5] 0 x [The definite integrl. 4 ] x e 1 (d) Clculte the speed of sound in grphene, -dimensionl solid, stting ny ssumptions you mke. [3] [Grphene hs the following properties: Debye temperture θd = 1070 K, nerest neighbour seprtion = 0.14 nm.] PHY51A 6 CONTINUED

7 PHY51A 6. () Stte two ssumptions tht re mde in the free electron theory of metls. [] Clculte the verge kinetic energy per electron nd hence show tht the kinetic energy of three dimensionl gs of N free electrons t 0 K is U N F, where F is the Fermi energy. [4] (c) In solid contining electrons in the reltivistic limit, the electron energy E is relted to the wvevector k by E pc kc, where p is the momentum, is Plnck s constnt divided by π nd c is the speed of light. In the reltivistic limit, show tht VE (i) the density of sttes g( E), where V is the 3 3 c volume of the 3-dimensionl solid, [5] (ii) N 3 the Fermi energy F c 3. [4] V 1 PHY51A 7 TURN OVER

8 PHY51A COMPULSORY QUESTION SECTION C THERMAL PHYSICS 7. () (c) When clculting the probbility tht system exists in prticulr microstte, wht is the role of the prtition function, Z? [] Explin how simple rguments bsed on the sttistics of lrge numbers led to the second lw of thermodynmics. [4] At microscopic level we cn sfely ssume tht processes obey the principle of detiled blnce. Wht is ment by the principle of detiled blnce? For wht sort of process must the principle be obeyed by definition? [3] (d) Given tht G(T,p) nd dg SdT Vdp derive Mxwell reltionship relting S nd p t constnt T, to V nd T t constnt p. [3] (e) Sketch nd lbel the p-t phse digrm for crbon dioxide. [3] PHY51A 8 CONTINUED

9 PHY51A ANSWER EITHER QUESTION 8 OR 9 8. () A system undergoes process under conditions of fixed volume nd temperture. Wht function will be t minimum when the system reches equilibrium? [] A simple quntum hrmonic oscilltor hs energy levels n = 0,1, 1 ( n ) where (i) Show tht the prtition function for the system is given by 1 where. kt B e Z 1 e 1, [4] (ii) Hence find n expression for the internl energy of the system. [5] (c) U Given tht CV, derive n expression for the het cpcity of the T V quntum hrmonic oscilltor in terms of β. [4] [You my wish to use.] 1 r n r r... r n 0 PHY51A 9 TURN OVER

10 PHY51A 9. () Wht re the primry ssumptions or simplifictions behind the idel gs? Under wht conditions re these most likely to be resonble pproximtion of relity? [3] A ten litre glss vessel contining 0.1 mol of helium is held t temperture of 30 C. (i) Wht is the verge seprtion between the helium toms in the gs? [] (ii) Derive n expression for the men free pth,, for helium tom in the vessel (i.e. the verge distnce between collisions) in terms of the effective rdius of helium tom nd the number of helium toms per unit volume. Stte the ssumptions you mke. [6] (iii) If the effective rdius of helium tom is m, wht is the men free pth for helium tom in the glss vessel? [] (iv) If the temperture of the vessel ws incresed how would the frequency of collisions between toms chnge? [] END OF EXAMINATION PAPER PHY51A 10 CONTINUED

11 PHYSICAL CONSTANTS & MATHEMATICAL FORMULAE Physicl Constnts electron chrge e = C electron mss m e = kg = MeV c proton mss m p = kg = MeV c neutron mss m n = kg = MeV c Plnck s constnt h = J s Dirc s constnt ( = h/π) = J s Boltzmnn s constnt k B = J K 1 = ev K 1 speed of light in free spce c = m s m s 1 permittivity of free spce ε 0 = F m 1 permebility of free spce µ 0 = 4π 10 7 H m 1 Avogdro s constnt N A = mol 1 gs constnt R = J mol 1 K 1 idel gs volume (STP) V 0 =.4 l mol 1 grvittionl constnt G = N m kg Rydberg constnt R = m 1 Rydberg energy of hydrogen R H = 13.6 ev Bohr rdius 0 = m Bohr mgneton µ B = J T 1 fine structure constnt α 1/137 Wien displcement lw constnt b = m K Stefn s constnt σ = W m K 4 rdition density constnt = J m 3 K 4 mss of the Sun M = kg rdius of the Sun R = m luminosity of the Sun L = W mss of the Erth M = kg rdius of the Erth R = m Conversion Fctors 1 u (tomic mss unit) = kg = MeV c 1 Å (ngstrom) = m 1 stronomicl unit = m 1 g (grvity) = 9.81 m s 1 ev = J 1 prsec = m 1 tmosphere = P 1 yer = s

12 Polr Coordintes x = r cos θ y = r sin θ da = r dr dθ = 1 ( r ) + 1r r r r θ Sphericl Coordintes Clculus x = r sin θ cos φ y = r sin θ sin φ z = r cos θ dv = r sin θ dr dθ dφ = 1 ( r ) + 1 r r r r sin θ ( sin θ ) + θ θ 1 r sin θ φ f(x) f (x) f(x) f (x) x n nx n 1 tn x sec x e x e x sin ( ) 1 x ln x = log e x 1 x cos 1 ( x sin x cos x tn ( 1 x cos x sin x sinh ( ) 1 x cosh x sinh x cosh ( ) 1 x sinh x cosh x tnh ( ) 1 x ) ) 1 x 1 x +x 1 x + 1 x x cosec x cosec x cot x uv u v + uv sec x sec x tn x u/v u v uv v Definite Integrls x n e x dx = n! (n 0 nd > 0) n+1 π e x dx = π x e x dx = 1 Integrtion by Prts: 3 b u(x) dv(x) dx dx = u(x)v(x) b b du(x) v(x) dx dx

13 Series Expnsions (x ) Tylor series: f(x) = f() + f () + 1! n Binomil expnsion: (x + y) n = (1 + x) n = 1 + nx + k=0 ( ) n x n k y k k n(n 1) x + ( x < 1)! (x ) f () +! nd (x )3 f () + 3! ( ) n n! = k (n k)!k! e x = 1+x+ x! + x3 x3 +, sin x = x 3! 3! + x5 x nd cos x = 1 5!! + x4 4! ln(1 + x) = log e (1 + x) = x x + x3 3 n Geometric series: r k = 1 rn+1 1 r k=0 ( x < 1) Stirling s formul: log e N! = N log e N N or ln N! = N ln N N Trigonometry sin( ± b) = sin cos b ± cos sin b cos( ± b) = cos cos b sin sin b tn ± tn b tn( ± b) = 1 tn tn b sin = sin cos cos = cos sin = cos 1 = 1 sin sin + sin b = sin 1( + b) cos 1 ( b) sin sin b = cos 1( + b) sin 1 ( b) cos + cos b = cos 1( + b) cos 1 ( b) cos cos b = sin 1( + b) sin 1 ( b) e iθ = cos θ + i sin θ cos θ = 1 ( e iθ + e iθ) nd sin θ = 1 ( e iθ e iθ) i cosh θ = 1 ( e θ + e θ) nd sinh θ = 1 ( e θ e θ) Sphericl geometry: sin sin A = sin b sin B = sin c sin C nd cos = cos b cos c+sin b sin c cos A

14 Vector Clculus A B = A x B x + A y B y + A z B z = A j B j A B = (A y B z A z B y ) î + (A zb x A x B z ) ĵ + (A xb y A y B x ) ˆk = ɛ ijk A j B k A (B C) = (A C)B (A B)C A (B C) = B (C A) = C (A B) grd φ = φ = j φ = φ x î + φ y ĵ + φ z ˆk div A = A = j A j = A x x + A y y + A z z ) curl A = A = ɛ ijk j A k = ( Az y A y z φ = φ = φ x + φ y + φ z ( φ) = 0 nd ( A) = 0 ( A) = ( A) A ( Ax î + z A ) ( z Ay ĵ + x x A ) x y ˆk

Data Provided: A formula sheet and table of physical constants is attached to this paper.

Data Provided: A formula sheet and table of physical constants is attached to this paper. PHY106 PHY47 Dt Provided: Formul sheet nd physicl constnts Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS & Autumn Semester 009-010 ASTRONOMY DEPARTMENT